Intracenter Transitions of Iron-Group Ions in II–VI Semiconductor Matrices

REVIEWS

Intracenter Transitions of Iron-Group Ions in IIÐVI Semiconductor Matrices V. F. Agekyan Institute of Physics, St. Petersburg State University, St. Petersburg, Petrodvorets, 198504 Russia e-mail: [email protected] Received December 27, 2001

Abstract—A review is made of studies of intracenter optical transitions in 3d shells of iron-group divalent (magnetic) ions. Attention is focused on the emission spectra of Mn2+ ions in CdTe, ZnS, and ZnSe crystals. An analysis is made of the structure of intracenter absorption and luminescence and of the effect that the elemental matrix composition, magnetic-ion concentration, temperature, hydrostatic pressure, and structural phase transitions exert on the intracenter transitions. The mutual influence of two electronic excitation relaxation mechanisms, interband and intracenter, is considered. The specific features of the intracenter emission of magnetic ions embedded in two-dimensional systems and nanocrystals associated with a variation in spÐd exchange interaction and other factors are discussed. Data on the decay kinetics over the intracenter luminescence band profile are presented as a function of temperature, magnetic ion concentration, and excitation conditions. The saturation of the luminescence and the variation of its kinetic properties under strong optical excitation, which are caused by excitation migration and the cooperative effect, as well as the manifestation of a nonlinearity in intracenter absorption, are studied. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION metal, its 3d shell can undergo charge transfer, thus imparting new properties to excitons bound to Cu2+, Crystals and glasses containing iron-group atoms Ni2+, Co2+, and Fe2+ in crystals of the wide-band-gap exhibit intracenter transitions in the unfilled 3d shells of IIÐVI compounds [2, 3]. DMSs and related quantum these atoms that manifest themselves strongly in structures exhibit specific properties associated with absorption and luminescence. The intracenter emission the large magnetic moment of the unfilled 3d shell, of these ions, particularly of the divalent manganese, 2+ such as giant Zeeman splitting of the electronic levels, has a potential use, because ZnS:Mn is the most effi- giant Faraday rotation, and the magnetic polaron effect. cient electroluminescent phosphor known to date [1]. DMSs reveal two types of electronic excitations and of The main subject of the present review is optically their relaxation, more specifically, the conventional excited intracenter luminescence (IL). While IL has semiconductor interband and the intracenter mecha- been studied on a broad range of objects, most reports 2+ nisms, which can affect each other substantially. The have dealt only with crystals containing Mn . Of most intracenter absorption and emission of light in the 3d interest is the investigation of the IIÐVI crystals, in 2+ shell of a magnetic ion become manifest in crystals which the relative concentration of the Mn cations with a broad enough band gap. The most popular DMSs can range from a few hundredths of a percent to tens of 2+ a percent. Considered over such a broad range of con- with Mn ions are model systems for studying mag- centrations, the manganese may act both as a paramagnetic and magnetooptical properties and intracenter netic impurity and an activator and as a solid-solution transitions. The application potential of DMSs contain- component capable of affecting the fundamental prop- ing various magnetic impurities lies in the area of pho- erties of a crystal. A large number of studies of the opti- toinduced magnetism, laser gain media, laser shutters, cal properties of IIÐVI solid solutions with a magnetic electroluminescent devices, Faraday cells, and magne- component and of the related quantum structures have tosensitive low-dimensional quantum structures. recently been reported. These compounds make up a Recently, electroluminescence has been studied in thin class of dilute magnetic semiconductors (DMS) some- epitaxial films and composites containing a polymer times referred to as semimagnetic semiconductors. and manganese-activated crystallites [4, 5]. Investiga- Typically, DMSs exhibit properties basically inherent tion of the properties of intracenter photoluminescence in crystalline solid solutions, such as a dependence of promotes our understanding of the electroluminescence the fundamental parameters on the solid-solution com- processes occurring in bulk crystals and quantum struc- position, inhomogeneous broadening, electron local- tures and shows the path to their optimization. There is ization associated with local compositional fluctua- a variety of reviews and papers that primarily describe tions, etc. If the isoelectronic impurity is a transition the band electronic states of DMSs and the effect of an

2014 AGEKYAN internal field of magnetic ions on them (electron–ion replaced by Dq) is commonly used to measure the crys- ∆ exchange interaction), as well as the interaction of mag- tal field. It is due to the comparatively small value of t netic ions with one another (ionÐion exchange interac- that the tetrahedral complexes in a IIÐVI matrix are tion) (see, e.g., [6Ð10]). The present review deals with classed among high-spin, weak-field complexes. In this the optical transitions occurring in the 3d shell of the case, the nearest neighbors (anions) forming the tetra- iron-group divalent ions, mainly of Mn2+, in IIÐVI crys- hedron do not affect the level diagram of a single man- tals and related nanostructures. Main attention is ganese atom noticeably. The tetrahedral type of the focused on the interaction between various mecha- anion environment also accounts for the small change nisms of electronic-excitation relaxation, on the effect that the fields acting on the Mn2+ ion undergo in the of external factors, elemental composition of the transition of a IIÐVI crystal from the wurtzite to zinc- matrix, and confinement, as well as on the kinetic prop- blende structure; indeed, the ligand field has the Td erties of the IL. symmetry and there is no inversion center. The level diagram for the Mn2+ ion in a tetrahedral field is presented in Fig. 1, with the energy reckoned from the 2. ENERGY LEVELS AND OPTICAL 6 TRANSITIONS IN THE 3d SHELLS ground state (the spin sextet A1 with spin S = 5/2 and OF IRON-GROUP IONS IN THE IIÐVI CRYSTAL orbital number L = 0), which is only weakly sensitive to MATRICES the field. The lowest excited states correspond to S = 3/2 and L = 1, 2, 3, 4, the 4G term with L = 4 lying at the The properties of the 3d shell of an iron-group ion in lowest energy. The 4G term splits into four levels in a a crystal and the methods to be used to calculate its tetrahedral field. The T level is the lowest, because its electronic levels are governed by a number of factors. 1 Among these are the electronÐelectron spin interaction energy decreases considerably with increasing field, V , the spinÐorbit interaction V , hybridization of the d whereas the A and E levels respond very weakly to a ss so field variation. Energy level calculations for iron-group states with s and p states of the band electrons, and the ions in crystal matrices (the ligand field) can be found crystal field Vc. The crystal field can be written in the in a number of monographs and papers (see, e.g., form [11, 12]). V s = V o ++V 1 V 2, (1) While the positions of the 3d-shell levels relative to where V is the spherically symmetric part of the field, the band-state extrema are difficult to derive from opti- o cal spectra, the absence of clearly pronounced transi- V1 provides the main contribution to the lowering of the 6 tions from the A1 level to the conduction band in all spherical field symmetry, and V2 contains small terms of an even lower symmetry. In the systems of interest to Mn-doped crystals of the IIÐVI compounds indicates us here, the strongest is the electronÐelectron interac- that this level lies below the top of the valence band tion, which corresponds to the case of a weak crystal derived from the chalcogenide p states. Photoelectron field. If the 3d shell is filled to one half or less, the emission spectra of Cd1 Ð xMnxTe were studied both weak-field complexes retain, in accordance with experimentally and theoretically in [13Ð18]. Two meth- Hund’s rule, the maximum possible total free-ion spin ods of photoelectron spectroscopy, integrated and moment (5/2 for Mn2+). In high-symmetry crystal angle-resolved, yielded contradictory results for the fields, the electron–electron interaction is described by extent of hybridization of various states. These contra- three Racah parameters, A, B, and C, with the first of dictions were finally removed, and it has been shown them yielding only the general shift of the whole that the observation of a strong photoemission peak 3d-level system. The ligand crystal field acting on the located a few electronvolts below the valence-band top cation in a zinc-blende-type lattice, to which IIÐVI and corresponding to the 3d-shell contribution to the cubic crystals belong, is tetrahedral. In this field, the d electron density of states is not at odds with a considerable spÐd hybridization. The energy level correspond- orbitals can be divided in two groups, with d 2 2 , d 2 ()x Ð y z ing to the removal of one 3d electron from Mn2+ lies belonging to the T2 term and dxy, dxz, dyz, to term E. The 3.5 eV below the valence-band top, whereas that due to tetrahedral and octahedral fields acting on the magnetic the addition of a sixth electron to the 3d shell is located ion located at the cube center are created by the ligands 3.0Ð3.5 eV above the valence-band top. This paper does occupying four positions at the cube vertices and six not consider the problems associated with charge trans- positions at the face centers, respectively. The crystal- fer in the 3d shell of the iron-group atoms. field splittings of the 3d levels in these fields are related through Absorption spectra corresponding to transitions in the Mn2+ 3d shell are observed in crystal matrices with ∆ ∆ t = Ð4 o/9. (2) a sufficiently broad band gap. Intracenter transitions with a threshold of about 2.13 eV are seen to appear in Here, the indices t and o relate to the tetrahedral and reflectance spectra of Zn1 Ð xMnxTe already at x = 0.02 octahedral cases for which the T1 and E terms are the [19]. At comparatively high manganese concentrations, lowest, respectively. The quantity ∆ (sometimes the intracenter absorption spectrum consists of bands

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2015

II I Mn–Mn

C 4 H SF 3 6 4 Α1 → Τ2 2.22 2.23 2.24 2.25 2.26 6 4 Α1 → Τ1 Energy, eV 2 4Τ → 6Α

Luminescence intesity, arb. units 1 1 Excitation of luminescence, arb. units 1

1.0 1.5 2.0 2.5 3.0 3.5 Energy, eV

2+ 2+ Fig. 1. (1) Intracenter absorption and (2) IL of Mn in a Zn0.96Mn0.04S crystal [1]. Inset presents Mn IL spectra in ZnS doped with manganese to concentrations of (3) 0.01 and (4) 1%. The C, H, and SF lines refer to zero-phonon emission of single ions from the cubic and hexagonal crystal regions and from the region containing stacking faults, respectively. The MnÐMn, I, and II peaks relate to emission of manganese pairs of minimum radius from a cubic region and its phonon replicas, respectively. T = 4 K. about 0.1 eV wide (Fig. 1). The intracenter absorption best studied of them being the ZnS:Mn2+ system. Zinc coefficient is approximately proportional to the manga- sulfide can contain regions with zinc-blende and wurtz- nese concentration, and the band width is governed by ite structure, as well as regions with stacking faults. At the 3d-electron interaction with phonons and by inho- Mn concentrations on the order of 10Ð4%, the IL spec- mogeneous broadening. trum of ZnS:Mn2+ exhibits several narrow zero-phonon The transitions between 3d shell levels are of the lines with energies lying in the interval 2.22Ð2.24 eV, intercombination type and are forbidden in the electric- the strongest of them belonging, according to [1], to the dipole approximation. The forbiddenness is partially emission of Mn2+ ions occupying cationic sites of the lifted by the noncentrosymmetric crystal field compo- cubic and hexagonal phases. The weaker lines can be nents, dynamic JahnÐTeller effect, and spinÐorbit cou- assigned to Mn2+ positions in the regions where the ZnS pling. The pronounced hybridization of the d states lattice has stacking faults. ZnS is a polymorphic mate- with the chalcogenide p states plays a significant part in rial with regular defect sequences forming the 4H, 6H, this class of compounds. As a result, the oscillator and 9R polytypes. In the case of irregular defects, there 2+ strength of the Mn intracenter transitions in II- to are inequivalent cationic positions of C3v symmetry. VI-type lattices is far larger than that, for instance, in For these reasons, the zero-phonon spectrum has a rich MnF2 and its analogs. The absorption coefficient at the structure and at low Mn2+ concentrations and a low 3 Ð1 band maxima of Cd0.5Mn0.5Te is 10 cm , which is one temperature, a system of closely spaced zero-phonon to two orders of magnitude smaller than the typical lines are observed (Fig. 1). Strong phonon replicas are value of the interband absorption coefficient. In sys- seen on the low-energy side of these lines. The excita- tems with a high Mn concentration, the IL band is struc- tion spectrum of the IL zero-phonon lines correlates tureless and about 120 meV wide and its maximum in with the absorption spectrum; the former spectrum Cd0.5Mn0.5Te lies at approximately 2.0 eV, so that the exhibits bands corresponding to transitions from the 2+ 6 4 4 Stokes losses relative to the peak of the first intracenter Mn ground state A1 to the excited states T1, T2, and 6 4 absorption band, A1Ð T1 (the maximum of the IL exci- 4E. At low Mn concentrations, the IL decay time tation spectrum), are quite large, 0.4 eV. exceeds 1 ms and increases for the zero-phonon lines The intracenter absorption, IL, and IL excitation toward lower energies, from 1.15 ms for the hexagonal spectra have a clearly pronounced structure for low phase to 1.77 ms for the cubic phase. In high-quality concentrations of atoms with unfilled 3d shells. Such matrices with no structural defects and at low Mn2+ spectra can be observed in wide-band-gap matrices, the concentrations, one succeeds in studying in consider-

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4 (a) the 1-meV doublet splitting of the T1 term, nearly 4 Ð4 A1 degenerate states differing by 10 eV were resolved in 4 each doublet component (Fig. 2). The doublet compo- E 2 4 É'(5/2) nents derive from the Γ , Γ and Γ , Γ states, respec- G 8 É6 6 8 7 8 2.5 4 T2 É É'(5/2) tively. It has been found that the crystal field model is 6 8 inadequate for determination of the multiplet splitting of terms of the 3d5 states. 4T 1 É É'(3/2) 7 8 While the covalent model provides a more accurate É'(3/2) 5 8 É7 description of the 3d level system, it is more compli- 2.0 cated and the determination of a number of its parameters requires experiments involving the application of a magnetic field and uniaxial strain. As follows from this ~ model, the Mn2+ IL fine structure in ZnS, ZnSe, ZnTe, and CdTe is largely governed by the spinÐorbit cou- 1 5 6 6 pling of the 3d and ligand states. As the relative man- S A1 0 É8 ganese concentration x is increased from 0.001 to 0.02, the IL and absorption bands shift toward higher ener- É7 gies at a rate of 3 meV/x and the spectrum becomes V V V V more complex because of the interaction between the o c so + JT magnetic ions. Already at an x value of about 0.01, Zn1 − xMnxS exhibits a zero-phonon IL line of the nearest neighbor MnÐMn pairs which is substantially stron- (b) ger than the zero-phonon emission line of single Mn ions (Fig. 1). The pair line has strong replicas with the emission of LA and LO phonons. The IL excitation

Luminescence intesity, arb. units bands still exhibit a phonon spectrum structure at x = 0.04; however, at x = 0.08 and larger, the bands become structureless. In the 2.2-eV region, the ZnS matrix reveals bands near 1.95 and 1.6 eV in addition to the Mn2+ IL. While at low temperatures these bands are weak, they have the same intensity as the 2.2-eV band at 60 K and are dominant at 100Ð200 K. The low- energy bands should be assigned, most likely, to Mn2+ ions occupying interstitial positions in the crystal 2.2180 2.2190 2.2200 matrix. These bands become relatively enhanced with Energy, eV increasing Mn2+ concentration and temperature, the conditions in which energy transfer between the vari- Fig. 2. Calculated energy level diagram and zero-phonon IL ous manganese centers becomes efficient. The relative spectrum of Mn2+ in a ZnS crystal free of structural defects weakening of the 2.2-eV IL band observed to occur in and having a low manganese concentration [20]. (a) Energy a ZnS:Mn2+ crystal heated above 60 K is accompanied levels of a free ion (Vo) and of an ion in a crystal field (Vc); by a fast decrease in the decay time as a result of the 6 insets (1) and (2) show, respectively, the A1 ground-state competing nonradiative mechanisms and energy trans- splitting (magnified 106 times) and two versions of addi- fer to lower lying states, whereas for the long-wave- tional splitting of the 4T excited state (magnified 102 times) length bands, this process starts at considerably higher 1 temperatures. with inclusion of the spinÐorbit (Vso) and JahnÐTeller (VJT) interactions. (b) Fine structure of the Mn2+ IL spectrum in Even at the limiting manganese concentrations in 4 ZnS obtained at T = 2 K and corresponding to the four T1 IIÐVI crystals, there is no indication of the properties of sublevels shown in inset 2. the 3d levels changing radically due to the nearest neighbor dÐd interaction. For high Fe2+ and Co2+ concentrations, this interaction is far stronger and a transition is possible from isolated deep levels to a contin- able detail the 3d level systems of single ions at low uum of hybridized d states. As a result, as the concen- temperatures. In such ZnS:Mn2+ crystals, the spectra of tration of iron or cobalt increases, the transitions polarized excitation of the Mn2+ terms were investi- involving the crystal-field-split 3d levels broaden and gated in [20]. The orbital-triplet fine structure is usually weaken in intensity. associated with the JahnÐTeller interaction of the 3d When cobalt is doped in low concentrations into states with the ⑀-type vibrational mode. In addition to ZnSe, numerous peaks about 0.01 eV and less in width

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2017 appear in the luminescence and subsequently extend simultaneous spin flip of two electrons in the 3d shells over a broad spectral region from 1.7 to 2.7 eV [21]. of neighboring Mn2+ ions provides a correct description Two bands, at 0.73 and 0.78 eV, are due to the transition of the low-temperature behavior of the IL band and of 4 from the A2(F) ground state of the 3d shell in the Td its weak sensitivity to an external magnetic field up to 4 field to the first excited state T1(F); the other bands 6 T [25]. The high-temperature shift toward higher involve higher excited states of the type of G, D, H, and energies is accounted for primarily by the lattice expan- 2+ P. The narrowness of these peaks and the absence of sion. Cd1 Ð xMnxTe also exhibits a shift of the Mn IL phonon replicas indicate a weaker interaction of Co2+ maximum from 2.0 to 2.15 eV in the interval 77Ð300 K. with lattice vibrations compared with Mn2+ [22]. The temperature-induced variation of the IL band width According to [23], the strengthening of the IL bands at below 80 K is consistent with the configurational model; however, in the high-temperature region, one 1.6, 1.9, and 2.4 eV in Zn1 Ð xCoxSe with x = 0.01Ð0.02 observed at 10 K is accompanied by a weakening and should also take into account the approach of the con- disappearance of the green edge luminescence of ZnSe. figurational coordinates of the Mn2+ ground and excited At x = 0.03, the IL also disappears, to be replaced by a states [26]. Note that the temperature-induced shifts of Raman continuum with a width larger than 0.5 eV, the manganese absorption threshold Et and of the Wan- which is due to electronic transitions in a d-type hybrid- nier exciton line Egx in Cd1 Ð xMnxTe (x > 0.4) are close ized impurity band. Zn1 Ð xFexSe exhibits one IL band at in magnitude, while their shifts observed under hydro- about 1.9 eV, which is more stable than the Co2+ IL static pressure differ even in sign. band. It starts to broaden at x = 0.2 and decays by an It was proposed to use the temperature-induced vari- order of magnitude only at x = 0.64, with the Raman 2+ continuum appearing at x = 0.3 and gaining in intensity ation of transition energies in the 3d shell of the Mn up to the maximum concentration x = 0.73. Interest- ions contained in various DMSs for determination of ingly, in IIÐVI crystals doped with cobalt and iron, no the temperature dependence of the lattice constant for crystals with a sufficiently wide band gap [27]. The transition from the nonconducting to metallic state 2+ occurs, despite the pronounced delocalization of the 3d possibility of using the IL of Mn as a probe rests on 4 states. This can be accounted for by the transition to the the strong dependence of the energy of the T1 term on conducting state being suppressed by disorder, i.e., by the crystal field and the weak field dependence of the 6 the formation of a continuum of localized electronic energy of the A1 ground state. Optical spectroscopy states. One can conceive of the following model for the has in this respect an edge over direct electronic and x- case of very high iron or cobalt concentrations: the ray methods, which are difficult to employ over a broad Coulomb energy for the 3d electrons is large compared temperature range covering the low-temperature to the characteristic energy variation occurring in one domain. The variation of the lattice constant can also be electron hop, as a result of which the electronic states studied by probing the band gap width Eg with exciton split into two subbands separated in energy, with one of radiation. However, the IL has an advantage over exci- them being completely filled and the other being empty, ton spectroscopy in its temperature stability and conve- so that the system retains its insulating properties. nient spectral range, because measurement of the exciton spectra of wide-band-gap crystals requires the use of sophisticated techniques. This appears to be an 3. DEPENDENCE OF INTRACENTER essential point in view of the interest focused currently LUMINESCENCE ON THE TEMPERATURE on IIÐVI crystals with a very wide band gap containing AND ELEMENTAL COMPOSITION light elements (Zn, Mg, Be, Ca). Furthermore, the exci- OF CRYSTALS ton in a solid solution is characterized by different The temperature dependence of the shape and width degrees of localization at different temperatures, which 2+ of the Mn IL band has been studied for Cd1 Ð xMnxTe makes accurate determination of Eg a difficult problem. (0.4 < x < 0.7) in the range 10Ð220 K [24]. The spectral The manganese IL in MnF2 crystals with a different lat- band width was found to comply with the configura- tice structure was used to determine the crystal-field tional-coordinate model up to 80 K. At higher temper- parameters and the degree of bond covalency [28]. atures, the experimental data are observed to deviate Variations in the elemental composition also strongly considerably from theoretical calculations because of affect the temperature-induced shift, which in the equilibrium positions of the manganese ion in the Cd1 − xMnxTe, for instance, increases by a factor of more excited and ground states approaching each other with than two with x increasing from 0.4 to 0.7 [29]. Molec- increasing temperature. This conclusion is buttressed ular beam epitaxy made it possible to prepare by the considerable decrease in the Stokes losses above Cd1 − xMnxTe films with x ranging from 0.6 to 1.0 and to 2+ 2+ 80 K. The Mn IL band in Zn0.5Mn0.5Se shifts toward study the temperature-induced variation of the Mn IL lower energies by 12 meV in the temperature interval at manganese concentrations inaccessible for bulk crys- from 4 to 45 K, after which a 40-meV shift in the oppo- tals [30Ð33]. The temperature sensitivity of the IL in site direction occurs within the region 45Ð300 K. The MnTe is far weaker than that in Cd0.3Mn0.7Te; however, 4 6 model based on the T1Ð A1 optical transition involving in both cases, the lowest energy of the IL maximum

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2018 AGEKYAN

2.2 the chosen pair of the substitution partners. The crystal 2.09 ﬁeld Dq is usually assumed to vary inversely proportional to the ﬁfth power of the distance between the manganese ion and the anion. If one uses the variation Energy, eV 2.1 4 0 0.5b 1.0 of the lattice constant a0(x) as a basis, estimates made 2.06 for x varying from 0 to 1 yield a considerably larger 2 shift of the dÐd transition bands than that observed experimentally. The reasons for this discrepancy may

Energy, eV 1 3 lie in that, first, the manganeseÐanion distance varies 2.03 more weakly than a0(x) does (this is supported by EXAFS measurements) and, second, one should take into account not only the geometric factor but also the change in the bond covalency upon cation substitution. 2.00 The concentration-induced shift varies sometimes non- 0 0.5 1.0 1.5 2.0 2+ b monotonically. It was shown in [37] that the Fe IL band in ZnS1 Ð xSex, which peaks at about 1.3 eV, shifts Fig. 3. Energy position of the Mn2+ IL profile maximum in toward lower energies up to x = 0.3; then, the shift IIÐVI solid solutions as a function of relative concentrations reverses sign, which apparently results from the com- b of the cation components (T = 77 K). Cd1 Ð x Ð yMnxMgyTe: bined effect of the crystal field variation and of the (1) b = y/(1 Ð x Ð y), x = 0.375 (Mg for Cd substitution); degree of the spÐd hybridization. (2) b = y/x, 1 Ð x Ð y = 0.25 (Mg for Mn substitution); (3) b = x/(1 Ð x Ð y), y = 0.25 (Mn for Cd substitution); A strong IL shift is expected to ensue from the intro- (4) b = x/(1 Ð x Ð y), y = 0.5 (Mn for Cd substitution) [35]. duction into a DMS of beryllium as a cation component Inset shows the position of the Mn2+ IL maximum in a (the Be2+ ionic radius is only 0.35 Å); beryllium has Zn1 − xMgxS crystal with a low manganese concentration, become a subject of considerable recent attention as a b = x/(1 Ð x) (Mg for Zn substitution) [36]. wide-band-gap component which would enhance the bond covalency in IIÐVI crystals and increase their microhardness [38]. The DMSs containing Be as one of corresponds to T = 50 K, where the interatomic dis- their components are of interest (both theoretically and tances are the smallest. Epitaxial films of other II–VI for application) in the form of both bulk crystals and as solid solutions containing iron-group elements can also layers in quantum well structures [39]. The currently be grown by molecular-beam epitaxy (MBE) within a available data are insufficient for a quantitative descrip- much broader relative cation concentration range than tion of the IL shift caused by Be introduction into vari- can be done with bulk crystals. ous DMSs. Investigation of the spectral position of the IL in Table 1 presents the positions of the intracenter 3d Cd Mn Te [34], Cd Mn Mg Te [35], Zn Mg S: absorption maxima for various group-VI elements 1 Ð x x 1 Ð x Ð y x y 1 Ð x x 2+ Mn2+ [36] and other systems performed at various x and determined for transitions to the four Mn excited y yielded information on the effect of the nearest cation states [40]. environment on the 3d transition energies in Mn2+. The We readily see that replacement of sulfur by sele- results obtained allow qualitative interpretation based nium and, all the more so, by tellurium shifts the on the change that the crystal field undergoes when one absorption bands noticeably, whereas substitution of cation is replaced by another and on the radii of the tellurium for selenium affects them only weakly. This divalent ions, which for Cd2+, Mn2+, Zn2+, and Mg2+ are does not correlate well with the simple theory taking 0.97, 0.80, 0.74, and 0.66 Å, respectively. As can into account only the ligand crystal field (the ionic radii be seen from Fig. 3, the shift rates depend strongly on of S2Ð, Se2Ð, and Te2Ð are 1.84, 1.91, and 2.11 Å, respectively). This disagreement can be removed, as in the case of cation substitution, if one includes the change in Table 1. Energy positions of the intracenter absorption the bond covalency. The Racah coefficient B, which is peaks in three types of DMS a measure of covalency, decreases in the order SÐSeÐTe with weakening valence bond ionicity; this decrease Final state (transition energy corresponds to the intracenter transition bands shifting from the 6A level in eV) 2+ DMS 1 toward higher energies. The Mn IL intensity varies little in the interval 4Ð60 K, but when heated to 100 K, the 4 4 4 4 4 4 2 2 T1( G) T2( G) E( G) T2( I) IL weakens by more than an order of magnitude. Note Mn Zn S 2.33 2.40 2.60 2.75 that the IL intensity in matrices with strongly differing 1 Ð x x parameters, for instance, in ZnS and CdTe, follows a Mn1 Ð xZnxSe 2.38 2.52 2.68 2.93 similar temperature behavior, provided the Mn2+ con- Mn1 Ð xZnxTe 2.38 2.54 2.70 2.93 centration exceeds a few percent. This similarity is

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2019 accounted for by the fact that the major factor here is Table 2. Dependence of the shift of the Mn2+ IL band max- the migration of intracenter excitation, a process which imum on hydrostatic pressure P measured on three DMS will be considered below. types with different magnetic component concentrations ÐdE /dP DMS x Mn (cmÐ1/GPa) 4. EFFECT OF HYDROSTATIC PRESSURE ± ON INTRACENTER TRANSITIONS Zn1 Ð xMnxS 0.05 224 4 0.3 286 ± 8 The effect of hydrostatic pressure P on the energy of ± transitions between the 3d-shell levels of iron-group 0.5 303 5 ± ions is connected intimately with the level shifts Zn1 Ð xMnxSe 0.15 169 16 induced by variations in temperature or DMS elemental 0.25 181 ± 16 ± composition. In the final count, the problem reduces to Zn1 Ð xMnxTe 0.1 500 40 the magnitude and symmetry of the local crystal field 0.3 419 ± 16 and the degree of spÐd hybridization. It was established ± that the pressure dependence of the shifts of the Wan- 0.6 395 40 nier exciton, dEgx/dP, and of the intracenter transition in 2+ the Mn ion, dEMn/dP, in Cd1 Ð xMnxTe measured at T = perature [42] for Zn- and Mn-based DMSs with differ- 2 K are 520 and Ð410 cmÐ1/GPa, respectively [41]. At ent Group-VI elements are listed in Table 2. T = 40 and 77 K, the manganese level shift rates grow to Ð450 and Ð600 cmÐ1/GPa, respectively; i.e., the tem- An analysis produced the following results [42]. perature dependence of the shift is nonlinear. The pres- The main characteristics of the crystal field are its force Dq and two Racah coefficients, B and C, for which the- sure-induced shifts of the band and intracenter transi- ory yields for the B/C ratio values ranging from four to tions have opposite signs, so that at certain values of P, five [43]. In the case of Zn Mn S, the theory can be depending on x, the threshold E of the intracenter tran- 1 Ð x x t reconciled with experiment for Dq = 500 cmÐ1 and B = sition turns out to be below the exciton transition Ð1 energy even for x < 0.4. After the level crossing, the IL 610 cm . The magnitude of Dq scales with the distance Ð5 band starts to grow in intensity and the exciton emis- a0 between the Mn and S ions as a0 , and calculations sion weakens. The low-energy shift correlates with suggest that a0 changes by 0.014 nm in the pressure inter- crystal-field theory for ions in that compression val from 0 to 10 GPa, whence dDq/dP = 18 cmÐ1/GPa. 4 2+ Ð1 enhances the field and lowers the first excited level T1 For ZnS:Mn , dB/dP = Ð3.5 cm /GPa. According to 6 relative to the A1 ground state; numerical calculations [44], the dependence of intracenter transition energy on of the pressure-induced shifts yield, however, pressure can be cast as −160 cm−1/GPa, which is considerably less than the () experimental value [42]. This disagreement is most dEMn/dP= rd Dq /dP (3) likely connected with the fact that the calculation used + []E Ð Dqr B Ð 1dB/dP, does not go beyond the usual crystal-field theory and Mn does not assume strong hybridization of the Mn2+ 3d where r determines the slope of the EMn/B dependence states with s- and d-type orbitals of other atoms. Later on Dq/B in the diagrams presented in [43]. For the 4T Ð calculations [43] wherein the hybridization was 1 6A transition, one obtains dE /dP = 300 cmÐ1/GPa, included yielded a larger pressure-induced shift of the 1 Mn which is in accord with the experimental value (267 ± 4T level. After the IL band has appeared with increas- 1 9) cmÐ1/GPa for Zn Mn S. ing P, interband exciton emission gradually decays and 1 Ð x x finally almost disappears. Two possible reasons were It was found that a hydrostatic pressure of up to suggested for the weakening of the exciton emission 13 GPa drives a phase transition in Zn1 Ð xMnxS (0.01 < with x increasing above 0.4: (i) enhanced relaxation via x < 0.1) from the zinc-blende to rocksalt structure [45]. the Mn2+ 3d levels and (ii) degradation of the solid- This transition is accompanied by a radical change in solution quality at high x, which reduces the quantum the IL spectrum, with the 2.0-eV band dying out and a yield of the exciton luminescence. Hydrostatic pressure new band peaking at 1.4 eV and having a considerably experiments indicate that the first reason is more essen- higher quantum yield that forms and rises in intensity tial, because the exciton luminescence is observed to (Fig. 4). These experiments permit one to estimate the weaken under elastic strains that do not degrade the change in the ligand field induced by the phase transi- crystal quality. tion, as well as to use the IL as a probe to detect structural transformations. IL studies have established that The pressure-induced shifts of the Mn2+ IL band temperature annealing carried out after stress removal maxima toward lower energies measured at room tem- restores the sphalerite structure practically completely.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2020 AGEKYAN

5. COMPETITION (a) BETWEEN THE INTRACENTER AND BAND CHANNELS OF RADIATIVE RECOMBINATION There is ample evidence supporting the existence of a connection between the interband and intracenter electronic states, for instance, the onset of ultrafast 1 phase relaxation of Wannier excitons in IIÐVI crystals 2 following their doping by iron-group ions, the presence of a peak corresponding to the exciton energy Egx in the IL excitation spectra, etc. In the cases where the DMS band gap width E 1 g (b) (more speciﬁcally, the exciton energy Egx) does not 2 exceed the threshold energy of the intracenter transition Et, the IL is weak. Under these conditions, energy transfer from the ﬁrst excited level of the 3d shell to band Luminescence intesity, arb. units states is more likely than a radiative transition to the ground state of a magnetic ion. If Egx > Et, the situation changes radically and the DMS may exhibit radiative relaxation of optical excitation of two types, conventional interband and intracenter. Their relative intensity 3 is governed by a number of factors, which we will dis- 4 cuss through the example of Cd1 Ð xMnxTe and 2.0 1.5 1.0 Zn1 − xMnxSe. Energy, eV The band gap Eg(x) in Cd1 Ð xMnxTe broadens with Fig. 4. Effect of hydrostatic pressure on the Mn2+ IL spec- increasing Mn concentration, so that at x = 0.4 it trum in a ZnxMg1 Ð xS crystal [45]. (a) IL spectra (1) in the reaches Egx(0.4) = Et. For x slightly in excess of 0.4, the original cubic crystal and (2) after application of a hydro- peak intensities of the IL, I3d, and of the Wannier exci- static pressure of 13 GPa; x = 0.07; (b) IL after removal of ton luminescence, Ix, are similar for the excitation hydrostatic pressure: (1) before annealing, (2, 3, 4) after Ð2 annealing for 2 h at 200, 400, and 500°C, respectively; x = intensity Ie = 1 W cm . A further increase in x results 0.04. The IL spectra were obtained at T = 77 K. in a weakening of the exciton luminescence; indeed, for x = 0.5, Ix is smaller than I3d by two orders of magnitude and for x = 0.6, by three orders of magnitude for the same value of Ie [41]. As x increases, Egx(x) falls into the 0.8 region of an ever increasing density of excited states of 2+ 0.7 the Mn 3d shell. The relaxation rate from these levels 2+ 0.6 to lower levels of the Mn ion is high, which leads to a 0.5 weakening of radiative relaxation via the conventional Reflectivity semiconductor mechanisms, thus enhancing the IL. 2.80 2.81 2.82 2.83 Photon energy, eV The excitation spectra of the Zn0.95Mn0.05Se reveal a nonmonotonic variation of Ix and I3d with increasing ν excitation photon energy h e [46]. As seen from Fig. 5, the values Exciton ν បω emission h e = Egx + k LO (4)

2+ ω Luminescence intesity, arb. units Mn emission (where LO is the longitudinal optical phonon fre- Egx +1 LO +2 LO +3 LO +4 LO quency and k = 0, 1, 2, …) are most favorable for the interband exciton luminescence. In this case, hot exci- 2.79 2.83 2.87 2.91 2.95 2.99 tons relax rapidly to the lowest exciton level upon emit- Excitaton energy, eV ting an integer number of LO phonons. If, however, condition (4) for hν is not upheld, the final stage of 2+ e Fig. 5. Excitation spectra of the Mn IL and exciton lumi- relaxation to the exciton level occurs slowly and with nescence in a Zn0.95Mn0.05Se crystal obtained at T = 4 K the emission of a large number of acoustic phonons; in [46]. Inset shows a reflectivity spectrum of a crystal with an excitonic structure near 2.803 eV. Arrows identify the posi- other words, the probability of excitation transfer to the tion of the exciton peak E and the positions spaced from manganese ion levels increases. The same effect is gx ν it by an integer number of LO phonon energies. observed at a fixed h e if the crystal is placed in an

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2021 external magnetic field H. Due to the giant Zeeman (a) (b) splitting of exciton levels, a relation similar to Eq. (4) holds for the lowest exciton Zeeman component at cer- ν 2 tain values of H and fixed h e. 1 Zn0.99Mn0.01Se was used to study the effect that resonant excitation of one of the Zeeman splitting compo- 12 nents of the n = 1 exciton exerts on the Mn2+ IL polar- ν Intensity, arb. units ization [47]. If h e and H are chosen such that the σÐ Transmission, arb. units energy of the photon is equal to that of the polarized 01020Excitation intensity, upper exciton Zeeman component, the IL band inten- t, ns arb. units sity in the σÐ polarization weakens and that in the σ+ polarization is enhanced. Such a resonant excitation of Fig. 6. Optical transmission of a 0.05-cm thick one of the polarized exciton components reduces the Zn0.99Mn0.01Se crystal obtained under strong pumping by YAG:Nd3+ laser second-harmonic pulses into the region of integrated I3d IL intensity, while the IL polarization 2+ 6 4 exceeds 50%. This experiment can be interpreted in the Mn A1Ð T1 intracenter transition. T = 280 K [51]. 2+ (a) Shape of the laser pulse on transmission through the terms of spin-dependent exciton capture by the Mn 3d Ð2 levels under the competition of exciton and intracenter crystal at pulse peak intensities Ie (W cm ): (1) 0.5 and radiative recombination channels, where the selection (2) 1; the dashed profile is the original laser pulse shape. rules for optical transitions in the 3d shell are weakened (b) Optical bistability hysteresis loops describing the dependence of transmission on Ie at laser pulse peaks for by the exchange interaction. cases (1) and (2) of item (a). Evidence of a redistribution of radiative recombination between the interband and intracenter channels, ciated with the acceptor concentration decreasing as a depending on the relative energies of the corresponding result of the Fe2+ÐFe3+ charge transfer [50]. transitions, is provided by the effect of hydrostatic pressure on the Cd1 Ð xMnxTe luminescence, which was considered in the preceding section. 6. NONLINEAR ABSORPTION AND BISTABILITY The luminescence intensity redistribution in favor of the interband exciton emission has been observed in The long excited-state lifetime of the Mn2+ ions pro- Cd0.6Mn0.4Te placed in a magnetic field [48]. At the vides a possibility of observation of the nonlinear level of nonselective excitation used in the experiment optical properties associated with intracenter transi- ν tions at comparatively low excitation levels. 7-ns-long (h e > Egx), the band exciton emission and the IL measured in a zero magnetic field had an approximately YAG:Nd3+ laser second-harmonic pulses have been equal intensity, but a field H = 5 T enhances the exciton used in pump-probe experiments on Zn1 Ð xMnxSe emission by an order of magnitude compared to the IL. (x = 0.01) [51]. At excitation levels Ie of the order of 4 Ð2 6 4 While this trend can be explained as being due to a shift 10 W cm , intracenter absorption (the A1Ð T1 and σ+ 6 4 of the exciton component toward lower energies A1Ð T2 transitions) is observed to weaken. For Ie > with increasing H, one does not succeed in fully 105 W cmÐ2 and at room temperature, the shape of the describing the intensity redistribution between the two laser pulse becomes distorted in passing through the emission channels in this way. The efficiency of excita- crystal and optical bistability simultaneously sets in tion transfer from band excitons to the 3d shell is appar- (Fig. 6). The up- and down-switching times are 9 and ently affected by the spin selection rules, which gain in 5 Ð2 1.8 ns (Ie = 10 W cm ) and 6 and 0.8 ns (Ie = significance under conditions where the intracenter and 6 Ð2 exciton levels are magnetic-field-split. A later study 10 W cm ), respectively, the optical hysteresis loop in [49] discusses the effect of the resonance between the the latter case being three times wider. The onset of band excitons and intracenter transitions in bistability is apparently accounted for by the positive feedback forming in the reflection of light inside the Cd1 − xMnxTe on the efficiency of the dipoleÐdipole energy transfer mechanism. crystal from its polished surfaces under increasing sample transparency. The parameters of optical bistability Besides creating a new recombination channel, ions in the region of 3d transitions in Zn1 Ð xMnxSe should be of the iron group are capable of acting directly on the determined by the concentration of the Mn2+ ions, the impurity states involved in various kinds of lumines- lifetime of their excited state with inclusion of excita- cence. Doping CdTe with Fe2+ suppresses the donorÐ tion migration, the optical pumping level, and tempera- acceptor emission, because iron associates with the ture. It should be pointed out that the absorption satura- components of the donorÐacceptor pairs. Increasing the tion and nonlinear properties in the region of intrac- iron concentration from 1017 to 1019 cmÐ3 weakens the enter transitions in iron-group ions present in IIÐVI emission of excitons bound to neutral acceptors by one crystal matrices remain inadequately understood; how- and a half orders of magnitude. This is most likely asso- ever, there is currently a renewed interest in this prob-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2022 AGEKYAN

30 decreasing nanocrystal size and reaches as high as 18% (Fig. 7), and the luminescence decay rate is several orders of magnitude lower than that in bulk crystals (one nanocrystal contains, on the average, one Mn atom) [55]. 20 The IL quantum yield is given by ητÐ1()τÐ1 τÐ1 Ð1 = R R + NR , (5) 10 τÐ1 τÐ1 where R and NR are the radiative and nonradiative

Quantum efficiency, % recombination rates, respectively. The magnitude of τÐ1 NR is determined by the number of surface atoms in a nanocrystal per unit volume, which is proportional to Ð1 τÐ1 0246810 D , and R depends on the degree of zinc substitution Diameter, nm by manganese. If a nanocrystal contains, irrespective of Fig. 7. IL quantum yield η of a manganese ion embedded in τÐ1 Ð3 a ZnS nanocrystal with diameter D. Points are experimental its size, a fixed number of Mn atoms, then R ~ D . In data for T = 4 K, and the line is a plot of the fitting relation this case, η ~ (1 + βD2)Ð1, where the parameter β η = (1 + βD2)Ð1[55]. τ τ depends on the ratio R/ NR. Estimates [55] show that the IL quantum efficiency in a nanocrystal with D = lem, particularly in what concerns the divalent ions of 3 nm increases by an order of magnitude compared chromium. with a bulk crystal, which is in accord with experiment. Confinement shifts the 3d levels into the region of the 4s band states, and the recombination of 4s electrons 7. INTRACENTER TRANSITIONS with holes can take place both directly and via the 3d IN DMS-BASED NANOSTRUCTURES states. In this case, transitions involving 3d electrons Quasi-two-dimensional DMS-based quantum struc- become parity-allowed. Thus, two properties of nanoc- tures, particularly CdTe/Cd1 Ð xMnxTe, are traditionally rystals appear to be essential: (i) fast excitation transfer used in optical studies. Recently, nanostructures includ- from band states to the 3d shell of the magnetic ion as ing wide-band-gap DMSs, where intracenter transitions a result of the excitation being localized within one τÐ1 are observed to occur already at low concentrations of nanocrystal and (ii) an increase in R by several orders the magnetic component, have received increasing of magnitude through enhanced spÐd mixing. Other- interest. Superlattices and nanocrystals doped by opti- τÐ1 wise, if R were to remain the same as in a bulk crystal cally active centers have potential application. In addi- Ð1 tion, such centers can be used as probes to study the (i.e., about 2 ms ), the fast excitation transfer to the properties of nanostructures. Self-organized DMS Mn2+ ion and the slow radiative recombination would nanocrystals 2 nm high and 20 nm in diameter have bring about IL saturation already at low interband optical excitation levels. Confinement produces the stron- been grown on the ZnTe(100) surface [52], Cd1 Ð xMnxS quantum wires 3.5 nm in diameter have been prepared gest effect on the sÐd mixing of excited 3d states in ZnS:Mn2+ nanocrystals, starting from D = 2Ð3 nm. by organometallic vapor deposition in pores of SiO2 (MCM-4) [53], and ZnS:Mn2+ has been embedded in a These nanocrystals were employed to develop the con- photonic crystal consisting of submicron-sized poly- cept of an atom under the conditions of quantum con- mer spheres [54]. One can presently prepare high-qual- finement [56] where the properties of an atom and its 2+ coupling with the matrix are modulated by the nanoc- ity IIÐVI nanocrystals containing one Mn atom per 3+ nanocrystal. Such a nanocrystal is a model object for rystal size. An analysis of Y2O3:Tb nanocrystals, use in studying the single magnetic moment in a quan- where the luminescence efficiency is also several times tum dot. Studies of the optical properties, including IL, higher than that in the bulk material, supports the valid- in various DMS-based nanostructures require investi- ity of this approach. gation of the interaction of the 3d states of magnetic The time interval between the interband excitation ions with the s and p states under the conditions of spa- pulse and the IL intensity maximum in ZnS:Mn2+ tial confinement of the electron wave functions. nanocrystals was experimentally shown to be less than Enhanced spÐd hybridization should increase the 0.5 ns; in other words, excitation transfer to a magnetic energy transfer rate from the electronÐhole pairs to the ion is indeed a fast process. The crystal field acting on 3d shell in interband optical excitation of DMS nanoc- this ion also changes somewhat when ones crosses over rystals. The IL quantum yield of ZnS:Mn2+ nanocrys- from a bulk crystal to a nanocrystal, which becomes tals with a diameter D of about 3 nm (the Bohr radius manifest in a small shift in the IL maximum; this factor of the Wannier exciton is 2.5 nm) increases with cannot, however, affect the spÐd coupling noticeably.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2023

The acceleration of excitation transfer to magnetic ions firmed by ESR measurements. These systems revealed was confirmed in a number of later studies, where spe- a shift in the IL maximum toward higher energies with cial measures were taken to ensure uniform manganese decreasing average nanocrystal size and an increase in distribution over the nanocrystals. It was shown theo- the IL quantum yield to 20% at room temperature and retically that the conduction band in a ZnS nanocrystal to 75% below 50 K. with saturated surface bonds and with a Mn2+ ion at its Doping IIÐVI nanocrystals with magnetic ions sup- center is modified by interaction with the 3d states of presses the impurityÐband and interimpurity emission, manganese and that this interaction becomes enhanced as well as the luminescence originating from surface with decreasing nanocrystal size, whereas the valence- states, which gain considerably in significance for band states remain virtually unchanged. CdS nanocrys- small crystal sizes. CdS:Mn2+ nanocrystals 4 nm in tals containing one Mn2+ each revealed a giant splitting average size exhibit strong IL that peak at 2.12 eV but of exciton spin sublevels in a zero external field [57]. lack the surface state emission characteristic of This splitting permits one to estimate the giant internal undoped CdS nanocrystals [62]. This confirms the effi- field induced by the manganese ion. The magnitude of ciency of excitation transfer from the band, impurity, the field is evidence of an enhancement of the short- and surface states to the 3d shell of magnetic ions under range spin–spin interaction in the case of confinement. the conditions of spatial confinement, where defects are The authors of [58] questioned the concept of rear- located close to the magnetic ion. rangement of the spÐd interaction connected with con- The strong influence of the lowered dimensionality finement and assigned the fast decay of the lumines- of a system on the IL kinetics has been demonstrated in cence observed to occur [55] at 2.1 eV to the superpo- a study of a bulk zinc sulfide crystal with planes doped sition of the tail of the conventional ZnS impurity by manganese [63]. The IL measured from two-dimen- luminescence on the IL band. Studies of CdS:Mn2+ sional layers with a Mn concentration of about 1014 cmÐ2 nanocrystals grown in a polymer matrix revealed that has a weaker temperature dependence, and its decay at τ the IL decays in them with R = 1.5 ms, which correlates low excitation levels proceeds faster than that in the with the Mn2+ IL kinetics in a bulk CdS:Mn2+ crystal three-dimensional crystal because of the spÐd hybrid- [59]. In all the manganese-containing nanocrystals ization enhancement discussed above, as well as due to studied, the temperature quenching of the IL is far the lattice strain caused by the Mn doping. This system weaker than that observed in bulk crystals. While in a was found to be convenient in determining the critical bulk CdS:Mn2+ crystal the IL intensity measured in the distance between doped layers at which interlayer 3d- interval 4Ð100 K decreases by a factor of ten, the IL in shell excitation transfer starts. This distance was esti- a nanocrystal weakens only by one half in the region 4Ð mated to be 9 nm [63]. This figure, equal to 30 cation 300 K. separations, appears to be overevaluated, because it There are apparently sound grounds, both theoreti- does not agree well with the minimum manganese con- cal and experimental, to believe that the excitation centration at which the migration of intracenter Mn2+ localization and the enhancement of 3d-state hybridiza- excitation sets in in uniformly doped bulk IIÐVI matrices. tion with band states in DMS nanocrystals strongly As found in the preparation of a bulk Zn1 Ð xMnxSe affect the quantum yield and kinetics of the IL. The crystal (0.15 < x < 0.25) containing CdSe nanocrystals, conflicting character of the data obtained in different manganese ions play an important part in CdSe nucle- studies originates most likely from technological fac- ation [64]. Application of a magnetic field of up to 5 T tors. The technique used in the preparation and subse- to this system enhances exciton luminescence in the quent treatment of samples containing nanocrystals of nanocrystals by an order of magnitude while suppress- IIÐVI compounds with a magnetic component strongly ing the IL of the Zn1 Ð xMnxSe matrix. The correlation influence the luminescent properties. Different systems between these effects, which has been established on a with ZnS:Mn2+ nanocrystals having the same original large number of samples, originates from a change in manganese concentrations but grown using different the efficiency of excitation transfer between the DMS techniques exhibit a strong scatter in the IL intensity. It and nonmagnetic nanocrystals in an external magnetic was shown in [60] that a microemulsion containing field; however, the details of this process remain 2+ ZnS:Mn nanocrystals subjected to hydrothermal unclear. Energy transfer from CdSe to Zn1 Ð xMnxSe is a treatment had a 60 times higher Mn2+ IL quantum yield process whose selection rules depend on the values of than a similar material obtained in an usual aqueous the initial and final spin states. For this reason, the mag- reaction. The factors responsible for the enhanced IL netic-field-induced splitting of band states and of the 3d quantum yield are the formation of nanocrystals with a levels can strongly affect the relative magnitude of the perfect zinc-blende structure in the course of growth excitation transfer time and the characteristic radiative and efficient passivation of the nanocrystal surface, times. Remarkably, this effect is critical to nanocrystal both during and after growth. An efficient method for size; more specifically, if the thickness of the CdSe uniformly distributing manganese was developed to interlayers of which the island nanocrystals form prepare ZnS:Mn2+ nanocrystals containing, on the increases from 1.5 to 2.5 monolayers, the redistribution average, four Mn atoms each [61], which was con- of the luminescence in a magnetic field disappears.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2024 AGEKYAN

tion of the barrier 3d shell is capable of considerably slowing down the decay of excitonic luminescence in

4 6 2+ 6 4 2+ the CdTe quantum wells because of the excitation being Τ1 → Α1(Mn ) Α1 → Τ1(Mn ) (Cd Mn Te) Eg 0.4 0.6 transferred from the barrier Mn2+ ions to the quantum wells. CdTe/Cd1 Ð xMnxTe quantum-well structures with x = 0.5 and 0.68 containing quantum wells from 1 to 10 nm in width were studied under a hydrostatic pres- QWCdTe sure of up to 3 GPa in [67]. The pressure-induced energy shifts obtained at 80 K constitute 640 cmÐ1/GPa 1 3 2 for the IL and 600 (x = 0.5) and 480 cmÐ1/GPa (x = 0.68) for the exciton luminescence of the quantum wells. Because of the pressure-induced shifts assuming oppo-

Luminescence intensity, arb. units site signs at certain values of pressure determined by x and the quantum-well width, the exciton energy starts 2+ 2.0 2.2 2.4 2.6 to transfer from quantum wells to the Mn levels in the Energy, eV barriers, a process detected from the threshold enhancement of the IL. 2+ Fig. 8. Mn IL profile (solid line) and IL excitation spectra A photonic crystal consisting of submicron-sized (dashed lines) obtained on (1) a bulk Cd Mn Te crystal 0.4 0.6 polymer spheres with voids filled by the ZnS:Mn2+ and (2) a quantum-well structure CdTe/Cd0.4Mn0.6Te [66]. Curve 3 is the exciton luminescence excitation spectrum DMS has become a new object for optical studies [54]. 6 4 In the case where the DMS IL band falls into the stop- from CdTe quantum wells. A1Ð T1 is the transition to the first excited state of the manganese ion; QWCdTe is the band region of the photonic crystal, the decay kinetics transition between the first levels of electrons and heavy over the band profile varies in an unusual manner. The 2+ holes in the CdTe quantum well; and Eg(Cd0.4Mn0.6Te) is ZnS:Mn IL decay is the slowest at the IL band center the edge of the interband transition in the barrier. T = 4 K. if the latter coincides in energy with the minimum of the photonic-crystal transmission. The reason for this is the low density of photon states in the stop band, which Suppression of the energy transfer from band excitons to is initiated by the strong difference in the dielectric the Mn2+ 3d shell by a magnetic field was observed to properties between the polymer spheres making up the occur in Zn1 Ð x Ð yCdxMnySe/ZnSe and Zn1 Ð xMnxSe/ZnSe photonic crystal and the ZnS:Mn2+ filler. The parame- epitaxial layers and in Cd1 Ð xMnxSe and Cd1 Ð xMnxS ters of the photonic-crystal stop band are an essential quantum wires [65]. A conclusion was drawn in [65] point. By properly selecting the diameter of the nano- that the efficiency of excitation transfer in a magnetic particles, one can control the width and energy position field is determined primarily not by the degree to which of the stop band and overlap, partially or completely, the excitonic and intracenter transitions are resonant the IL profile, to entrap, in this way, the radiation in the but rather by an enhancement of the role played by the photonic crystal. Moreover, by introducing defects of spin-dependent selection rules. This is in accord with an appropriate type and concentration into a photonic the above data on bulk Cd1 Ð xMnxTe crystals. crystal, one can also control the formation of transparency regions in the stop band. Such systems combine a The Mn2+ IL at T = 2 K was studied in high efficiency of electro- and photoluminescence with CdTe/Cd0.4Mn0.6Te quasi-two-dimensional quantum- a possibility of controlling the spectral composition of well structures in [66]. Time-resolved experiments per- the emission. ∆ formed with a t = 0 delay reveal exciton emission The influence of confinement on the properties of 3d from the CdTe quantum well with a maximum at luminescence is still not fully clear. This phenomenon, 2.02 eV, and only the IL of barrier Mn2+ ions peaking at while being of interest in itself, is connected intimately 2.0 eV is left in the spectrum for ∆t = 2.5 µs. Because with the properties of electroluminescence in nano- of the small thickness of the structure, all of the Mn- structures and has potential application. One can iden- containing layers are excited, even under optical band- tify at least three reasons accounting for the specific to-band-type pumping occurring with a high absorption features of the IL in nanostructures; these reasons are coefficient. The energy is subsequently transferred the change in the degree of spÐd hybridization, excita- from the barrier band states to the Mn2+ ions, thus pro- tion transfer between the nanocrystal and the matrix (or viding a more efficient excitation of the 3d shells than between a quantum well and the barrier), and spatial obtained in direct intracenter excitation. For this rea- confinement of migration. son, the maximum of the IL excitation spectrum lies at A conclusion was drawn in [68, 69] that the 3d lev- the fundamental edge of the barrier with Eg approxi- els of transition ions correlate with the corresponding mately equal to 2.6 eV rather than near the excitation matrix levels. In particular, it was maintained that the 2+ threshold of the Mn 3d shell at Et = 2.15 eV, as is the difference between the intraion transition energies in case with bulk crystals (Fig. 8). The long-lived excita- isovalent semiconductors is equal to that between the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2025 valence-band top energies. This provides a possibility ing will not be completely overcome. Moreover, under of using intracenter transitions as probes to determine the conditions of excitation migration, inhomogeneous the height of the heterojunction barriers for the broadening should depend on the optical pumping electrons and holes separately. This point was studied level, because the lowest levels of an inhomogeneous on the heterostructures Zn1 Ð xCdxSe/ZnSe and distribution saturate with increasing excitation. 3 2+ ZnS1 − xSex/ZnSe with an Ni impurity [the T1(F)Ð Studies of the Mn IL in Cd1 Ð xMnxTe show that in 1 T2(F) transition] over a broad range of x, up to 0.5 [70]. the interval 4Ð20 K, the inhomogeneous and homoge- ν While the measurements yielded good agreement with neous broadenings for h e > 2.2 eV are approximately theory, the results obtained in the case of equal. Within the concentration interval 0.46 < x < 0.70, ZnS1 − xSex/ZnSe are at odds with the data derived from the inhomogeneous broadening varies from 70 to measurements of optical transitions between size quan- 85 meV, with the IL proﬁle shape being Gaussian. tization levels. There is nothing strange in there being an absence of strong concentration dependence of the inhomoge-

2+ neous broadening at such high manganese concentra- 8. SELECTIVE EXCITATION OF Mn tions. The inhomogeneous broadening should even INTRACENTER LUMINESCENCE decrease with increasing x, because the maximum dis- IL in Cd1 Ð xMnxTe with x > 0.4 is excited by photons order in a solid solution sets in usually at x = 0.5. The ν increase in H with increasing x in this interval is con- with h e > 2.14 eV. In crystals with x = 0.5 pumped in ν nected most likely with an increasing concentration of selectively in the interval 2.14 < h e < 2.25 eV, the IL maximum shifts, accordingly, from 1.98 to 2.005 eV structural defects, because x = 0.7 is almost the highest ν possible concentration for bulk Cd Mn Te crystals. [71, 72]. As h e is increased further from 2.25 eV up, 1 Ð x x the IL maximum shifts again toward lower energies The assumption [72] of the temperature-induced varia- [72, 73]. This behavior of the IL remains unclear; one tion of Hin being somehow connected with magnetic can, however, put forward several possible reasons for phase transitions in Cd1 Ð xMnxTe is not supported by its occurrence. Nonmonotonic variation of the intrac- the results of studies on the effect of temperature on the enter absorption coefficient within the scanned region IL profile. No changes in the profile shape of the IL or (several absorption bands) affects the optical excitation in its integrated intensity have been found at the para- density. Another possible reason consists in the onset of magnetÐspin-glass and paramagnetÐantiferromagnet resonance conditions for excitation of the states in an phase transitions for x in the interval 0.4Ð0.7. inhomogeneously broadened IL profile whose energy At liquid-helium temperatures, anti-Stokes IL is ν ν differs from h e by the LO phonon energy. A similar excited in crystals with x > 0.4 by photons with h e < Et effect is observed for the inhomogeneously broadened in a two-stage process, which is suppressed as the sam- bound-exciton emission band in Cd0.5Mn0.5Te [74]. ple is warmed up to 80 K, apparently, because of the A configurational model was proposed in [72] to decreasing electron lifetime in the intermediate state account for the IL spectrum obtained under selective [71, 73]. There are grounds to believe that it is this excitation within an inhomogeneously broadened pro- parameter that is associated with temperature-driven file. In this model, the dependence of the position of the magnetic phase transitions [48]. ν IL maximum, EL, on h e can be written as dE /dh()ν = H2 /()H2 + H2 , (6) 9. KINETIC PROPERTIES OF INTRACENTER L e in h in LUMINESCENCE where Hh and Hin are the homogeneous and inhomoge- The kinetic properties of the IL depend on many fac- neous broadenings. Hh, Hin, and the full width at half tors, most of all on the concentration of the magnetic maximum of the IL profile, H, can be cast in the form component. Study of the concentration dependences is associated with certain difficulties. In II–VI wide-band- 2 2[]2 ()2 2 H = Hh 1 + Hin/ Hh + Hin , (7) gap matrices, it is often difficult to dope iron-group elements in high enough concentrations, while in the com- []()()1/2 Hh = H 1 + R /1+ 2R , (8) paratively narrow band-gap materials of the type of

1/2 CdTe, IL is observed only for high cadmium substitu- Hin = Hh R , (9) tions by manganese. Combining of the data obtained on different matrices is hampered by the fact that the spe- 2 2 ν where R = Hin /Hh . Thus, by sweeping h e and measur- cific properties of a matrix affect the IL kinetics. This ing the shift of the IL maximum, one can calculate the point can be exemplified by the observation that as the value of R. A comparison of the model with experiment sulfur concentration in the ZnSxSe1 Ð x matrix is suggests that for T > 55 K, the inhomogeneous broad- increased from 0.001 to 0.3, the decay time of the Co2+ ening disappears [72]. It is difficult to conquer with this IL L band (2.36 eV) decreases by more than five times conclusion. If an excitation can relax during its lifetime [75]. Partial substitution of cadmium for zinc in ZnSe with a lowering of energy, the inhomogeneous broaden- brings about a similar result. These changes may be

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2026 AGEKYAN associated with the system becoming disordered in the tem of excited states is generally completed during the formation of a solid solution, with changes in the 3d- excitation lifetime. According to current models, in the state interaction with ligands and in the spinÐorbit case of excitation near the intracenter absorption τ interaction. The characteristic IL decay time R can be threshold, the relaxation of excited states occurs in two written as stages. More specifically, it occurs due to fast relaxation by 0.2 eV in 1 ps (via the electronÐphonon cou- τ []()αρ ρ 1/ R = p/ 3d + C + A , (10) pling) to the minimum of the configurational curve for the 4T level of the Mn2+ photoexcited ion, followed by where α is determined by the spÐd hybridization, ρ is 1 p a slow relaxation within a few hundredths of an elec- the valence electron LS coupling, ρ is the LS coupling 3d tronvolt over the inhomogeneously broadened 4T lev- for the 3d electrons in a magnetic ion, and C and A are 1 constants. The valence band in IIÐVI crystals is formed els of the manganese ion ensemble in a few tens of ρ microseconds. When excited substantially above the by the p electrons of the anions, and the magnitude of p ρ intracenter absorption threshold, an intermediate pro- varies strongly in the SÐSeÐTe sequence. As for 3d, this cess with a characteristic time of less than 0.1 µs quantity depends very little on the actual IIÐVI matrix for becomes operative, which corresponds to the onset of a a given magnetic ion. One should also bear in mind the “preliminary” equilibrium in the system of Mn2+ decrease in the spinÐlattice relaxation time with increas- excited ions possibly involving band states in Cd1 Ð ing concentration of the magnetic component [76]. ν xMnxTe [if h e > Eg(x)], after which subsequent varia- 2+ The Mn IL kinetics in Cd1 Ð xMnxTe can be studied tion of the population over the inhomogeneous profile only in crystals with a high manganese concentration, occurs only through excitation migration over the ions. wherefore the relaxation dynamics of the intracenter excitation is determined largely by its migration [73, 77]. Nonradiative excitation transfer occurs due to the 10. SATURATION OF INTRACENTER Coulomb interaction of electrons of the excited and LUMINESCENCE AND ITS KINETICS nonexcited atoms [78, 79]; this interaction has a UNDER STRONG OPTICAL EXCITATION dipoleÐdipole, dipoleÐquadrupole, and quadrupoleÐ Studies of the IL kinetics are carried out in a pulsed quadrupole component. In ions in which the dipole mode assuming a sufficiently high level of optical exci- transitions are at least partially forbidden, the nondi- tation. The behavior of a system of ions with an unfilled pole components may play an important role by provid- 3d shell under strong pumping is of interest in more ing a strong dependence of the excitation transfer prob- than one aspect. Excitation of an Mn2+ ion initiates an ability on the separation between magnetic ions. Fol- intracenter intercombination transition, with the spin of lowing [73], while the IL decay curves in Cd0.5Mn0.5Te the 3d shell decreasing from 5/2 to 3/2. This affects the measured at half maximum of the profile on both sides exchange ionÐion coupling, which is of particular sig- of the peak differ strongly from each other in the inter- nificance for nearest neighbor Mn2+ ions. Thus, on the val 2Ð60 K, they coincide at already 77 K. It is believed one hand, excitation reduces the total spin of the system that the kinetics becomes the same over the IL band of magnetic ions, while on the other, a weakening of profile in the case where thermal energy kBT exchange coupling between nearest neighbors destroys approaches the magnitude of the average jump in the magnetic order. These changes affect the paramag- energy occurring as the excitation transfers from one netic properties of a crystal (and thus, the strength of manganese ion to another. It was concluded that (i) the the internal field created in an external magnetic field) average energy change in a jump is 40 cmÐ1 and (ii) at in opposite ways. The decrease in the spin of 3d shells low temperatures (migration with a decrease in energy), influences the carrier–ion exchange interaction and, the excitation can transfer, on the average, over three hence, the magnetic polaron effect, the giant Zeeman manganese ions in the characteristic IL decay time splitting, and the Faraday effect. Thus, we deal here (about 25 µs). with a complex photoinduced variation of the magnetic More comprehensive studies of the IL kinetics per- properties of a crystal depending on the concentration of magnetic ions and of their fraction residing in the formed on Cd0.4Mn0.6Te have shown that the IL profile shape measured under selective excitation near the excited state. A comparison of the properties of IL 2+ ν under strong optical pumping with those of electrolu- Mn intracenter absorption threshold (h e = 2.175 eV) minescence at high carrier injection levels is also of depends strongly on temperature [80]. This dependence interest with respect to device applications. is determined by the efficiency of upwise excitation redistribution over the inhomogeneously broadened Under weak optical pumping of Cd1 Ð xMnxTe crys- excitation levels of an ensemble of manganese ions in tals with x > 0.4, the IL intensity I3d is higher than the energy during the excitation lifetime. At T = 2 K, this Wannier exciton luminescence intensity even at low process does not operate and the IL decay is exponen- temperatures and for the value most favorable for exci- tial. At T = 60 K, a broad spectrum of inhomogeneously ton emission, x = 0.4. However, already at excitation Ð2 broadened states is populated and the IL decay is essen- levels Ie on the order of a few W cm , the IL undergoes tially nonexponential. Energy relaxation within a sys- noticeable saturation [74], so that as Ie increases, the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2027 exciton emission becomes dominant. The dependence 1.0 of I3d on Ie at low manganese concentrations can be studied in wide-band-gap systems (ZnS:Mn2+ and oth- (a) ers). In such a system, saturation can be driven by a 2+ transition of a sizable fraction of Mn ions to the 1 excited state, which is favored by the long lifetime in 2 4 0.5 the T1 level. The IL saturation becomes manifest particularly strongly, however, at high manganese concentrations [74, 81], whereat excitation migration becomes a major factor. The history of the investigation of migration dates back to the cited publications [78, 79], with more comprehensive studies on the migration of excitation over the 3d and 4f shells of magnetic ions and of the 1.0 role played by this migration in spectral diffusion (b) reported in later communications (see, e.g., [82Ð84]). The integrated saturation of the IL 2-eV band in Luminescence intensity Cd1 Ð xMnxTe under excitation into the intracenter ν 0.5 2 absorption region [h e < Eg(x)] is presented graphically 1 in Fig. 9 [85, 86]. The fraction of excited Mn2+ ions can be estimated if one knows the value of x, the intracenter 4 absorption coefficient, the lifetime in the T1 excited state, and the pumping parameters. Estimates show that saturation is observed to occur already at comparatively low excited-ion concentrations. Thus, the saturation 0 0.5 1.0 Excitation level occurring at high Mn2+ concentrations is indeed con- 2+ nected with the excitation migration. This conclusion is Fig. 9. Mn IL saturation curves obtained on Cd1 Ð xMnxTe corroborated by the temperature dependence of satura- for (a) x = 0.7 and (b) 0.4 at (1) T = 4 and (2) 77 K [85, 86]. tion, which is far more efficient at T > 60 K than at 4 K The IL intensity is normalized against Ie = 1 (the value cor- at concentrations 0.45 < x < 0.75. It should be pointed responding to the excitation level 4 MW cmÐ2). out that for x = 0.4, the saturation becomes enhanced with decreasing temperature, provided the excitation proceeds into the interband transition region (Fig. 10a). (a) (b) The reason for this lies in that for x = 0.4, the Wannier 2 exciton level coincides at 80 K with the intracenter 1 absorption threshold (where the density of electronic states of Mn2+ is low), whereas at 4 K, this level shifts 1.8 2.0 2.2 into the region of higher densities of states, toward the E, eV 1 4 maximum of the A6Ð T1 intracenter transition; this facilitates excitation transfer from band states to the manganese ions. The quenching of IL accompanying excitation (c) (d) migration can be caused by relaxation on defects and by the cooperative effect (CE) [87]. While the CE (up-conversion) is usually associated with a manifestation of anti-Stokes luminescence, in our case it should be treated instead as the transfer of an additional excitation Luminescence intensity, arb. units to the already excited Mn2+ ion, a process in which anti- Stokes luminescence cannot be observed at all. In contrast to the defect mechanism, the CE is, first, a nonlinear effect and, second, practically cannot be saturated. 02460246 µ The Frenkel exciton concentration at a level of energy t, s ν h at time t for a concentration x can be written in the 2+ form Fig. 10. Mn IL kinetics obtained on a Cd0.5Mn0.5Te crystal at different excitation levels Ie and temperatures [89]. , ν , ν ()τ ν,, nt()= n()0 exp Ðt/ R f (),xt (11) (a, b) and (c, d) refer to points 1 and 2 on the IL profile (see inset), respectively; solid and dashed lines relate to the exci- Ð2 where function f determines the rate of excitation tation levels Ie = 4 and 0.06 MW cm , respectively; (a, c) transfer to other ions. Until recently, only transfer to T = 4 and (b, d) 77 K.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2028 AGEKYAN neighboring nonexcited ions was taken into account; in Cd1 Ð xMnxTe). In the opposite case (for instance, in 4 other words, the CE and the effect of the optical pump- Mg1 Ð xMnxTe), the T1 level is lower at the cluster center ing level were left out of consideration. The CE is also and the excitation will therein be localized at low tem- disregarded in Monte Carlo simulations of the IL kinet- peratures. ics in Cd Mn Te, even under the conditions of strong 1 Ð x x The system of excited 3d states in a DMS with a pumping [88]. This effect should dominate, however, in high magnetic-ion concentration is similar in some DMSs with a not too high defect concentration. There respects to localized Wannier excitons in semiconduc- is a substantial difference between excitation transfer to tor solid solutions. The similarity relates to the behavior a nonexcited and to an excited ion. Relaxation involv- of migration with temperature, the existence of a mobil- ing vibrational states is a fast process, so that energy ity edge, the kinetics of emission migration over the transfer to a neighboring ion occurs from the equilib- 4 inhomogeneously broadened band profile, etc. Natu- rium state of the T1 electronic level. Therefore, excita- rally, the energy and temporal characteristics of these tion transfer to a nonexcited ion requires that the systems differ strongly from one another. phonon system of the crystal donate an energy corre- 6 4 An interesting type of DMS is three-cation solid sponding to the Stokes losses of the A1Ð T1 transition. solutions in which two cations are nonmagnetic and When excitation is transferred to an already excited ion, one is magnetic. It has been established that the IL sat- practically no energy goes into the Stokes losses, uration in Cd Mn Mg Te with increasing I is much because the transition from 4T to higher states in an 1 Ð x x y e 1 less pronounced than that in Cd1 Ð xMnxTe [45]. This is acceptor ion entails a small Stokes shift. Adequate accounted for by the excitation migration being sup- description of the IL kinetics requires analysis of at pressed by doping Cd1 Ð xMnxTe with magnesium. Mag- least two important points, namely, (i) estimation of the nesium enhances the fluctuations of the local fields act- difference between the excitation transfer probabilities ing on the manganese ions, thus impeding intracenter to a nonexcited and an excited ion and (ii) determina- excitation transfer. The pronounced effect exerted on tion of the possible contribution of the CE to IL under the crystal field by the presence of magnesium among specific experimental conditions. Because the CE and the ligands is corroborated by the noticeable depen- relaxation at defects are significant only under efficient dence of the energy position of the Cd1 Ð x Ð yMnxMgyTe migration, a strong enhancement of intracenter excitation IL band on magnesium concentration (Fig. 3). transfer for T > 60 K entails as strong a decrease in the IL quantum yield; this exactly is observed in experiment. Considerable interest has been aroused recently in connection with studying the lasing effect involving 3d Let us turn to the dependence of the IL kinetics in magnetic ion transitions in semiconductor matrices, CdxMn1 Ð xTe on the optical-pumping level. Because of which might have application potential, in particular, in the excitation being redistributed over the inhomoge- medicine. Cr2+ ions doped into the ZnS, ZnSe, and 5 2 neously broadened excitation levels of the ensemble of Cd1 − xMnxTe matrices (the T2Ð E transition) in concen- Mn2+ ions, the IL intensity reaches a maximum on the trations of 1018Ð1019 cmÐ3 were studied as active lasing high-energy wing of the 2-eV band faster than it does centers [90Ð93]. The maxima of the Cr2+ absorption on the low-energy one. As follows from the IL kinetics and IL bands lie near 0.70 and 0.53 eV, respectively, (Fig. 10), the spectral response of decay at high pump- and the IL band halfwidth is 0.2 eV. Under sufficiently ing levels Ie manifests itself strongly even at a high tem- strong selective pumping by a pulsed laser, the Cr2+ IL perature. The variation of the IL kinetics with increas- transforms to a narrow band whose energy can be swept ing Ie is most pronounced at a high temperature and on from 0.50 to 0.45 eV depending on the actual resonator the high-energy wing, where the decay accelerates con- mirror characteristics. In the best ZnSe:Cr2+ samples siderably [89]. This implies the development of a fast prepared by thermal diffusion, lasing is achieved process associated with the CE. As follows from an already at 12 µJ with passive losses of 7% if the excita- analysis of the kinetics, one can isolate localized and tion is produced in the 0.7-eV region with an delocalized states in a system of Mn2+ ions. The local- YAG:Nd3+-pumped Ba(NO ) Raman laser [94]. ized states should most probably be identified with the 3 2 Mn2+ ions at the boundaries of manganese clusters, 4 where the T1 level is lower. Strong pumping saturates 11. CONCLUSION localized states, and, as a result, excitation migration The optical investigation of intracenter transitions in over states lying above the mobility threshold occurs iron-group ions embedded in crystal matrices, which even at low temperatures. Whether the inner or inter- has been pursued over the past two decades, allows one face regions in a manganese cluster correspond to local- to conclude that the properties of these transitions ized states depends on the elemental composition of the exhibit a rich diversity, depending on the actual type of matrix. In the case where the predominance of Mn in magnetic ion and crystal matrix used, the magnetic ion the nearest cation environment of Mn reduces the crys- concentration, the system dimensionality, excitation 4 tal field, the T1 level rises at the center of the cluster conditions, and temperature. Despite the numerous and localization occurs at the boundaries (the case of studies performed, the mechanism by which electronic

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 INTRACENTER TRANSITIONS OF IRON-GROUP IONS 2029 excitation is transferred among the ions and between 11. W. Busse, H.-E. Gumlich, and D. Theiss, J. Lumin. the crystal matrix and the 3d shell of a magnetic ion 12/13, 693 (1976). remains poorly understood. This makes investigation of 12. J. N. Murrel, S. F. A. Kettle, and J. M. Tedder, Valence crystal matrices containing more than one type of ion of Theory (Wiley, London, 1965; Mir, Moscow, 1968). the iron group and lanthanides an interesting field. Nev- 13. P. Oelhagen, M. P. Vecchi, J. L. Treeong, and ertheless, optics of the iron-group ions in bulk IIÐVI V. L. Moruzzi, Solid State Commun. 44, 1547 (1982). semiconductor matrices is fairly well known and we are 14. W. Zahorowski and E. Gilberg, Solid State Commun. 52, presently witnessing a shift in interest to nanostructures 921 (1984). doped by iron-group ions. The effect of confinement on 15. M. Taniguchi, L. Ley, R. L. Johnson, et al., Phys. Rev. B the degree of spÐd hybridization and the rate of excita- 33, 1206 (1986). tion transfer between the band and 3d states, on the 16. A. Balzarotti, M. De Crescenzi, R. Messi, et al., Phys. magnitude of electronÐphonon and electronÐion cou- Rev. B 36, 7428 (1987). pling, and on other characteristics of such systems will 17. M. Taniguchi, A. Fujimori, M. Fujisawa, et al., Solid become a major area of research in the years to come. State Commun. 62, 431 (1987). ∨ Confinement provides a possibility of purposefully 18. J. Mas ek and B. Velicky, Phys. Status Solidi B 140, 135 controlling the characteristics of the IL, in particular, of (1987). strongly raising its quantum yield at high temperatures. 19. A. Bonanni, K. Hingerl, H. Sitter, and D. Stifner, Phys. Technological progress in growing nanostructures Status Solidi B 215, 47 (1999). assumes precise doping of magnetic ions within one 20. D. Boulanger, R. Parrot, U. W. Pohl, et al., Phys. Status nanocrystal and their controlled arrangement over the Solidi B 213, 79 (1999). volume or on the surface, as well as development of 21. J. Dreyhsig, K. Klein, H.-E. Gumlich, and J. W. Allen, new composite materials with the use of DMS nanoc- Solid State Commun. 85, 19 (1993). rystals. Naturally, the performance of electrolumines- 22. C. Chen, X. Wang, Z. Qin, et al., Solid State Commun. cent devices will also be improved by basing them on 87, 717 (1993). nanocrystals and quantum planes containing optically 23. C.-L. Mak, R. Sooryakumar, M. M. Steiner, and active ions, primarily Mn2+. Investigation of the IL sat- B. T. Jonker, Phys. Rev. B 48, 11743 (1993). uration in the same activated media under strong optical 24. M. M. Moriwaki, W. M. Becker, W. Gebhardt, and and injection pumping likewise deserves attention. The R. R. Galazka, Solid State Commun. 39, 367 (1981). use of semiconductor matrices with magnetic ions as 25. J. F. MacKay, W. M. Becker, J. Spalek, and U. 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PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2031Ð2033. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1940Ð1942. Original Russian Text Copyright © 2002 by Prikhod’ko, Shibanova.

METALS AND SUPERCONDUCTORS

Microwave Field Effects in HTSC Single Crystals of Bi(2212) A. V. Prikhod’ko and N. M. Shibanova St. Petersburg State Technical University, Politekhnicheskaya ul. 29, St. Petersburg, 195251 Russia Received November 13, 2001

Abstract—Microwave-radiation absorption of single-phase Bi(2212) single crystals in a near-2-mm wavelength range is studied as a function of microwave ﬁeld overvoltage. A characteristic plateau is detected for the electric-ﬁeld orientation along superconducting planes. A possible mechanism for this behavior is considered on the basis of the concepts of percolation network and pinchlike current distributions. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION length range, i.e., under conditions when size-quantiza- Analysis of microwave-radiation absorption by tion effects are not manifested. Samples were in the high-temperature superconductors was found to be form of single-phase and multiphase single crystals effective in determining the properties of intrinsic which display, first, coherent conduction effects [4] Josephson junctions. The discovered oscillatory depen- and, second, weak-coupling effects [5]. dence of power on the radiation absorbed for different values of a constant magnetic field is associated with the existence of such junctions [1]. An analysis of the 2. SAMPLES AND EXPERIMENTAL experimental data makes it possible to single out the TECHNIQUE characteristic features of absorption. This primarily concerns the scales of mw fields which do not lead to a We used Bi(2212) single crystals prepared by spon- local loss of superconductivity. These features can be taneous crystallization of melt in air. The sample size explained [2] by size quantization effects in small loops was 0.8 × 1.7 × 3.7 mm. The crystal growth technology with an area on the order of hundreds of square does not differ from that described in [5, 6]. micrometers. It follows from [2] that the frequency dependence of the absorbed power has a peak at a fre- Microwave experiments are made using a strongly quency which is a periodic function of the reciprocal emitting slit in a wide wall of a rectangular waveguide. amplitude of the ac magnetic field. A further increase in We constructed an original setup for studying the pro- power leads to a more complex form of absorption. The file of a standing wave in an open dielectric resonator periodicity of absorption as a function of the field van- (ODR) loaded with a sample. The effectiveness of such ishes in this case, and mesoscopic fluctuations start experiments for an analysis of superconducting proper- being manifested [3]. Microwave probing of HTSC sin- ties was demonstrated in [4]. Here, we use a single- gle crystals revealed such features as Josephson har- mode teflon ODR with an inner radius of 0.325 mm, monics generation and coherent conductivity peaks. In outer radius of 0.470 mm, and length of 4 mm. Radial the former case, an analogue of Shapiro steps is also electric and azimuthal magnetic components of the observed. The experiments were carried out in the electromagnetic field at the resonator end face interact range of weak microwave fields for which mesoscopic with the sample surface. The resonator is pumped effects are not observed. through a 0.1 × 8 mm slit. The sample was cooled to In this work, we analyze the microwave properties 78 K. We used a microwave oscillator (operating in a of Bi(2212) single crystals (in a near-2-mm wavelength near-2-mm wavelength range) whose maximum elec- range), which depend on the magnitude of the micro- tric power P0 reached 4 mW (100%) and whose pulse wave field used for sample probing. repetition frequency was 103 Hz. The resonance fre- An analysis of the known features of absorption quency of the ODR was 143.52 GHz. Temperature shows that, according to [2], the maximal frequency at measurements of the signal power were carried out at which size-quantization effects are still observed the standing-wave antinode closest to the sample, while extends approximately to a few gigahertz. It is conceiv- the dependence on the normalized incident power able that such effects will not be manifested at frequen- (overvoltage) P/P0 was measured with the sample cies higher than the critical value. In addition, the arranged immediately above the slit. The sample orien- effects of Josephson harmonics generation are tation was specified by the mutual orientation of the observed only for multiphase samples. In this study, we electric field vector E in the slit and the normal n to the analyze microwave features in a near-2-mm wave- base plane of the sample.

2032 PRIKHOD’KO, SHIBANOVA

(a) (b) ×3

20 60 100 20 60 100

, arb.units (d) I (c)

20 60 100 20 60 100 P/P0, %

Fig. 1. Dependence of the output microwave power I on the overvoltage P/P0 for (a, b) single-phase and (c, d) multiphase samples for the ﬁeld orientation (a, c) E ⊥ n and (b, d) E || n.

3. EXPERIMENTAL RESULTS shows the dependence of the superconducting transi- AND DISCUSSION tion temperature Tc on the overvoltage for the ﬁeld orientation E || n in a single-phase sample. This depen- The main result is the following experimental fact: dence indicates that the superconducting state exists the dependence of the power on the overvoltage in a over the entire range of overvoltage values (0Ð100%). single-phase sample in ﬁelds corresponding to a 70% Furthermore, the change in the form of this dependence overvoltage displays a characteristic plateau that occurs at the same value of overvoltage for which the depends on the orientation. For a multiphase sample, a kink in Fig. 1b is observed. characteristic kink is detected at an overvoltage of 40%. Figure 1 shows the dependence of the output micro- It is well known that nonlinear effects associated wave power on the overvoltage for single-phase and with the emergence of electrical instabilities can be multiphase samples oriented in different ways at liquid- manifested in small volumes [7]. In this case, a current nitrogen temperatures. At room temperature, the fea- pinch can be formed, which leads to a change in elec- tures mentioned above are not manifested; i.e., these trophysical parameters. For example, a study of micro- features characterize the superconducting state. Figure 2 wave noise in the regime of current instability revealed saturation of the current noise of the low-resistivity state [8]. This effect is attributed to the pinch expan- 84.5 sion, which is analogous to the conservation of the cur- 84.0 rent density in the pinch observed earlier [7]. In accordance with Kirchhoff’s law, the peculiarities of radia- 83.5 tion must be associated with the peculiarities of 83.0 absorption at the same temperature and wavelength.

, K Actually, the absorption of microwave power in the c

T 82.5 electric-instability conditions [9] correlates with the 82.0 behavior of microwave noise. It is conceivable that the decisive factor for the manifestation of the observed 81.5 plateau is the emergence of electrical-instability 81.0 regions in the bulk of the sample, followed by the for- 0 20406080100 mation of pinchlike distributions of the mw current. P/P , % The centers for the emergence of such regions can be 0 the regions of nonuniform field distribution over the volume. It is well known that, in samples of a nonellip- Fig. 2. Dependence of the superconducting-transition tem- soidal shape, a magnetic field is distributed nonuni- perature Tc on overvoltage P/P0 for a single-phase sample; the field orientation E || n. formly in the bulk of a sample. The nonuniform field

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

MICROWAVE FIELD EFFECTS IN HTSC SINGLE CRYSTALS OF Bi(2212) 2033 distribution in the high-temperature superconductors channels with a higher transition temperature in the measured in [10] indicates the emergence of a geomet- percolation network and requires further investigations. ric barrier: the screening current is smaller than the critical current at the center of the sample and larger at the periphery. The geometrical barrier may determine the ACKNOWLEDGMENTS distribution of mw current. The beginning of the forma- This study was supported by the Ministry of Indus- tion of a current pinch can be described in terms of the try, Science, and Technology of the Russian Federation, model of percolation transition [11] as follows. Under grant “Superconductivity.” the action of mw current flowing in the bulk, a network of percolation channels forms, and a part of the conducting segments can be destroyed upon a subsequent REFERENCES increase in current. The onset of the destruction process 1. V. F. Masterov, in High-Temperature Superconductivity: suppresses the superconducting transition, which is Fundamental and Applied Investigations, Ed. by manifested in a decrease in the superconducting-transi- A. A. Kiselev (Mashinostroenie, Leningrad, 1990), tion temperature (Fig. 2). By increasing the field, we Vol. 1, p. 405. bring the system closer to the percolation threshold, 2. V. F. Masterov and V. A. Kharchenko, Sverkhprovodi- since the network of percolation channels becomes pro- most: Fiz., Khim., Tekh. 4 (4), 629 (1991). gressively coarser. As the system approaches the perco- 3. V. F. Masterov, I. L. Likholit, V. V. Potapov, and lation threshold (critical concentration of conducting N. M. Shibanova, Supercond. Sci. Technol. 6, 593 channels), the percolation network is ruptured com- (1993). pletely. The effective conductivity of such a system, as 4. M. N. Kotov, V. F. Masterov, V. V. Potapov, et al., Int. J. well as the density of percolation channels, is deter- Infrared Millim. Waves 14 (8), 1679 (1993). mined numerically as a function of the breakdown field 5. A. V. Prikhod’ko and N. M. Shibanova, Fiz. Tverd. Tela in [11]. In our case, the breakdown field is the mw field (St. Petersburg) 42 (6), 992 (2000) [Phys. Solid State 42, corresponding to the onset of electric instability in the 1023 (2000)]. percolation network. 6. N. M. Shibanova, V. V. Potapov, N. M. Baranova, and G. A. Nikolaœchuk, Sverkhprovodimost: Fiz., Khim., Tekh. 6 (3), 597 (1983). 4. CONCLUSION 7. S. A. Kostylev and V. A. Shkut, Electronic Switching in The main result obtained in this work is the detec- Amorphous Semiconductors (Naukova Dumka, Kiev, tion of the superconducting transition coexisting with 1978). electric instabilities associated with a pinchlike current 8. A. V. Prikhod’ko, A. A. Chesnis, and V. A. Bareœkis, Fiz. Tekh. Poluprovodn. (Leningrad) 15 (3), 536 (1981) [Sov. distribution over the percolation network. It can be con- Phys. Semicond. 15, 303 (1981)]. cluded that for fields strong enough for the emergence 9. Zh. I. Alferov, A. T. Gorelenok, V. V. Mamutin, et al., of electric instability, the physics of the processes Fiz. Tekh. Poluprovodn. (Leningrad) 19 (11), 2004 change radically in comparison with the case of weak (1985) [Sov. Phys. Semicond. 19, 1233 (1985)]. mw fields. In the latter case, mw fields change the struc- 10. E. Zeldov, Phys. Rev. Lett. 73, 1428 (1994). ture of a percolation network, while strong fields initiate the so-called strong-field effects. The supercon- 11. A. P. Vinogradov, A. V. Gol’dshteœn, and A. K. Sarychev, Zh. Tekh. Fiz. 59 (1), 208 (1989) [Sov. Phys. Tech. Phys. ducting state is destroyed only for critical fields close to 34, 125 (1989)]. the percolation threshold. It should be noted that the increase in the superconducting-transition temperature may be associated with stimulation of superconducting Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2034Ð2038. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1943Ð1947. Original Russian Text Copyright © 2002 by Dolidze, Tsekvava.

SEMICONDUCTORS AND DIELECTRICS

On the Model of Divacancies in Germanium N. D. Dolidze and B. E. Tsekvava Dzhavakhishvili State University of Tbilisi, pr. Chavchavadze 3, Tbilisi, 380028 Georgia e-mail: [email protected] Received February 28, 2002

Abstract—A model of a divacancy which explains a number of experimental facts for Ge exposed to a ﬂow of accelerated electrons at temperature T = 77 K is proposed. The model explains the absence of photoconduction associated with photoexcitation of infrared absorption bands at 0.44 and 0.52 eV corresponding to divacancies; the difference in the values of activation energy for divacancy energy levels determined by optical and electrical methods; and the existence of the limiting position of the Fermi level in the forbidden gap in the case of irradiation with large radiation ﬂuxes. © 2002 MAIK “Nauka/Interperiodica”.

It was shown in [1] that so-called light-sensitive be used for germanium [7]. According to this model, at defects are formed in p-type germanium, as well as in the initial stage (before JahnÐTeller distortions appear), n-type germanium converted into p-Ge as a result of there are two vacancies at adjacent atomic sites c and c' irradiation with accelerated electrons (1Ð5 MeV). Pho- (dashed circles in Fig. 1) located along a spatial diago- tosensitivity is manifested only in samples possessing nal of the cube and six neighboring atoms (a, d, a', d', p-type conductivity after irradiation [1Ð6]. It is shown b, b') with bent bonds aÐd, a'Ðd', and bÐb'. The corre- in [4, 6, 7] that light-sensitive defects are responsible sponding polyatomic nonlinear molecule possesses the for infrared (IR) absorption bands at 0.44 and 0.52 eV. D3d symmetry. The group D3d has four one-dimensional These defects are manifested electrically (in experi- and two two-dimensional irreducible representations. mental temperature dependences of the Hall coeffi- Consequently, all electron terms of the molecule belong cient) as energy levels EV + 0.16 eV and EV + 0.08 eV to one of the six irreducible representations of the group in the forbidden gap, depending on the charge state of a D3d. Thus, in addition to nondegenerate energy levels defect [5]. At the same time, the electrically active lev- corresponding to one-dimensional representations, els EV + 0.52 eV and EV + 0.44 eV have not been there also exist doubly degenerate electron energy lev- observed in experiments. It should also be noted that els corresponding to two-dimensional irreducible rep- photoconduction was not observed in experiments in resentations (if we disregard spin). A degenerate elec- the case of absorption of photons with energy 0.52 and tron energy level occupied by electrons is unstable to 0.44 eV. In [4, 7], these defects in germanium were the JahnÐTeller effect. This effect leads to a distortion called divacancies. of the configuration of the nonlinear molecule, and its A similar situation (the difference between the ener- symmetry is lowered to C2h; the group C2h contains gies of the optical absorption bands ~0.69 and ~0.32 eV only three one-dimensional irreducible representations and the energies of the electrically active levels EC Ð [14], and, hence, only nondegenerate electron energy 0.39 eV and EC Ð 0.54 eV, respectively [8]) is observed for a divacancy in silicon [9Ð11]. In this case also, the [100] absorption of photons corresponding to energies of – optical absorption bands does not lead to photoconduc- [111] tion. a' At the same time, it is well known [12, 13] that irra- d' diation of n- and p-type germanium and silicon samples b c' leads to a decrease in the concentration of the majority – charge carriers; i.e., the Fermi level is displaced to the [011] c b' middle of the forbidden gap and attains a certain limit- d ing position, which is the same for both types of conduction (the so-called limiting Fermi level) and a changes insignificantly upon a further increase in the integrated radiation flux. A unified theory explaining all these experimental facts has not yet been developed. At the present time, there exists a model of diva- Fig. 1. 3D model of a divacancy in silicon (according to cancy in silicon [9] (Fig. 1). The same model can also [9]).

ON THE MODEL OF DIVACANCIES IN GERMANIUM 2035

' levels are present. The spatial model of a divacancy and b b' a' its wave functions as given by the LCAO model without a' and with the JahnÐTeller effect included (according to d d [9]) are presented in Figs. 1 and 2. According to this d' d' model, a divacancy is a multiply charged center with a a four energy levels: the two lower levels (1, 2) lie b b -dipoles X–Y- extremely close to each other in the immediate vicinity Z dipoles (b ) a – d + a' – d' of the top of the valence band, while the other two lev- d g els (3, 4) lie deep in the forbidden gap. The charge state (au) a – d – a' + d' of the divacancy is determined by the number of elec- a – d + a' – d' (eg) trons executing the bÐb' bond between the atoms. When {2b – a – d + 2b' – a' – d'} a – d – a' + d' this bond is free of electrons (levels 3, 4 in Fig. 3), the (e ) u {2b – a – d – 2b' + a' + d'} λ divacancy is in the state with double positive charge. (bu) b – b' – 2(a + d – a' – d') c λ (alu) a + b + d – a' – b' – d' (ag) b + b' – 1(a + d + a' + d') When one or two electrons (filling level 3) execute this (a ) a + b + d + a' + b' + d' lg ab bond, we have a singly positively charged or neutral λ (bu) a + d – a' – a' + 2 2(b – b') e λ divacancy, respectively. This bond can accommodate (ag) a + d + a' + d' + 2 1(b + b') one or two more electrons at level 4. In this case, the D3d Jahn–Teller C2h distortion divacancy acquires a single negative or double negative ab charge. This model of the divacancy used in [7, 9Ð11] correctly describes some experimental results but fails Fig. 2. Simple molecular-orbital LCAO model of the elec- to explain all the above-mentioned experimental facts tronic structure of a divacancy (a) before JahnÐTeller distor- for germanium and silicon. tion (D3d) and (b) after the distortion C2h. Solid arrows denote electrons and their spins for a singly charged positive In order to solve this problem, we propose, in con- state, while dashed arrows denote extra electrons for a sin- trast to [7, 9Ð11], a model of divacancies taking into gly charged negative state. account the bending of energy bands of a semiconductor by fluctuation electric fields produced by charged defects. We assume that there is no correlation in the distribution of defects in a semiconductor and fluctua- 4 tions in their concentrations are of the Gaussian type c [15Ð17]. 3 Let us apply the method of bent bands to our problem. It was shown in [16] that the fluctuation field of the spatial charge bending the bands is created by chemical ab impurities. In our case, such impurities are the initial impurities (donors) compensating radiation-induced 2 point defects and their simple complexes, formed in n- e Ge as a result of irradiation by an integrated flux of 1 accelerated electrons (E = 2Ð6 MeV, Φ = 1016Ð5 × 1017 cmÐ2) at temperature T = 77 K. This conclusion follows from the fact that conversion of the conduction Fig. 3. Fragment of Fig. 2 showing the levels of a divacancy type (overcompensation) was attained as a result of determining its charge state. irradiation of the n-Ge samples under investigation with an initial concentration of conduction electrons n ~ 1014Ð1016 cmÐ3 and that the samples displayed p- screening is weak. However, a low concentration of type conduction after irradiation. The equilibrium hole mobile carriers and a high density of the fluctuation concentration in this case did not exceed ~1010 cmÐ3. charge give rise to difficulties associated with the inclusion of screening [18Ð20]. It should be noted that irradiation gives rise to randomly distributed charged centers (defects). In the case The difficulties associated with screening in the case of a strong compensation, it is necessary to take into of a strong compensation were overcome in [16] with account Gaussian fluctuations of the concentrations of the help of the hypothesis that the arrangement of impu- charged impurities and radiation defects, which create rities is correlated (for shallow centers), which is deter- a fluctuating large-scale electrostatic potential. The mined by the crystal growth technology. In this case, a effect of this potential on band bending and on the shift specific nonelectron mechanism of screening (mutual of the Fermi level toward the middle of the forbidden screening of donors and acceptors) takes place. In the gap must be taken into consideration even for moderate case of radiation defects, such a screening model is not concentrations of defects on the order of 1015Ð1016 cmÐ3. adequate to the actual situation. Since the experimen- Under these conditions, the concentration of charged tally determined energy spacing between the lowest defects creating strong fluctuation fields is high, while and highest position of the top of the bent valence band

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2036 DOLIDZE, TSEKVAVA

Ei where ε is the permittivity of the medium and e is the 4 electron charge. As R ∞, we have γ(R) ∞ in complete accord with the well-known result according 0.16 eV 0.36 eV 3 to which the rms Coulomb energy fluctuation 0.08 eV 2 b [N ()e2/εr dr ]1/2 diverges at large distances. This a t ∫ 0.44 eV 0.52 eV result is physically meaningless, which indicates that screening must be taken into account in one way or another for a random distribution of Coulomb centers. 2 1 E In our case of large-scale fluctuations of randomly dis- V tributed charged centers, the only possible mechanism of nonlinear screening that bounds the radius of a sphere is the electron (hole) screening, in spite of the Fig. 4. Diagram explaining the difference between optically low concentration of free charge carriers. We introduce and electrically determined values of the activation energy the radius of a sphere R = R in such a way that the fluc- of a divacancy in germanium: 0.52 eV is the energy of opti- 0 cal excitation of an electron from the lower level of the diva- δ 3 tuation charge density N = 0.5Ze Nt/R0 is equal to cancy to level EV + 0.16 eV, while 0.44 eV is the same energy corresponding to level E + 0.08 eV (intracenter the hole charge density. Obviously, the holes (elec- V trons) in this volume neutralize the excess density of transitions); Ei is the middle of the forbidden gap of the unexposed crystal. The curve shows the bending of the top the negative (positive) charge of the sphere. Conse- of the valence band EV by the fluctuation potential of quently, we have charged defects, while the solid straight line corresponds to the top of the unperturbed valence band. 3 3 2 2 p ==0.5ZNt/R0,orR0 Z Nt/4p . (2) In our experiments, n-type germanium is compensated (potential relief) in germanium is approximately equal by irradiation: Nt = Nd + Na (Nd and Na are the concen- to 0.3 eV, the linear-screening approximation is inappli- trations of radiation defects of the donor and acceptor cable. The reason for this is that such a high fluctuation type, respectively). Considering that compensation has potential leads to a deep spatial modulation of the elec- been attained, we can assume that, approximately, Nt = tron density. However, the linear-screening theory is 2Nd. based on linearization of the Poisson equation, which is justified only in the case of a small depth of the poten- It follows that fluctuations of a size R > R0 are neu- tial well and a weak inhomogeneity of the electron tralized completely by free carriers, while fluctuations density. with a size R < R0 remain unscreened. Consequently, an approximate value of γ is obtained from Eqs. (1) and (2) We will now use the basic concepts of the theory of to be nonlinear screening formulated in [18Ð20]. In this the- γ 3 4 6 2 ε3 ory, one of the main difficulties is associated with deter- = 4Z e Nt / p. (3) mining the form of distribution of impurities in a semiconductor. Following [18Ð20], we assume that there is For low concentrations of holes, the potential well is no correlation in the distribution of defects, i.e., that the found to be quite deep. The electron levels of a divacancy lie in this well (Fig. 4). distribution is of the Poisson type and fluctuations are 10 Ð3 15 Ð3 ≈ Gaussian (small fluctuations). In order to estimate For p = 10 cm and Nd = 10 cm (Nt 2Nd), esti- potential-energy fluctuations, we will use the approximating the amplitude γ of potential-energy oscillations mation of a uniformly charged sphere [20]. We assume for Ge from Eq. (3) gives γ = 0.16 eV. The condition for that fluctuations have the form of homogeneous spher- the location of the ground-state energy levels in the ប2 2 Ӷ γ ical defect pileups having radius R and characterized by well, /2mR0 , can be written, in accordance with a Poisson distribution of defects in them. We denote the Eqs. (2) and (3), in the form (p/N )5/3 Ӷ aÐ1 NÐ1/3 , where average concentration of defects by Nt. Then, the aver- t t age number of particles in a sphere is N = 4πN R3/3 ≈ a = εប2/me2 (m is the effective electron mass). Since t Ӷ 4N R3 and the root-mean-square fluctuation of particles p Nd, this inequality holds with a large margin and t energy levels 1 and 2 of the divacancy lie near the bot- ∆ 3 is N = N = 2Nt R . A typical charge of a sphere is tom of the well (the top of the valence band in the given part of the crystal). The energy interval between the equal to 2Ze N R3 (Z = ±1, ±2), and the rms fluctua- bottom and the maximum (hump) of the potential relief t ∆ γ ≈ tion potential energy γ(R) of an electron is given by (bent band) is U = 2 0.32 eV, while the corresponding experimental value, determined with the help of optical and electrical measurements, is ∆U = (0.52 Ð γ () 2 3 ε 2 ε exp R ==2Ze Nt R / R 2Ze Nt R/ , (1) 0.16) eV = (0.44 Ð 0.08) eV = 0.36 eV (Fig. 4). Consid-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ON THE MODEL OF DIVACANCIES IN GERMANIUM 2037 ering that the above values of parameters are typical of ference between optically and electrically determined our experiments, the agreement with the theoretical values of activation energy of a divacancy in germa- model can be regarded as satisfactory. nium: the optically determined value of the activation As regards the range of applicability of formulas (2) energy for level 4 (0.52 eV) corresponds to the electri- and (3), it should be noted that the characteristic scales cally active level EV + 0.16 eV, while the electrically of fluctuations (and, accordingly, the amplitude of active level for the activation energy of level 3 oscillations of the potential energy γ(R)) in a compen- (0.44 eV) is EV + 0.08 eV. sated semiconductor with a low concentration of mobile charge carriers can become, in accordance with (3) The existence of the limiting position of the formulas (2) and (3), indefinitely large (e.g., can exceed Fermi level in the forbidden gap. It was noted above the forbidden gap width). It was proved in [19], how- that the existence of the limiting Fermi level in both ever, that if the bottom of the conduction band is low- types of semiconductors has been established experimentally. It has been shown that the limiting Fermi ered by 0.5Eg relative to its position in an unexposed crystal, it will cross the Fermi level in this region of the level is different for different semiconductors: it lies crystal, intrinsic carriers (electrons) will appear in a near the middle of the forbidden gap in silicon and is number sufficient for screening of the potential well closer to the lower edge of the gap in germanium. In formed, and further lowering of the bottom of the con- n-Ge, a conversion of the conduction type is observed. duction band will stop. Similarly, in the crystal region Different authors give different positions of the limiting with an excess of negative charge (potential well for Fermi level in the forbidden gap of germanium (from holes), the number of holes formed upon the crossing of EV + 0.07 eV to EV + 0.24 eV) for various temperatures the Fermi level will be sufficient for preventing a fur- and types of irradiation. In [12], the limiting position of ther rise of the top of the valence band. Consequently, the Fermi level for p-Ge and for n-Ge converted to the formulas (2) and (3) are inapplicable for the case of low p type by irradiation by fast electrons at T = 77 K was γ ≥ found to be E + 0.07 eV. It was assumed in [12] that an concentrations of charge carriers when (R) 0.5Eg. V γ amphoteric level belonging to a complex radiation Under our experimental conditions, (R) < 0.5Eg, and formulas (2) and (3) hold. defect exists in this region. The idea of pinning of the Fermi level near an amphoteric level appears attractive, Let us now apply the proposed divacancy model to and we adhere to the same point of view. However, in explain the following experimental facts. contrast to [12], we proved that such an amphoteric (1) The absence of photoconduction in the case of center is a multielectron center (divacancy), whose optical excitation of levels 3 and 4 (transitions a, b in energy level diagram is given in Fig. 4. Since γ ~ Fig. 3). The above estimates indicate that the ground- 0.16 eV, according to our estimates, the optically deter- state levels of the divacancy (1, 2) lie near the bottom of mined levels 0.52 and 0.44 eV must be separated by the potential well. When photons with energy 0.44 or intervals ~0.16 and 0.08 eV from the hump of the bent 0.52 eV are absorbed, electron transitions from these valence band, respectively, which corresponds to a dis- levels to levels 3 and 4 (intracenter transition) take tance ~0.1 eV from the middle of the forbidden gap in place. In this case, electrons must perform transitions the unexposed crystal. During irradiation of n- and from the valence band to the lower vacant levels, 1 and p-Ge, divacancies (which are compensating centers in 2; i.e., holes are generated near the bottom of the well both cases) are introduced. In p-type samples, the diva- in the valence band (consequently, photoconduction cancy formed can be in the doubly positive (no elec- must appear). However, the divacancy in the well is in trons on level 3), singly positive (one electron on level 3), an unstable charge state, and the surplus electron from or neutral (two electrons on level 3) charge states. In the the upper level instantaneously recombines with a near- former two cases, the divacancy is a positively charged est neighbor hole. Therefore, the hole disappears before ionized center and behaves as a donor. Therefore, the it can participate in photoconduction. Fermi level moves upwards to the middle of the forbid- (2) Difference in the values of activation energy den gap upon irradiation of the sample (i.e., upon an of divacancy levels 3 and 4 (Fig. 3) determined by increase in the divacancy concentration). When the optical and electrical methods. The activation ener- Fermi level lies between level 3 and 4 and is separated gies for energy levels determined from experimental from these levels by at least (2Ð3)kT, the charge state of temperature dependences of the Hall coefficient indi- the divacancy changes and the divacancy becomes neu- cate that the Fermi level is shifted towards the middle tral (level 3 is occupied by two electrons, while level 4 of the forbidden gap upon an increase in temperature. is empty). This situation comes about, because the In this case, electrons can occupy levels 3 and 4; i.e., the energy interval between level 3 and 4 is ∆E ӷ kT, divacancy can change its charge state. Electrons per- ∆E/kT ≥ 10 (T = 77 K). A further upward displacement form transitions from the hump of the bent valence of the Fermi level as a result of irradiation leads to a band to these levels. The energy required for the excita- transition of the divacancy to the singly or doubly tion of such electrons corresponds to the spacing charged state (filling of level 4), which means that the between the hump of the bent valence band and levels divacancy (negatively charged ionized center in this 3 and 4. Figure 4 shows the diagram explaining the dif- case) becomes an acceptor. As a result, the Fermi level

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2038 DOLIDZE, TSEKVAVA starts moving towards the valence band again, and, 8. V. S. Vavilov, N. P. Kekelidze, and L. S. Smirnov, Effects after it attains the lower level, the divacancy again of Radiation on Semiconductors (Nauka, Moscow, transforms into a donor, after which the process is 1988). repeated. The Fermi level turns out to be ﬁxed between 9. G. D. Watkins and J. W. Corbet, Phys. Rev. 138, A543 the two levels, which is the reason for the existence of (1965). the limiting Fermi level. The same pattern is also 10. L. J. Cheng, J. C. Corelli, J. W. Corbet, and G. D. Wat- observed for n-Ge, the only difference being that the kins, Phys. Rev. 152, 761 (1966). Fermi level moves, as a result of irradiation, toward the 11. A. H. Kalma and J. C. Corelli, Phys. Rev. 173, 734 middle of the forbidden gap from the conduction band. (1968). 12. A. B. Gerasimov, Fiz. Tekh. Poluprovodn. (Leningrad) 12 (6), 1194 (1978) [Sov. Phys. Semicond. 12, 709 REFERENCES (1978)]. 13. V. S. Vavilov, Effects of Radiation on Semiconductors 1. A. B. Gerasimov, N. D. Dolidze, N. G. Kakhidze, et al., (Fizmatgiz, Moscow, 1963; Consultants Bureau, New Fiz. Tekh. Poluprovodn. (Leningrad) 1 (7), 982 (1967) York, 1965). [Sov. Phys. Semicond. 1, 822 (1968)]. 14. L. D. Landau and E. M. Lifshitz, Course of Theoretical 2. H. Saito, N. Fukuoka, H. Hattori, and J. H. Crawford, in Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Radiation Effects in Semiconductors, Ed. by F. L. Vook Theory (Nauka, Moscow, 1974; Pergamon, New York, (Plenum, New York, 1968), p. 232. 1977). 3. T. M. Flanagan and E. E. Klontz, Phys. Rev. 167, 789 15. V. L. Bonch-Bruevich, Fiz. Tverd. Tela (Leningrad) 4 (1968). (9), 2660 (1962) [Sov. Phys. Solid State 4, 1953 (1962)]. 4. H. J. Stein, in Radiation Damage and Defects in Semi- 16. L. V. Keldysh and G. P. Proshko, Fiz. Tverd. Tela (Len- conductors, Ed. by J. E. Whitehouse (The Institute of ingrad) 5 (12), 3378 (1963) [Sov. Phys. Solid State 5, Physics, London, 1973), p. 315. 2481 (1963)]. 17. E. O. Kane, Phys. Rev. 131, 79 (1963). 5. A. R. Basman, A. B. Gerasimov, N. G. Kakhidze, et al., Fiz. Tekh. Poluprovodn. (Leningrad) 7 (7), 1347 (1973) 18. B. I. Shklovskiœ and A. L. Éfros, Zh. Éksp. Teor. Fiz. 60 [Sov. Phys. Semicond. 7, 903 (1973)]. (2), 867 (1971) [Sov. Phys. JETP 33, 468 (1971)]. 19. B. I. Shklovskiœ and A. L. Éfros, Zh. Éksp. Teor. Fiz. 62 6. A. B. Gerasimov, N. D. Dolidze, B. M. Konovalenko, (3), 1156 (1972) [Sov. Phys. JETP 35, 610 (1972)]. and M. G. Mtskhvetadze, Fiz. Tekh. Poluprovodn. (Len- ingrad) 11 (7), 1349 (1977) [Sov. Phys. Semicond. 11, 20. A. L. Éfros, Usp. Fiz. Nauk 111 (3), 451 (1973) [Sov. 793 (1977)]. Phys. Usp. 16, 789 (1973)]. 7. A. B. Gerasimov, N. D. Dolidze, R. M. Donina, et al., Phys. Status Solidi A 70, 23 (1982). Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2039Ð2049. From Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1948Ð1957. Original English Text Copyright © 2002 by Lemanov, Sotnikov, Smirnova, Weihnacht.

SEMICONDUCTORS AND DIELECTRICS

Giant Dielectric Relaxation in SrTiO3ÐSrMg1/3Nb2/3O3 1 and SrTiO3ÐSrSc1/2Ta1/2O3 Solid Solutions V. V. Lemanov*, A. V. Sotnikov*, **, E. P. Smirnova*, and M. Weihnacht** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia ** Leibniz Institute of Solid-State and Materials Research, Dresden, D-01069 Germany Received April 11, 2002

Abstract—Ceramic samples of (1 – x)SrTiO3ÐxSrMg1/3Nb2/3O3 and (1 Ð x)SrTiO3ÐxSrSc1/2Ta1/2O3 were prepared, and their dielectric properties were studied at x = 0.005Ð0.15 and 0.01Ð0.1, respectively, at frequencies 10 HzÐ1 MHz and at temperatures 4.2Ð350 K. A giant dielectric relaxation was observed in the temperature range 150Ð300 K, and not so strong but well-developed relaxation was found in the temperature range 20Ð90 K. τ Ð11 Ð12 The activation energy U and the relaxation time 0 were determined to be 0.21Ð0.3 eV and from 10 to 10 s for the high-temperature relaxation and 0.01Ð0.02 eV and 10Ð8Ð10Ð10 s for the low-temperature relaxation, respectively. The additional local charge compensation of the heterovalent impurities Mg2+ and Nb5+ (or Sc3+ and Ta5+) by free charge carriers or the host ion vacancies is suggested to be the underlying physical mechanism of the relaxation phenomena. On the basis of this mechanism, the MaxwellÐWagner model and the model of reorienting dipole centers Mg2+ (or Sc3+) associated with the oxygen vacancy are proposed to explain the high- temperature relaxation with some arguments in favor of the latter model. The polaron-like model with the Nb5+−Ti3+ center is suggested as the origin of the low-temperature relaxation. The reasons for the absence of ferroelectric phase transitions in the solid solutions under study are also discussed. © 2002 MAIK “Nauka/Interperiodica”.

1 1. INTRODUCTION about 0.002 and is almost the same for all these impu- Strontium titanate SrTiO is known to be an incipi- rities, with some specific features for the case of Ba. 3 Isovalent B-impurities (Zr, Sn, Ge) have a much smaller ent ferroelectric and a quantum paraelectric [1]. The effect on the dielectric properties of SrTiO3. Simulta- SrTiO3 crystal has a polar soft mode but never exhibits 2+ 4+ a ferroelectric phase transition down to T = 0 due to neous replacement of the host Sr and Ti ions by quantum fluctuations. At low temperatures, the dielec- impurity ions yields some specific effects. For example, in the SrTiO3ÐPbMg1/3Nb2/3O3 (PMN) solid solution, tric constant in SrTiO3 attains very high values. According to [2], ε = 41900 and ε = 9380 (4 K, 1Ð the transition to a ferroelectric phase (with relaxor a c properties) was observed only at x > 0.2 with a linear 100 kHz). It should be noted that these remarkable val- dependence of Tc on x, which was associated with ran- ues were obtained by extrapolation of the inverse sus- 2+ 5+ ceptibility versus stress to zero stress and nobody has dom fields due to the disordered Mg and Nb distri- been able to reproduce this result directly. A more or bution [9]. ε less typical experimental value of a in direct measure- Quite a different situation takes place for heterova- ments is around 20000 instead of 41900, and even this lent impurities. In this case, instead of induced ferro- value is remarkable. electric phase transitions, distinct dielectric relaxation is observed. (The only exception is perhaps The SrTiO3 crystal can be considered a marginal 3+ system that is near the limit of its paraelectric phase sta- SrTiO3:Bi ; in [10], it was claimed that the Bi impu- bility. Small external perturbations such as elastic stress rity induces a ferroelectric phase transition with xc = or impurities can destroy the stability and induce a fer- 0.0005.) There is a very long history of studying dielec- roelectric phase transition. Various impurities substi- tric relaxation in SrTiO3 with various heterovalent tuted for the host ions in SrTiO3 both in the A- and B- impurities, such as Bi [11Ð16], La [17Ð21], La and a position have been studied [3, 4]. It was shown that wide range of other trivalent rare-earth ions [22], and divalent impurities substituted for Sr2+, such as Ca [5], Fe [23, 24]. Ba [6], Pb [7], and Cd [8], induce a ferroelectric phase In this paper, we studied some special cases of het- 4+ transition with the transition temperature Tc propor- erovalent substitution when the host Ti ion is substi- 1/2 tuted by two heterovalent ions whose average charge is tional to (x Ð xc) , where the critical concentration xc is equal to that of the Ti4+ ion. As an example of such sys- 1 This article was submitted by the authors in English. tems, the solid solutions of SrTiO3 with SrMg1/3Nb2/3O3

2040 LEMANOV et al.

(SMN) and with SrSc1/2Ta1/2O3 (SST) were chosen. The The symmetry of the SrSc1/2Ta1/2O3 crystal, as far as SrTiO3ÐSMN system, contrary to SrTiO3ÐPMN, does we know, has not been determined earlier, but the sym- 2+ not contain ferroelectrically active Pb ions. A giant metry of the closely related compound SrSc1/2Nb1/2O3 dielectric relaxation and no ferroelectric phase transi- (SSN) is known [27] to be cubic with the O5 space tion have been found in these solid solutions. Prelimi- h nary results of this study have been published else- group; the compound has ordered perovskite structure where [25]. In the present paper, the giant relaxation in with the doubled-unit-cell parameter a = 8.057 Å. (1 Ð x)SrTiO3ÐxSMN and (1 Ð x)SrTiO3ÐxSST is stud- According to our measurements, the lattice parame- ied in detail. ter of SST is equal to a = 8.054 ± 0.003 Å, which is very close to the SSN lattice parameter. This allows us to conclude that these compounds are isomorphic with the 2. EXPERIMENTAL PROCEDURE 5 Oh space group. Ceramic samples of SMN, SST, and (1 Ð x)SrTiO3Ð X-ray diffraction measurements demonstrated that xSMN and (1 Ð x)SrTiO3ÐxSST solid solutions were the lattice parameter in (1 Ð x)SrTiO3ÐxSST solid solu- prepared by a standard ceramic technology. A stoichio- tions also follows a linear Vegard law between a = metric mixture of strontium carbonate and of Ti, Mg, 3.905 (SrTiO3) and 4.027 Å (SST perovskite reduced Nb, Sc, and Ta oxides of a special purity were used to cell) with the slope da/dx = 0.12 Å. The lattice-param- prepare the appropriate compounds and solid solutions. eter difference between SrTiO and SST is 3.1%. Pure compounds of SMN and SST were synthesized 3 through a columbite (MgNb O ) and a wolframite Though this difference is a little bit higher than that 2 6 between SrTiO and SMN, this solid solution exists in (ScTaO ) route, respectively, according to the reactions 3 4 a much broader range of SST concentrations due to the MgO + Nb2O5 = MgNb2O6, similarity in crystal structure. SrCO + 1/3MgNb O = SrMg Nb O + CO ; All the samples had a density between 92 and 97% 3 2 6 1/3 2/3 3 2 with regard to the theoretical x-ray density. Pure SMN

Sc2O3 Ta2O5 = 2ScTaO4, and SST samples had a density of about 87%. The samples for dielectric measurements had a diameter of SrCO3 + 1/2ScTaO4 = SrSc1/2Ta1/2O3 + CO2. 8 mm and a thickness of 1Ð0.4 mm. For the measurements, the samples were coated with silver-burnt, gold- The columbite and wolframite were synthesized at evaporated, and InÐGa-alloy electrodes. In all cases, we 1000 and 1200°C, respectively, for 20 h. After calcining ° obtained absolutely the same (within small experimen- the mixture at about 1200 C for several hours, the tal errors) temperature and frequency dependences of material was reground, and pellets were formed by the dielectric constant. The dielectric constant was pressing 9-mm diameter disks at 200 MPa. The ﬁnal ° measured using a Solartron SI 1260 Impedance/Gain- sintering was conducted at 1450 C for 1.5 h. X-ray dif- Phase Analyzer interfaced with a computer. The mea- fraction study indicated that the samples had single- surements were performed at frequencies between phase cubic perovskite structure for concentrations up 10 Hz and 1 MHz in a temperature range between 4.2 to x = 0.15 in the case of SMN and more than x = 0.2 in and 300 K by cooling at a constant rate of 1 K/min. The the case of SST. amplitude of the ac electric ﬁeld was 1 V/cm. The symmetry of the SrMg1/2Nb2/3O3 crystal is 3 known [26] to be rhombohedral with the D3d space 3. EXPERIMENTAL RESULTS AND ANALYSIS group with lattice parameters a = 5.66 and c = 6.98 Å. ε The parameter of the reduced perovskite pseudocubic The temperature dependence of the real part ' of the dielectric constant for SMN and SST ceramic samples unit cell is a = 4.01 Å. With the SrTiO3 parameter a = 3.905 Å, one obtains a 2.7% difference in the lattice is shown in Fig. 1. For both materials, the dielectric parameter between SrTiO and SMN. At such a small constant as a function of temperature behaves as that of 3 ordinary nonferroelectric dielectrics: the dielectric con- difference, one might expect the possibility of obtain- stant decreases with decreasing temperature with a ing the SrTiO ÐSMN solid solutions in the whole con- 3 slope of (1/ε)dε/dT = +0.8 × 10Ð4 and +1.0 × 10Ð4 KÐ1 centration range. However, experiment shows that there for SMN and SST, respectively. Similar values of is a solubility limit, with the limiting SMN concentra- (1/ε)dε/dT are characteristic of nonferroelectric oxides, tion lying between 0.15 and 0.20. This low solubility alkali halides, and other conventional dielectrics. This may be attributed to the difference in crystal structure. is in great contrast with SrTiO3, where the quantity The lattice parameter of the SrTiO3ÐSMN solid (1/ε)dε/dT < 0 and its absolute value in ceramics and solutions was measured, and it appeared that the lattice single crystals is two and three orders of magnitude parameter follows a linear Vegard law between a = larger, respectively, as compared to (1/ε)dε/dT in SMN 3.905 (SrTiO3) and 4.01 Å (SMN perovskite pseudocu- and SST. There is no frequency dispersion of the dielec- bic unit cell) with the slope da/dx = 0.105 Å. tric constant in both SMN and SST, as well as in

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 GIANT DIELECTRIC RELAXATION 2041

21.0 4500 (a) 4000 10 Hz 100 Hz 3500 1 kHz 20.5 10 kHz 3000 100 kHz 1 MHz

' 2500 ' ε

20.0 ε 2000 1500 19.5 SrMg1/3Nb2/3O3 1000 SrSc Ta O 1/2 1/2 3 500 19.0 0 0 50 100 150 200 250 300 T, K (b) 400 Fig. 1. Temperature dependence of the dielectric constant ε' in SrMg1/3Nb2/3O3 and SrSc1/2Ta1/2O3. 300 ''

SrTiO3, in the frequency range under study, 10 HzÐ ε 1 MHz. The situation drastically changes in the 200 SrTiO3ÐSMN and SrTiO3ÐSST solid solutions. At the SMN concentration x(SMN) = 0.005, the ﬁrst hints of a frequency dispersion (dielectric relaxation) appear both in the ε' and ε'' frequency spectra and in the ε' and ε'' 100 temperature dependences. At x = 0.01, this relaxation becomes quite distinct (Fig. 2) and the relaxation features in this case are superimposed on the dielectric- 0 50 100 150 200 250 300 constant temperature dependence characteristic of pure T, K SrTiO3 ceramics. It should be noted that the maximum value of the dielectric constant at 4.2 K in SrTiO3 Fig. 2. Temperature dependence of (a) the real ε' and ceramic samples is usually several times less than that (b) imaginary ε'' parts of the dielectric constant of (1 Ð in single crystals; in our samples, this value is about x)SrTiO3ÐxSrMg1/3Nb2/3O3 at x = 0.01. 5000. The relaxation strength attains a maximum at x = 0.03 (Fig. 3) and then decreases with increasing x. Along with this (“high-temperature”) relaxation, which develops between 100 and 300 K, a low-temperature 14000 relaxation appears as x increases (Fig. 4) and the high 10 Hz value of the dielectric constant of SrTiO becomes sup- 12000 100 Hz 3 1 kHz pressed at x ≥ 0.03. One can follow the evolution of this 10 kHz low-temperature relaxation in Fig. 5, where the ε'(T ) 10000 100 kHz dependences are shown at x between 0.05 and 0.15 at a 1 MHz ' frequency of 1 MHz. ε 8000 Certainly, the dielectric relaxation reveals itself not only in the ε' and ε'' temperature dependences but also 6000 in the ε' and ε'' frequency spectra. As an example, these spectra are shown for the high-temperature relaxation 4000 at x = 0.03 in Fig. 6 and for the low-temperature relaxation at x = 0.15 in Fig. 7. 2000 High-temperature dielectric relaxation was also 0 50 100 150 200 250 300 observed in (1 Ð x)SrTiO3ÐxSST solid solutions T, K (Figs. 8, 9). Samples with x = 0.01, 0.05, and 0.1 were ε measured. The strongest relaxation occurs at x = 0.05, Fig. 3. Temperature dependence of ' for (1 Ð x)SrTiO3Ð and the relaxation disappears at x = 0.1. xSrMg1/3Nb2/3O3 at x = 0.03.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2042 LEMANOV et al.

3000 3000 x = 0.05 x = 0.07 x = 0.1 2500 10 Hz 2500 x = 0.15 100 Hz 1 kHz 2000 10 kHz 100 kHz 2000 1 MHz ' ' ε 1500 ε 1500

1000 1000

500 500

0 50 100 150200 250 300 T, K 0 20 40 60 80 100 120 140 T, K ε ε Fig. 4. Temperature dependence of ' for (1 Ð x)SrTiO3Ð Fig. 5. Temperature dependence of ' for (1 Ð x)SrTiO3Ð xSrMg1/3Nb2/3O3 at x = 0.15. xSrMg1/3Nb2/3O3 at a frequency of 1 MHz.

In SrTiO3ÐSMN samples, the relaxation is much From the experimental data, it follows that the relax- stronger than that in SrTiO3ÐSST. ation time τ obeys an Arrhenius relation, The most remarkable feature of the high-tempera- ττ () ture relaxation in Figs. 2Ð4 and 6 is the very high value = 0 exp U/kT . (2) of the dielectric constant. At the SMN concentration x(SMN) = 0.03, the dielectric constant ε attains a value Fitting the experimental data to Eqs. (1) and (2), one 0 obtains U and τ , which are presented in Tables 1 and 2. of 14000 at 150 K and is roughly 1000 at 300 K. The 0 ε ε relaxation strength ( 0 Ð ∞) is very high and varied In Table 1, one can see that, in general, there is a roughly proportional to 1/T, as can be seen in Figs. 2 small but systematic increase in the activation energy U and 3. with increasing x. Interestingly, a similar behavior of All the temperature and frequency dependences in U(x) was observed in the case of SrTiO3 doped with Figs. 2Ð9 look like classical textbook relaxation depen- trivalent rare-earth ions [22]. dences. These dependences demonstrate a typical A detailed analysis shows that the experimental Debye relaxation behavior of the dielectric properties: results can be described better by a ColeÐCole complex ()ε ε function instead of Eq. (1): 0 Ð ∞ ε' = ε∞ + ------, ω2τ2 ()ε ε 1 + ∗ 0 Ð ∞ (1) ε = ε∞ + ------, (3) ()ωτε ε β 0 Ð ∞ 1 + ()iωτ ε'' = ------. ω2τ2 1 + where ε* = ε' Ð iε''. As an example, one can refer to the ColeÐCole τ Table 1. Activation energy U and relaxation time 0 for the graph at x(SMN) = 0.03 and various temperatures in high-temperature relaxation in (1 Ð x)SrTiO3ÐxSrMg1/3Nb2/3O3 Fig. 10. The best fit of the experimental data to Eq. (3) and (1 Ð x)SrTiO3ÐxSrSc1/2Ta1/2O3 is obtained with β = 0.7. xU, eV τ , 10Ð11 s Fitting the experimental data to Eq. (3) does not sig- 0 τ nificantly change the values of U and 0 presented in SMN 0.01 0.23 5 Table 1. 0.03 0.21 0.9 Concerning the low-temperature relaxation, we 0.05 0.23 0.3 should note that the experimental data can be fairly well 0.07 0.24 0.4 fitted to a simple Arrhenius relation [Eq. (2)] only at 0.1 0.26 0.2 temperatures T ≥ 20 K. The best fit in the whole temper- 0.15 0.3 0.2 ature range can be obtained using a VogelÐFulcher relation, SST 0.01 0.27 1.5 ττ []() 0.05 0.29 0.7 = 0 exp U/kTÐ Tg , (4)

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 GIANT DIELECTRIC RELAXATION 2043

1400 214 K 204 K 193 K 14000 185 K 1200 175 K 165 K 12000 155 K 144 K 1000 10000 135 K 125 K ' ε 800

' 8000 ε 600 71 K 6000 60 K 50 K 40 K 4000 400 30 K 24 K 2000 200 0 4000 300 3500 3000 200

2500 '' ε '' ε 2000 1500 100 1000 500 0 0 101 102 103 104 105 106 101 102 103 104 105 106 Frequency, Hz Frequency, Hz

ε ε ε ε Fig. 6. Frequency spectra of ' and '' for (1 Ð x)SrTiO3Ð Fig. 7. Frequency spectra of ' and '' for (1 Ð x)SrTiO3Ð xSrMg1/3Nb2/3O3 at x = 0.03. (The high-temperature relax- xSrMg1/3Nb2/3O3 at x = 0.15. (The low-temperature relaxation at various temperatures.) ation at various temperatures.)

≈ ε with U 0.05 eV and Tg < 0. Since such values of Tg ∞), which at x = 0.03 and a frequency of 10 Hz attain have no physical meaning, we used the Arrhenius rela- 14000 and 12000, respectively. Such a large value of ε τ 0 tion with the U and 0 values given in Table 2. (larger than the dielectric constant at 4.2 K in nominally We also tried to observe P(E) hysteresis loops in our pure SrTiO3 ceramic samples) is surprising. These val- samples at low temperatures. This attempt failed, and ues are, to our knowledge, the highest yet reported for we may conclude that there are no ferroelectric phase SrTiO with heterovalent impurities in this temperature transitions in SrTiO ÐSMN and SrTiO ÐSST solid 3 3 3 range. In the literature, there are only two examples [16, solutions. ε ε ε 22] of a very high value of 0 and ( 0 Ð ∞), which are close to but less than that mentioned above. One of 4. DISCUSSION

Thus, in the (1 Ð x)SrTiO3ÐxSMN and (1 Ð τ Table 2. Activation energy U and relaxation time 0 for the low- x)SrTiO3ÐxSST solid solutions, we observed strong temperature relaxation in (1 Ð x)SrTiO ÐxSrMg Nb O high-temperature (150Ð300 K) dielectric relaxation; 3 1/3 2/3 3 τ Ð9 much smaller low-temperature (20Ð90 K) relaxation xU, eV 0, 10 s was also observed in the case of SrTiO3ÐSMN. As mentioned above, the observed high-temperature relaxation 0.07 0.02 0.3 0.1 0.01 0.2 in SrTiO3ÐSMN is characterized by a very high dielec- ε ε tric constant 0 and very high relaxation strength ( 0 Ð 0.15 0.02 15

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2044 LEMANOV et al.

1200 1200 10 Hz 10 Hz 100 Hz 100 Hz 1000 1 kHz 1000 1 kHz 10 kHz 10 kHz 100 kHz 100 kHz 800 1 MHz 800 1 MHz ' ' ε 600 ε 600

400 400

200 200

0 50 100 150 200 250 300 0 50 100 150 200 250 300 350 T, K T, K

ε ε Fig. 8. Temperature dependence of ' for (1 Ð x)SrTiO3Ð Fig. 9. Temperature dependence of ' for (1 Ð x)SrTiO3Ð xSrSc1/2Ta1/2O3 at x = 0.01. xSrSc1/2Ta1/2O3 at x = 0.05.

ε these examples is SrTiO3:2 at. % Er with 0 = 10000 well as in SrTiO3ÐSST, the lattice constant follows the ε ε and ( 0 Ð ∞) = 9400 [22]. linear Vegard law between SrTiO3 and SMN (SST). 2+ 5+ Very strong dielectric relaxation has also been Thus, we make the conclusion that the Mg and Nb observed in another incipient ferroelectric, KTaO . In (or Sc3+ and Ta5+) ions substitute for the host Ti4+ ions 3 in SrTiO . Though the impurity ions have a charge dif- KTaO3 with 2.3 at. % Nb (KTN) and 0.055 at. % Ca, 3 dielectric relaxation was measured between about 50 ferent from that of the host Ti4+ ions, they are “self- ε 5+ and 100 K, with 0 being about 16000 [28]. A similar compensating;” i.e., the excess charge of two Nb ions ε 2+ value of 0 in the same temperature range was found in is compensated by the deﬁcient charge of one Mg ion, and their average charge is equal to the charge of the KTaO3 with 2.5% Nb and 0.1% Li (KLTN) [29]. How- 4+ ever, in both cases [28, 29], the dielectric relaxation host Ti ion. A similar situation occurs in SrTiO3ÐSST, develops close to a ferroelectric phase transition, which where the excess charge of a Ta5+ ion is compensated by certainly affects the relaxation strength strongly. the deﬁcient charge of an Sc3+ ion. Very large values of the dielectric constant are usu- However, at small x, when the Mg2+ and Nb5+ ions ally observed in SrTiO at ferroelectric phase transi- 3 (Sc3+ and Ta5+) are far from each other, the Mg2+ ion tions induced by impurities at an impurity concentra- does not “know” that somewhere there are Nb5+ ions tion x near the critical concentration xc. For example, ε with compensating charge (the same is the case for the m = 110000 in SrTiO3:Ca single crystals at x = 0.01 [5] 5+ ε 18 Nb ions). Therefore, the impurity ions need some and m = 170000 in SrTiO3: O single crystals at x = 0.37 [30]. In ceramic samples of Sr1 Ð xBaxTiO3, accord- ε ing to our measurements, m = 35000 at x = 0.02. How- 8000 ever, in the systems under study (SrTiO3ÐSMN and SrTiO ÐSST), there are no ferroelectric phase transi- 214 K 3 6000 193 K tions and the observed giant dielectric constant in them 175 K should be determined by some other mechanisms. 155 K

'' 144 K

ε 4000 In discussing possible mechanisms of the giant dielectric relaxation, the ﬁrst question to be answered is 5+ 2+ 2000 what positions Nb and Mg occupy in the SrTiO3 host lattice. The Nb5+ ions should substitute for the Ti4+ host ions due to the size and charge factors. As for the 0 Mg2+ ions, they can occupy, in principle, either Sr2+ or 2000 6000 10000 14000 ε' Ti4+ positions. Two experimental facts prove the latter 2+ possibility. First, the Mg ions being added to SrTiO3 Fig. 10. ColeÐCole plot for (1 Ð x)SrTiO3ÐxSrMg1/3Nb2/3O3 alone (in the form of MgTiO3) do not lead to any dielec- at x = 0.03. (The high-temperature relaxation at various tric relaxation. Second, in the SrTiO3ÐSMN system, as temperatures.)

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 GIANT DIELECTRIC RELAXATION 2045

≅ 2 ε ≅ 2 additional local charge compensation. This hypothesis grain boundary. With d1/d2 10 and 2 10 , Eq. (7) is a basic point for explanation of dielectric relaxation gives ε ≅ 104. in the systems under study. We believe that without this eff hypothesis of the additional local charge compensation, The MaxwellÐWagner mechanism was studied in it is impossible to suggest any models for the observed detail for SrTiO3 ceramics in [31]. Ceramic samples of dielectric relaxation. The additional local charge com- stoichiometric SrTiO3 and those with up to 1% excess pensation may proceed through the formation of either of Ti were used. It was shown that the dielectric con- free charge carriers or the host ion vacancies in the fol- stant reached a value of about 104 at 150°C at a fre- lowing ways. quency of 10Ð2 Hz and that this value of the dielectric Substituting for the host Ti4+ ion, Nb5+ (and Ta5+) constant was due to the MaxwellÐWagner relaxation. ε ω serves as a donor and Mg2+ (and Sc3+) acts as an accep- Temperature-dependent maxima in ''( ) were 5+ 2+ observed; they shifted to higher frequencies with tor. Being fully ionized, an Nb and an Mg provide ° ε ω one electron and two holes, respectively. If the electron increasing temperature. At 150 C, ''( ) attained a and hole mobilities are equal, the electrical conductiv- maximum value at a frequency of about 1 Hz. Assum- ing reasonable (though rather arbitrary) values of the ity will be compensated and the sample will exhibit a µ ε high resistivity. In the opposite case, the resistivity may parameters (d1 = 10, d2 = 0.1 m, 2 = 200, carrier con- 18 Ð3 µ 2 be relatively low. centration n0 = 10 cm , mobility = 6 cm /V s, acti- A second possible mechanism of local charge com- vation energy U = 0.7 eV), the authors of [31] obtained satisfactory agreement between the experimental pensation is the formation of one Sr2+ vacancy (V ) per Sr results and the MaxwellÐWagner model. two Nb5+ ions and of one O2Ð vacancy (V ) per Mg2+ O The value of U = 0.7 eV is determined by the energy ion. In the case of SST, one Sr2+ vacancy should be 5+ 2Ð levels of donors or acceptors in the band gap [31]. This formed per two Ta ions, and one O vacancy is energy is quite typical for SrTiO . To determine 3+ 3 formed per two Sc ions. whether the relaxation in our samples that develops in These two means of local charge compensation may the 150Ð300 K range can be described by the MaxwellÐ lead to two mechanisms of the high-temperature (150Ð Wagner model, we have to set this energy equal to U ~ ε 300 K) dielectric relaxation. 0.25 eV. Then, using Eqs. (5)Ð(7) with d1, d2, and 2 High electrical conductivity may be a reason for the taken from [31] and the charge-carrier mobility from MaxwellÐWagner relaxation in ceramic samples, and it [32], we find that the Maxwell–Wagner mechanism can is well known that strong dielectric relaxation in semi- satisfactorily describe the experimental results; the conducting ceramic samples can always be attributed to conductivity and carrier concentration at 150 K are σ ~ Ð5 Ω Ð1 12 Ð3 × 20 Ð3 the MaxwellÐWagner mechanism. This relaxation is 10 ( cm) , n ~ 10 cm , and n0 ~ 5 10 cm , and due to the different properties of ceramic grains and the quantities σ and n vary exponentially with temper- grain boundaries (see [31] and references therein). In a ature, following Eq. (6) with U = 0.25 eV. Note that at very simplified model, the grain is considered as a x = 0.03, the SMN concentration is n ≅ 5 × 1020 cmÐ3, resistor R with the grain boundary as an insulating layer which is not quite consistent with the value of n0. Nev- with the capacity C in series with the resistor. A ceramic ertheless, it is possible to conclude that the high-tem- sample represents a system of such RC elements, and perature dielectric relaxation can be associated with the the relaxation time (which is the effective Maxwell MaxwellÐWagner mechanism. relaxation time of the whole system) is given by However, the following experimental facts contra- τ × Ð14()ε σ = 8.8 10 eff/ , (5) dict the MaxwellÐWagner model of the dielectric relaxation in the temperature range 150Ð300 K. where the relaxation time τ is in seconds and the elec- σ Ω Ð1 According to Eq. (7), the dielectric constant is inde- trical conductivity is in ( cm) . pendent of temperature, whereas the experiment dem- The temperature dependence of conductivity σ = onstrates a 1/T dependence. enµ is mainly determined by the temperature dependence of the charge-carrier concentration At about 250 K and a frequency of 10 Hz, one observes an increase in ε' with increasing temperature, () nn= 0 exp ÐU/kT . (6) which may be the onset of an additional relaxation, which is especially distinct at x = 0.15 (Fig. 4). A crude The MaxwellÐWagner relaxation mechanism can ≥ ε ωτ estimate shows that for this relaxation, U 0.5 eV. This give a very high value of eff at < 1. relaxation can be ascribed to the MaxwellÐWagner The effective dielectric constant of a ceramic sam- model. Though nothing forbids the occurrence of two ple is approximately given by [31] MaxwellÐWagner relaxations in different temperature ε ≅ ()ε regions (say, at T > 300 K and T < 300 K) in one sample, eff d1/d2 2 , (7) such an event seems to be rather accidental. where d1 is the grain diameter, d2 is the thickness of the The observed dependence of the activation energy U ε grain boundary, and 2 is the dielectric constant of the on concentration x is difficult to explain in terms of the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2046 LEMANOV et al. oxygen octahedron, in which one of six oxygen ions is 2+ 2Ð absent (VO). The Mg ÐO distance is a/2 (a is the lattice parameter), and the distance between the nearest neighbor O2Ð ions is a/2 . The dipole moment p = 2ea/2 = ea = 18.7 × 10Ð18 CGSE units = 18.7 D is associated with this center. Thermally activated reorientation of this dipole moment via the vacancy jumping (or, more correctly, the oxygen ion jumping through the oxygen vacancy) is suggested as the origin of the τ dielectric relaxation with U = 0.21Ð0.3 eV and 0 = 10−11Ð10Ð12 s. These relaxation parameters seem to be quite reasonable. Indeed, in the model under discussion, τÐ1 = ω should be on the order of the lattice Sr2+ Mg2+ O2– V 0 0 O Debye frequencies. As for the activation energy, U ~ 2+ 0.25 eV seems to be too low, at first sight, since the acti- Fig. 11. Structure of the [MgTi ÐVO] center. vation energy for the oxygen vacancy diffusion in the perovskite is not less than 1 eV [34, 35]. However, it is well known that the activation energy for the oxygen- MaxwellÐWagner mechanism, as has been emphasized ion movement near a defect (as in our case) can be earlier [22]. much lower. For example, in KTaO3, where the activa- Application of a dc electric field E = 1 kV/cm does tion energy for dc ionic conductivity is also not less not change the relaxation under discussion but leads to than 1 eV, the activation energy for an oxygen vacancy great changes in ε'' and tanδ at higher temperatures. At hopping around a defect is very low: U = 0.08, 0.11, the SMN concentration x = 0.05, T = 275 K, and a fre- and 0.36 eV for the defect centers [Ca2+ ÐV ], [Mn2+ Ð ε Ta O Ta quency of 1 kHz, '' increases by more than an order of 2+ magnitude under the action of the dc electric field. VO], and [CoTa ÐVO], respectively (see [28] and refer- The MaxwellÐWagner mechanism of the dielectric ences therein). Even for the reorientation of the Fe3+Ð relaxation in SrTiO with rare-earth ions [16, 17, 22] in 2Ð 3 Oi center in KTaO3 with an interstitial oxygen ion the same temperature range as discussed above has also 2Ð been denied. In SrTiO3:La ceramics, only relaxation at Oi , the activation energy is as low as 0.34 eV [36, 37]. temperatures around 200°C has been found to be associated with the MaxwellÐWagner mechanism [17]. One may suppose that a similar situation takes place Relaxation in the range 150Ð300 K was ascribed to the in SrTiO3; in this case, the value of U = 0.21Ð0.3 eV formation of the host ion vacancies. One of the argu- seems to be quite reasonable. It is important to note that ments against the MaxwellÐWagner mechanism for this an activation energy around U = 0.25 eV was found ear- relaxation was that the relaxation holds even in single lier to be characteristic of the dielectric relaxation in crystals [17]. SrTiO3 ceramics doped with Bi [11, 12], Mn [20], La [17, 20Ð22], and other trivalent rare-earth ions [22]. Now, we will discuss an alternative model of the The dielectric relaxation in all these systems was giant dielectric relaxation based on the local charge ascribed to the host ion vacancies. compensation of impurities by the host ion vacancies as described above. In the model under discussion, the relaxation ε ε Due to electrostatic interaction, the most favorable strength ( 0 Ð ∞) decreases with increasing concentra- configuration is that in which the impurity ions and tion x of SMN, since the Mg2+ and Nb5+ ions are pro- related vacancies are nearest neighbors. As a result, the gressively closer to each other and their charge self- following impurity centers are formed: compensation takes place instead of their compensation by the vacancies. If the observed dielectric relaxation is 5+ 2+ ε ε [2NbTi Ð VSr] and [MgTi Ð VO] in SrTiO3ÐSMN, due to this mechanism, the relaxation strength 0 Ð ∞ should first increase with increasing x, then reach a 5+ 3+ [2TaTi Ð VSr] and [2ScTi Ð VO] in SrTiO3ÐSST. maximum at a certain value of x, and then decrease and completely vanish at large x. Exactly this behavior is It is widely accepted that in the ABO3 perovskites, demonstrated by the high-temperature relaxation in our the mobility of the oxygen vacancies is higher than that samples. of the A ions [33]; therefore, we shall discuss below Another important point is the unusually high relax- only the centers with the oxygen vacancies, namely, ε ε 2+ 3+ 2+ ation strength 0 Ð ∞. Let us estimate the relaxation [ÐMgTi VO] and [2ScTi ÐVO]. The [MgTi ÐVO] complex strength in the framework of the proposed model. An is shown in Fig. 11. The Mg2+ ion is in the center of the electric field E induces a polarization P due to the reori-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 GIANT DIELECTRIC RELAXATION 2047

2+ entation of [MgTi ÐVO] complexes with dipole moment p = ea and concentration n: Pp= ()2n/kT E . (8) Therefore, the dielectric susceptibility is χ = p2n/kT. (9) For x = 0.03, assuming that all the Mg2+ impurity ions 2+ form [MgTi ÐVO] centers, we obtain (in CGSE units) χ ≅ 10 and ε ≅ 4πχ ≅ 102. This value is too small to explain the experimental results. However, the local electric field Eloc should be taken in Eq. (8) instead of Nb5+ Ti3+ the applied electric field E. Ti4+ (possible positions of Ti3+) As a crude estimation of the local field, one can use the expression [38] 5+ Fig. 12. Structure of [Nb ÐTi3+] center. ()()ε Ti Eloc = ∞ + 2 /3 E. (10) For the dielectric relaxation at x = 0.03, we have ε ≈ and hops as a “polarization-dressed” electron, i.e., as a ∞ 2000 (Fig. 6); i.e., the local electric field is almost polaron. In such a case, an accepted point of view [39] three orders of magnitude larger than the applied field. is that the relaxation time τ should be in the range As a result, one obtains ε ≈ 2.5 × 104. Certainly, this 0 τ ωÐ1 ≅ value should be considered as an upper limit and only between the characteristic lattice times ( 0 = 0 demonstrates that the proposed model can provide a 10−12Ð10Ð13 s) and electronic times (10Ð14Ð10Ð15 s). very high dielectric constant. However, we believe that for a polaron bound to the It is interesting to note that from experiment [37] it 5+ impurity ion (NbTi in our case), this relaxation time 3+ follows that the local electric field at the [FeK ÐOi] cen- can be much longer, because it can take much time for ter in KTaO3:Fe is an order of magnitude larger than the lattice to come to the equilibrium after the electron the applied field. This means that the local field in hopping from one Ti ion to another. Indeed, for example, in TiO reduced crystals, three relaxation processes SrTiO3 is much larger than that in KTaO3. 2 associated with polaron hopping were observed [40] Thus, we may say that the proposed model is consis- Ð3 Ð2 tent with the experimental data: the model can, in prin- with activation energies between 10 and 10 eV and ε ε ∝ τ between 10Ð8 and 10Ð6 s. Thus, our values of U and τ ciple, give a high value of 0; the experimental 0 1/T 0 0 temperature dependence is described by Eq. (9); and an seem to be reasonable. experimental activation energy U of about 0.25 eV is in Now, let us turn to the SrTiO3ÐSST solid solution. the range of typical values of the activation energy for The high-temperature relaxation is also observed in this an oxygen vacancy jumping around an impurity ion in system, but the relaxation strength is much smaller than perovskites. that for the SrTiO ÐSMN. The [2Sc3+ ÐV ] center has a Along with the high-temperature relaxation in 3 Ti O rather complicated structure to provide easy movement SrTiO3ÐSMN, a low-temperature relaxation is τ Ð8 Ð10 of the oxygen vacancy. However, some small part of observed with U ~ 0.01Ð0.02 eV and 0 ~ 10 Ð10 s. 3+ We believe that such an activation energy is too low for these centers can exist as a simple [ScTi ÐVO] center. the ion movement and should be attributed to the elec- Therefore, the relaxation mechanism will be the same tronic system. The following model may be suggested as for the [Mg2+ ÐV ] centers but with a much smaller to explain the low-temperature relaxation. Ti O relaxation strength due to the small concentration of the Some part of the Nb5+ ions is not compensated with [ÐSc3+ V ] centers. 2+ 5+ 3+ Ti O the Sr vacancies but forms a new center [NbTi ÐTi ] When discussing the dielectric relaxation associated (Fig. 12). The Nb5+ ion is surrounded by six Ti4+ ions, 3+ with the thermally activated movement of the host ion and one of them is in the Ti state; i.e., there is an elec- vacancies, one should certainly bring to mind Skanavi’s tron localized on the Ti ion. This electron jumps over 4+ relaxation model [11, 12]. This model postulates that six Ti ions. The distance between the Nb and Ti ions the Sr2+ ion vacancies distort the neighboring oxygen and between the Ti ions is a and a 2 , respectively. The octahedra and, as a result, several off-center equilib- activation energy for this electron hopping can be suffi- rium positions for the Ti4+ ion appear. The dielectric ciently low. The excess electron polarizes the lattice relaxation in this model is associated with thermally

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2048 LEMANOV et al. activated motion of the Ti4+ ion between these equiva- using only dielectric measurements, one cannot deter- lent off-center positions. In SrTiO3 doped with La [17], mine the structure of defect centers. La and Mn [20], and with a wide range of the rare-earth Finally, and most importantly, both models are ions [22], the observed dielectric relaxation was founded on the hypothesis of the additional local explained in terms of Skanavi’s model. However, this charge compensation of the heterovalent B ions by the purely qualitative model is difficult to accept, since it is host ion vacancies or by free charge carriers. This sce- doubtful that an asymmetric distortion of the oxygen nario seems to be inevitable for explaining the dielec- octahedron can lead to several equivalent off-center tric relaxation in the solid solutions under study. How- positions for the Ti4+ ion. Theoretical microscopic cal- ever, in this context, more experimental proofs are culations must be made to support this model. This is desirable. Such proofs might be provided by experi- why we did not discuss this model as a possible reason ments with SrTiO3ÐSMN single crystals and/or using for the dielectric relaxation. Furthermore, the authors SrMn1/2Nb2/3O3 instead of SrMg1/3Nb2/3O3 to obtain the of a recent paper [16] argue that the experimental data possibility of ESR study of Mn2+ centers. These and on the dielectric relaxation in SrTiO3 are not consistent other more complicated experiments are now in with Skanavi’s model. progress. In conclusion, in SrTiO ÐSrMg Nb O and in In connection with the model of the additional local 3 1/3 2/3 3 charge compensation, it is worth noting that the ques- SrTiO3ÐSrSc1/2Ta1/2O3 solid solutions, the giant high- temperature (100Ð350 K) dielectric relaxation and a not tion of whether this charge compensation can play any role in dielectric properties of disordered relaxor ferro- so strong, but well-developed, low-temperature (20Ð electrics such as PbMg Nb O (PMN) is not ruled 90 K) relaxation were observed instead of a ferroelec- 1/3 2/3 3 tric phase transition induced by impurities in the incip- out. ient ferroelectric and quantum paraelectric SrTiO3. This means that not any impurity can disturb the stabil- ACKNOWLEDGMENTS ity of the paraelectric phase in SrTiO3 and induce a ferroelectric phase transition (otherwise, one could speak V.V.L. would like to thank O.E. Kvyatkovskiœ and about an “impurity trigger effect”). In all the incipient V.K. Yarmarkin for useful discussions. ferroelectric-based solid solutions studied earlier [3Ð9], At the Ioffe Institute, this work was partly supported the second-end members of the solid solutions were by the program “Physics of Solid State Nanostructures” ferroelectrics (BaTiO3, PbTiO3, CdTiO3, PbMg1/3, and NWO, grant no. 16-04-1999. E.P.S. is grateful to Nb2/3O3 in SrTiO3 and LiTaO3, KNbO3 in KTaO3) or at Sächsisches Staatsministerium für Wissenschaft und Kunst for financial support. least incipient ferroelectrics such as KTaO3 and CaTiO3 in SrTiO3 [5, 41, 42]. In all these cases, a transition to the ferroelectric (or polar) phase inevitably occurred. In REFERENCES our case, SMN and SST are not ferroelectrics, which may be the reason for the absence of a ferroelectric 1. K. A. Müller and H. Burkhard, Phys. Rev. B 19, 3593 (1979). phase transition in their solid solutions with SrTiO3. This point of view may be supported by the data from 2. H. Uwe and T. Sakudo, Phys. Rev. B 13, 271 (1976). [22], where SrTiO3 doped with all the rare-earth ions 3. V. V. Lemanov, Fiz. Tverd. Tela (St. Petersburg) 39, 1645 except promethium was studied and no evidence of the (1997) [Phys. Solid State 39, 1468 (1997)]. ferroelectric state was found. This result may imply 4. V. V. Lemanov, Ferroelectrics 226, 133 (1999). that, in the incipient ferroelectrics with impurities, the 5. J. G. Bednorz and K. A. Müller, Phys. Rev. Lett. 52, ferroelectric phase transition is not due to the impurity- 2289 (1984). trigger effect but is associated with a Vegard-type law 6. V. V. Lemanov, E. P. Smirnova, E. A. Tarakanov, and for the Tc(x) dependence at medium and large values of P. P. Syrnikov, Phys. Rev. B 54, 3151 (1996). the concentration x of the second-end member of the 7. V. V. Lemanov, E. P. Smirnova, and E. A. Tarakanov, Fiz. incipient ferroelectric-based solid solution. At low x, Tverd. Tela (St. Petersburg) 39, 714 (1997) [Phys. Solid the transition to the quantum ferroelectric state occurs State 39, 628 (1997)]. with no ferroelectric phase transition at x values lower than the critical concentration x . 8. M. E. Guzhva, V. V. Lemanov, P. A. Markovin, and c T. A. Shaplygina, Ferroelectrics 218, 93 (1998). Returning to the dielectric relaxation in the solid 9. V. V. Lemanov, A. V. Sotnikov, E. P. Smirnova, et al., Fiz. solutions under study, one may conclude that both the Tverd. Tela (St. Petersburg) 41, 1091 (1999) [Phys. Solid MaxwellÐWagner mechanism and the reorienting State 41, 994 (1999)]. dipole-center model can explain the main features of 10. Chen Ang, Zhi Yu, P. M. Vilarinho, and J. L. Baptista, the high-temperature relaxation. There are some argu- Phys. Rev. B 57, 7403 (1998). ments in favor of the model of the reorienting dipole 11. G. I. Skanavi and E. N. Matveeva, Zh. Éksp. Teor. Fiz. centers, but this conclusion is anything but final since, 30 (6), 1047 (1956) [Sov. Phys. JETP 3, 905 (1957)].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 GIANT DIELECTRIC RELAXATION 2049

12. G. I. Skanavi, I. M. Ksendzov, V. A. Trigubenko, and 27. Powder Diffraction File, Inorganic Phases (Interna- V. G. Prokhvatilov, Zh. Éksp. Teor. Fiz. 33 (2), 321 tional Center for Diffraction Data, Park Lane, Swarth- (1957) [Sov. Phys. JETP 6, 250 (1958)]. more, 1989). 13. Chen Ang, J. F. Scott, Zhi Yu, et al., Phys. Rev. B 59, 28. G. A. Samara and L. A. Boatner, Phys. Rev. B 61, 3889 6661 (1999). (2000). 14. Chen Ang, Zhi Yu, J. Hemberger, et al., Phys. Rev. B 59, 29. V. A. Trepakov, M. E. Savinov, S. Kapphan, et al., Ferro- 6665 (1999). electrics 239, 305 (2000). 15. Chen Ang, Zhi Yu, P. Lunkenheimer, et al., Phys. Rev. B 30. M. Itoh and R. Wang, Appl. Phys. Lett. 76, 221 (2000). 59, 6670 (1999). 31. H. Neumann and G. Arlt, Ferroelectrics 69, 179 (1986). 16. Chen Ang, Zhi Yu, and L. E. Cross, Phys. Rev. B 62, 228 32. H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 161, 822 (2000). (1967). 33. A. Seuter, Philips Res. Rep. Suppl. 3, 1 (1974). 17. T. Y. Tien and L. E. Cross, Jpn. J. Appl. Phys. 6, 459 (1967). 34. G. Arlt and H. Neumann, Ferroelectrics 87, 109 (1988). 35. I. Sakaguchi, H. Honeda, S. Hishita, et al., Nucl. 18. J. Bouwma, K. J. De Vries, and A. J. Burggraaf, Phys. Instrum. Methods Phys. Res. B 94, 411 (1994). Status Solidi A 35, 281 (1976). 36. V. S. Vikhnin, A. S. Polkovnikov, H. J. Reyher, et al., 19. Chen Ang and Zhi Yu, J. Appl. Phys. 71, 6025 (1992). J. Korean Phys. Soc. 32, S486 (1998). 20. E. Iguchi and K. J. Lee, J. Mater. Sci. 28, 5809 (1993). 37. L. S. Sochava, V. E. Bursian, and A. G. Razdobarin, Fiz. 21. Zhi Yu, Chen Ang, and L. E. Cross, Appl. Phys. Lett. 74, Tverd. Tela (St. Petersburg) 42, 1595 (2000) [Phys. Solid 3044 (1999). State 42, 1640 (2000)]. 22. D. W. Johnson, L. E. Cross, and A. Hummel, J. Appl. 38. C. Kittel, Introduction to Solid State Physics (Wiley, Phys. 41, 2828 (1970). New York, 1976; Nauka, Moscow, 1978). 39. M. Fisher, A. Lahmar, M. Maglione, et al., Phys. Rev. B 23. Chen Ang, J. R. Jurado, Zhi Yu, et al., Phys. Rev. B 57, 49, 12451 (1994). 11858 (1998). 40. L. A. K. Dominik and R. K. MacCrone, Phys. Rev. 163, 24. Chen Ang, Zhi Yu, Zhi Jing, et al., Phys. Rev. B 61, 3922 756 (1967). (2000). 41. V. V. Lemanov, V. A. Trepakov, P. P. Syrnikov, et al., Fiz. 25. V. V. Lemanov, E. P. Smirnova, A. V. Sotnikov, and Tverd. Tela (St. Petersburg) 39, 1838 (1997) [Phys. Solid M. Weihnacht, Appl. Phys. Lett. 77, 4205 (2000). State 39, 1642 (1997)]. 26. F. S. Galasso, Structure, Properties and Preparation of 42. V. V. Lemanov, A. V. Sotnikov, E. P. Smirnova, et al., Perovskite-Type Compounds (Pergamon, Oxford, 1969). Solid State Commun. 110, 611 (1999).

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2050Ð2054. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1958Ð1962. Original Russian Text Copyright © 2002 by Zaripov, Tarasov, Ulanov, Shakurov.

SEMICONDUCTORS AND DIELECTRICS

JahnÐTeller Chromium Ions in CdF2 and CaF2 Crystals: An EPR Spectroscopic Study in the Frequency Range 9.3Ð300 GHz M. M. Zaripov, V. F. Tarasov, V. A. Ulanov, and G. S. Shakurov Kazan Physicotechnical Institute, Russian Academy of Sciences, Sibirskiœ trakt 10/7, Kazan 29, 420029 Tatarstan, Russia e-mail: [email protected] Received January 9, 2002

Abstract—The bivalent chromium impurity centers in CdF2 and CaF2 crystals are investigated using electron paramagnetic resonance (EPR) in the frequency range 9.3Ð300 GHz. It is found that Cr2+ ions in the lattices of 6Ð these crystals occupy cation positions and form [CrF4F4] clusters whose magnetic properties at low temperatures are characterized by orthorhombic symmetry. The parameters of the electron Zeeman and ligand interactions of the Cr2+ ion with four ﬂuorine ions in the nearest environment are determined. The initial splittings in the system of spin energy levels of the cluster are measured. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION (EPR) spectra of CdF2 : Cr crystals. These authors The ground state of a free Cr2+ ion is characterized examined the ligand hyperfine structure of the EPR by the 5D term. Upon incorporation into crystals of the spectra and determined the molecular structure of the impurity paramagnetic complex [3]. It turned out that fluorite structural family (CdF2, CaF2, SrF2, and BaF2), 2+ bivalent chromium ions occupy cation sites whose the Cr ion is located at the center of a coordination coordination polyhedron is represented by a regular polyhedron that is a right-angle orthorhombic prism Ð cube with FÐ ions in the vertices. For such a structure, whose vertices are occupied by eight F ions. according to the crystal field theory, the quintuply The same conclusions regarding the structure of 5 degenerate ground D term of the chromium ion is split Cr2+ complexes were drawn on the basis of the EPR into the excited orbital doublet 5E and the ground trip- g data for crystals of CaF2 : Cr [4, 5] and SrF2 : Cr [6]. In 5 6Ð let T2g. In this regard, the impurity complex [CrF8] our earlier work [6], we performed an EPR investiga- formed in a crystal is of considerable interest as an tion over a wide frequency range. This made it possible object exhibiting JahnÐTeller properties. to determine virtually all the spin Hamiltonian parame- Ulrici [1] investigated the optical spectra of CdF2 : ters with sufficient accuracy. Crystals of CdF2 : Cr [3] 5 Cr crystals and revealed transitions between the T2g and CaF2 : Cr [4, 5] were examined at frequencies of 5 and Eg states. The splitting of the levels of the ground 9.3 [3, 4] and 34 GHz [5]. For this reason, the spin 5 Ð1 Hamiltonian parameters were derived from the angular triplet T2g was estimated at 5000 cm . This indicates | 〉 | 〉 considerable distortions of the crystal lattice in the dependences of the EPR transitions +1 Ð1 and |+2〉 |Ð2〉. Since the transitions |0〉 |Ð1〉 and vicinity of a paramagnetic impurity. The characteristic |± 〉 |± 〉 temperature dependence of the optical spectra observed 1 2 could not be observed at low frequencies, in [1] suggests a JahnÐTeller nature of these distortions. the majority of the parameters obtained in [3Ð5] were It should be noted that, according to [2], the vibronic determined only approximately. For example, in [5], 5 the splittings between the spin levels |0〉, |±1〉, and |±2〉 interaction of the orbital triplet T2g with eg vibrations should lead to a tetragonal distortion of the coordina- were evaluated from the temperature dependence of the tion cube of the impurity ion and the interaction with t population of the spin energy levels of the impurity 2g complex. Note that the spin Hamiltonian parameters vibrations should result in a trigonal distortion. How- obtained in [3] are questionable. Their values differ ever, the orthorhombic anisotropy is observed in the 2+ severalfold from those determined in [5, 6]. This seems optical properties of Cr centers [1]. Therefore, in the to be unlikely considering the similarity between the case under consideration, the interactions with vibra- molecular structures of the complexes in these three tions of tetragonal and trigonal symmetries are equally same-type crystal matrices. efficient. These inferences were confirmed by Jablonski et al. In this respect, the main purpose of the present work [3], who studied the electron paramagnetic resonance was to refine the data obtained in [3–5].

2. SAMPLE PREPARATION (a) (c) AND EXPERIMENTAL TECHNIQUE | 〉 | 〉 270 –1 +2 Crystals of CdF2 : Cr were grown by the Bridgman method in a helium atmosphere with small fluorine additives. The crucible used for a CdF : Cr melt was |+1〉 |+2〉 2 260 |–1〉 |+2〉 produced from chemically pure graphite. Chromium impurities in the form of a well-dried chromium triflu- |+1〉 |+2〉 |–1〉 |–2〉 oride powder were introduced into the melt. Fluorine 250 was added to the helium atmosphere used in the crystal growth for the purpose of preparing a melt of nonsto- |–1〉 |–2〉 |+1〉 |–2〉 ichiometric composition with an excess fluorine con- 240 tent. This encouraged an increase in the equilibrium |+1〉 |–2〉 concentration of chromium ions dissolved in the melt. (b) (d) It turned out that the presence of excess fluorine ions in , GHz the melt was a necessary condition for the incorpora- ƒ tion of chromium into the lattice of the growing crystal. 90 |0〉 |+1〉 |0〉 |+1〉 Crystals of CaF2 : Cr were grown under similar conditions. However, since the vapor pressure of the CaF2 melt was substantially less than that of CdF2 melt, the CaF : Cr crystals were grown by the Czochralski |0〉 |–1〉 2 80 |0〉 |–1〉 method. The grown crystals were examined by EPR spectroscopy. It was found that, depending on the growth conditions, the grown crystals contained either predom- 0 0.10.2 0 0.1 0.2 inantly bivalent chromium centers (the temperature B0, T gradient in the vicinity of the crystallization front was greater than 50 K/cm, and the velocity of the crystalli- Fig. 1. Dependences of the frequencies of the resonance transitions |0〉 |±1〉 and |±1〉 |±2〉 on the external zation front was higher than 20 mm/h) or trivalent chro- 2+ mium centers of trigonal symmetry (under different magnetic field B0 for (a, b) CdF2 : Cr and (c, d) CaF2 : conditions). The latter chromium centers were Cr2+ crystals. described in [7]. The spectra of certain CdF : Cr crystals exhibited 2+ 2 the CdF2 : Cr crystals and the present work, for the weak EPR signals whose angular dependences indi- most part, was devoted to examination of these crystals. cated orthorhombic symmetry of the relevant centers. Figure 1 displays the dependences of the frequencies of Repeated recrystallization of these samples led to the resonance electron transitions of the types |0〉 |±1〉 disappearance of these centers. On the other hand, the |± 〉 |± 〉 and 1 2 on the external magnetic field B 0 for addition of oxygen in small amounts to the atmosphere the CdF : Cr2+ and CaF : Cr2+ crystals studied in the used in the crystal growth resulted in an increase in the 2 2 concentration of the centers under investigation. Simi- submillimeter-wavelength region. The points are the lar oxygen-containing centers were observed in CaF : experimental data, and the solid lines correspond to the 2 results of calculations. The type of resonance transition Cr crystals. The parameters of the EPR spectra of these 6Ð is indicated near the corresponding curve in Fig. 1. To crystals coincided with those obtained for [CrF6O] avoid overloading of the figures by too large a number impurity complexes in [5]. of graphs, Fig. 1 shows the dependences for transitions In this work, we performed an EPR investigation of in the system of the spin energy levels only for the CdF2 : Cr and CaF2 : Cr samples predominantly con- structurally equivalent centers for which the Z axis is 6Ð parallel to the vector B of the external constant mag- taining [CrF4F4] complexes. The measurements were 0 carried out at frequencies of 9.3 and 37 GHz (on an E- netic field. In actual fact, there occur also transitions 12 Varian EPR spectrometer) and in the frequency between the states of five magnetically nonequivalent range 65Ð300 GHz (on a quasi-optical spectrometer centers with other orientations with respect to the described in our previous work [8]). The use of the fre- vector B0. quency-tuned spectrometer in the submillimeter-wave- The angular dependences of the resonance external length region made it possible to determine directly all magnetic field measured upon rotation of the vector B the initial splittings in the system of the spin energy lev- 0 2+ in the (110) plane of the crystal at T = 4.2 K and f = els for the Cr impurity ion. 37 GHz are plotted in Fig. 2. The dependences shown Since some doubts were cast upon the spin Hamilto- in this figure correspond to transitions within the pairs nian parameters obtained in [3], our prime concern was of the spin levels |±1〉 and |±2〉 and between the levels

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2052 ZARIPOV et al.

B0, T B0, T

0.5 2.0

0.4

1.5

0.3

1.0 0.2

0.5 0.1

0306090 0306090 [110] Angle, deg [001] [110] Angle, deg [001]

Fig. 2. Angular dependences of the resonance external magnetic ﬁeld upon rotation of the vector B0 in the (110) plane Fig. 3. Angular dependences of the resonance external mag- of the CdF : Cr2+ crystal (T = 4.2 K, f = 37 GHz) for the netic ﬁeld upon rotation of the vector B0 in the (110) plane 2 2+ electron transitions |+1〉 |Ð1〉 (closed circles), of the CdF2 : Cr crystal (T = 4.2 K, f = 9.3 GHz) for the |+2〉 |Ð2〉 (open circles), and |0〉 | Ð1 〉 (closed electron transitions |+1〉 |Ð1〉 (closed circles) and squares). |+2〉 |Ð2〉 (open circles).

|0〉 and |Ð1〉. The transitions |Ð1〉 |+1〉 are observed tra exhibited a ligand hyperfine structure. The best res- for all six magnetically nonequivalent centers. The olution was achieved for the transitions |+2〉 |Ð2〉. transitions |0〉 |Ð1〉 occur only for one group of For the transitions |+1〉 |Ð1〉, the ligand hyperfine 2+ these centers, in which the vector B0 is oriented at small structure was revealed only in the CdF2 : Cr samples ≤ ° ° || | 〉 angles ( 12 Ð15 ) with respect to the Z axis. This can with the orientation B0 Y. For the transition 0 be explained by the fact that the states |0〉 and |±1〉 are |−1〉, a partly resolved ligand hyperfine structure was characterized by a considerable energy splitting, which 2+ observed in the CdF2 : Cr samples at the orientation 0 || is determined primarily by the parameter b2 of the ini- B0 Z. In the majority of cases, the ligand hyperfine 0 ӷ β 2 structure was represented by five resonance lines with tial splitting (because 3b2 egB0 ~ 6b2 ). relative intensities approximately equal to the ratio 1 : In the case when the magnetic vector Bf of the elec- 4 : 6 : 4 : 1. Moreover, the dependence of the line width tromagnetic field in the spectrometer cavity is oriented of the ligand hyperfine quintet on the direction of the | 〉 perpendicularly to the quantization axis of spin vector B0 was observed for the transitions +2 moments of the centers under investigation, the transi- |−2〉 in both matrix crystals. | 〉 | 〉 tions 2 +2 are forbidden. However, the vector B f The angular dependences of the resonance external during rotation of the vector B in the (110) crystal 0 magnetic field upon rotation of the vector B0 in the plane has a nonzero component along the quantization (110) plane at T = 4.2 K and f = 9.3 GHz are depicted axes of five (out of six) groups of the magnetically non- in Fig. 3. It can be seen that the line attributed to the equivalent centers. Therefore, the possibility appeared transitions |+1〉 |Ð1〉 should be observed at the ori- of recording the resonance lines corresponding to these || 〈 〉 ≈ entation B0 001 in a magnetic field of 0.4 T. In [3], transitions. In reality, this could be done only for the this line was erroneously assigned to the transition four groups of these centers for which the angle |+2〉 | Ð1 〉 . between the Z axis and the vector B0 did not exceed 50°Ð60°. 3. RESULTS AND DISCUSSION For certain orientations of the vector B0 of the external magnetic field with respect to the principal axes of The theoretical positions of the resonance lines in 2+ 2+ the CdF2 : Cr and CaF2 : Cr crystals, the EPR spec- Figs. 1Ð3 were determined from the equality between

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 JAHNÐTELLER CHROMIUM IONS 2053 the corresponding differences of the eigenvalues of the (1), which is represented with the use of the eigenfunc- spin Hamiltonian tions of the spin Hamiltonian (2), are taken to be equal to the photon energy of electromagnetic oscillations at β 0 0 2 2 H = e B0gS++ b2O2 b2O2 the resonance frequency. (1) 0 0 2 2 4 4 It turned out that the determinant of the matrix of the + b4O4 ++b4O4 b4O4 coefficients is close to zero. This indicates that the and the energy of electromagnetic-field quanta in the occurrence of unavoidable experimental errors prevents spectrometer cavity. Here, the spin Hamiltonian param- unique determination of the parameters of the spin Hamiltonian (1). In particular, it is found that, in the 0 ± eters are as follows: b2 = Ð27700 50 MHz, matrix, the column of the coefficients of the unknown ()2 2 ± 0 ≈ 4 ± 2 b2 Ð 4b4 = 900 30 MHz, b4 5 MHz, b4 = 45 b2 linearly depends on the column of the coefficients of ± ± 5 MHz, gx = 1.978 0.005, gy = 1.995 0.005, and gz = 2 the unknown b4 . This linear dependence can be ± 0 ± 1.946 0.005 for the CdF2 crystal and b2 = Ð28 400 2 2 approximated using the quantity (b2 Ð 4b4 ). Further- ()2 2 ± 0 ≈ 100 MHz, b2 Ð 4b4 = 1800 50 MHz, b4 3 MHz, more, it is revealed that, when the set of experimental 4 ± ± ± points is limited to those corresponding to the transi- b4 = 25 10 MHz, gx = 1.97 0.01, gy = 1.98 0.01, | 〉 | 〉 | 〉 | 〉 ± tions +1 Ð1 and +2 Ð2 , the solutions of and gz = 1.94 0.01 for the CaF2 crystal. the system of equations become extremely sensitive to The spin Hamiltonian (1) is represented in a coordi- experimental errors. Actually, these calculations make nate system whose axes are oriented with respect to the it possible to determine a number of important parame- || 〈 〉 || 0 2 principal axes of the crystal as follows: X 001 , Y ters of the spin Hamiltonian (1), namely, b , (b Ð 〈1Ð10〉, and Z || 〈110〉. The choice of the Z axis was gov- 2 2 2 erned by the requirement for a maximum magnitude of 4b4 ), gx, gy, and gz, with a satisfactory accuracy. How- 0 0 ever, the other parameters can be obtained with a con- the coefficient b2 of the spin operator O2 in Hamilto- nian (1). The Hamiltonians of the other five centers can siderably lower accuracy. be obtained through transformation of the cubic sym- Analysis of the splittings in the ligand hyperfine structure of the EPR spectra enabled us to derive the metry group. parameters of the Hamiltonian of the ligand hyperfine In the course of calculations, some doubts were cast interaction between the electron magnetic moment of on the possibility of uniquely determining all eight the studied center and the nuclear magnetic moments of parameters of the spin Hamiltonian (1). In order to elu- fluorine ions in the nearest environment of the chro- cidate this problem, we constructed a system of approx- mium impurity ion. In the case under consideration, 0 2 imate linear equations in which the parameters b2 , b2 , this Hamiltonian has the form b0 , b2 , b4 , g , g , and g were unknown; i.e., we used F()i F()i 4 4 4 x y z Hshfi = ∑SA I , (3) the least-squares method to the first order in the pertur- i bation theory. For this purpose, the matrix of the spin Hamiltonian (1) was determined using the eigenfunc- where S is the electron spin moment operator, AF(i) is the tions of the truncated Hamiltonian tensor for the ligand hyperfine interaction with FÐ(i) ions, and IF(i) is the nuclear spin moment operator for ()0 β ()0 ()()0 ()0 0 ()2 ()0 2 − H = e B0g Sb++2 O2 b2 O2, (2) F (i) ions. In order to describe the ligand hyperfine splittings in the EPR spectrum, Hamiltonian (3) should ()()0 ()0 where ()b 0 and ()b2 are the approximate values be considered simultaneously with the nuclear Zeeman 2 2 interaction Hamiltonian [representing the interaction of 0 2 of the parameters b2 and b2 , respectively. These the external magnetic field with the nuclear magnetic approximate parameters can be obtained, for example, moment of the FÐ(i) ion]; that is, from the splittings of the spin energy levels at B = 0 0 β F()i ()0 HNZ = Ð∑gN N B0 I , (4) (Fig. 1). Then, the approximate components gi (i = x, i y, z) of the tensor g can be found from the EPR spectra β measured at a frequency of 37 GHz. Thereafter, for a where gN and N are the g factor and the nuclear mag- necessary set of experimental points in the angular neton of this ion, respectively. dependences of the resonance external magnetic field As follows from the experimental data, each center B0, we write the initial approximate equalities deter- interacts only with four equivalent fluorine ions. These mining the coefficients of the system of simultaneous ions are located in the same plane as the Cr2+ impurity equations in the least-squares method. In the approxi- ion. This plane coincides with the XOY coordinate mate equalities, the corresponding differences between plane and is aligned parallel to one out of the six (110) the diagonal matrix elements of the spin Hamiltonian planes in the crystal. The ligand hyperfine interaction

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2054 ZARIPOV et al. with four fluorine ions each is described by the symme- prime indicates the local coordinate system) for CdF2 : 2+ try group Cs, and the symmetry of the paramagnetic Cr crystals was determined from the ligand hyperfine complex, as a whole, corresponds to the D2h group. This structure of the EPR spectrum for the electron transi- | 〉 | 〉 || inference is confirmed by the angular dependences of tion +1 Ð1 at the orientation B0 Y. It was found the resonance magnetic field B0 measured at frequen- that |A||| = 20 ± 8 MHz. A similar tensor component for 2+ cies of 9.3 and 37 GHz and in the submillimeter-wave- CaF2 : Cr crystals cannot be obtained because the length region. It is known (see, for example, [5]) that, ligand hyperfine structure of the EPR spectrum is in the case of large initial splittings of the spin energy observed only for the transition |+2〉 |Ð2〉. levels, the ligand hyperfine splitting of the EPR lines As regards the interaction with the other four nearest for the electron transition |+2〉 |Ð2〉 is predomi- ligands, its influence on the EPR spectra manifests nantly determined by three components (AZX, AZY, and F(i) itself in an angular dependence of the line width in the AZZ) of the tensor A and very weakly depends on the ligand hyperfine structure of the EPR spectra for the direction of the external magnetic field. If the ligands electron transitions |+2〉 |Ð2〉. are located in the XOY plane, we have AZY = AZX = 0 and the measured splittings in the ligand hyperfine structure of the EPR spectra permit us to determine only one 4. CONCLUSIONS component of the tensor AF(i) [i.e., the component A , ZZ Thus, the results of the above investigation allow us which corresponds to the perpendicular component A⊥ to make the inference that the models of the studied F(i) of the same tensor A represented in a local coordinate centers in CdF2 and CaF2 crystals are consistent with system with the Z' axis aligned along the Cr2+ÐFÐ(i) the model of a bivalent chromium impurity complex in bond]. An examination of the ligand splittings in the the SrF2 crystal [6]. The hyperfine structure parameters EPR spectra recorded at frequencies of 9.3 and 37 GHz determined from analyzing the EPR spectra of CdF2 for the transition |+2〉 |Ð2〉 demonstrates that crystals in the present work and in [3] differ signifi- nearly identical ligand hyperfine splittings of the cantly. This can be explained by the misinterpretation nuclear energy levels are observed under all experimen- of the EPR spectra in [3], most likely, as a result of a tal conditions. The splitting of the electron level |M〉 can lack of necessary experimental data. be determined from the approximate relationship δ ≅ {}()〈〉2 2 2 〈〉 α Ð1/2 REFERENCES EM SZ A⊥ + f L Ð 2A⊥ f L SZ M cos . (5) 〈 〉 1. W. Ulrici, Phys. Status Solidi B 84, K155 (1977). Here, SZ M is the mean of the electron operator SZ in the | 〉 2. I. B. Bersuker and V. Z. Polinger, Vibronic Interactions state M that corresponds to either of two electron in Molecules and Crystals (Nauka, Moscow, 1983; levels between which the resonance transition occurs β Springer-Verlag, New York, 1989). (M = +2 or Ð2), fL = gN NB0 (resonance) is the Larmor 3. R. Jablonski, M. Domanska, and B. Krukowska-Fulde, precession frequency of fluorine ions in the external Mater. Res. Bull. 8, 749 (1973). magnetic field, α is the angle between the Z axis and the 〈 〉 ӷ 4. J. M. Baker, W. Hayes, and D. A. Jones, Proc. Phys. Soc. vector of the external magnetic field, and A⊥ SZ M fL. London 73, 942 (1959). Relationship (5) holds with a high accuracy in a 5. P. V. Oliete, V. M. Orera, and P. J. Alonso, Phys. Rev. B weak external magnetic field (the EPR spectra mea- 53 (6), 3047 (1996). sured at a frequency of 9.3 GHz). Moreover, according 6. M. M. Zaripov, V. F. Tarasov, V. A. Ulanov, et al., Fiz. to calculations, this relationship in our case can also be Tverd. Tela (St. Petersburg) 37 (3), 806 (1995) [Phys. used for analyzing the ligand hyperfine structure of the Solid State 37, 437 (1995)]. EPR spectra measured at a frequency of 37 GHz. The 7. R. Alcala, P. J. Alonso, V. M. Orera, and H. V. den Har- calculations performed using relationship (5) give the tog, Phys. Rev. B 32 (6), 4158 (1985). | | ± 2+ components A⊥ = 40 4 MHz for CdF2 : Cr crystals 8. V. F. Tarasov and G. S. Shakurov, Appl. Magn. Reson. 2 ± 2+ (3), 571 (1991). and 42 5 MHz for CaF2 : Cr crystals (the latter value coincides with the result obtained in [5]). The ligand ≈ hyperfine interaction tensor component A'ZZ A|| (the Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2055Ð2062. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1963Ð1969. Original Russian Text Copyright © 2002 by Novikov, Wojciechowski.

SEMICONDUCTORS AND DIELECTRICS

Frequency Dependences of Dielectric Properties of MetalÐInsulator Composites V. V. Novikov* and K. W. Wojciechowski** * Odessa State Polytechnical University, Odessa, 65044 Ukraine e-mail: [email protected] ** Institute of Molecular Physics, Polish Academy of Sciences, Poznan, 60-179 Poland Received March 11, 2002

Abstract—Based on the fractal model of an inhomogeneous medium with a chaotic structure and the iteration method of averaging, frequency dependences of the dielectric properties of metalÐinsulator composites were determined. In the low-frequency limit, the considered methods of the investigation of two-component media were shown to permit one to obtain detailed information on the metalÐinsulator transition. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION changes upon the propagation of the electromagnetic field periodically with a frequency ω as Theoretical investigations of the dielectric proper- (), (), ω ()ω ties of inhomogeneous media stem from works pub- Ert = E0 r exp i t , (3) lished as far back as the 1870sÐ1930s [1Ð3]. Based on these works, the method of an effective medium was then Ohm’s law may be written in the form [10] developed [4], which in essence represents the replace- jr()σ, ω = *()r, ω Er(), ω , (4) ment of an inhomogeneous medium consisting of two σ σ ω σ ω composites with conductivities 1 and 2 by a continu- where j(r, ) is the current density and *(r, ) is the ous medium with an effective conductivity σ. Note that complex coefficient of conductivity: such an approximation is only applicable for the case σ ()σ, ω (), ω ωε(), ω where the length of the electromagnetic wave interact- * r = r + i r . (5) ing with the medium is much greater than the sizes of For an inhomogeneous medium with a chaotic struc- inhomogeneities and the spacings between them. ture, the dielectric constant ε(r, ω) and the conductivity σ(r, ω) are random (stochastic) functions of the coordi- The effective-medium method has been sufficiently nates r. widely used for the description of physical properties of inhomogeneous media [4]; however, it did not permit Note that from Eq. (5) we can determine the scaling one to predict the behavior of the system upon the expressions for the conductivity of a lattice of resistors metalÐinsulator transition near the percolation thresh- of finite dimensions l, which were first obtained in old [5Ð9]. For the case where the ratio of the conductiv- [11, 12]: ities of the phases σ /σ tends to zero, scaling depen- () 2 1 σσξÐt/v ()σ σ ξ ts+ /v, ξ ∆ > dences were obtained in terms of the percolation theory = 1 G+ 1/ 2 /l , p 0, (6) for the effective dc conductivity σ (see [5, 6]), which σσξs/v ()σ σ ξ()ts+ /v, ξ ∆ < reflect the behavior of the conductivity near the geo- = 2 GÐ 2/ 1 /l , p 0, (7) metrical phase transition (near the metalÐinsulator transition) when the disconnected region of the conducting where G+(x, y) and GÐ(x, y) are the functions of two phase passes into a connected region: variables that describe the frequency and scale dependences of the conductivity of the function above and below the percolation threshold. σ∆∼ pÐs, ∆p = ppÐ < 0, (1) c In recent years, great attention has been paid to an

T analysis of the dependences of the properties of metalÐ σ∆∼ p , ∆p > 0, (2) insulator composites on frequency [11Ð31], which is related to the difficulties in the description of the anom- where T = s ≈ 1.1 for two-dimensional systems (d = 2) alous behavior of dielectric properties in the low-fre- and T ≈ 1.6, s ≈ 1 for three-dimensional systems (d = 3); quency limit. The nature of the anomalous behavior of pc is the percolation threshold. If we assume that for frequency dependences of dielectric properties can be local regions the strength of the electric field E(r, t) clarified if we consider a model medium consisting of

2056 NOVIKOV, WOJCIECHOWSKI

(a) imaginary part of the effective dielectric constant of the A medium strongly increases in the vicinity of the fre- B ω quency p/3 [11, 12]. A B For inhomogeneous media such as a metalÐinsulator composite with a chaotic structure, the behavior becomes even more complex. (b) In the majority of numerical calculations of the anomalous frequency behavior of the metalÐinsulator composites (in particular, near the percolation threshold pc) under the action of an alternating current, lattice (discrete) models have been used, which were studied in terms of the transfer-matrix method [11, 12] and using the FrankÐLobb algorithm [13]. The numerical calculations and the theoretical analysis of the properties of metalÐinsulator composites performed in [11Ð 31] led to significant progress in the understanding of this phenomenon, but the dielectric properties of metalÐinsulator composites with fractal structures virtually have not been considered in the literature. Below, we report the results of calculations of the dielectric properties of metalÐinsulator composites with a chaotic hierarchic self-similar structure on the basis of a fractal model in the entire concentration range of inhomogeneities at various frequencies of an external field. In the work, the iteration method of averaging was employed [24, 32Ð36]. Ω Fig. 1. Obtaining the n(l0, p0) set for l0 = 2 at the fourth iteration step (k = 4): (a) p0 = 1 and (b) p0 = 0.75 (schematic). 2. MODEL Simulation of a chaotic structure of an inhomogeneous medium was performed on the basis of a lattice small spherical metallic particles described by the with a random distribution of its parameters. The sites Drude dielectric function of the lattice simulated microinhomogeneities (compo- ω2 nents of the system) in space, and the bonds between ε ()ω p1 the sites simulated their contacts with neighbors [32, 1 = 1 Ð ωω------()τ (8) + i/ 1 35, 36]. and embedded into an insulating matrix with a dielec- The basic set of bonds Ω was obtained with the help ε ω tric constant equal to unity ( 2 = 1). In Eq. (8), p1 is the of an iteration process in which at the first step (k = 0), τ plasma frequency and 1 is the relaxation time of the there is a finite lattice in d = 3 space with a probability metallic phase. p0 that the bond between neighboring nodes of the lat- If such a medium is subjected to the action of an tice is unbroken (is “painted” a definite color). Bonds of electric field (3), then, with allowance for the fact that the same color possess the same properties. At the next the solution to the electrostatic problem for a spherical step (k = 1, 2, …, n), each bond in the lattice is replaced inclusion in a homogeneous medium yields for the by the lattice obtained at the previous step (Fig. 1). The electric field E inside a spherical metallic particle a iteration process is terminated when the properties of 1 the lattice stop being dependent on the number of the dependence of the form Ω iteration k. The set of bonds n(l0, p0) obtained using ()ε E1 = 3E0/ 1 + 2 , (9) the iteration procedure depends on the size of the initial lattice l0 and on the probability p0. we obtain that the electric field E1 tends to infinity at frequencies close to The probability of forming a set of bonds of the same color connecting two opposite faces of the lattice ω ≅ ω p1/3. (10) Y(l0, p0) was calculated as the ratio of the number of connecting sets (CS) to the number of all possible con- At such frequencies, the applied field is in resonance σ* with the mode of the small metallic particle; as a result, figurations of bonds of the same color (1 ) at a given a strong absorption appears at this frequency; i.e., the p0 and l0 [32].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 FREQUENCY DEPENDENCES OF DIELECTRIC PROPERTIES 2057

At the ﬁrst iteration step, the length of the lattice (a) (b) 2 edge is equal to l1 = l0 and the density of bonds of the σ same color, e.g., black (1* ), is equal to p1 = Y(l0, p0). At the next iteration steps, the length of the lattice edge is 2 l2 = l1 and the density of black bonds is ()…, ,, (), p2 ==Yl1 p1 pk Ylk Ð 1 pk Ð 1 .

Here, pk = Y(lk Ð 1, pk Ð 1) is the density of CSs at the kth iteration step. Ω The growth of the chaotic fractal set n(l0, p0) is terminated at the nth step at the ﬁxed point 0 or 1 for the Fig. 2. Simulation of (a) a connecting set (CS) and (b) a nonconnecting set (NCS) (schematic). function pn = Y(ln Ð 1, pn Ð 1): > 1, p0 pc pn =  δ 0, p < p . where (x) is the Dirac delta function, p0 is the proba- 0 c bility that a given local region possesses the property σ ()0 σ The functions Y(l0, p0) for various rectangular lat- 1* = 1* , and 1 Ð p0 is the probability that this region tices were calculated using the Monte Carlo method in σ ()0 σ [32]. possesses the property 2* = 2* . The unstable critical point pc (percolation threshold) In this case, after k iteration steps, the density func- was determined from the equality pc = Y(l0, pc). tion takes on the form

Then, the probability function pk for the event that σ ()δσσ ()k δσ σ ()k the set of bonds at the kth iteration step is a CS was Pk(* ) = 1 Ð pk ()* Ð 2* + pk ().* Ð 1* (15) determined using the formula [37] In what follows, we will distinguish two types of 2 ( 2 3 pk = pk Ð 1 48+ pk Ð 1 Ð4014 pk Ð 1 Ð pk Ð 1 sets of bond conﬁgurations: connecting sets (CSs) and 4 5 6 nonconnecting sets (NCSs). +16pk Ð 1 + 288 pk Ð 1 Ð 655 pk Ð 1 (11) To determine the dielectric properties of the CSs and 7 8 9 10 ) + 672 pk Ð 1 Ð376 pk Ð 1 + 112 pk Ð 1 Ð 14 pk Ð 1 , NCSs, we used a cell of the cube-in-cube type (Fig. 2); i.e., at each step of the iteration process of the calcula- which agrees satisfactorily with numerical calculations tion of the properties, the structures of the CSs and [32]. NCSs were simulated by a cube-in-cube cell as follows: The percolation threshold pc, according to [11], is CS, a continuous body of a well conducting phase equal to 0.2084626828…; i.e., the nonconnecting set including a cube of a poorly conducting phase (Fig. 2a); ≈ (NCS) passes into the connecting set (CS) at pc NCS, a continuous body of a poorly conducting phase 0.208462. including a cube of a well conducting phase (Fig. 2b). Ω Each kth bond in the set n(l0, p0) possesses a com- The dc conductivity of the cube-in-cube cell (Fig. 2) ω plex resistance (impedance) Zk( ) which consists of was determined in [7, 38]. The results obtained in [7, σ an active resistance Rk, an inductance Lk, and a capac- 38] for a cell in which a cube with a conductivity 2 is σ itance Ck: in the center of a cube with a conductivity 1 can be written in the form Ð1()ω ()ω Ð1 ω Zk = Rk + i Lk + i Ck. (12) σ σ 1()ψ ψ In what follows, each bond will be characterized by = ----- 1 + 2 , (16) σ 2 the complex conductivity k* with allowance for the fact that the equality where Ð1 σ* ()ω 2/3 k = Zk (13) σ + ()σ Ð σ ()1 Ð p ψ = ------1 2 1 -, (17) is fulﬁlled. 1 σ ()σ σ ()2/3[]()1/3 1 + 2 Ð 1 1 Ð p 11Ð Ð p Consider a two-phase system with a distribution function σ + ()σ Ð σ ()1 Ð p 1/3[]11Ð ()Ð p 2/3 ψ = ------2 1 2 -. (18) σ ()δσσ ()0 δσ σ ()0 2 σ ()σ σ ()1/3 P0(* ) = 1 Ð p0 ()* Ð 2* + p0 (),* Ð 1* (14) 2 + 1 Ð 2 1 Ð p

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2058 NOVIKOV, WOJCIECHOWSKI

Here, p is the volume concentration of the phase with a With allowance for Eqs. (16)Ð(18), the complex σ conductivity 1. conductivity of CSs at the kth step of the calculations According to Eqs. (4) and (5), the problem of the was determined using the formulas determination of the effective characteristics of a σ ()k Ð 1 medium in a quasi-stationary approximation differs () * ()k Ð 1 ()k Ð 1 σ* k = ------c ()ψ + ψ , (19) from the static case only in the replacement of the con- c 2 1 2 ductivity σ (dc conductivity) by the complex conductivity σ*. where

()k Ð 1 ()k Ð 1 ()k Ð 1 2/3 () σ* + ()σ* Ð σ* ()1 Ð p ψ k Ð 1 = ------c n c k Ð 1 ------, (20) 1 σ ()k Ð 1 ()σ ()k Ð 1 σ ()k Ð 1 ()2/3[]()1/3 c* + n* Ð c* 1 Ð pk Ð 1 11Ð Ð pk Ð 1

()k Ð 1 ()k Ð 1 ()k Ð 1 1/3 2/3 () σ* + ()σ* Ð σ* ()1 Ð p []11Ð ()Ð p ψ k Ð 1 = ------c c n k Ð 1 k------Ð 1 -, (21) 2 σ ()k Ð 1 ()σ ()k Ð 1 σ ()k Ð 1 ()1/3 n* + c* Ð n* 1 Ð pk Ð 1

σ ()0 σ σ ()0 σ 1 c* = 1* , n* = 2* , and p0 = p. The magnitudes ε ==1; ε 10; γ =------; ω τ =30; 1 2 30 p 1 of pk were determined using formula (11). σ σ Ð2 τ ≤≤ω ω In Eqs. (19)Ð(21), the subscripts n and c denote that 2/ 1 ==10 ; 1 1; 0.001 / p 1.5. a given quantity refers to the NCS and CS, respectively, Figures 3 and 4 display the dependences of the and the index k indicates the order number of the itera- ε σ ω tion step. effective dielectric constant = Im( *)/ and the effective conductivity σ = Re(σ*) on the concentration of the To determine the complex conductivity of an NCS ω ω metallic phase p and the relative frequency / p. The σ ()k ε n* , it is necessary to replace the indices in Eqs. (19)Ð zeros of the effective dielectric constant determine the (21) as follows: n c, 1 2, and (1 Ð p) p. plasma frequencies of the system, i.e., the metalÐinsu- lator transition. It follows from the calculations (Figs. 3, 4) that at 3. CALCULATION RESULTS low frequencies, a divergence arises in the effective The calculations were performed for a two-phase dielectric constant and in the effective conductivity (a (two-component) medium. sharp increase in losses). This is explained by the fact that in this case, in the system, finite clusters of the The calculations of the dielectric properties of inho- metallic phase arise which are separated by thin insulat- mogeneous media at various frequencies and concen- ing interlayers. Such structures form a hierarchic self- trations of the phases using Eqs. (11) and (19)Ð(21) similar chaotic capacitance net which generates a sys- showed that the iteration process converges; i.e., tem of resonance frequencies. σ ()k σ ()k σ In addition, the frequency dependences of the effec- lim c* ==lim n* *. (22) k → ∞ k → ∞ tive properties are affected by the configurations of the finite clusters [39]. This may be illustrated by the con- The complex local conductivity for the metallic phase sideration of a pair of inclusions that have the shape of with allowance for the Drude dielectric function (8) a circle with the associated set of discrete frequencies: was determined as ω2 ω2 ()ξ 1 1m = p tanh m 0 , σ*()ω = σ+ iωεÐ ------, (23) (25) 1 1 1 2 2 ω2 ω2 ()ξ ,,… x + γ 2m ==p cothm 0 , m 1 2 , where where ω ω γ ω τ 2 1/2 x ==/ p, 1/ p 1. ξ ρρ+ ()Ð 4R 0 = ln ------, The complex local conductivity of the insulating 2R phase was determined as ρ is the spacing between the centers of the circles, and σ ()ω σ ωε R is the radius of the circles. 2* = 2 + i 2. (24) Thus, if such regions are formed in a composite, It was assumed in the calculations that they create circuits with resonance frequencies.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 FREQUENCY DEPENDENCES OF DIELECTRIC PROPERTIES 2059

(a) (b)

0 20

2 0

ε /

ε –500 0.10

/ 0.010 –20 ε ε –1000 –40

0.006 0.06 p ω p ω 0 / 0 / 0.2 ω 0.2 ω p 0.6 0.002 p 0.6 0.02 1.0 1.0 (c) (d)

2 1.0

2 /

0 0.5 ε 0.5

/ ε 1.4 –2 0 0.3 1.0 0 p 0 p ω ω 0.2 / 0.2 / ω ω 0.6 p p 0.6 0.6 1.0 0.1 1.0

ε ε Fig. 3. Variation of the dielectric constant / 2 of a metalÐinsulator composite as a function of the concentration of the metallic phase ω ω p and the frequency / p.

(‡) (b)

1 2 0 σ

σ 0.010 0 0.10 –1 log

log –2 –2 0.006 0 0.06 p ω p ω 0 / 0.2 / 0.2 ω ω 0.6 0.002 p 0.6 0.02 p 1.0 1.0 (c) (d)

2 0 σ

σ 1.4

0 0.5 log

log –2 –2 0 1.0 ω p 0.3 0.2 / 0 ω p 0.2 / ω ω 0.6 0.6 0.6 p p 0.1 1.0 1.0

Fig. 4. Variation of the logarithm of the conductivity logσ of a metalÐinsulator composite as a function of the concentration of the ω ω metallic phase p and the frequency / p.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2060 NOVIKOV, WOJCIECHOWSKI

(a) (b)

500 0 0 0.010 0.010

–500 –500 h

h 0.06 ω p 0.006 p ω 0 / 0 / 0.2 ω 0.2 ω 0.6 0.02 0.6 p p 0.002 1.0 1.0

50 20 0.5 10

h 0 1.4

h 0 –50 0.3 1.0 0 p 0 ω ω p 0.2 / 0.2 / ω ω 0.6 0.6 p 0.6 1.0 0.1 p 1.0

Fig. 5. Variation of the ratio of the capacitance conductivity to the active conductivity h = |εω/σ| of a metalÐinsulator composite as ω ω a function of the concentration of the metallic phase p and the frequency / p.

It also follows from Eq. (25) that, at ρ 2R, we the system at hand are located chaotically, hierarchi- ξ have 0 0 and frequencies (25) form a quasi-con- cally, in a self-similar way, and also lead to the appear- tinuous spectrum [39]. ance of peaks in the conductivity. It was shown in [11, 12] that ring-shaped structures Figure 5 displays the dependences of the modulus of the ratio of the capacitance conductivity to the active (ring clusters) generate double peaks in the frequency |εω σ| ω ω conductivity h = / on p and / p. Calculations dependence of the conductivity. Such ring structures in show that the displacement current in the region of ω ω small frequencies ( / p < 1) behaves nonmonotoni- ω ω cally. In the range of high frequencies ( / p > 1) and at (a) (b) concentrations of the metallic phase below the percola- 2 4 tion threshold (p < pc), the displacement current 1 2 3 exceeds the current through the active conductors (h ӷ 2 2 1) and the surface of the dielectric properties becomes 2 0 1 1 ε / smooth. At p > pc, the current through the active con- ε 0 –1 1 ductors exceeds the displacement current (h Ӷ 1). –2 –1 –2 Now, we discuss one of the possible applications of –3 –3 the above model concepts of the dielectric properties of 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 fractal systems. p The optical properties of colloid systems have not ε ε yet been explained in terms of the classical theory (e.g., Fig. 6. Variation of the dielectric constant / 2 of a metalÐ insulator colloid solution as a function of the metallic phase in terms of the Mie theory [40Ð42]). In the framework (silver) p: (a) calculation based on the Lorentz model; (b) of this theory, the change in the color of sols was ω ω calculation based on the fractal model; (1) / p = 0.25 and assumed to be due to the appearance of metallic (silver) ω ω (2) / p = 0.3. particles of various sizes in the solution; the change in

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 FREQUENCY DEPENDENCES OF DIELECTRIC PROPERTIES 2061 the color was ascribed to the dependence of the reso- 6. M. Sahimi, Applications of Percolation Theory (Taylor nance (plasma) frequency on the particle radius. How- and Francis, London, 1994). ever, the experimental investigations showed that the 7. V. P. Privalko and V. V. Novikov, The Science of Hetero- spectral dependences of colloid solutions do not corre- geneous Polymers: Structure and Thermophysical Prop- late with the statistical particle-size distribution func- erties (Wiley, Chichester, 1995). tion; i.e., the role of the particle size seems to be insig- 8. I. Webman and J. Jortner, Phys. Rev. B 11 (8), 2885 nificant [43]. The appearance of a long-wavelength (1975). wing in the spectrum of the colloid solution can be 9. I. Webman and J. Jortner, Phys. Rev. B 13 (2), 713 explained by the aggregation of particles into fractal (1976). structures. 10. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course Indeed, a small silver particle has a frequency of of Theoretical Physics, Vol. 8: Electrodynamics of Con- λ π ω tinuous Media (Nauka, Moscow, 1982; Pergamon, New plasma vibrations with a wavelength = 2 c/ p = 140 nm. To explain the presence of a peak at 650 nm, York, 1984). the classical (Lorentz) theory [40Ð42] requires the pres- 11. X. Zeng, P. Hui, and D. Stroud, Phys. Rev. B 39 (2), 1063 ence in the colloid solution of silver with a volume con- (1989). centration of p ≅ 0.86 (Fig. 6a), whereas the experiment 12. X. C. Zeng, P. M. Hui, D. J. Bergman, and J. D. Stroud, yields p values that are much smaller [44], which agrees Phys. Rev. B 39 (18), 13224 (1989). with our calculations (Fig. 6b). Thus, the shift of the 13. D. J. Frank and C. J. Lobb, Phys. Rev. B 37 (1), 302 peak in colloid solutions toward the region of small (1988). concentrations of metal can be explained by the forma- 14. J. Abel and A. A. Kornyshev, Phys. Rev. B 54 (9), 6276 tion of fractal structures in these solutions. (1996). We indicate some other objects that have fractal 15. D. Bergman and Y. Imry, Phys. Rev. Lett. 39 (19), 1222 structures. For example, using sputtering regimes that (1977). correspond to the model of diffusional aggregation [5], 16. A. Bug, G. Grest, M. Cohen, and I. Webman, J. Phys. A thin films consisting of metallic fractal clusters can be 19 (1), 323 (1986). obtained. Fractal structures are also characteristic of 17. A. Bug, G. Grest, M. Cohen, and I. Webman, Phys. Rev. percolation clusters near the percolation threshold, as B 36 (7), 3675 (1987). well as some binary solutions and polymer solutions. 18. T. W. Noh and P. H. Song, Phys. Rev. B 46 (7), 4212 The dielectric properties of all these objects can be pre- (1992). dicted using the above-considered fractal model. 19. T. B. Schroder and J. C. Dyre, Phys. Rev. Lett. 84 (2), 310 (2000). 4. CONCLUSION 20. X. L. Lei and J. Qzhang, J. Phys. C 19, L73 (1986). 21. R. Koss and D. Stroud, Phys. Rev. B 35 (17), 9004 Thus, calculations of the dependences of the con- (1987). ductivity and the dielectric constant of chaotic hierarchic self-similar structures of metalÐinsulator compos- 22. D. Stroud and P. Hui, The Physics and Chemistry of ites were performed on the basis of a fractal model in Small Clusters (Plenum, New York, 1987), pp. 547Ð565. the entire range of concentrations of inhomogeneities at 23. A. Jonscher, Dielectric Relaxation in Solids (Chalsea various frequencies of an external field. The metalÐ Dielectric Press, London, 1983). insulator transition was shown to occur not only near 24. V. V. Novikov, O. P. Poznansky, and V. P. Privalko, Sci. the percolation threshold. It was also shown that the Eng. Compos. Mater. 4 (1), 49 (1995). transition depends on the concentration of the metallic 25. V. Raicu, Phys. Rev. E 60 (4), 4677 (1999). phase and the frequency of the external field. 26. V. E. Dubrov, M. E. Levinshteœn, and M. S. Shur, Zh. Éksp. Teor. Fiz. 70 (5), 2014 (1976) [Sov. Phys. JETP 43, 1050 (1976)]. REFERENCES 27. Y. Yagil, P. Gadenne, C. Julin, and G. Deutscher, Phys. 1. J. C. Maxwell, A Treatise on Electricity and Magnetism Rev. B 46 (4), 2503 (1992). (Clarendon, Oxford, 1873); J. C. Maxwell, A Treatise on 28. F. Brouers, Phys. Rev. B 47 (2), 666 (1993). Electricity and Magnetism (Dover, New York, 1973). 29. A. K. Sarychev, D. J. Bergman, and J. Yagil, Phys. Rev. 2. J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London B 51 (8), 5366 (1995). 203, 385 (1904). 30. F. Brouers, S. Blacher, A. N. Lagrkov, et al., Phys. Rev. 3. D. A. G. Bruggeman, Ann. Phys. 24, 636 (1935). B 55 (19), 13234 (1997). 4. R. Landauer, in Proceedings of the 1st Conference on the 31. V. M. Shalaev and A. K. Sarychev, Phys. Rev. B 57 (20), Electrical Transport and Optical Properties of Inhomo- 13265 (1998). geneous Media, Ohio State University, 1997, edited by J. C. Garland and D. B. Tanner; R. Landauer, AIP Conf. 32. V. V. Novikov and V. P. Belov, Zh. Éksp. Teor. Fiz. 106 Proc. 40, 2 (1978). (3), 780 (1994) [JETP 79, 428 (1994)]. 5. D. Stauffer, Introduction to the Percolation Theory (Tay- 33. V. V. Novikov, Teplofiz. Vys. Temp. 34 (5), 688 (1996). lor and Francis, London, 1985). 34. V. V. Novikov, Phys. Met. Metallogr. 83 (4), 349 (1997).

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35. V. V. Novikov and K. W. Wojciechowski, Fiz. Tverd. 41. C. F. Bohren and D. R. Huffman, Absorption and Scat- Tela (St. Petersburg) 41 (12), 2147 (1999) [Phys. Solid tering of Light by Small Particles (Wiley, New York, State 41, 1970 (1999)]. 1983; Mir, Moscow, 1986). 36. V. V. Novikov, K. W. Wojciechowski, V. P. Privalko, and 42. Yu. I. Petrov, Clusters and Small Particles (Nauka, Mos- D. V. Belov, Phys. Rev. E 63, 036120 (2001). cow, 1986). 37. J. Bernasconi, Phys. Rev. B 18 (5), 2185 (1978). 43. S. H. Heard, F. Griezer, G. G. Barrachough, and 38. S. R. Coriell and J. L. Jackson, J. Appl. Phys. 39 (10), J. V. Sanders, J. Colloid Interface Sci. 93 (3), 545 4733 (1968). (1983). 39. B. Ya. Balagurov, Zh. Éksp. Teor. Fiz. 88 (15), 1664 44. V. I. Emel’yanov and I. I. Koroteev, Usp. Fiz. Nauk 135 (1985) [Sov. Phys. JETP 61, 991 (1985)]. (2), 345 (1981) [Sov. Phys. Usp. 24, 864 (1981)]. 40. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957; Inostrannaya Literatura, Mos- cow, 1961). Translated by S. Gorin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2063Ð2066. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1970Ð1973. Original Russian Text Copyright © 2002 by Zaœnullina, Leonidov, Kozhevnikov. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Specific Features of the Formation of Oxygen Defects in the SrFeO2.5 Ferrate with a Brownmillerite Structure V. M. Zaœnullina, I. A. Leonidov, and V. L. Kozhevnikov Institute of Solid-State Chemistry, Ural Division, Russian Academy of Sciences, Pervomaœskaya ul. 91, Yekaterinburg, 620219 Russia e-mail: [email protected] Received December 4, 2001

Abstract—The electronic structure and chemical bonding in the Sr2Fe2O5 strontium ferrate are investigated in the framework of the ab initio linear-mufﬁn-tin-orbital tight-binding representation and extended Hükcel calculations. Models of defect formation (oxygen vacancies and anti-Frenkel defects) in the brownmillerite structure are considered. A model of ion transfer in strontium ferrate is proposed reasoning from the results of the calculations. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tight-binding method. The theoretical concepts of this approach were thoroughly described in [5, 7]. Here, we Oxide compounds with a brownmillerite structure dwell on some computational details that affect the of the general formula A2B2O5 have attracted consider- accuracy of the results obtained, especially for the total able attention, because they are promising materials for energy of the crystal. use in designing high-temperature electrochemical devices [1, 2]. In particular, strontium ferrate exhibits The Bloch functions for the studied crystal are con- high characteristics of ion transfer. Recently [3], we structed using a basis set of atomic orbitals, including the valence 5s and 4d orbitals of strontium atoms; the revealed that the oxygen ion conductivity in Sr2Fe2O5 ° 4s, 4p, and 3d orbitals of iron atoms; and the 2p orbitals reaches 0.1 S/cm in the temperature range 850Ð900 C. of oxygen atoms. The Sr 5p orbitals are not included in The structure of the Sr2Fe2O5 ferrate can be considered a derivative of the SrFeO3 perovskite structure in which 1/6 of the oxygen sites are empty. The orthorhombic unit cell of this compound is shown in Fig. 1. b The unit cell consists of alternating layers of Fe(1)O6 octahedra and Fe(2)O4 tetrahedra, whereas oxygen vacancies are ordered in the [101] direction. As a rule, ordering of oxygen vacancies leads to suppression of ion transfer [4]. In this respect, it is of interest to eluci- Fe(2) date the mechanism of the disordering of the Sr2Fe2O5 E structure that is favorable to electrical conduction O(3) through oxygen ions. In the present work, the use of the O(2) ab initio linear-mufﬁn-tin-orbital tight-binding (TB- Sr LMTO) method [5] in combination with extended O(1) Fe(1) Hükcel calculations [6] made it possible for the ﬁrst time to investigate the electron energy spectrum, chemical bonding, and the energy of formation of anti-Fren- kel defects in strontium ferrate. The mechanism of ion transfer in Sr2Fe2O5 was proposed from analyzing the results of quantum-chemical calculations.

2. CALCULATION TECHNIQUE a In order to calculate the electronic structure and the c total energy, we used one of the most time-efficient and sufficiently exact methods in the electron density functional theory, namely, the linear-muffin-tin-orbital Fig. 1. Supercell of strontium ferrate.

2064 ZAŒNULLINA et al.

Table 1. Atomic parameters used in the extended Hükcel spheres with a basis set of s orbitals (extraspheres [5, calculations: the ionization potentials of valence orbitals Hii, 7]); the p and d states of extraspheres are taken into ξ the exponents i, and the weight coefficients Ci of exponents account only within the down-folding approach. in the expressions for atomic orbitals of the Slater type The calculations are performed using 256 k vectors ξ 1(C1) in the Brillouin zone (75 k vectors per irreducible part Atom Orbital Hii, eV ξ 2(C2) of the Brillouin zone). The radii of atomic spheres are determined from the condition of filling the defect-free O2s 32.30 2.275 crystal volume with these spheres according to the pro- 2p 14.80 2.275 cedure described in [10]. The accuracy in the determi- Fe 4s 9.10 1.90 (1.00) nation of the energies of defect formation depends on 4p 5.32 1.90 (1.00) the appropriate choice of the radii of the extraspheres and real atoms, which interchange sites with one 3d 12.60 5.35 (0.5505) another to form a defect. We assume that the oxygen 2.00 (0.6260) atom occupying an extrasphere site takes on the extras- Sr 5s 6.62 1.214 (1.00) phere radius and vice versa; at the same time, the radii 5p 3.92 1.214 (1.00) of atoms and extraspheres that are not involved in the defect formation remain unchanged. This procedure of choosing the atomic radii was detailed and approved for the basis set of the Bloch functions used to construct the alkaline-earth fluorides and zirconium oxides in our Hamiltonian matrices with the use of the down-folding earlier work [11]. The lattice parameters for the method [8], which is based on the Löwdin perturbation Sr2Fe2O5 crystal are taken from [12]. theory [9]. The calculations are performed with a super- The characteristics of the chemical bonding in the cell of composition Sr8Fe8O20E4, where E are empty ferrate under investigation were analyzed in the frame- interstitial spheres. work of semiempirical extended Hükcel calculations Since the TB-LMTO method is developed and [6, 13]. The standard parameters used in the extended offers the most exact results for close-packed lattices, Hükcel calculations are tabulated in [14] and presented in our calculations, interstices are filled by empty in Table 1.

3. RESULTS AND DISCUSSION 80 EF A (a) Figure 2 shows the total and partial densities of B states for Sr2Fe2O5, which were obtained in the frame- 40 C work of ab initio LMTO calculations. The broad band A in the energy range from Ð9.87 to Ð5.27 eV corre- 0 sponds to a hybrid O 2pÐFe 3d band with an admixture (b) of Sr 5s and Sr 5p states. The high-energy band B in the 40 energy range from Ð5.27 to Ð0.75 eV consists predominantly of Fe 3d orbitals with an admixture of O 2p 20 states. The metallic band C, which is composed of Sr 5s and Sr 5p states, is located at even higher energies. The 0 Fermi level corresponds to a maximum of the B band. ), 1/eV

E The absence of the band gap at the Fermi level is a con- ( 30 (c) N sequence of incomplete inclusion of correlation effects in the calculation. Note that the correlation effects are responsible for the splitting of the B band and the tran- 10 sition of the ferrate to a MottÐHubbard insulator. This 0 assumption is conﬁrmed by the fact that a band gap of (d) ~1.4 eV appears in the electronic spectrum in the 30 course of simulation of the electronic structure of Ba2In2O5 with a similar crystal lattice. This band gap is 10 underestimated with respect to the experimental value of 2.7 eV [15]. It should be noted that traditional ab ini- 0 –15 –10 –5 0 5 10 tio calculations (TB-LMTO [5], full-potential LMTO E, eV [16], and HartreeÐFock [17] methods) give sufﬁciently exact values of the total crystal energy and the energy Fig. 2. Densities of states N(E) for a perfect crystal of of formation of point defects [18, 19]. More correct cal- SrFeO2.5: (a) the total density of states and (b) Fe 3d, (c) O culation procedures of the electronic band structure and 2p, and (d) Sr 5s and Sr 5p partial densities of states. the total energy of MottÐHubbard insulators must take

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

SPECIFIC FEATURES OF THE FORMATION OF OXYGEN DEFECTS 2065

Table 2. Populations of overlap of crystal orbitals, the total energies Etot of the crystal lattice with one oxygen vacancy per supercell, the total populations of overlap of crystal orbitals for different-type oxygen atoms, and the energies of formation of anti-Frenkel defects EAFD in the SrFeO2.5 crystal Type of oxygen atom O(1) O(2) O(3) Populations of overlap Fe(1)ÐO 0.274Ð0.254 0.163 Ð of crystal orbitals Fe(2)ÐO Ð 0.422 0.465Ð0.292 SrÐO 0.050Ð0.022 0.079Ð0.043 0.156

Etot, Ry Ð74011.00176 Ð74010.8814 Ð74010.9569 Total populations of overlap of crystal orbitals 0.674 0.767 0.915

EAFD, Ry (eV) 0.225 (3.067) 0.264 (3.597) Ð into account nonlocal exchange-correlated interactions confirm the preferential formation of oxygen vacancies and their dependence on the energy of skeletal states of in the equatorial plane of the Fe(1)O(2)2O(1)4 oxygen the crystals. However, these methods are still under octahedron. The total population of the overlap of crys- development (see, for example, [20]). tal orbitals for different types of oxygen atoms charac- Apart from analyzing the electronic spectrum of terizes the covalent contribution to the chemical bond strontium ferrate, we investigated chemical interactions and appears to be minimum for the O(1) atom (Table 2). in the Sr2Fe2O5 structure in the framework of the The results obtained allow us to assume that the ion semiempirical Hükcel method. The results of the calcu- migration in ferrate occurs through vacant oxygen sites lations of chemical bonding are given in Table 2. The in the equatorial plane of the FeO6 octahedra. strongest covalent interactions are observed between the Fe(2) and O(3) atoms (the bond order is equal to The above assumption is confirmed by the calcu- 0.466) in layers formed by the FeO4 tetrahedra. The FeÐ lated energies of formation of anti-Frenkel defects in O(1) bonds, which involve oxygen atoms located in the the Sr2Fe2O5 crystal. The lowest energy of formation of equatorial plane of the FeO6 octahedra, prove to be anti-Frenkel defects corresponds to a transition of the somewhat weaker. Small populations of the overlap of O(1) atom to a tetrahedral interstice (E) (Table 2). The the crystal orbitals (0.022) correspond to a substantial transition of the O(2) atom to the nearest tetrahedral contribution of the ionic component and a small contri- vacancy is less energetically favorable. Analysis of the bution of the covalent component of the SrÐO(1) chem- calculated data demonstrates that the formation of anti- ical bond. Therefore, the structure of strontium ferrate Frenkel defects proceeds through the direct interstitial is characterized by a rigid framework consisting of mechanism. As a result, the interstitial oxygen atom ironÐoxygen octahedra and tetrahedra with strong FeÐ occupies a tetrahedral position and the oxygen vacancy O covalent bonds. The strontium atoms are more weakly bonded to the nearest oxygen environment, and is located in the equatorial plane of the oxygen octahe- the SrÐO interactions exhibit a mixed ionicÐcovalent dron. The lower bonding energy of the equatorial oxy- nature. gen atoms suggests that the dominant mechanism of ion migration in the ferrate is the positional exchange of In order to elucidate the mechanism of ion transfer oxygen ions and oxygen vacancies in the equatorial in the brownmillerite structure, we simulated the plane of FeO octahedra forming perovskite-like struc- defect-containing phases of the strontium ferrate. All 6 tural units of the Sr Fe O structure. the calculations of the total energy for crystals with one 2 2 5 oxygen vacancy or one anti-Frenkel defect per compu- Moreover, we evaluated the degree of interaction tational supercell were carried out in the framework of between oxygen vacancies in the case when their con- the ab initio TB-LMTO representation. We considered centration increases to two vacancies per computa- three possible variants of the arrangement of a particular oxygen vacancy in the brownmillerite structure, tional supercell of the studied ferrate. The results namely, at the O(1), O(2), and O(3) sites. The configu- obtained indicate a preferential arrangement of vacancies in the form of VOÐFeÐVO chains in the equatorial ration with an oxygen vacancy in the Fe(1)O(1)4 equa- plane of the perovskite-like structural unit. The forma- torial plane of the Fe(1)O(2)2O(1)4 octahedra, which form perovskite-like structural units, turned out to be tion of these associates leads to an effective elimination the most energetically favorable (Table 2). The forma- of vacancies from the ion transfer even at sufficiently tion of an oxygen vacancy in the O(2) apical position of high temperatures. This explains the decrease in the ion these octahedra is less probable. The independent band conduction in ferrate at extremely low oxygen pres- calculations within the extended Hükcel approximation sures in the gas phase [3].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2066 ZAŒNULLINA et al.

ACKNOWLEDGMENTS 9. P. O. Löwdin, J. Chem. Phys. 19 (11), 1396 (1951). We would like to thank V.P. Zhukov for his partici- 10. G. Krier, O. Jepsen, A. Burkhardt, and O.-K. Andersen, pation in discussions of the results. The TB-LMTO-ASA Program (MPI für Festkorperfors- chung, Stuttgart, 1996). This work was supported in part by the Russian 11. N. M. Zainullina, V. P. Zhukov, and V. M. Zhukovsky, Foundation for Basic Research, project RFBR-Ural 01- Phys. Status Solidi B 210, 145 (1998). 03-96519. 12. B. C. Greaves, A. J. Jacobson, B. C. Toﬁeld, and B. E. Fender, Acta Crystallogr. B 31, 641 (1975). REFERENCES 13. R. Hoffmann, Manuals for Extended Hückel Calcula- tions (Cornell Univ. Press, Ithaca, 1989). 1. J. B. Goodenough, J. E. Ruiz-Diaz, and Y. S. Zhen, Solid State Ionics 44, 21 (1990). 14. S. Alvarez, Tables of Parameters for Extended Hückel Calculations (Universitat de Barcelona, Barcelona, 2. M. Schwartz, J. White, and A. Sammels, Int. Patent 1989). Application PCT. WO 97/41060 (1997). 15. G. B. Zhang and D. M. Smyth, Solid State Ionics 82, 161 3. V. L. Kozhevnikov, I. A. Leonidov, M. V. Patrakeev, (1995).

et al., J. Solid State Chem. 158, 320 (2000). 16. M. Methfessel, Phys. Rev. B 38 (2), 1537 (1988). 4. H. J. M. Bouwmeester and A. J. Burggraaf, in Funda- 17. C. Pisani, R. Dovesi, and C. Roetti, in Lecture Notes in mentals of Inorganic Science and Technology Series, Ed. Chemistry (Springer-Verlag, Berlin, 1988), Vol. 48. by A. J. Burggraaf and L. Cot (Elsevier, Amsterdam, 1996), Issue 4. 18. H. M. Polatoglou, M. Methfessel, and M. Schefﬂer, Phys. Rev. B 48 (3), 1877 (1993). 5. O.-K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 (1984). 19. T. Brudevoll, E. A. Kotomin, and N. E. Christensen, Phys. Rev. B 53 (12), 7731 (1996). 6. M.-H. Whangbo and R. Hoffmann, J. Am. Chem. Soc. 100, 6093 (1978). 20. B. Holm and F. Aryasetiawan, Phys. Rev. B 62 (8), 4858 (2000). 7. O.-K. Andersen, Phys. Rev. B 34 (8), 5253 (1986). 8. W. R. L. Lambrecht and O.-K. Andersen, Phys. Rev. B 34 (4), 2439 (1986). Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2067Ð2071. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1974Ð1978. Original Russian Text Copyright © 2002 by Lisitsyn, Yakovlev.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Relaxation Kinetics of Primary Pairs of Radiation Defects in Ionic Crystals V. M. Lisitsyn and A. N. Yakovlev Tomsk Polytechnical University, Tomsk, 634034 Russia e-mail: [email protected] Received October 23, 2001; in ﬁnal form, January 8, 2002

Abstract—The relaxation kinetics of primary pairs of radiation defects in ionic crystals with a face-centered lattice is investigated using the Monte Carlo method. The dependence of the relaxation kinetics of an FÐH pair on the parameters of the interaction potential between the components of the pair is studied. The obtained kinetic dependences are analyzed to determine the factors responsible for the relaxation processes. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION jumps. The origin of the coordinate system is taken to be at the F center, which corresponds to the zeroth state. The action of ionizing radiation on ionic crystals Each state corresponds to a coordination shell around results in the creation of electron excitations decaying the F center in which the H center can be arranged as it into pairs of Frenkel defects [1]. The decay occurs over moves away from the F center. The H center is in the time periods of the order of several or tens of picosec- field of the F center, which is taken into account in the onds [2, 3]. The primary Frenkel pairs (F and H pairs) model by introducing the interaction potential between formed in this way remain stationary up to 0.1Ð10 ns. the H and F centers and, therefore, by a difference in Subsequent motion of the mobile component (H center) the barrier height for the motion of the H center towards of a pair leads, as a result of a series of consecutive ther- the F center and away from it. The fall of the H center mally activated jumps, either to the annihilation of the to the zeroth state means its annihilation with the F cen- pair or to spatial separation of its components. In this ter and the vanishing of the pair. In the case when the H case, the number of pairs decreases with time and the center is beyond the twelfth shell of its possible movable component is transformed into a stable hole arrangement around the F center, the H center is center. A model of relaxation of primary defects with assumed to be transformed into a center that is stable time, taking into account the sample temperature, was under the experimental conditions and cannot be anni- proposed in [4, 5]. This model was confirmed experi- hilated during its subsequent motion. This corresponds mentally and developed in [6, 7]. to the actual situation upon irradiation: at temperatures This study aims at analyzing the primary processes higher than the delocalization temperature, an H center of relaxation of defect pairs created during the decay of is transformed into a stable Vk center after several electron excitations by using methods of computer sim- jumps over the lattice. We calculated the probability of ulation and determination of the dependence of the survival of the H center (and, accordingly, of the FÐH kinetics on the form of the interaction potential and on pair) by a given instant of its migration as a function of the initial mutual distribution of pair components. the crystal temperature. Any H center was regarded as having survived irrespective of its location (inside or 2. COMPUTATIONAL METHOD outside the twelve shells around the F center). We calculated the motion of a mobile component (H We assumed that, in the course of decay of electron center) of a pair in the field of the stationary component excitations, the initial mutual arrangement of the com- (F center) in the face-centered lattice of an alkali halide ponents of pairs is created in the form of a compact metal. We specified the initial position of the H center function with a maximum in the first shell around the F at one of the lattice sites nearest to the F center. Using center, which is taken as the starting point for calcula- the Monte Carlo method [8], we then determined the tions. probabilities of the transition of the H center from the The effect of the stationary defect on the motion of given state to any other state possible for its arrange- the mobile defect was presented by the superposition of ment as a result of a series of consecutive jumps as a the periodic potential of the crystal lattice and the inter- function of the interaction potential between the com- action potential of the form E(r) = Ðaexp(Ðbr). The ponents of the defect pair. The H center moves over the periodic variation of the potential during migration of anionic sublattice as a result of thermally activated the H center is associated with the fact that the H center,

2068 LISITSYN, YAKOVLEV

Table 1. Starting probabilities of arrangement of an H center over the shells of its possible location around the F cen- in different shells around the F center used in the calculations ter. The values 1 and 0.5 in the table indicate the starting (initial distribution function, IDF) probability of the location of the H center in the corre- No. First shell Second shell Third shell sponding shell. The starting probability of the location of the H center indicates the initial distribution function 11 0 0for primary pairs formed as a result of the decay of elec- 2 0.5 0.5 0 tron excitations. 30 1 0 The parameters of the interaction potential were 40 0 1chosen such that the H center could be in a metastable state even at a site nearest to the F center. This means that we presumed the existence of a barrier for a jump of the H center towards the F center from the site near- when moving from a lattice site to a neighboring site, est to the F center. The forms of the potential relief for must pass through the constriction formed by cations in the migrating H center in the vicinity of the F center are a plane perpendicular to the direction of motion of the presented in Fig. 1. The values of the parameters of the H center. The form of the interaction potential is interaction potential were set in the intervals a = 0.1Ð unknown. However, its representation in exponential 0.25 and b = 0.5Ð2.0. Calculations were made for the form ensures the possibility of wide variation of its same sample temperature, 100 K. In this case, an H cen- form, which is determined by the type of interaction ter with an activation energy for its motion equal to (electrostatic, elastic, etc.). 0.04 eV is mobile. The transition of an H center to a state more distant from the F center indicates spatial separation of the pair components. We calculated the migration of an H cen- 3. RESULTS OF CALCULATION ter in the bulk of a crystal containing 1000 anionic lat- Typical relaxation kinetics curves calculated for var- tice sites. ious initial states of mutual distribution of pair compo- At present, neither the form of the interaction poten- nents are presented in Fig. 2. All the curves have three tial of the components of a primary pair nor the form of clearly manifested segments on which the probability the function describing their initial mutual distribution of pair survival decreases with time (the first segment is known. It is known, however, that the components of corresponds to the time interval 10Ð12Ð10Ð8 s, the sec- a genetically connected primary pair of defects interact ond to 10Ð8Ð10Ð7 s, and the third to a time longer than with each other. This leads to the assumption that the 10Ð7 s). The first segment corresponds to 1Ð3 single components of a pair created during the decay of an jumps of the H center. The probability decrease on this electron excitation are in the nearest shells of their pos- segment is significant if the H center starts from the sible arrangement. To establish the dependence of the sites closest to the F center. As the distance between the relaxation kinetics on the starting state, the initial starting point of the H center and the F center increases, mutual distribution of the pair components was speci- the role of the first stage in the decrease becomes much fied as follows. We assumed that the H center is in the smaller. For example, the results of calculations pre- first, second, or third shell of its possible arrangement sented in Fig. 2 (curves 4) show that in the case when around the F center. These shells correspond to location the H center starts from the third shell, the contribution of the H center at sites with coordinates equivalent to of the first stage is insignificant. The contribution of the 〈110〉, 〈200〉, and 〈211〉 relative to the F center. Table 1 first stage to the kinetics of decay depends not only on gives the probabilities of distribution of the H center the initial mutual arrangement of the pair components

0.05 0.05 1 0 0 4 2 –0.05 –0.05 5 , eV 3 , eV E E –0.15 –0.10 6

–0.15 –0.25 0 1 2 3 4 5 0 1 2 3 4 5 r/a r/a

Fig. 1. Interaction potential reliefs for the components of a pair of defects for different values of the model parameters. Curves 1Ð 3 correspond to b = 2.0, 1.0, and 0.5, respectively, and a = 0.15; curves 4Ð6 correspond to a = 0.15, 0.2, and 0.25, respectively, and b = 1.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 RELAXATION KINETICS OF PRIMARY PAIRS OF RADIATION DEFECTS 2069

(a) (d) 0 0 3 –0.5 4 4 3 –0.5

) –1.5 H P 2 2 –2.5 –1.5 1 –3.5 1

–4.5 –2.5 0 1 2 3 0 1 2 3 0 (b) 0 (e) 3 –0.5 3 –0.5 4 4

2 ) log( H –1.5 2 –1.5 P

log( 1 –2.5 –2.5 1

–3.5 –3.5 0 1 2 3 0 1 2 3 (c) (f) 0 0 3 3 –0.5 4 –0.4 4

) –1.5 H

P 2 –0.8 –2.5 log( 2 1 –1.2 1 –3.5

–1.6 –4.5 0 1 2 3 0 1 2 3 t, 10–7 s

Fig. 2. Relaxation kinetics curves of FÐH pairs for different initial distributions of pair components in the nearest three coordination shells for the following values of interaction potential parameters: (a) a = 0.15, b = 0.5; (b) a = 0.15, b = 1.0; (c) a = 0.15, b = 2.0; (d) a = 0.15, b = 1.5; (e) a = 0.2, b = 1.5; and (f) a = 0.25, b = 1.5. Ea = 0.04 eV. Numbers 1 to 4 correspond to serial numbers of the IDF from Table 1. but also on the parameters of the interaction potential The results of our calculations show that the defect (Fig. 2). As the interaction potential (parameter a) relaxation on the second segment is described satisfac- increases, the fraction of pairs annihilating during the torily in semilogarithmic coordinates by a linear func- τ ﬁrst jumps of the H center increases sharply. A depen- tion with the characteristic relaxation time 2. Table 2 τ dence on the rigidity of the interaction potential contains the calculated values of 2 for the second stage (parameter b) can also be seen. of Frenkel defect relaxation. At this stage, the characteristic relaxation time slightly decreases with increas- The decay at the second stage is most interesting. At ing interaction parameter a and exhibits a noticeable this stage, the main decrease in the concentration of dependence on the interaction parameter b: the relax- pairs with time from the beginning of motion of the ation time increases with b. mobile component is observed for all the ranges of the τ parameters of the interaction potential and for all start- The results presented in Table 2 show that 2 weakly ing mutual arrangements of the pair components. The depends on the starting arrangement of the H center rel- length of the second segment corresponds to the time ative to the F center and is independent of the starting required for performing from three to several dozens of state in the case when the interaction potential is jumps of the H center. extended (the value of b is small). For any other ver-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2070 LISITSYN, YAKOVLEV

Table 2. Dependence of the characteristic relaxation time for an FÐH pair (s) on the parameters of interaction potential for different starting states (see Table 1) Potential parameters Starting state ab1234 0.15 0.5 1.1 × 10Ð7 1 × 10Ð7 1 × 10Ð7 1 × 10Ð7 0.15 1 1.25 × 10Ð7 1.43 × 10Ð7 1.43 × 10Ð7 1.43 × 10Ð7 0.15 2 2.5 × 10Ð7 3.3 × 10Ð7 3.3 × 10Ð7 3.3 × 10Ð7 0.15 1.5 1.67 × 10Ð7 2.5 × 10Ð7 2.5 × 10Ð7 2.5 × 10Ð7 0.2 1.5 1.43 × 10Ð7 2 × 10Ð7 2 × 10Ð7 2 × 10Ð7 0.25 1.5 1.43 × 10Ð7 2 × 10Ð7 2 × 10Ð7 1.67 × 10Ð7

τ sions of interaction potentials, 2 is independent of the eter b) decreases, the fraction of preserved pairs starting state for any initial states except the first; the decreases. τ relaxation time 2 upon a start from the first state is shorter than that for starts from all other states (but not 4. DISCUSSION by more than 30%). In this study, we used computer simulation methods The last, third, segment, on which the number of for analyzing the relaxation kinetics of correlated inter- pairs remains unchanged with time, corresponds to the acting Frenkel pairs. The calculations were made for case where all of the mobile components of pairs different starting arrangements of correlated pair com- escape from the rated volume and transform into stable ponents relative to each other, i.e., for situations with hole color centers. The results of investigations of the different extents of correlation of pairs. The degree of third relaxation stage are presented in Table 3 in the correlation in our calculations varied from extremely form of a dependence of the fraction of preserved pairs high, when the components of a pair were at nearest on the parameters of the interaction potential and on the neighbor sites, to that corresponding to the location of starting mutual arrangement of the components of the the H center in the third shell around the F center. pairs. It follows from the results of calculations pre- Of the two stages of the decay of correlated pairs sented in the table that the survival probability for a pair due to annihilation during thermally activated motion depends on the initial mutual arrangement of the pair of the mobile component of a pair, the second stage is components. The shorter the distance between the pair more interesting. Obviously, this stage can be observed components in the initial state, the higher the probabil- experimentally as the shortest stage. In fact, it follows ity of their annihilation; in other words, the longer the from the results of calculations that the maximum distance between the starting H center and the F center, decrease is observed in the second stage in most ver- the larger the fraction of surviving (accumulated, long- sions of the interaction potential and starting states that lived) pairs by the beginning of the third stage. As the we used. The first stage of the calculated relaxation interaction potential (parameter a) increases and, espe- kinetics is of too short a duration and is noticeable only cially, as the rigidity of the interaction potential (param- when the first state is predominantly the starting state. The results of calculations showed that the relax- Table 3. Dependence of the probability of survival of an FÐH ation kinetics depends on the starting state only slightly pair after completion of the relaxation process on the parame- (Table 2). This means that from the relaxation kinetics ters of interaction potential for different starting states (see we can only judge whether or not a pair is correlated; Table 1) i.e., kinetic curves provide no information on the degree of correlation and, in particular, on the type of Potential Starting state starting state. parameters The experimental results on the decay kinetics of FÐ ab1234H centers in alkali halide crystals available at present [5, 7] show that at least two stages of defect relaxation 0.15 0.5 0.03 0.09 0.12 0.17 exist after the application of a short radiation pulse in 0.15 1 0.06 0.22 0.28 0.38 the nanosecond and microsecond time ranges. The first 0.15 2 0.26 0.45 0.58 0.64 (nanosecond) stage is correctly described by monomo- 0.15 1.5 0.14 0.35 0.45 0.55 lecular kinetics, while subsequent stages are described by bimolecular kinetics. The ratios of decay rates and 0.2 1.5 0.06 0.3 0.39 0.5 of the decreases in the pair concentration at the two 0.25 1.5 0.02 0.28 0.34 0.45 (monomolecular and bimolecular) stages depend on the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 RELAXATION KINETICS OF PRIMARY PAIRS OF RADIATION DEFECTS 2071 temperature of the sample during irradiation. In the two groups is carried out arbitrarily. In connection with temperature range from liquid-nitrogen to room tem- this, we can propose the following division of pair com- perature, the initial concentration of primary defects ponents into short- and long-lived centers. Short-lived induced by the radiation pulse depends on temperature pairs should be regarded as correlated and long-lived only slightly. However, the ratio of concentrations of pairs as uncorrelated, irrespective of their origin. correlated and uncorrelated pairs changes signiﬁcantly Uncorrelated pairs can be formed both due to primary in favor of the latter pairs. processes of decay of electron excitations and as a The ratio of concentrations of correlated and uncor- result of separation of the correlated pairs formed. related pairs created by a radiation pulse also depends There can be several long-lived stages. Relaxation can on the energy of the exciting pulse. The fraction of result from reactions between uncorrelated pairs, as uncorrelated pairs increases with the excitation power. well as from a combination of reactions involving secondary affects. Thus, in the range of short times following excitation by a nanosecond radiation pulse, the relaxation of primary defects has several stages. The shortest stage is 5. CONCLUSIONS associated with annihilation of correlated pairs. Uncor- The theoretical analysis carried out in this paper related pairs of defects have a longer lifetime and are, proved that experimentally measured relaxation kinet- hence, referred to as long-lived. The relation between ics curves for primary pairs created as a result of a radi- the relaxation stages associated with correlated and ation pulse can provide information on the presence of uncorrelated pairs is determined by the experimental correlated and uncorrelated pairs and on the ratio of conditions. their concentrations. However, the relaxation kinetics The results of experimental investigation at the ini- curves give no information on the degree of correlation tial stages of breakdown of FÐH pairs formed in alkali and, in particular, on the function of the initial mutual halide crystals as a result of the decay of electron exci- arrangement of the components of primary pairs. tations are described in [6]. The experimentally measured relaxation curves for F centers in the nanosecond time interval exhibit two clearly manifested stages: a ACKNOWLEDGMENTS short stage, in time intervals from tens to hundreds of This study was supported by the Russian Founda- nanoseconds, and a long stage, for which no noticeable tion for Basic Research, project no. 01-2-18035. time variation is observed in these time intervals [6]. At the ﬁrst relaxation stage, at least half of the created pairs of centers decay at 80 K. REFERENCES A comparison of the results of theoretical and exper- 1. Ch. B. Lushchik and A. Ch. Lushchik, Decay of Electron imental investigations of the initial stages of the break- Excitations with Defect Formation in Solids (Nauka, down of FÐH pairs leads to the following conclusions. Moscow, 1989). The second relaxation stage on the theoretical curves 2. T. Sugiyama, H. Fujiwara, T. Suzuki, and K. Tanimura, obviously corresponds to the shortest experimentally Phys. Rev. B 54 (21), 15109 (1996). observed relaxation stage [6]. The results of calcula- 3. H. Fujiwara, T. Suzuki, and K. Tanimura, J. Phys.: Con- tions show that only correlated and uncorrelated pairs dens. Matter 9, 923 (1997). can be distinguished in the relaxation kinetics of the 4. V. M. Lisitsyn, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 2, components of FÐH pairs during thermally activated 86 (1979). migration of the H centers. The degree of correlation is 5. P. V. Bochkanov, V. I. Korepanov, and V. M. Lisitsyn, Izv. manifested too weakly in the kinetics. Consequently, Vyssh. Uchebn. Zaved., Fiz., No. 3, 16 (1989). we can conclude that only one (the shortest) relaxation 6. V. M. Lisitsyn, V. I. Korepanov, and V. Yu. Yakovlev, Izv. stage observed in the experiments is associated with Vyssh. Uchebn. Zaved., Fiz., No. 11, 5 (1996). correlated pairs. Any subsequent stage is not the result 7. V. M. Lisitsyn, L. A. Lisitsyna, and E. P. Chinkov, Izv. of annihilation of correlated pairs. Vyssh. Uchebn. Zaved., Fiz., No. 3, 13 (1995). 8. I. M. Sobol’, Numerical Monte Carlo Methods (Nauka, The concepts of short- and long-lived pairs are often Moscow, 1973), pp. 7Ð9. encountered in the description and analysis of the results of investigations. Such pairs are often referred to as unstable and stable, respectively. The division into Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Kinetic Mechanism of the Formation of Fragmented Dislocation Structures upon Large Plastic Deformations G. A. Malygin Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received January 22, 2002; in ﬁnal form, January 29, 2002

Abstract—The mechanism of formation of fragmented (banded, block) dislocation structures (FDSs) in crystals subjected to large plastic deformations are discussed. The theoretical analysis is based on the kinetic equations for the density of geometrically necessary dislocations (GNDs). The equations include the processes of multiplication, immobilization, annihilation, and diffusion of GNDs. The formation of an FDS is considered a synergetic process of self-organization of GNDs obeying the principle of similitude of dislocation structures at various degrees of plastic deformation. Conditions for the formation of the structures at hand have been determined, as well as their parameters and the dependence of these parameters on the degree of deformation. The theoretical results are compared with the available experimental data. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION plication. As to FDSs, there are good grounds to assume that geometrically necessary dislocations Experiments show that after large plastic deforma- γ (GNDs) take part in their formation [12Ð14]. These dis- tions ( > 0.5Ð10), a fragmented dislocation structure locations are formed as a response of the crystal to its (FDS) is formed in crystalline materials [1Ð14]. Such a elastic bending (twisting) due to the nonuniformity of structure is also called a banded or block dislocation plastic deformation and the related geometrical distor- structure according to the shape of the fragments that tions in the shape of the crystal or its local regions [15]. are formed after large deformation. The formation of Thus, the geometrically necessary dislocations differ such a structure is determined by the amount of defor- from the statistically random dislocations in their ori- mation and is independent of the way in which this gin. deformation has been reached, i.e., irrespective of whether it is formed in the neck of a tensile sample [2] The concentration of GNDs in fragment boundaries as a result of twisting [4], drawing [1], rolling [3], or decreases the energy of a dislocation ensemble as a multiple extrusion through a knee-shaped die (equal- result of a decrease in the energy of dislocation interac- channel angular pressing technique [11]). tions [16, 17]. However, this cause alone cannot lead to the formation of an FDS. Experiments show that GNDs The formation of an FDS is accompanied by the can be distributed in the crystal randomly. Such a disappearance of strong crystallographic misorientations tribution is observed if dislocations, during their in the crystal, since the boundaries of fragments contain motion, suffer strong friction from the side of the a large density of like dislocations (of the same sign). Peierls crystal relief or, in the case of alloys, friction As the degree of deformation increases, the fragment due to a high concentration of impurities or disperse dimensions decrease from several microns to several inclusions in the lattice. In this case, continuous rather hundreds of nanometers and the misorientation of the than localized misorientations arise in the crystal [5, lattice between neighboring fragments changes from 14]. A similar situation takes place when a cellular dis- fractions of a degree of arc to several tens of degrees. location structure is formed: such a structure has not Straining to large degrees of plastic deformation is at been observed in strongly doped and dispersion-hard- present one of the most efficient techniques for the pro- ened alloy [6, 18]. duction of fine-crystalline (nanostructured) materials. Thus, the formation of an FDS has geometrical (dis- The formation of an FDS begins at the third stage of clinational [2]), energetical [16], and kinetic aspects. the strain-hardening process and continues at the fourth The last aspect was considered in [12, 13, 19]. It fol- and fifth stages. At the second and third stages of strain lows from these works that the FDSs, like cellular dis- hardening, a cellular dislocation structure is known to location structures, obey the principle of similitude of be formed in the crystal. Its formation is related to the dislocation structures. According to this principle, the formation and accumulation in the crystal of statisti- dislocation structure changes in a self-similar way with cally random dislocations (SRDs) as a result of their increasing plastic deformation; i.e., it retains the rela- generation at dislocation sources and subsequent multi- tionship between its parameters during its evolution.

KINETIC MECHANISM OF THE FORMATION 2073

The self-similar character of the evolution of FDSs dislocations [7, 22], as in the case of cell boundaries in indicates that their formation is due to the process of cellular dislocation structures [23]. This means that the self-organization of dislocations. walls of the fragments are formed as a result of disloca- This paper is aimed at a theoretical analysis of the tion slip and contain dislocations of various slip sys- kinetic mechanism of formation of FDSs on the basis of tems. equations that describe the evolution of the density of According to the Ashby relation, the rate of genera- GNDs at large plastic deformations. In Section 2, the tion of GNDs is ρú = (1/bl)γú , where γú = bρu is the rate corresponding kinetic equations are formulated. Sec- of plastic deformation and u is the dislocation velocity. tion 3 is devoted to an analysis and solution of these With the above in mind, we can write equations of evo- ρ equations. In Section 4, a comparison of the theoretical lution of the densities of GNDs of different signs ( + results with available experimental data on the evolu- ρ and Ð) that are analogous in their structure to the equation of parameters of fragmented structures with tion of evolution of the density of SRDs [20, 21]: increasing deformation is performed. ∂ρ± 1 1 ------+ ∇J± = --- Ð ---- uρ± ∂t l λ 2. EQUATION OF THE EVOLUTION i (1a) OF THE DENSITY OF GEOMETRICALLY 1 + δ ---()ρ + ρ uρ± Ð h uρ ρ , NECESSARY DISLOCATIONS f 2 + Ð a + Ð What is the mechanism of generation of GNDs and where J± are the fluxes of unlike dislocations and t is the the size (scale) of the regions of bendÐtwist that lead to time. The first term on the right-hand side describes the the formation of these dislocations upon large plastic rate of generation of dislocations caused by the nonuni- deformations in the crystal? Since the formation of formities of plastic deformation with characteristic fragmented structures is independent of the loading dimensions of order l; the second term describes the conditions, it is obvious that it is due to internal causes. immobilization of dislocations by obstacles; the third According to Ashby [15], the density of GNDs is deter- ρ γ γ term is responsible for the multiplication of disloca- mined by the relationship = /bl, where is the degree tions on the dislocation forest; and the fourth term of plastic deformation and l is the size of the region of allows for dislocation annihilation. The coefficients bending (it determines the radius of the lattice curva- λÐ1 δ ture R = l/γ). The radius of curvature R is linked with the l , f and ha determine the intensity of the corres- dislocation density as ρ = 1/bR, wherefrom we imme- ponding processes. When writing Eq. (1a), we assumed diately obtain the Ashby relation. that there occurs multiple slip in the crystal and that the A cellular dislocation structure is related to a non- dislocation densities in intersecting slip planes differ uniformity of plastic deformation of the crystal on a only insignificantly. scale of l3 (the size of a dislocation cell) due to a non- Assuming that, in the process of deformation at the uniform distribution of dislocations, i.e., their high den- fourth and fifth strain-hardening stages, the relationship ρ sity in the cell boundaries and a small density in the between the unlike dislocations remains constant ( Ð = βρ β bulk of the cells. Therefore, at the second and third +, where < 1), we obtain, instead of Eq. (1a), the stages of the strain-hardening curve, the size of the equation of evolution of the density of dislocations of ρ region of bending is on the order of several l3, i.e., is on one sign ( +) the order of the length of slip lines in the cellular dislo- ∂ρ cation structure. At the fourth and fifth stages of the + ∇ 1 1 ρ δ ρ3/2 δ ρ2 ------∂ + J+ = --- Ð λ---- u + + 4u + Ð 5u +, (1b) strain-hardening curve, the size of the slip lines can t l i apparently be much greater than the size of fragments δ β δ δ β l4, since the GNDs are pileups of like dislocations. where 4 = (1 + ) f/2 and 5 = ha. In what follows, ρ In their kinetic properties, the geometrically neces- we omit the plus sign at and J for the time being. sary dislocations do not differ from the statistically ran- According to [21], the appearance of spatially non- dom dislocations. Therefore, the evolution of their uniform drift- and diffusion-related dislocation fluxes J ensemble should be described by a kinetic equation of in the crystal can be caused by the nonuniform multipli- the same type as in the case of SRDs; i.e., it should cation of dislocations by double cross slip of screw dis- include the processes of multiplication, immobiliza- locations: tion, annihilation, and diffusion of dislocations [20, ∂ρ ∂3ρ 21]. As was said above, the main feature of GNDs is ρ J x = u ++D1------D2------, (2) that these are dislocations that are not compensated in ∂x ∂x3 the Burgers-vector sign. Therefore, they form their own δ 2 λ δ mesostructure in the crystal, which differs from the where D1 = 1(M Ð 1)(hc / s)u and D2 = 2(M Ð mesostructure of statistically random dislocations com- 4 λ pensated in the Burgers-vector sign. Experiments show 1)(hc / s)u are the diffusion coefficients for dislocations δ ≈ δ ≈ that the fragment boundaries lie in the slip planes of of the first and second orders, respectively; 1 2 1

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2074 MALYGIN are numerical parameters; M = Ð∂lnu/∂lnρ > 1 is the 3. FRAGMENTED DISLOCATION λ coefficient of dislocation-related strain hardening; s is STRUCTURES the free path of screw dislocations between the events of double cross slip; and hc is the characteristic distance The above linear analysis shows that the distribution for the escape of a dislocation segment into a parallel of GNDs in the crystal is unstable to spatial fluctuations slip plane upon double cross slip of screw dislocations. of the dislocation density near the critical value of the ρ Since M = Vτµ/2kT ӷ 1, the diffusion fluxes (2) change density 2. What is the subsequent evolution of the fluc- their sign due to local strain hardening of the crystal in tuations of dislocation density and of the ensemble of the regions of an enhanced dislocation density. (Here, GNDs on the whole? To answer this question, we V = b2ρÐ1/2 is the activation volume upon multiple slip; should solve nonlinear equation (1b), in which the dis- 1/2 τµ = αµbρ , the dislocation-induced strain-hardening location flux J in the general case depends on the inte- of the crystal; α, the coefficient of dislocation interac- gral of the dislocation density [21]. Since this is an tion; µ, the shear modulus; k, the Boltzmann constant; unsolvable problem, we consider, as the first approxi- and T, the temperature.) mation, the solution to Eq. (1b) using Eq. (2) for the dislocation flux and restricting ourselves to the first dif- Substituting Eq. (2) into Eq. (1b), we can investigate fusion term in it. the spatial stability of the dislocation distribution determined using this equation. Let us denote, for brevity, As a result, we obtain the equation the right-hand side of Eq. (1b) by uΦ(ρ). Equation Φ ρ ( ) = 0 has three roots, namely, ∂ρ ∂ρ ∂2ρ λ ------++u------Du------δ ∂t ∂x ∂x2 ρ1/2 4 []± ()η 1/2 (6) 12, = ------δ- 11Ð , 2 5 ()β uρδρ3/2 δ ρ2 = Ð i Ð 1 --- + 4u Ð 5u , δ (3) l ρ η ()β 5 3 ==0, 4 i Ð 1 ------2-, δ 2 4l λ λ where D = (M Ð 1)hc / s. Taking into account the con- where β = l/λ > 1 is the coefficient of immobilization vective instability of density fluctuations, we seek the i i solution to Eq. (6) in the form ρ(z, t), where z = x Ð Ut of dislocations. Near the critical values ρ , ρ , and ρ , 1 2 3 and U is a constant velocity. Taking into account that the stability of the dislocation density toward fluctua- ∂ρ ∂ ∂ ρ ∂ ∂ρ ∂ tions of the form δ(x, t) ~ exp(ωt + iqx) is determined / t = ' / 't Ð U / z, we have, instead of Eq. (6), the ω ω ω equation by the equation (q) = i(q) + i 2(q), where 2 ∂Φ 2 4 ∂'ρ ∂ρ ∂ ρ ω ()q ==u + D q Ð D q , ω uq. (4) ------++()uUÐ ------λ u------1 ------1 2 2 ∂ ∂ D 2 ∂ρ ρ 't z ∂z 123,, (7) ∂ω ∂ ()β uρδρ3/2 δ ρ2 The condition 1/ q = 0 yields critical values of the = Ð i Ð 1 --- + 4u Ð 5u . ω l wave vectors q1, 2, 3 and increments 1(q1, 2, 3): Assuming then that U = u, we transform Eq. (7) by tak- D 1/2 1 ing into account that ∂'ρ/∂'t = (∂ρ/∂γ)γú , where γú = bρu. q123,, ==------1/ 2hc, 2D2 ρ γ (5) As a result, we obtain for the density of GNDs (z, ) ∂Φ an equation of the form ω ()M Ð 1 1 q123,, = u ------+ ------. ∂ρ ρ 4λ 123,, s ∂ρ λ ∂2ρ ρ D ()β ρ ρ3/2 ρ2 ------+Ð------= i Ð 1 k3 + k4 Ð k5 , (8) η Ӷ ρ ≈ δ δ 2 ∂γ b ∂ 2 At 0 < 1, it follows from Eq. (3) that 1 ( 4/ 5) z and ρ ≈ [(β Ð 1)/δ l]2. An analysis shows that at the 2 i 4 δ δ ρ ρ where k3 = 1/bl, k4 = 4/b, and k5 = 5/b. Equation (8) critical points 2 and 3 under the condition (M Ð 1)l/(β Ð 1)λ > 1, the increments ω (q ) > 0 are posi- describes the spatial evolution of the dislocation density i s 1 2, 3 γ tive and, consequently, the dislocation density is unsta- of GNDs depending on the degree of deformation . λ π ble toward fluctuations of size c = 2 2 hc. At the crit- To find this density, we write Eq. (8) in the dimen- ρ ρ ical point 1 > 2, the dislocation density is stable to sionless form such fluctuations if (M Ð 1)δ < 2δ2λ . In addition, 5 4 s ∂Ψ ∂2Ψ since ω (q ) ≠ 0, the critical fluctuations are convec- Ψ------+ ------= Ψ[]ÐΨ + ()Ψ1 + Ψ 1/2 Ð Ψ (9a) 2 1, 2 ∂Γ 2 0 0 tively unstable. ∂Z

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 KINETIC MECHANISM OF THE FORMATION 2075 by introducing the following designations: where z0 is the integration constant and

ρ 2 2 Ψ Γ γ z 4ψ ρ ν ()γ ==-----, k5 , Z =------, ρ+ ()γ 0 1 ρ Λ max = ------, 1 0 4 2 (9b) ---()∆ψ ν λ 1/2 ρ 1/2 η 1 + 0 Ð Λ D Ψ 2 11Ð Ð 5 0 ===------ρ- , 0 ρ------. bk5 1 1 11+ Ð η 1/2 Λγ()= Λ∞/ν ()γ , We seek the solution to Eq. (9a) in a self-similar form ()1/2 Ψ(Z , Γ) = ν2(Γ)ψ(Z ), where Z = Zν1/2(Γ). As a result, Λ π M Ð 1 l5 1 1 1 ∞ = 4 ------()λβ - hc , (14b) we obtain i Ð 1 5

∂ν ρ ()γ 1/2 2------+ ν max ≈ ψÐ1 ∂Γ f 0 = ------ρ ()γ - 0 . min 2 (10) 1 ∂ ψ Ð1 1/2 = ------Ð------+ ψ[]ÐΨ ν + ()ψ1 + Ψ It follows from relations (13) that the stationary (ν = 1) ψ2 ∂ 2 0 0 ψ Z1 fragmented structure is formed if 0 < 0 < 2/3. Equa- tion (14a) describes a spatially periodic dislocation Λ from which we see that the variables ν(Γ) and ψ(Z ) are structure with fragment dimensions and with a dis- 1 location density in the fragment boundaries and in the not completely separated. To separate the variables, we ψ Ӷ Ψ ψ ν bulk of fragments equal (at 0 1) to should assume that 0 = 0 . This means that in η Ӷ ν β λ Eq. (9b) we have 1 and that l = l5/ , i = l/ i = 2 ρ+ ()γ ρ 1 γ ψ β δ δ2 max = ∞ 1 Ð expÐ---k5 , const, and 0 = ( i Ð 1) 5/4l5 , where l5 is the length of 2 (14c) slip lines at the fifth stage of strain-hardening. The ρ+ ()γ ≈ ψ2ρ+ ()γ ρ ()2ρ β min 0 max , ∞ = 6/5 1, coefficient i > 1 has in this case the meaning of the coefficient of transparency of the fragment boundaries respectively. for GNDs. Then, equating the left-hand and right-hand sides of Eq. (10) to unity, we obtain an equation for ν, Taking into account (see Section 2) that the density the solution to which has the form of dislocations with the opposite sign of the Burgers ρ γ βρ γ vector is Ð(z, ) = +(z + zÐ, ), we have the following νΓ()= 1Ð exp[] 1Ð ()Γ 1/2 . (11) expression for the distribution of the total density of GNDs: ψ For the function (Z1), we have the equation ρ()ρ, γ ()βρ, γ (), γ z = + z Ð + zz+ Ð , (15) ∂2ψ ψ[]ψ ()ψψ ν 1/2 ψ ------= Ð 0 + 1 + 0 Ð . (12) where z = (1/2)Λ. Figures 1a and 1b show the distribu- ∂ 2 Ð Z1 tion of the density of GNDs in the crystal according to Eq. (15) under deformations Γ = k γ ∞ and ψ = 4 × ψ ν Ӷ 5 0 Since 0 1, this equation describes a stationary dis- Ð2 β β ≠ 10 and f0 = 25 in the cases where = 0 and 0 tribution of GNDs in the crystal. (β = 0.5). Integrating Eq. (12) under the conditions Since the density of GNDs and the angle of lattice rotation (misorientation between fragments) θ are 16 2 8 Ð1 ∆ = ------()1 + ψ ν Ð ---ψ >>0, ψ 0 (13) related as ρ = b ∂θ/∂z, the distribution of the rotation 25 0 3 0 0 angles θ(z, γ) corresponding to the density of GND given by (15) is described by the integral leads to a tabulated integral and a solution to Eq. (9a), which, after simple transformations, may be written in z the following compact form (now, we restore the plus θ()z, γ = b∫ρ()z, γ dz. (16a) sign at ρ): 0 ρ+ ()γ Figures 1c and 1d display the distribution of rotation ρ ()z, γ = ------max , (14a) + zz+ 2 angles corresponding to the distribution of the density ()π2 0 of GNDs given in Figs. 1a and 1b. It is seen that the 1 + f 0 Ð 1 sin ------Λγ() fragment boundaries, in which most GNDs are concen-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2076 MALYGIN

(a) (b) 1.0 1.0

0.5 ∞ ρ

/ 0.5 ρ 0 12345

–0.5 0 12345 (c) (d) 5 5

4 4

3 3 ∞ θ / θ 2 2

1 1

0 123450 12345 z/Λ

Fig. 1. Distributions of (a, b) the density of geometrically necessary dislocations (GNDs) ρ and (c, d) misorientation angles θ in the fragmented dislocation structure according to Eqs. (15) and (16a) at γ ∞ with (a, c) β = 0 and (b, d) β = 0.5. trated, lead to misorientations of the fragment lattices misorientations of both the same sign (Fig. 1c) and with by the angles alternating signs (Fig. 1d) can occur [7]. f + 1 θ γ ------0 Λγ()ρ+ γ θ ν3/2 γ +()==3/2 b max() ∞ (), 4. COMPARISON WITH EXPERIMENT 2 f 0 (16b) It was found while treating data for Ni [12] that a f 0 + 1 correlation exists between the width (thickness) of frag- θ ()γ == βθ()γ , θ∞ ------bΛ∞ρ∞, Ð + 3/2 ∆Λ γ 2 f ment boundaries ( ) and the dimensions of the frag- 0 ments Λ(γ). This correlation is illustrated in Fig. 2 where ν(γ) is determined by Eq. (11). Experiments (straight lines 1, 2). Line 1 refers to the narrowest show that in the fragmented structures, fragments with boundaries (∆Λ(γ)/Λ(γ) = 0.04); line 2, to the widest boundaries (∆Λ/Λ = 0.13). Assuming that in Eq. (14a) ∆Λ the boundary width 1/2 corresponds to a dislocation 0.6 ρ+ γ ρ+ γ 3 density (z, ) = (1/2)max ( ), we ﬁnd that 2 ∆Λ ()γ 1/2 1/2 2 21Ð ≈ ψ1/2 ------Λγ()- = π--- arcsin------0.43 0 . (17) 0.4 f 0 Ð 1 m µ

, At the average width of the boundaries in Fig. 2 equal ∆Λ Λ ∆Λ to / = 0.085, we obtain, according to Eq. (17), that 0.2 ψ × Ð2 1 0 = 4 10 . Line 3 in Fig. 2 illustrates the correlation between the width of cell boundaries and cell dimensions in the cellular dislocation structure of nickel (processing of the data from [2]). It is seen that, as com- 0 1 2 3 4 Λ µ pared to fragment boundaries, the cell boundaries are , m much wider (∆Λ/Λ = 0.32). The ratio ∆Λ/Λ is independent of the degree of Fig. 2. Correlation between the width of fragment boundaries ∆Λ and fragment size Λ in (1, 2) fragmented and (3) deformation; this means that the evolution of the FDSs cellular dislocation structures in Ni [2, 12]. at the fourth and ﬁfth stages of strain hardening occurs

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 KINETIC MECHANISM OF THE FORMATION 2077 in a self-similar manner, with the retaining of a constant 300 150 relation between the FDS parameters, as in the case of a cellular dislocation structure at the second and third stages of strain hardening [21]. To illustrate this, Fig. 3 200 100 b b displays the results of treatment of the data of [12] on / 1 / –– θΛ 2 –– θΛ the misorientation angles of the fragments (line 1) and 100 50 the cells (line 3) in the fragmented and cellular dislocation structures of nickel, respectively. It is seen that the 3 product of the average angles of misorientation by the 0 0 1 2 3 4 5 6 average dimension of the fragments or the cells ε θγ()Λγ()/b remains constant in the process of the formation and evolution of the corresponding structures, Fig. 3. The independence of the product of the average mis- which indicates a correlated, synergistic character of orientation angle θ and the average fragment size Λ from dislocation processes occurring in the crystal at all the amount of deformation ε in (1, 2) fragmented and (3) stages of plastic deformation. The fact that this correla- misoriented cellular dislocation structures; (1, 3) Ni [12] tion has a regular rather than accidental character is also and (2) α-Fe [1]. conﬁrmed by the results of treatment of the data on the evolution of the parameters of the fragmented dislocation structure in α-Fe [1] depending on the amount of deformation (line 2 in Fig. 3). 1 According to the above-developed theory (Eqs. (14b), (16a)), the product θ(γ)Λ(γ)/b = (θ∞Λ∞/b)ν(γ) does not remain constant in the process of Λ / 1 evolution of a fragmented dislocation structure. The ∞ Λ cause of this is likely that the initial kinetic equation for –– 10–1

the density of GNDs (Eq. (1b)) has an excessively 1 – deterministic character and does not take into account 2 the statistically random mechanism of the fragment formation, which, as was established in [12, 13], leads to a distribution of fragments in size and misorientation 10–2 angle of the type 0 1 2 3 4 5 ε 2 P()Λ ∼ () Λ/Λγ() exp[]Ð3Λ/Λγ(). (18) Fig. 4. Variation of the average size of (1) fragments [12] For the product θγ()Λγ() /b to remain constant, it is and (2) cells [2] in Ni as a function of the amount of deformation ε in semi-logarithmic coordinates necessary that the average size and the misorientation log()1 Ð Λ∞/Λε() Ð ε. angle of fragments change as Λγ() = Λ∞ /ν(γ) and

θγ() = θ∞ ν(γ). It follows from the above experimental and theoret- The experiment confirms such dependences. ical results that, as in the case of cellular dislocation Straight line 1 in Fig. 4 demonstrates a dependence structures [25], the average density of dislocations ρ(γ) Ð1 and the flow stress τ(γ) at the fourth and fifth stages of Λε() = Λ∞ [1 Ð exp(Ð(1/2)mk ε)] for nickel [12] in 5 crystal strengthening may be described by the relations the theoretical coordinates log[]1 Ð Λ∞/Λε() Ð ε, ε γ Λ µ ργ() ()β []ρ+ ()γ ()ρ+ ()γ where = /m, m is the Taylor factor, and ∞ = 0.1 m. = 1 + f G max + 1 Ð f G min For comparison, Fig. 4 also demonstrates an analogous (19a) ≈ ()ρβ + ()γ dependence for the average size of dislocation cells in f G 1 + max , nickel [2] (line 2, Λ∞ = 0.21 µm). At m = 3, we find, from the slope of line 1, the effective coefficient of 1/2 1 ταµ==bρ τ 1 Ð,exp Ð---k γ (19b) annihilation of dislocations at the fifth stage of strain- 5 2 5 hardening k5 = 0.2; the slope of line 2 yields the coefficient of dislocation annihilation at the third stage ka = where f = ∆Λ/Λ ≈ 0.43ψ1/2 is the volume fraction ha/b = 2.5. Thus, the ratio between the densities of dis- G 0 locations of different signs in nickel is β = k /k = 0.08. τ α 1/2 µρ1/2 5 a occupied by fragment boundaries, 5 = ∗ f G b ∞ , This value is close to the value found in [19] for copper α α β 1/2 (β = 0.1, according to [24]). and ∗ = (1 + ) . The density of dislocations in the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2078 MALYGIN

12 Ð1 Fig. 3 (line 1), we have for nickel K2K3 = f G = 183, × Ð3 α K2 = K3 = 13.5, and fG = 5.4 10 . Assuming that ∗ = 0.5, we obtain the following estimates for the other ≈ × Ð2 8 scale coefﬁcients: K1 7 and K4 = 3.7 10 . Note for

–3 comparison that in the case of a misoriented cellular

10 dislocation structure in nickel (line 3 in Fig. 3), we have × Ð1 N K K = f = 64 and K = K = 8. 4 2 3 S 2 3 In a similar way, in the case of α-Fe (line 2 in Fig. 3), Ð1 we find that K2K3 = f S = 70 and K2 = K3 = 8.4. It follows from the relation σ = A/Λ between the flow stress 0 1 2 3 4 5 σ and the size of fragments Λ in iron (see [26]) that at ε µ m = 3 and A = 120 MPa m, the coefficient K1 in (20a) is equal to 2. The estimation of this coefficient accord- Fig. 5. Variation of the number of fragments N in the trans- ing to Eq. (21) yields a twice as large value at α∗ = 0.5. verse section of a nickel sample with the amount of defor- α mation ε [12]. The curve corresponds to a calculation It was also found in [1] that in -Fe at mean misorien- according to Eq. (23). tation angles and fragment sizes, a relation θ/Λ = Bε2 is held, where B = 50 mmÐ1. It follows from Eqs. (20a) and (20b) that ρ+ ()γ boundaries max is determined by Eq. (15). An analysis of the curves of strain hardening in copper at the 1 2 θ/Λ ==bρθ()∞/Λ∞ 1 Ð exp Ð---mk ε . (22) fourth and fifth stages confirms the above relations 2 5 [19]. Thus, in accordance with the principle of similitude Therefore, according to Eq. (22), we have θ/Λ ~ ε2 at of dislocation structures, we have the following rela- the fourth stage of the strain-hardening curve (at ε Ӷ tionships between the parameters of a fragmented dis- 2/mk5), which is in agreement with the results of [1]. location structure: The kinetic mechanism of formation of fragmented µb Ð1/2 Λ∞ dislocation structures permits us to clarify the paradox Λγ( ) = K ------= K ρ = ------, (20a) 1τγ() 2 1Ð exp[]Ð() 1/2 k γ related to the evolution of the sizes and shapes of frag- 5 ments at large plastic deformations, namely, the viola- 1/2 tion of the TaylorÐPolanyi (TP) law in a fragmented θγ()==K bρ θ∞[]1Ð exp()Ð() 1/2 k γ , (20b) 3 5 dislocation structure [26]. According to this law, upon θγ()Λγ() the deformation of a polycrystalline aggregate, the τγ() µθ() γ ε ------==K2K3, K4 . (20c) grain sizes d with increasing plastic deformation b should increase in the direction of the dislocation slip and decrease in the transverse direction: d = d exp(Ðε), Here, K1, K2, and K3 are coefficients that are indepen- 0 dent of deformation but depend on the relative volume where d0 is the initial grain size. For example, at d0 = µ ε ≈ Ð4 Ð2 µ of fragment boundaries in the volume of the crystal fG 100 m and = 10, we have d 10 d0 = 10 m. How- and, consequently, on the magnitude of kinetic coeffi- ever, experiments show that in a strongly misoriented cients that determine the intensity of the processes of dislocation structure that arises instead of the initial multiplication, immobilization, and diffusion of dislo- polycrystalline structure, the fragment size, after defor- cations in Eqs. (1b) and (6). mations to ε > 1, changes only a little and lies in the µ Assuming that the average dislocation density is ρ = range of 0.1Ð0.5 m irrespective of the initial grain size, even upon the deformation of single-crystal sam- Λ ρÐ1/2 1/2 ρÐ1/2 1/h , where h = max = f G is the separation ples. between dislocations in fragment boundaries, we find Upon drawing or rolling, the transverse dimensions that of the samples subjected to deformation decrease by a K ==f Ð1/2, K α f Ð1/2, K =α /K . (21) factor of a few tens or even hundreds. In this case, 2 G 1 * G 4 * 3 because of the violation of the TaylorÐPolanyi law, a ≈ ε It follows from Eqs. (16b), (17), and (19) that at f0 second paradox arises; namely, at deformations > 1, ψÐ1 ӷ the number of dislocation fragments in the transverse 0 1, the average angle of misorientation between section of the sample begins decreasing with increasing θ ≈ Λρ degree of plastic deformation. Figure 5 illustrates this the fragments is b and, consequently, K3 = K2 = Ð1/2 according to the data of [12] for nickel. The experimen- f G . According to Eq. (20c) and to the data given in tal points show the change in the number of fragments

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 KINETIC MECHANISM OF THE FORMATION 2079

N(ε) = H(ε)/Λε() in the transverse section of a planar REFERENCES sample with initial thickness H0. Since the sample 1. G. Langford and M. Cohen, Metall. Trans. A 6 (4), 901 thickness changes with increasing degree of deforma- (1975). ε ε tion as H( ) = H0exp(Ð ), we have, with allowance for 2. A. S. Rubtsov and V. V. Rybin, Fiz. Met. Metalloved. 44 (3), 618 (1977). the dependence of Λ on ε (Eq. (20a)), 3. A. S. Malin and M. Hatherly, Met. Sci. 13 (8), 463 1 (1979). N()ε = N 1 Ð exp Ð---mk ε exp()Ðε , (23) 0 2 5 4. D. A. Hughes and W. D. Nix, Mater. Sci. Eng., A 122 (2), 153 (1989). where N = H /Λ∞ . The curve in Fig. 5 demonstrates 5. N. A. Koneva and É. V. Kozlov, Izv. Vyssh. Uchebn. 0 0 Zaved., Fiz. 33 (2), 89 (1990). × 4 this dependence for N0 = 6.6 10 and mk5 = 1.06. It is seen that at deformations ε < 0.8, the number of frag- 6. D. A. Hughes, Acta Metall. Mater. 41 (5), 1421 (1993). ments in the section increases because of the decrease 7. J. H. Driver, D. J. Jensen, and N. Hansen, Acta Metall. in their dimensions, whereas at large deformations, this Mater. 42 (9), 3105 (1994). number decreases because of the thinning of the sample 8. N. Hansen and D. A. Hughes, Phys. Status Solidi A 149 (1), 155 (1995). at an approximately constant size of the fragments Λ∞ . 9. D. A. Hughes, Q. Liu, D. C. Chrzan, and N. Hansen, Acta A question that arises is where do the fragments dis- Mater. 45 (1), 105 (1997). appear to from the section. According to [12], their dis- 10. N. Hansen and D. J. Jensen, Philos. Trans. R. Soc. Lon- appearance is related to the coalescence of low-angle don, Ser. A 357, 1447 (1999). boundaries of fragments with the formation of high- 11. Y. Iwahashi, Z. Horita, M. Nemoto, and T. G. Langdon, angle boundaries, as well as to the annihilation of Acta Mater. 46 (9), 3317 (1998). boundaries of opposite signs. Although such processes 12. D. A. Hughes and N. Hansen, Acta Mater. 48 (11), 2985 can take place if the fragment sizes and misorientations (2000). obey a distribution of type (18), upon deformation, the 13. A. Godfrey and D. A. Hughes, Acta Mater. 48 (8), 1897 number of “medium” fragments in the section of the (2000). deformed sample decreases according to Fig. 5 and the law of similitude of fragmented structures (Fig. 3). 14. A. N. Tyumentsev, M. V. Tret’yak, Yu. P. Pinzhin, and A. D. Korotaev, Fiz. Met. Metalloved. 90 (5), 44 (2000) Therefore, the only way in which the number of frag- [Phys. Met. Metallogr. 90 (5), 461 (2000)]. ments can decrease in the section is through the escape of fragment boundaries to outside the sample, i.e., their 15. M. F. Ashby, Philos. Mag. 21 (170), 399 (1970). annihilation with the surface, as in the case of indi- 16. N. Hansen and D. Kuhlmann-Wilsdorf, Mater. Sci. Eng. vidual dislocations emerging onto the surface of the 81 (1/2), 141 (1986). sample. 17. G. A. Malygin, Fiz. Tverd. Tela (St. Petersburg) 44 (7), This means that the fragment boundaries are mobile, 1249 (2002) [Phys. Solid State 44, 1305 (2002)]. as follows from the solution (14a) to Eqs. (1) and (6), 18. M. A. Lewis and J. W. Martin, Acta Metall. 11 (11), 1207 rather than static, as may be supposed based on elec- (1963). tron-microscopic investigations of samples unloaded 19. G. A. Malygin, Fiz. Tverd. Tela (St. Petersburg) 43 (10), after deformation. Since the formation of fragments is 1832 (2001) [Phys. Solid State 43, 1909 (2001)]. a dynamic process of self-organization of dislocations 20. G. A. Malygin, Fiz. Tverd. Tela (Leningrad) 31 (7), 43 caused by internal factors, the emergence of fragment (1989) [Sov. Phys. Solid State 31, 1123 (1989)]. boundaries onto the surface of the crystal does not dis- 21. G. A. Malygin, Usp. Fiz. Nauk 169 (9), 979 (1999) turb this dynamic process until the fragment dimen- [Phys. Usp. 42 (9), 887 (1999)]. sions become comparable with the sample thickness. 22. J. Kim, I. Kim, and D.-H. Shin, Scr. Mater. 45 (4), 421 The dynamic, synergetic character of the process of for- (2001). mation of FDSs explains why the TaylorÐPolanyi law is 23. L. Kawasaki, J. Phys. Soc. Jpn. 36 (1), 142 (1974). violated for such structures. 24. M. Zehetbauer and V. Seumer, Acta Metall. Mater. 41 In conclusion, we note that the analysis of experi- (2), 577 (1993). mental data on the basis of equations of dislocation 25. H. A. Mughrabi, Mater. Sci. Eng. 85 (1), 15 (1987). kinetics makes it possible to consider the processes of 26. G. Langford and M. Cohen, Trans. Am. Soc. Met. 62 (3), deformation strengthening [19, 21] and the formation 623 (1969). of dislocation structures [21] at all ﬁve stages of the curves of deformation strengthening of crystals from common positions. Translated by S. Gorin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2080Ð2082. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1987Ð1989. Original Russian Text Copyright © 2002 by Gruzdkov, Petrov, Smirnov.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

An Invariant Form of the Dynamic Criterion for Yield of Metals A. A. Gruzdkov, Yu. V. Petrov, and V. I. Smirnov St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 191119 Russia e-mail: [email protected] Received December 28, 2001

Abstract—The known dynamic criteria for yield of metals are generalized to the case of any stressedÐstrained state and an arbitrary loading duration. The criteria for uniaxial tension and simple shear are a special case. The dependences of the yield point on the strain rate are calculated using the proposed criterion and are compared with the experimental data. © 2002 MAIK “Nauka/Interperiodica”.

For slow uniaxial tension or compression, it is gen- criterion and Tresca yield criterion. The latter criterion erally assumed that the material transforms into a plas- turns out to be more adequate for the majority of mate- tic state under a critical load (yield point). In this case, rials and satisﬁes the condition the yield criterion can be written in the form σ()t ≤ σ, (1) σ Y T ()t ≤ ------Y-, (2) σ k where Y is the yield point for a slow uniaxial tension. 3 The classical generalizations of criterion (1) to the case of an arbitrary stressed state are the Mises yield where

1 ()σ σ 2 ()σ σ 2 ()σ σ 2 ()σ2 σ2 σ2 T k = ------x Ð y +++y Ð z z Ð x 6 xy ++yz zx 6 1 ()σ σ 2 ()σ σ 2 ()σ σ 2 = ------1 Ð 2 ++2 Ð 1 3 Ð 1 6

σ σ σ is the intensity of shear stresses ( 1, 2, and 3 are the under loading at a constant rate). This hampers compar- principal stresses). This criterion makes it possible, ison with the results obtained using other load schemes. ﬁrst, to analyze the complex stressed state and, second, Second, in many cases, the strain rate cannot be consid- to compare the results of testing with the use of differ- ered to be constant even approximately. This is particu- ent schemes (for example, testing for bending and ten- larly true in regard to pulsed (impact) load and tests for sion). For a stressed state, criterion (2) transforms into yield delay under a permanent load applied instantly. criterion (1). For a simple shear, criterion (2) trans- Finally, the construction of the experimental diagram forms into the condition σ ()ε Y ú presents considerable technical difficulties, τ()≤ τ t Y , (3) especially if account is taken of the strong effects of temperature, structural features of the material (for τ τ σ where is the shear stress and Y = Y /3 is the shear example, grain size), etc. yield point. The problem of constructing a dynamic yield crite- It is well known that, under the conditions of instan- rion is associated with the experimental determination taneous loads, criteria (1) and (2) cease to describe ade- of a certain invariant characteristic of the process that quately the behavior of the material. Traditional must be independent of the type of action. Suvorova [1] attempts to introduce corrections to these criteria by analyzed the results of numerous experimental investi- substituting a strain rate function for the statistical yield gations and proved that, for high-rate loading, the point have a number of material disadvantages. First, dependence of the yield stress (σ∗) on the time of tran- the invariance is lost; i.e., in this case, we can deal only with a certain type of testing (for example, testing sition to a plastic state (t∗) is well approximated by a

AN INVARIANT FORM OF THE DYNAMIC CRITERION 2081

500 Model 350 Experiment 450 (Campbell and Ferguson [7]) 300 400 250 350 200 300

Yield point, MPa Yield 150 Yield point, MPa Yield 250 100 200 –4 –2 0 2 –2 0 2 4 Logarithm of tension rate (1/s) Logarithm of shear rate (1/s)

Fig. 1. Dependence of the yield point on the logarithm of Fig. 2. Dependence of the yield point on the logarithm of the tension rate. The curve represents the results of our cal- the shear rate. The curve represents the results of our calcu- culations, and the points are the experimental data taken lations, and the points are the experimental data taken from from [7]. [7]. straight line in logarithmic coordinates, i.e., by the rela- the transition to a plastic state is associated with the tionship development of the dislocation structure, it is reasonable to assume that this quantity is related to the time ασlog +logt = const. * * characteristics of the dislocation motion; i.e., tinc is We considered variants of the generalization of this inversely proportional to the dislocation velocity v. The relationship to the case of an arbitrary law of variation dislocation velocity v can be represented using the in the stress with time. Analysis demonstrates that the expression proposed by Gilman [6]; that is, best qualitative agreement with the experimental data ∆ on instantaneous loads is achieved with the criterion G vv= 0 expÐ------. proposed by Campbell [2]: kT t ∆ σ()α Here, G is the activation free energy, k is the Boltz- s ≤ () mann constant, and T is the absolute temperature. Con- ∫------σ - ds CC= const . (4) 0 sequently, the relationship for the incubation time can 0 be written in the following form: σ σ Here, is the current stress, 0 is the yield point at α ∆G absolute zero temperature, and is the dimensionless t = γ exp ------. (6) parameter. The time t* of transition to the plastic state inc kT corresponds to the shortest time t at which relationship (4) transforms into an equality. The advantage of Criterion (5) agrees well with the experimental data this criterion is its applicability to arbitrary variations in over a wide range of strain rates and temperatures [4]. the stress with time. The disadvantages of this criterion In order to generalize criterion (5) to the case of any are its applicability only to uniaxial tension or compres- complex stressedÐstrained state, we can represent it in sion and only to very short-term loading. In actual fact, the form criterion (4) is contradictory to criterion (1) and, starting with a certain loading duration, leads to consider- t ()α 3T k s ≤ ably underestimated threshold values of the yield stress ∫ ------σ ds tinc. (7) Y as compared to the experimental data. The change-over ttÐ to the quasi-static case becomes impossible in terms of inc the limiting condition (4). As a consequence, the posi- Now, we consider two special cases. Let t0 be the tion of the dynamic branch in the time dependence of characteristic time of loading. It is easy to verify that, the yield remains uncertain. For uniaxial tension, these for uniaxial load, criterion (7) coincides with condi- disadvantages can be eliminated under the condition tion (5). For a slowly varying load, the limiting transi- proposed in [3Ð5], that is, tion tinc/t0 0 satisﬁes the static criterion (1). Figure 1 t depicts the calculated dependence of the yield point on σ()α α σ s ≤ the tension rate ( = 17, tinc = 0.1 s, and Y = 215 MPa) ∫ ------σ - ds tinc. (5) upon uniaxial loading at a constant rate in comparison Y ttÐ inc with the experimental data taken from [7]. σ τ σ σ Here, t is the incubation time, which characterizes the For simple shear, we have 1 = , 2 = 0, and 3 = inc −τ τ rate of structural transformation in the material. Since ; consequently, we obtain Tk(t) = (t). As a result, ine-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2082 GRUZDKOV et al. quality (7) takes the form REFERENCES

t 1. Yu. V. Suvorova, Prikl. Mekh. Tekh. Fiz., No. 3, 55 τ()α (1968). s ≤ ∫ ------τ - ds tinc. 2. J. D. Campbell, Acta Metall. 1 (6), 64 (1953). Y ttÐ inc 3. A. A. Gruzdkov and Yu. V. Petrov, in Proceedings of the 1st All-Union Conference “Technological Problems of The change-over to the static case leads to criterion (3). Strength of Load-Carrying Structures,” Zaporozhye, Figure 2 presents the calculated dependence of the 1991, Vol. 1, Part 2, p. 287. α × Ð4 yield point on the shear rate ( = 11, tinc = 5 10 s, 4. A. A. Gruzdkov and Yu. V. Petrov, Dokl. Akad. Nauk 364 τ and Y = 110 MPa) and the experimental data taken (6), 786 (1999) [Dokl. Phys. 44, 114 (1999)]. from [7]. As can be seen from Figs. 1 and 2, the pro- 5. N. V. Morozov and Yu. V. Petrov, Problems of Fracture posed model is in good agreement with the experiment. Dynamics of Solids (St. Petersburg Univ., St. Petersburg, 1997). 6. M. A. Meyers, Dynamic Behavior of Materials (Wiley, ACKNOWLEDGMENTS New York, 1994), Chap. 13, p. 323. 7. J. D. Campbell and W. G. Ferguson, Philos. Mag. 21 This work was supported by the Russian Foundation (169), 63 (1970). for Basic Research (project no. 99-01-00718 and project TsKP 00-01-05020) and the Federal Program “Integration.” Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2083Ð2086. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1990Ð1993. Original Russian Text Copyright © 2002 by Zakrevskiœ, Pakhotin, Shul’diner. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

On the Possible Effect of a Magnetic Field on the Breaking of Mechanically Loaded Covalent Chemical Bonds V. A. Zakrevskiœ, V. A. Pakhotin, and A. V. Shul’diner Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia e-mail: [email protected] Received February 8, 2002; in ﬁnal form, February 15, 2002

Abstract—It was established using the EPR method that a magnetic field (B = 0.6 T) does not affect the process of the appearance of free radicals that are formed as a result of the breaking of covalent chemical bonds in macromolecules of mechanically loaded fibers of polycaproamide. On this basis, as well as on the basis of some other data, doubts were cast on the possibility of using the assumption (frequently discussed in the literature) on the influence of a magnetic field on the rate of transformation of defect complexes due to singletÐtriplet transitions upon the fluctuational lengthening of covalent bonds in such complexes for explaining some manifestations of the magnetoplastic effect in ionic crystals. © 2002 MAIK “Nauka/Interperiodica”.

Al’shits et al. [1, 2] revealed an increase in the concrete centers, proceeding from only the general con- mobility of individual dislocations under the effect of a cepts of spin-dependent free-radical reactions. How- relatively weak dc magnetic field (B < 1 T) in alkali- ever, works have appeared recently [11Ð13] in which halide crystals (AHCs). Later, it turned out that the concrete transformations in the systems of point defects magnetic field also affects the macroplastic properties were considered. To be more precise, a scheme of mag- of these crystals. A decrease in the yield stress in a magnetosensitive reactions in complexes of point defects netic field was observed [3, 4], as well as an increase in containing several impurity ions was suggested. In par- the rate of straining after switching-on a field [5–7] and ticular, the effect of a magnetic field on the transforma- a decrease in the microhardness of AHCs in a magnetic tions in trimers of impurityÐvacancy dipoles was stud- field [8]. ied. The heart of the scheme is as follows. Various types The change in the plastic properties of crystals in a of trimers differ from one another in binding energy magnetic field was called the magnetoplastic effect and the spacing between impurity ions (bond lengths). (MPE). In recent years, several dozen works were Changes in the structure of impurity complexes may devoted to the MPE. The main results of these investi- affect the plasticity of crystals. The magnetic field may gations were generalized in a comprehensive review [9] affect the spin-dependent stages of the transition of the containing a complete list of papers related to studying complexes into more stable configurations with the MPE. Based on the data obtained in those works, a changes in the separation between impurity ions. It is conclusion was derived on the occurrence in ionic crys- assumed that a magnetic field may accelerate the tran- tals of magnetosensitive reactions in the subsystem of sition of a complex of defects from the configuration in structural defects, namely, of reactions between para- which the spacing between the impurity ions is mini- magnetic centers (PMCs) at dislocations and point mum and, consequently, the contribution of exchange defects. Thus, a hypothesis was suggested on the elec- forces to the interaction is large into a configuration in tron nature of the MPE. which the separation between impurities is enhanced as According to [9–11], a magnetic field acts not only compared to the initial one. Thus, it is assumed that in on the process of interaction of dislocations with obsta- the initial configuration, there exists a covalent bond cles but also on the structure of the dislocations them- between some impurity ions in the paramagnetic state, selves, as well as on the structure of point defects and i.e., possessing unpaired electrons. their complexes. A conclusion was made on the impor- A transformation of a defect complex, as was noted tant role of short-range exchange forces (formation and in [12, 13], is a thermoactivational process. It may be opening of covalent chemical bonds between PMCs) in supposed that it also can occur in a field of mechanical the formation of plastic properties of crystals. forces [11]. At sufficiently large thermofluctuational Because of the lack of data on the structure of increments in the lengths of the covalent interatomic PMCs, the discussion of the mechanism of the effect of (“interion”) bonds in a magnetic field, singlet–triplet a magnetic field on the dislocation plasticity of crystals (SÐT) transitions occur, which are accompanied by an has frequently been performed in a general form, with- increase in the interion spacings. The complexes loos- out consideration of reactions with the participation of ened due to the presence of broken covalent bonds then

2084 ZAKREVSKIŒ et al. using the EPR method can be accumulated upon tension [14, 15]. In this work, we used monofibers of polycaproamide (nylon-6) 0.8 mm in diameter. The EPR spectra NS1 were recorded after applying a mechanical load to the fiber. The load decreases the energy of activation for bond breaking and accelerates the thermofluctuational breaking of bonds. The concentrations of free macroradicals (concentrations of broken bonds) formed in the 2 fiber under the effect of a dc magnetic field with an induction B = 0.6 T and without a magnetic field were compared. Load The scheme of the experiment is illustrated in Fig. 1. A piece of a fiber was loaded with a force of Fig. 1. Schematic of the loading of a polymer fiber. Hori- 27 kg (a stress of 54 kg/mm2). Part of the fiber was zontal bars mark fiber segments (1 and 2) in which free rad- located between the poles of a dc magnet, i.e., in a mag- icals were registered. netic field, and part of the fiber was outside the field. After holding under a load for 2 min, the fiber was unloaded and placed in a cylindrical resonator of a V4500-10A EPR spectrometer operating in the Q range. Then, the spectra of the fiber segments 1 and 2 marked by horizontal bars in Fig. 1 were recorded sequentially (during loading, segment 1 was located in the magnetic field; segment 2 was outside the magnetic field). The tensile tests and the recording of the spectra were performed at room temperature. The time of recording a spectrum was 2.5 min; therefore, the time between two successive recordings of spectra (with allowance for the time necessary for the translation of the fiber along the resonator axis when selecting a proper position of a segment) was 3 min and the time between two sequential recordings of spectra from the a × –3 same segment (1 or 2) was 6 min. Twelve spectra were 5 10 T recorded for each sample (six spectra from each of the selected segments). All the spectra had the same shape, shown in Fig. 2. The spectra mainly belong to secondary middle macroradicals ÐNHÐCHÐCH2Ð (I) that are formed as a result of interaction of active primary end macroradi- Fig. 2. An EPR spectrum of free radicals formed in a loaded cals with neighboring macromolecules [16]. The con- fiber of nylon-6. centration of the radicals was about 1017 cmÐ3. Since in the center of the spectrum there is a weak singlet belonging to free radicals that are more stable than rad- suffer a sufficiently fast rearrangement due to a facili- icals of type I, we selected the amplitude of the high- tated motion of dipoles. field line of the hyperfine structure of the spectrum (a in Fig. 2) as a measure of the concentration of radicals. It was of interest to experimentally test the validity The dependence of the amplitude of this line on the of the basic assumption on the acceleration of the ther- time of recording of the spectrum (reckoned from the mofluctuational breaking of stressed covalent chemical moment of the beginning of the recording of the first bonds under the effect of a magnetic field. To detect the spectrum) is shown in Fig. 3. The decrease in the con- events of the breaking of covalent bonds in loaded centration of free radicals in time is caused by their materials, the EPR method has long been successfully recombination. It is seen that the points corresponding applied. Suitable objects for studying the processes of to both segments fall well on the same curve. This breaking of interatomic chemical bonds in mechani- means that the number of radicals formed under a load cally loaded materials are high-strength polymer fibers, in the segment located in a magnetic field is equal to the in which high concentrations of rather stable (long- number of radicals that arise in the segment that did not lived) free macroradicals that can reliably be registered suffer the field effect. Therefore, the magnetic field

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

ON THE POSSIBLE EFFECT OF A MAGNETIC FIELD 2085 does not affect the process of the breaking of covalent 30 bonds. This fact indicates that the scheme of transformation of defect complexes suggested in [11Ð13] is insuffi- 20 ciently substantiated. Note one more obvious circum- , arb. units stance concerning the main supposition of the authors a of those works: the mixing of spin states (SÐT conver- 10 sion) upon thermofluctuational excitations of the 0 10 20 30 40 bonds. To realize transitions between the spin states of Time, min a pair of paramagnetic particles, it is necessary that the exchange energy (spin contribution to the energy of Fig. 3. Time dependence of the amplitude of the EPR spec- interaction of the particles) be comparable with the tra. Open circles correspond to segment 1 (B ≠ 0); solid cir- energy of transitions stimulated by the magnetic field; cles, to segment 2 (B = 0). i.e., the difference between the energies of the singlet Ð6 and triplet states should be approximately 10 eV. This unsuccessful [18]. The references to [19] sometimes is possible only at very large bond lengthenings. occurring in the literature are somewhat unconvincing, The estimates can be made using one of the known since this work contains no data on PMCs at disloca- potentials, e.g., the Morse potential, to characterize the tions in unirradiated crystals. Moreover, in view of the interatomic interaction: results obtained in [20Ð22], the conclusions made in {}[]() []() [19] apparently require corrections. UD= expÐ2brÐ r0 Ð 2exp ÐbrÐ r0 , The paramagnetism of point defects is frequently where D is the energy of dissociation of the bond, b = attributed to the presence of calcium as an impurity in 2F/D, F is the maximum force of interatomic interac- AHCs (e.g., in NaCl crystals that are widely used as a tion, and r0 is the equilibrium interatomic separation. suitable object for the investigation of various proper- ӷ ++ At r r0, we have U ~ Ð2Dexp(Ðbr). Assuming D = ties of ionic crystals) [9]. The Ca ions enter into the 1 eV, U = Ð2 × 10Ð6 eV, and b = 2 × 107 cmÐ1, we obtain composition of impurityÐvacancy dipoles, in which r = 7.0 nm. This is a formal estimate, since it makes no vacancies compensate their charge. There are no data sense to speak of chemical bonding when |U| Ӷ kT. on the observation of EPR signals from paramagnetic Nevertheless, it is clear that the above-considered bond Ca+ ions in AHCs even at a high concentration of the lengthenings in crystals are unrealistic. impurity [23], which appears to be due to the small Note also that the expected times of SÐT transitions, magnitude of the second ionization potential of cal- which are, according to [12, 13], 10Ð9 to 10Ð8 s, exceed cium, for which reason the impurity calcium is present the lifetime of a fluctuationally extended bond (equal to in AHCs in the form of double-charged ions. 10Ð13 s, according to [17]) by several orders of magni- The assumption on the presence in the complexes of tude. dimers of single-charged ions participating in the for- Thus, the scheme of transformation of complexes mation of covalent bonds seems insufficiently (trimers) of defects that was discussed in the literature grounded, in particular, since, with the change of the apparently cannot be regarded as a basis for explaining charged state of the impurity (e.g., with the appearance the spin origin of the magnetoplastic effect (MPE). In of a single-charged instead of a double-charged cal- this connection, we consider this problem in some more cium ion), the reason for which the impurity ion can detail. hold a vacancy disappears; the reason for the formation of a dipole also disappears. In addition, the separation As was noted above, the MPE is attributed to the between the cations, e.g., in NaCl, exceeds 0.4 nm. At occurrence of spin-dependent reactions between para- the same time, it is known that the lengths of even weak magnetic centers (PMCs). Therefore, the problems of covalent bonds do not exceed 0.25 nm. At large dis- substantiating the presence of PMCs in alkali-halide tances, the unpaired electrons form the so-called radical crystals (AHCs) subjected to no preliminary actions pairs, which have a characteristic doublet EPR spec- (e.g., to ionizing radiations), as well as of clarifying the trum. Naturally, the possibility of a sharp (approxi- structure of the centers possessing unpaired electrons mately twofold) change in the interatomic separation and determining the minimum concentration of PMCs (upon the formation of a covalent bond) requires spe- sufficient for the observation of the MPE against the cial substantiation. background of “nonmagnetic” defects, seems to be very important. In our opinion, these problems have Finally, when discussing the nature of the MPE, we been given insufficient attention. However, these are by should take into account that there are data on the no means simple problems. Indeed, there are no reliable absence of the effect of impurity paramagnetic Mn++ data on the behavior of PMCs at dislocations in AHCs ions on the plasticity of AHCs in a magnetic field [24] in the literature. The attempts to record EPR signals and on the insensitivity of MPE to the presence of F upon severe deformation and fracture of AHCs were centers in the crystal [25].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

2086 ZAKREVSKIŒ et al.

From the above-said, we conclude that the scheme 10. Yu. I. Golovin and R. B. Morgunov, Zh. Éksp. Teor. Fiz. of transformation of defect complexes in alkali-halide 115 (2), 605 (1999) [JETP 88, 332 (1999)]. crystals that was discussed earlier in the literature can- 11. Yu. I. Golovin and R. B. Morgunov, Izv. Ross. Akad. not be considered to be well grounded. To clarify the Nauk, Ser. Fiz. 61 (5), 850 (1997). mechanism of the magnetoplastic effect and, above all, 12. Yu. I. Golovin, R. B. Morgunov, V. E. Ivanov, and to determine the concrete type of spin-dependent reac- A. A. Dmitrievskiœ, Zh. Éksp. Teor. Fiz. 117 (6), 1080 tions in the system of defects in ionic crystals, further (2000) [JETP 90, 939 (2000)]. investigations are necessary. 13. R. B. Morgunov, Author’s Abstract of Doctoral Disserta- tion (Voronezh. Gos. Tekhn. Univ., Voronezh, 2000). 14. S. N. Zhurkov, A. Ya. Savostin, and É. E. Tomashevskiœ, ACKNOWLEDGMENTS Dokl. Akad. Nauk SSSR 159 (2), 303 (1964) [Sov. Phys. We are grateful to É.E. Tomashevskiœ for the oppor- Dokl. 9, 986 (1965)]. tunity to perform experiments on the investigation of 15. B. Ya. Levin, A. V. Savitskiœ, A. Ya. Savostin, and electron paramagnetic resonance of free radicals in É. E. Tomashevskiœ, Vysokomol. Soedin., Ser. A 13 (4), ﬁbers subjected to tension. 947 (1971). 16. V. A. Zakrevskiœ, V. V. Baptizmanskiœ, and É. E. Toma- shevskiœ, Fiz. Tverd. Tela (Leningrad) 10 (6), 1699 REFERENCES (1968) [Sov. Phys. Solid State 10, 1341 (1968)]. 1. V. I. Al’shits, E. V. Darinskaya, T. M. Perekalina, and 17. A. I. Slutsker, A. I. Mikhaœlin, and I. A. Slutsker, Usp. A. A. Urusovskaya, Fiz. Tverd. Tela (Leningrad) 29 (2), Fiz. Nauk 164 (4), 357 (1994) [Phys. Usp. 37, 335 467 (1987) [Sov. Phys. Solid State 29, 265 (1987)]. (1994)]. 2. V. I. Al’shits, E. V. Darinskaya, and E. A. Petrzhik, Fiz. 18. V. V. Boldyrev, V. A. Zakrevskiœ, and F. Kh. Urakaev, Izv. Tverd. Tela (Leningrad) 33 (10), 3001 (1991) [Sov. Akad. Nauk SSSR, Neorg. Mater. 15 (12), 2154 (1979). Phys. Solid State 33, 1694 (1991)]. 19. S. Z. Shmurak and F. D. Senchukov, Fiz. Tverd. Tela 3. Yu. I. Golovin and R. B. Morgunov, Izv. Ross. Akad. (Leningrad) 15 (10), 2976 (1973) [Sov. Phys. Solid State Nauk, Ser. Khim., No. 4, 739 (1997). 15, 1985 (1973)]. 4. A. A. Urusovskaya, V. I. Al’shits, A. E. Smirnov, and 20. A. A. Kusov, M. I. Klinger, and V. A. Zakrevskiœ, Fiz. N. N. Bekkauér, Pis’ma Zh. Éksp. Teor. Fiz. 65 (6), 470 Tverd. Tela (Leningrad) 31 (7), 67 (1989) [Sov. Phys. (1997) [JETP Lett. 65, 497 (1997)]. Solid State 31, 1136 (1989)]. 5. Yu. I. Golovin and R. B. Morgunov, Fiz. Tverd. Tela 21. A. V. Shuldiner and V. A. Zakrevskii, Radiat. Prot. (St. Petersburg) 37 (7), 2118 (1995) [Phys. Solid State Dosim. 65 (1Ð4), 113 (1996). 37, 1152 (1995)]. 22. V. A. Zakrevskiœ and A. V. Shul’diner, Fiz. Tverd. Tela 6. Yu. I. Golovin, R. B. Morgunov, S. V. Zhulikov, and (St. Petersburg) 42 (2), 263 (2000) [Phys. Solid State 42, A. M. Karyakin, Izv. Ross. Akad. Nauk, Ser. Fiz. 60 (9), 270 (2000)]. 173 (1996). 23. C. Hoentzsch and J. M. Spaeth, Phys. Status Solidi B 88 7. Yu. I. Golovin, R. B. Morgunov, and S. V. Zhulikov, Fiz. (2), 581 (1978). Tverd. Tela (St. Petersburg) 39 (3), 495 (1997) [Phys. 24. Yu. I. Golovin and R. B. Morgunov, Pis’ma Zh. Éksp. Solid State 39, 430 (1997)]. Teor. Fiz. 61 (7), 583 (1995) [JETP Lett. 61, 596 (1995)]. 8. Yu. I. Golovin, R. B. Morgunov, D. V. Lopatin, and 25. Yu. I. Golovin and R. B. Morgunov, Fiz. Tverd. Tela A. A. Baskakov, Phys. Status Solidi A 160 (2), R3 (St. Petersburg) 35 (9), 2582 (1993) [Phys. Solid State (1997). 35, 1280 (1993)]. 9. Yu. I. Golovin and R. B. Morgunov, Materialovedenie, No. 3, 2 (2000); No. 4, 2 (2000); No. 5, 2 (2000); No. 6, 2 (2000). Translated by S. Gorin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2087Ð2090. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1994Ð1997. Original Russian Text Copyright © 2002 by Melnichuk, Malanushenko.

MAGNETISM AND FERROELECTRICITY

Effect of Irradiation with Ne+ Ions on the Transformations of Domain Structures of a Uniaxial Magnetic Film in an AC Magnetic Field I. A. Melnichuk and E. L. Malanushenko Donetsk National University, Universitetskaya ul. 24, Donetsk, 83055 Ukraine e-mail: [email protected] Received November 2, 2001

Abstract—Effects of irradiation with Ne+ ions on the transformations of domain structures (DSs) that occur in a uniaxial magnetic film under the action of an ac magnetic field are investigated. Transitions of a DS from an amorphous state into a hexagonal lattice and a labyrinthine structure are considered. The irradiation is found to lead to a change in the amplitudes of the ac field at which phase transformations of the DS occur. The effect of the magnitude of the ac field on the number of domains in a block with a hexagonal lattice has been studied. It is shown that the process of annealing of defects in a DS consisting of blocks with a hexagonal lattice can be described by the equation of a first-order reaction. The irradiation-induced change in the energy of activation for the annealing of defects in the DS has been found. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION formations in DSs in samples with various concentrations of defects makes it possible to test the validity of In recent years, large attention has been paid to the the application of the concept of effective temperature investigation of phase transformations of domain struc- for describing the behavior of DSs in an ac magnetic tures (DSs) in thin magnetic films. The interest in the field. investigation of DSs in the presence of a low-frequency ac magnetic field is, on the one hand, due to the specific This work was aimed at the study of the effect of features of new dynamic stable domain configurations irradiation with accelerated particles on the conditions [1, 2] and, on the other hand, to the effect of an ac mag- of the transition from an amorphous domain structure netic field on the phase transformations in DSs, which (ADS) of bubble domains into the states of the hexago- are described using the concept of an effective temper- nal lattice (HL) and labyrinthine domain structure ature related to the ac magnetic field [3–5]. (LDS) and also at describing the ADSÐHL transition. The experimental investigations of the transformations of DSs in an ac magnetic field show that the use 2. EXPERIMENTAL of the effective temperature makes it possible to qualitatively correctly describe such features of the behavior Domain structures were observed on a magnetoop- of DSs in ac magnetic fields as the times of relaxation tical device using the Faraday effect. The investigations of DSs in ac magnetic fields toward the equilibrium were performed on a uniaxial epitaxial film of µ state [6] and the changes in the average spacings (YEuTmCa)3(FeGe)5O12 with a thickness h = 2.9 m between the bends in stripe domains [7]. In addition, and a saturation magnetization 4πM = 250 G. This sam- the velocity distribution for the segments of domain ple was chosen for two reasons. First, the DSs in such walls (DWs) that move in an ac field is Maxwellian, films are easily observable, and the technology of their which indicates a sufficient “chaotization” in the sys- growth permits one to obtain samples of quite varying tem of DWs [8]. degrees of perfection. Second, the main features of the The effective temperature is defined through the effect of irradiation on the magnetic and structural average kinetic energy of domains [4] or domain walls properties of iron garnet films have been very well stud- [5]; therefore, the presence of defects, which contribute ied (see, e.g., [9]). To introduce additional defects into to the pinning or retardation of DWs, affects the condi- the sample, it was irradiated with Ne+ ions (60 keV) to tions of phase transformations in the DSs. To break a dose of 0.8 × 1014 cmÐ2. This dose is smaller than the away a DS from the site of pinning and to overcome dose 2 × 1014 cmÐ2 necessary for the formation of friction forces, the ac field must do additional work; anisotropy of the easy-plane type over the whole thick- therefore, the introduction of additional defects into the ness of the irradiated layer; therefore, the properties of sample increases the amplitudes of the ac field at which the DWs are mainly affected by radiation defects. phase transformations occur in DSs. The study of trans- Under given irradiation conditions, the depth of pene-

2088 MELNICHUK, MALANUSHENKO

1000 DS. As a nucleus of the HL, a block with a hexagonal HL1 HL2 packing containing no less than 30 domains was considered. This criterion is very close to that used in [11]; 800 Polycrystallic structure 1 in addition, it is precisely 30 domains that are required ADS1 to construct the second coordination shell of the HL in bubble domains with allowance for the fact that its unit 600 Polycrystallic structure 2 cell contains two domains. Since no HL blocks with a ≥ ADS2 domain number N 30 were observed in the ADS, we 400 used the formation of blocks with this number of ), arb. units domains as a criterion for the transition of the ADS into H ( 1 LDS2 N a polycrystalline structure. At H1 < H0 < H2, the DS rep- 2 ≥ 200 LDS1 resented a set of HL blocks with N 30.

3. RESULTS AND DISCUSSION 0 20 40 60 80 100 H, Oe Figure 1 displays the dependence of N on H0. As is seen, the average number of domains in an HL block Fig. 1. Variation of the number of domains in an HL block grows with increasing H0, and irradiation leads to a as a function of the amplitude of the ac magnetic field: (1) shift of this dependence along the H0 axis toward larger unirradiated sample; (2) irradiated sample; ADS, amor- amplitudes of the ac field. The fields of existence of the phous domain structure; LDS, labyrinthine domain structure; and HL, hexagonal lattice. ADS, PS, HL, and LDS for the irradiated and unirradiated samples are separated in Fig. 1 by dashed vertical lines. It is seen that the irradiation of the sample tration of Ne+ ions into the sample is about 0.1 µm; con- increases H1 by 25% and H2 by 17%. The changes are sequently, for this thickness of the film, all the defects large, since the phase transition at H = H2 is observed are localized near its surface. very clearly and occurs in a rather narrow range of fields. In addition, the criterion for the determination of An ADS was obtained by converting the sample H1 used in this case also permits us to reliably deter- from a saturated state into a single-domain state by mine the field for the transition from the ADS to the PS. decreasing the external magnetic field applied parallel The errors in the measurements of critical fields are no to the film plane. The ADSÐHLÐLDS transitions were more than 3%. Since the effective temperature of the realized by applying an ac magnetic field H oriented ω DS is unambiguously related to the amplitude of the ac along the normal to the sample surface (H = H0sin t, field [4, 5], an increase in the critical fields upon irradi- ω π where /2 = 300 Hz and H0 = 0Ð80 Oe). The applica- ation may be interpreted as being due to the influence tion of an ac magnetic field led to transformations of the of irradiation on the effective temperature of phase DS, which manifested themselves in changes in the transformations in the DS. domain size and in domain displacements. In a certain An analysis of the ADSÐHL transition for the irradi- time, the changes were stopped and the DS passed into ated and unirradiated samples was performed using the a stable state (for a given magnitude of H0). The trans- effective temperature with allowance for the N(H0) formations of the DS were considered to be completed dependences shown in Fig. 1. This transition has many after the termination of domain displacements. Further, common features with the process of crystallization of to diminish the diffuse character of the DWs in the DS solids; therefore, the analysis of the effect of an ac mag- images, the amplitude of the ac magnetic field was netic field on the behavior of the DS was based on the decreased to zero. model of defect annealing in the crystal lattice of a The intermediate state of the DS between ADS and solid. According to [12, 13], the process of defect elim- HL represents a polycrystalline structure (PS). This ination may be described by the following equation: structure consists of blocks of HL separated by defects dc/dt ==ÐKcn, KKexp()U /kT , (1) and regions of an ADS. To quantitatively describe the 0 0 DS in this state, we used the average number of where c is the concentration of defects, n is the order of domains that compose a single block with a hexagonal the reaction, K0 is a constant, U0 is the activation packing (N). Other quantities to be measured were the energy, k is the Boltzmann constant, and T is the tem- critical fields of the transitions ADS–PS (field H1) and perature. The order of the reaction is determined by the HL–LDS (field H2). Note that while the field of the HL– mechanism of defect elimination [12, 13]. According to LDS transition can easily be determined from the fact [14], the main mechanism of elimination of defects in of the formation of stripe domains in the DS, no strict DSs in the range H1 < H0 < H2 is connected with their criterion for distinguishing between the ADS and PS migration to sinks or with the annihilation of close exists. In this work, as in [10], the magnitude of H1 was paired defects. Therefore, the process of defect elimina- determined based on the formation of HL nuclei in the tion was described using the equation of a first-order

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

EFFECT OF IRRADIATION WITH Ne+ IONS 2089 reaction. It is seen that Eq. (1) qualitatively correctly 0.5 reﬂects the behavior of the N(H0) dependence shown in Fig. 1. In order to obtain a quantitative correspondence of 0 the N(H0) dependence to Eq. (1), it is necessary to use the dependence of the effective temperature on the –0.5

amplitude of the ac ﬁeld and unambiguously relate the )), rel. units H

number of domains in an HL block to the concentration (

N 1 of defects in the DS. According to [4, 5], the effective ( α 2 Q –1.0 temperature may be determined as T eff = H0 , where 2 α is a constant depending on the parameters of the sample. –1.5 0 0.1 0.2 0.3 0.4 The analytical relation between the concentration of (H/H )–2, rel. units defects in the DS and the number of domains in an HL 1 block was obtained under the assumption that the num- Fig. 2. Q(N(H)) dependence plotted for (1) unirradiated and ber of domains remained constant with increasing (2) irradiated samples. Points correspond to experimental amplitude of the ac magnetic field and that at H0 > H1, values of N(H); solid lines, to calculated values of N(H). all the defects were localized at the boundaries of these blocks. The width of the interface between the blocks of the HL was assumed to be equal to one unit cell of this nism of annihilation of paired defects is realized in the lattice. middle of the interval of variation of the fields H0. It is The relative concentration of defects was deter- for this reason that in Fig. 2, we chose only these values mined as follows: C(N) = N /(N + N), where N is the of the amplitude of the ac field. As is seen from Fig. 2, d d d both dependences are linear; note that irradiation led to number of domains in the cells localized at the block ()Ð2 boundaries and N is the total number of domains in an a change in the slope of the QH0 dependence. This HL block. On the assumption that the block has a hex- means that the model of the annealing of defects in a DS upon transition from the amorphous state into an HL is agonal symmetry, we obtained C(N) = 4/(4 + N ). described by the equation of a first-order reaction. This relation permits comparison of the measured ()Ð2 N(H0) dependence with the C(N) dependence calcu- The slope of the straight line QH0 in the depen- lated using Eq. (1). dence on H yields the energy of activation for the pro- Note that, in accordance with the model used, the cess of defect annealing. A comparison of the QH()Ð2 number of defects in the lattice exponentially depends 0 on time. Therefore, the defect concentration decreases dependences for the irradiated and unirradiated sam- sharply with increasing time of residence of the DS in ples shows that an irradiation with ions not only leads an ac field. Such a behavior corresponds to the way of to a change in the first and second critical fields but also estimation of the moment of termination of the trans- changes the entire process of crystallization of the DS formations of the DS in an ac field assumed in the on the whole, which manifests itself in a change in the model. activation energy, which, in turn, causes a change in the slope of the QH()Ð2 dependence. The comparison To compare the experimental values of N(H0) with 0 calculated values of C(H0), it is convenient to use the shows that in this case, the irradiation led to an increase Ð1 in the activation energy of the process by 50%. Note function Q(H0) = lnln(C(H0) ). Then, in accordance with Eq. (1) and the definition of the effective tempera- that the changes that occurred in the sample are related ture, we obtain to the appearance of additional defects as a result of irradiation with Ne ions. () α 2 QH0 = AUÐ 0/k H0, (2) where A is a constant that is independent of the mag- 4. CONCLUSION netic field. Therefore, if the experimental N(H0) depen- Thus, in this work, we showed that irradiation dence corresponds to the model under consideration, affects the magnitudes of the critical fields for the Ð2 then the magnitude of Q linearly depends on H0 and ADSÐHLÐLDS transitions. It was found that N(H0) the magnitude of the activation energy determines the increases with increasing H0, while the irradiation leads slope of this dependence. to a shift of this dependence along the H axis toward larger amplitudes of the ac magnetic field. It is shown ()Ð2 The QH0 dependences for the irradiated and that the change in N(H0) may be described in terms of unirradiated sample are given in Fig. 2. We emphasize the model of annealing of DS defects with the use of an that, in accordance with the results of [14], the mecha- equation of the first-order reaction. In this case, the irra-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2090 MELNICHUK, MALANUSHENKO diation leads to an increase in the energy of activation netic Materials for Microelectronics,” Moscow, 2000, for the process of annealing of structural defects of the p. 494. DS. The effective temperature related to the ac ﬁeld can 8. D. A. Chumakov and I. A. Melnichuk, in Abstract of the be used for the description of the ADSÐHL transition. 8th European Magnetic Materials and Applications Conference (EMMA), Kiev, 2000, p. 279. 9. Bubble-Based Elements and Devices: Reference Book, ACKNOWLEDGMENTS Ed. by N. N. Evtikhiev and B. N. Naumov (Radio i This work was supported by the Ministry of Educa- Svyaz’, Moscow, 1987). tion and Science of the Ukraine. 10. Yu. I. Gorobets, E. L. Malanushenko, and I. A. Mel- nichuk, in Abstracts of the Euro-Asian Symposium “Trends in Magnetism” (EASTMAG), Ekaterinburg, REFERENCES 2001, Vol. 17, p. 336. 1. G. S. Kandaurova and A. É. Sviderskiœ, Zh. Éksp. Teor. 11. Yu. I. Gorobets, I. A. Melnichuk, and I. V. Komissarov, Fiz. 97 (25), 1218 (1990) [Sov. Phys. JETP 70, 684 Vestn. Donetsk. Univ., Ser. A: Estestv. Nauki, No. 2, 98 (1990)]. (1999). 2. I. E. Dikshteœn, F. V. Lisovskiœ, and E. G. Mansvetova, 12. L. Pranyavichyus and Yu. Dudonis, Modiﬁcation of Solid Zh. Éksp. Teor. Fiz. 100 (5), 1606 (1991) [Sov. Phys. Properties by Ion Beams (Mokslas, Vilnius, 1980). JETP 73, 888 (1991)]. 13. A. N. Goland, in Atomic Structure and Mechanical 3. K. L. Babcock and R. M. Westervelt, Phys. Rev. Lett. 64, Properties of Metals (Proceedings of the International 2168 (1990). School of Physics “Enrico Fermi,” Course LXI, Ed. by 4. V. A. Zablotskii and Yu. A. Mamalui, J. Phys.: Condens. G. Gaglioti) (North-Holland, New York, 1976), pp. 1Ð Matter 7, 5271 (1995). 119. 5. E. S. Denisova, Physica B (Amsterdam) 253 (3Ð4), 250 14. Yu. I. Gorobets and I. A. Melnichuk, in Proceedings of (1998). the XVII International School-Workshop “Novel Mag- 6. V. A. Zablotskii, Yu. A. Mamalui, and E. N. Soika, in netic Materials for Microelectronics,” Moscow, 2000, Proceedings of the 7th International Conference on Fer- p. 491. rites, Bordeux, France, 1996, p. 274. 7. Yu. I. Gorobets and I. A. Melnichuk, in Proceedings of the XVII International School-Workshop “Novel Mag- Translated by S. Gorin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2091Ð2094. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 1998Ð2000. Original Russian Text Copyright © 2002 by Guanghua, Eremin, Kolmakova, Lagutin, Levitin.

MAGNETISM AND FERROELECTRICITY

Magnetic Properties of the HoMn2Ge2 Intermetallic Compound Guo Guanghua*, M. V. Eremin**, N. P. Kolmakova**, A. S. Lagutin***, and R. Z. Levitin* * Moscow State University, Vorob’evy gory, Moscow, 119899 Russia ** Bryansk State Technical University, Bryansk, 241035 Russia *** Institute of Molecular Physics, Russian Research Centre Kurchatov Institute, pl. Akademika Kurchatova 1, Moscow, 123182 Russia e-mail: [email protected] Received December 11, 2001

Abstract—The magnetic properties of a tetragonal intermetallic compound, namely, HoMn2Ge2, are investigated experimentally and theoretically. The experimental temperature dependences of the initial magnetic susceptibility and the lattice parameters are obtained in alternating and static magnetic fields. The magnetization curves are measured in strong magnetic fields up to 50 T. The parameters of the crystal field and Ho–Mn and MnÐMn exchange interactions are determined, and the temperature dependence of the magnetic field of the phase transition from an antiferromagnetic phase to a ferromagnetic phase in a magnetic field aligned along the tetragonal axis is calculated. © 2002 MAIK “Nauka/Interperiodica”.

1. Crystals of intermetallic compounds of the gen- et al. [3] investigated the magnetic phase transitions in eral formula RMn2Ge2 (R is a rare-earth element) have RMn2Ge2 intermetallic compounds in ultrastrong mag- a tetragonal structure of the ThCr2Si2 type (space group netic fields and revealed that, at the liquid-helium tem- I4/mmm) that consists of a set of ÐRÐGeÐMnÐGeÐRÐ perature, the holmium-containing compound under- layers aligned perpendicularly to the c axis. As is goes a first-order phase transition from an antiferro- known, RMn2Ge2 intermetallic compounds are ideal magnetic phase to a ferromagnetic phase in a magnetic natural superlattices. Over the last two decades, the field of approximately 90 T. In the present work, we discussed the magnetic properties of the HoMn Ge physical (including magnetic) properties of RMn2Ge2 2 2 have been intensively investigated, especially, as intermetallic compound reasoning from the aforemen- regards the numerous phase transitions revealed in tioned findings and the results of our measurements of the initial magnetic susceptibility, lattice parameters, these compounds. In RMn2Ge2 intermetallic compounds with heavy rare-earth elements, the magnetic and magnetization curves in magnetic fields up to 50 T. moments of manganese are ordered in layers due to Moreover, we determined the parameters of the main MnÐMn exchange interactions (these interactions interactions revealed in this compound. appeared to be the strongest of the interactions 2. Polycrystalline samples of the HoMn2Ge2 inter- involved). It is found that MnÐMn and RÐMn interac- metallic compound were prepared by melting the initial ≈ tions exhibit antiferromagnetic nature. At T < TN 460 K components (with a purity of 99.9%) in an induction (where TN is the Néel temperature), the manganese sub- furnace under quasi-levitation conditions in an argon system is antiferromagnetically ordered, whereas the atmosphere. In order to achieve a better homogeniza- rare-earth subsystem is disordered. In the HoMn Ge tion, the samples were remelted three times and were 2 2 ° compound, unlike the intermetallic compound with then annealed at a temperature of 750 C in a dynamic other rare-earth elements (R = Gd, Tb, or Dy), the rare- vacuum for 170 h. The phase homogeneity of the pre- earth subsystem remains disordered to very low tem- pared samples was checked using x-ray diffraction peratures [1]. Szutula and Leciejewicz [2] demon- analysis. The initial magnetic susceptibility was mea- strated that this phenomenon is associated with the sured both in an alternating magnetic field (in the tem- weak RÐMn exchange interaction. According to neu- perature range from 4.2 to 270 K) and in a static mag- tron diffraction data [1], the magnetic moments of hol- netic field (in the temperature range from 300 to Ho 500 K). The temperature dependence of the lattice mium at temperatures below TN = 2.5 K are character- parameters was measured using the x-ray diffraction ized by a superposition of two sinusoidally modulated technique on a Geigerflex diffractometer (Japan) in the magnetic structures with different wave vectors. The temperature range from 10 to 800 K. Figure 1 shows the manganese subsystem whose magnetic moments are temperature dependences of the initial magnetic sus- collinear with respect to the tetragonal axis remains ceptibility and the lattice parameters of the HoMn2Ge2 ≈ antiferromagnetically ordered down to 1.3 K [1]. Kirste compound. It can be seen from Fig. 1 that, at TN 460 K,

2092 GUANGHUA et al. the manganese subsystem undergoes a transition from (a) the paramagnetic state to the antiferromagnetic state and the temperature dependence of the lattice parameter a(T) exhibits a negative anomaly. At low tempera- 0.16 /g 3 tures, the magnetic susceptibility increases drastically. cm

, This can be associated with the appearance of precur- 1 2 3 arb. units sors of magnetic ordering in the holmium subsystem. 10 , ac χ χ × The magnetization of HoMn2Ge2 was measured on a pulsed induction magnetometer [4] in the temperature 0.08 range from 8 to 50 K in strong pulsed magnetic fields up to 50 T with a pulse duration of 26 ms. The magne- (b) 10.96 T tization measurements were performed with powders N whose particles could rotate in a magnetic field. At T = 4.00 10.92 8 K, the magnetization was also measured on a vibrat- ing-coil magnetometer in magnetic fields up to 12 T.

Å 3.98 c 10.88 Å Figure 2 depicts the magnetization curves of the , , c a a HoMn2Ge2 compound. As can be seen from this figure, the results of magnetization measurements carried out 3.96 10.84 using the two techniques are in good agreement (cf. curves a and b). The magnetization of HoMn2Ge2 in 10.80 strong magnetic fields is approximately equal to 3.94 µ 8 B/f.u. This value is less than that for a free ion µ (10 B/f.u.) and indicates a considerable effect of the Fig. 1. Temperature dependences of (a) the initial magnetic crystal field. susceptibility in (1) an alternating magnetic field and (2) a static magnetic field and (b) the lattice parameters of the Ho 3. As was noted above, at temperatures below T N = HoMn2Ge2 compound. 2.5 K, the holmium subsystem of the HoMn2Ge2 intermetallic compound is in the paramagnetic state, whereas the manganese subsystem is antiferromagnetically ordered and jumpwise transforms into the ferro- 10 magnetic state in a critical magnetic field oriented along the tetragonal axis. Let us now consider the magnetic properties of this compound in a magnetic field 1 b 8 aligned parallel to the tetragonal axis with allowance a made for the smallest (but sufficiently large for describ- 2 100 F ing the experimental data) number of exchange interac- 6 tions in the molecular field approximation. For the AF /f.u. 3 80 manganese subsystem, we take into account both the B

µ exchange interactions in a layer (the exchange parame- , 60 , T 0 M 4 λ H ter 22 ) and the exchange interactions between adja- 0

µ 40 λ cent layers (the parameter 22). The HoÐHo exchange 2 20 interaction is disregarded, because, in this case, it is relatively weak [1] and is of no significance in the temper- 0 100 200 300 400 T, K ature range under investigation. The effect produced by 0 all the layers involved in the manganese subsystem on 0 20 30 40 50 the rare-earth subsystem is summed and can be µ H, T λ 0 described by the parameter 12 (see [5]). In these calculations, we also take into account the known depen- λ dence of the parameter 22 (characterizing the MnÐMn Fig. 2. Experimental (solid lines) and calculated (dashed lines) curves of the magnetization at temperatures of (1) 8, exchange interactions in adjacent layers) on the inter- (2) 27.5, and (3) 49 K. The inset shows the HÐT phase dia- atomic distance in the layer, i.e., the lattice parameter a gram for the transition from the antiferromagnetic (AF) [6]; this dependence leads to the temperature depen- phase to the ferromagnetic (F) phase in a magnetic field dence of the exchange parameter and can be repre- directed along the tetragonal axis. The closed circle indi- sented by the relationship cates the experimentally determined field of the phase transition at T = 5 K [3]. For explanation of curves a and b, see λ ρ() the text. 22 = aaÐ c , (1)

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 MAGNETIC PROPERTIES 2093 where ac = 4.045 Å for RMn2Ge2 intermetallic com- range under investigation, the molecular fields acting pounds. on each holmium layer from the two adjacent layers The effective Hamiltonian for an Ho3+ ion subjected cancel each other. The magnetization curves were used to a tetragonal crystal field in a magnetic field directed to determine the crystal field parameters. For lack of along the tetragonal axis (the z axis) has the form reliable spectroscopic information, we restricted our consideration to the case of three parameters (the sixth- 0 4 m m ()Ho order parameters B and B were taken to be equal to HB= ∑ O Ð g µ J ()HH+ , 6 6 n n J B z z (2) zero, as was done by Venturini et al. [7] in their deter- nm, mination of the crystal field parameters from the tem- n ==246;,, m 0.4, perature dependences of the hyperfine field and the

m m quadrupole interaction in the case of DyMn2Ge2). The where Bn stands for the crystal field parameters, On fitting was performed with the crystal field parameters are the equivalent operators, gJ is the Landé splitting of DyMn2Ge2, which were taken as the initial values. factor (gJ = 5/4), and Jz is the z component of the angu- These parameters were refined using available mag- lar momentum operator for the Ho3+ ion. The molecular netic data in our recent work [5]. The best fitting of the field of the manganese subsystem can be represented by magnetization curves (Fig. 2) was achieved with the ()Ho λ µ 〈 〉 Ð1 0 0 the relationship Hz = 12(m1z + m2z), mkz = Bg Skz , following parameters (in cm ): B2 = 169, B4 = Ð72, where k = 1 and 2, g is the Landé splitting factor for 4 and B4 = Ð556. These parameters differ significantly manganese, and Skz is the z component of the operator from those determined with the use of only the param- of the kth spin moment for manganese. 0 The thermodynamic potential of the system in a eter B4 for DyMn2Ge2. magnetic field along the tetragonal axis per formula 5. The parameters of MnÐMn and HoÐMn exchange λ λ unit is determined by the expression interactions ( 22 and 12, respectively) were obtained using the experimentally determined field of the transi- Φ 1 ()Ho = ÐkBTZln + ---Mz Hz tion from the antiferromagnetic phase to the ferromag- 2 netic phase in a magnetic field directed along the tetrag- 2 2 (3) []() onal axis: HAF → F = 91 T at T = 5 K [3]. At low temper- sinh 2S + 1 xk/2 1 Ð kBT ∑ ln------() + --- ∑ mk Hmk. atures, from the expansion of the thermodynamic sinh xk/2 2 k = 1 k = 1 potential (3) (see also [5]), we obtain λ λ The partition function Z for the holmium magnetic HAF → F = Ð 12M Ð 22m. (4) moment was determined using numerical diagonaliza- µ 〈 〉 The magnetic moment M for holmium, according to tion of Hamiltonian (2), Mz = BgJ Jz , and xk = µ our experimental data, is taken to be equal to 8.3 B µ ()Mn ()Mn ϕ BgHk /kBT, where Hk = Hcos k + Hmk, Hmk = (Fig. 2), and the magnetic moment m for manganese in () the antiferromagnetic phase is assumed to be 2.3µ . λ n m cos()ϕ Ð ϕ + λ M cosϕ , ϕ is the B ∑nkk= , ± 1 22 n n k 12 z k k The latter value was found in [1] by averaging the mag- polar angle of the kth magnetic moment of manganese netic moments over a series of RMn2Ge2 compounds ()n λ (in our case, ϕ = 0 or π), and λ are the parameters of (R = Tb, Ho, Er, and Tm). The exchange parameters 12 k 22 λ the MnÐMn exchange interaction between atoms of the and 22 were determined from their linear combination (4) using two methods. In the first method, the kth and nth layers. The second and fourth terms in λ expression (3) are conventional correcting terms in the exchange parameter 12 for HoMn2Ge2 was calculated λ molecular field theory. from the parameter 12 for DyMn2Ge2 [5] with the use λ Thus, we calculated the magnetization curves M(H) of the relevant Landé factors; then, the parameter 22 was obtained from expression (4). In the second and the HÐT phase diagram for the magnetic field of the λ first-order phase transition from the antiferromagnetic method, the exchange parameter 22 for HoMn2Ge2 was λ state to the ferromagnetic state in the manganese sub- calculated from the parameter 22 for DyMn2Ge2 [5] system in a magnetic field directed along the tetragonal with the use of formula (1) (for the lattice parameter axis. The lattice parameters of the HoMn Ge com- aDy, we used our data obtained earlier in [5]); then, the 2 2 λ pound were determined by comparing the calculated parameter 12 was determined from expression (4). It is and experimental data. worth noting that both methods lead to close values of λ ± µ 4. The experimental magnetization curves for the the exchange parameters: 12 = Ð4.5 0.5T/ B and λ ± µ HoMn2Ge2 compound at three temperatures (solid lines 22 = Ð23 2T/ B. A similar calculation of the in Fig. 2) indicate that the crystal field substantially exchange parameters for HoMn2Ge2 from the parame- affects the magnetic moment of the holmium sub- ters for GdMn2Ge2, which were determined in [8], system. In this case, the holmium subsystem resides in gives a larger spread of the results obtained. This can be the paramagnetic state, because, in the temperature explained by the fact that, compared to dysprosium,

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2094 GUANGHUA et al. gadolinium is more distant from holmium in the row of REFERENCES rare-earth elements in the periodic table. 1. G. Venturini, B. Malaman, and E. Ressouche, J. Alloys The temperature dependence of the critical ﬁeld Compd. 240 (1), 139 (1996). H → of the transition from the antiferromagnetic AF F 2. A. Szutula and J. Leciejewicz, in Handbook on the Phys- phase to the ferromagnetic phase was calculated using ics and Chemistry of Rare Earths, Ed. by K. A. Gschne- the thermodynamic potential (3), which accounts for idner, Jr. and L. Eyring (Elsevier, Amsterdam, 1989), the temperature and ﬁeld dependences of all the charac- Vol. 12, p. 133. teristics of the system under consideration. Moreover, we took into account the temperature dependence of the 3. A. Kirste, R. Z. Levitin, M. von Ortenberg, et al., Fiz. λ Tverd. Tela (St. Petersburg) 43 (9), 1661 (2001) [Phys. MnÐMn interlayer exchange parameter 22 according Solid State 43, 1731 (2001)]. to formula (1) with the use of the measured temperature dependence of the lattice parameter a(T) (Fig. 1b). The 4. A. S. Lagutin, J. Vanacken, N. Harrison, and F. Herlach, Rev. Sci. Instrum. 66 (8), 4267 (1995). parameter λ0 characterizing MnÐMn exchange inter- 22 5. Guo Guanghua, M. V. Eremin, A. Kirste, et al., Zh. Éksp. actions in a layer was found from the temperature TN = Teor. Fiz. 120 (10), 910 (2001) [JETP 93, 796 (2001)]. λ λ0 × 460 K and the known parameter 22: 22 = 2.1 6. C. Kittel, Phys. Rev. 120 (2), 335 (1960). 3 µ 10 T/ B. The calculated phase diagram is depicted in 7. G. Venturini, B. Malaman, K. Tomala, et al., Phys. Rev. the inset to Fig. 2. B 46 (1), 207 (1992). 8. A. Yu. Sokolov, Guo Guanghua, S. A. Granovskiœ, et al., ACKNOWLEDGMENTS Zh. Éksp. Teor. Fiz. 116 (4), 1346 (1999) [JETP 89, 723 (1999)]. This work was supported in part by the Russian Foundation for Basic Research, project no. 99-02- 17358. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2095Ð2097. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2001Ð2003. Original Russian Text Copyright © 2002 by Granovskiœ, Sato, Aoki, Yurasov. MAGNETISM AND FERROELECTRICITY

Tunneling Thermopower in Magnetic Granular Alloys A. Granovskiœ*, H. Sato**, Y. Aoki**, and A. Yurasov* * Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: [email protected] ** Department of Physics, Tokyo Metropolitan University, Tokyo, 192-0397 Japan Received December 25, 2001

Abstract—The nature of the field dependence of thermopower in the CoÐAlÐO and FeÐAlÐO magnetic granular alloys with tunneling conduction is shown to be related to tunneling thermopower. The tunneling thermopower is small and depends approximately linearly on temperature and squared magnetization, and its field dependence is described by a relation of the type S(H)/T = a + bρ(0)/ρ(H), where ρ is the alloy electrical resistivity and the parameters a and b are field-independent. © 2002 MAIK “Nauka/Interperiodica”.

Recently, Sato et al. [1, 2] found that thermopower Kohler rule [5], we can present the thermopower of S in the CoÐAlÐO and FeÐAlÐO magnetic granular such a conductor in the form alloys of the metalÐinsulator type with tunneling magnetoresistance (TMR) is negative and considerably SmetWmet + StunWtun smaller than that in bulk Co and Fe and that its ﬁeld S = ------met tun . (2) dependence is described by the relation W + W

SH() ρ()0 Kohler’s rule is a consequence of a temperature gradi------= ab+ ρ------()-, (1) ent distribution in the inhomogeneous system. If all the T H elements of the conductor are metallic and obey the ρ WiedemannÐFranz law, Eq. (2) reduces to the well- where a and b are ﬁeld-independent and is the electri- known NordheimÐGorter rule [5]. In the case under cal resistivity. A relation of the type of Eq. (1), found consideration, Eq. (2) is a rough approximation, earlier to hold for metallic multilayers and granular because it is assumed at the very beginning that all the alloys with giant magnetoresistance (see, e.g., [3]), has ρ elements of the electrical circuit are identical. In a more been shown to be due to the fact that both (H) and consistent analysis, one should take into account the S(H) are governed in these metallic systems by spin- difference in the height of the tunneling barriers and dependent scattering in the grain volume and from perform an averaging over the optimum electron trans- interfaces. In granular metalÐinsulator alloys, however, port trajectories, as is done, for instance, in the theory the magnetoresistance is connected with electron tun- of magnetoresistance [6]. However, this would be neling transport through the insulating spacer between beyond the scope of the present communication, which adjacent grains; therefore, Eq. (1) requires adequate tun substantiation. Moreover, one can readily show that the deals primarily with calculation of S . effective-medium theory developed for metalÐinsulator Because the thermal resistance of the insulating composites fails to account for this relation [4]. spacer W tun is of the order of that of the insulator and is met We present here a calculation of tunneling ther- certainly larger than W , we have tun mopower S in granular metalÐinsulator alloys and met tun tun metW tun met γ show that the thermopower in these systems near the SS≈ + S ------= S + S ------, (3) percolation threshold is primarily of tunneling nature Wtun γ met and can be approximated by Eq. (1). As far as we know, tunneling thermopower has not yet been discussed in where γtun and γmet are the thermal conductivities of the the literature. insulator and the metal, respectively. Near the percolation threshold, it can be assumed The tunneling process does not involve a change in that the conducting channel in the metalÐinsulator sys- the electron energy; therefore, S tun can be calculated tem is made up of series-connected elements of two using Mott’s relation: types, more speciﬁcally, of metallic grains and tunneling barriers. Let each of these elements be character- 2 2 met tun tun Ðπ k 1 ∂GE() ized by the values of their thermopower (S , S ) and S = ------B T ------, (4) met tun 3 e GE()∂E thermal resistance (W , W ). Following the obvious EF

2096 GRANOVSKIŒ et al.

1 ρ()0 where G(E) is the tunneling conductance. In terms of According to Eq. (11), m2 = ------1Ð ; Eq. (9) can 2ρ() the TMR theory, the tunneling conductance can be writ- P H ten in the form [7] now be recast in the form of Eq. (1), where

Ð2 κ() () []2()2 2 E C/kBT 1 ∂P GE = G0 1 + P E m e , (5) 2 ------2 2 1/2 ∂ Ðπ k T m P E E where G = const, C = const, m is the relative magneti- a = ------B -----0 ------eff Ð,------F (12) 0 3 e T 2 2 2 zation, 2ប κ 1 + P m

D↑()E Ð D↓()E PE()= ------, (6) D↑()E + D↓()E 1 ∂P 2 ------π2 2 ∂ Ð kB P E EF Dσ(E) (σ = ↑, ↓) is the local density of states at the tun- b = ------2 2 . (13) neling junction interfaces for the corresponding energy 3 e 1 + P m E,

κ ()ប2 Equations (9), (12), and (13) sum up the main result of = 2meff VEÐ / , (7) our study. Because, in all the systems studied, the TMR does not exceed 9% [1, 3] and the spin polarization is meff is the effective mass of a tunneling electron, and V is the height of the barrier. In accordance with Eq. (5), P ~ 0.3 [7, 9], one can limit oneself without any loss of the temperature dependence of the conductance is generality to a linear approximation in P. Furthermore, the second term in brackets in Eq. (9) is small compared ()1/2 to the first one and should be taken into account only in ∼ Ð T0/T 8kc κ ≥ Ge , T 0 = ------, (8) the case of paramagnetic alloys. Indeed, because kF kB (where kF is the Fermi wave vector [7]) and near the ≈ which is in good agreement with experiment [6]. Equa- percolation threshold T0 10 K [6], the second term in tion (5) also quite well describes the dependence of Eq. (9) is certainly smaller than 1/4EF. However, the TMR on m2 and the weak temperature dependence of quantity the TMR. We note that Eqs. (5) and (8) are valid within a fairly broad but limited temperature interval T < ∂P 1 1 ∂D↑ 1 ∂D↓ min ------= ------Ð ------(14) T < T [6] and although these equations were derived ∂ ∂ ∂ max E 2 D↑ E D↓ E EF using the approximation of Sheng et al. [8] for conductance averaging over intergrain distances, relations of may be considerably larger than P/EF for transition the same kind can be obtained based on general consid- metals if the spÐd hybridization is included. As we shall erations concerning the character of grain distribution see below, a comparison with experiment shows that with respect to size [6]. For high temperatures, the T1/2 the neglect of the second term in Eq. (9) is indeed jus- law in Eq. (8) fails and is replaced by a relation describ- tified, particularly in the temperature interval 77Ð300 K. ing an activated behavior. We shall restrict ourselves to In this case, Eqs. (9), (12), and (13) become simplified: the region where Eq. (8) holds, because the high-temperature case is not realized in ferromagnetic systems 2 2 tun Ðπ k ∂P 2 even if the Curie temperature is high. S = ------B T 2P------m , (15) 3 e ∂E After substituting Eq. (5) into Mott’s relation (4) EF and performing some straightforward calculations, we 2 2 obtain Ðπ kB 2 ∂P ba==Ð --------- ∂------. (16) 3 e P E EF ∂P 2 2P------m 2 2 ∂ 1/2 tun Ðπ k E E m T For the FeÐAlÐO system, these relations show good S = ------B T ------F - +.------eff -----0 (9) 2 2 2 2 agreement with the experimental data b = (0.08 ± 3 e 1 + P m 2ប κ T exp µ 2 ± µ 2 0.01) V/K and aexp = Ð(0.09 0.01) V/K [10]. ≈ Indeed, bexp Ðaexp and both parameters are practically It follows from Eq. (5) that temperature-independent. Moreover, it is commonly accepted that tunneling is dominated by sp-like elec- ∆ρ GÐ1()0 Ð GÐ1()H ρ()H P2m2 ------= ------= 1 Ð------= ------, (10) trons and that their relatively large polarization (P ≈ ρ GÐ1()0 ρ()0 1 + P2m2 0.3) is related to spÐd hybridization. For the free-elec- − ∆ 1/2 ∆ tron model D↑(↓) = A↑(↓)(E + ), where is the sub- ρ()0 2 2 ------1= + P m . (11) band spin splitting, or for the case of a semielliptical ρ()H curve of density of states with a nearly half-filled sub-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 TUNNELING THERMOPOWER IN MAGNETIC GRANULAR ALLOYS 2097 band with spin σ = ↑, which corresponds to Fe, we have experimental data on tunneling thermopower for the ∂P 1 ∂P ∂P ------< 0. Because P > 0, we have ------< 0 and b > 0, determination of ------of the electrons taking part in tun- ∂E P∂E ∂E which is in agreement with experiment. We can reason- neling appears to be a promising approach. 1 ∂D↑ It should be pointed out that, because tunneling ther- ably assume, for a rough estimate, that ------Ð D↑ ∂E mopower is associated neither with scattering processes, including interband sÐd transitions, nor with 1 ∂D↓ 1 ------= Ð---- , where ω is the band half-width or the phonon or magnon drag, this thermopower is substan- D↓ ∂E ω tially simpler and, in this sense, more informative than Fermi energy. Then, for ω = 1 eV [7], we obtain from conventional thermopower. In addition, tunneling ther- µ 2 mopower should be taken into account when describing Eq. (16) that bcalc = 0.08 V/K . While S for the CoÐAlÐO system is of the same thermoelectric phenomena in composites in terms of order of magnitude, here, in contrast to FeÐAlÐO, we the percolation theory [4]. Alongside elastic tunneling have b < 0 and a > 0, with |b | ≠ |a |, with the processes, phonon- or magnon-assisted tunneling may exp exp exp exp take place. Since Mott’s relation is invalid for descrip- parameter a being temperature-dependent [10]. This tion of the corresponding contribution to thermopower, ∂P this problem requires additional analysis. can be readily understood if we assume that P∂------> 0 E Thus, the nature of the field dependence of the ther- and take into account the second term in Eq. (3). That mopower in CoÐAlÐO and FeÐAlÐO magnetic granular the second term in Eq. (3) must be retained follows alloys is related to the tunneling thermopower. The tun- clearly from the fact that the room-temperature ther- neling thermopower is small and depends approxi- mopower in Co is Ð30 µV/K, which is substantially mately linearly on temperature, and its field depen- γ tun 1 dence is described by Eq. (1). larger in magnitude than S tun; because ------≥ --- , one γ met 5 cannot neglect the metallic contribution to the ther- REFERENCES mopower in the system in question when comparing 1. H. Sato, Y. Kabayashi, K. Hashimoto, et al., J. Magn. theory with experiment. We believe that in order to iso- Soc. Jpn. 23, 73 (1999). tun late S from experimental data, one should make use 2. H. Sato, Y. Kabayashi, K. Hashimoto, et al., J. Phys. Soc. of the correlation S tun ~ m2, which follows from Jpn. 67, 2193 (1998). Eq. (15). The situation with the sign of the quantity 3. J. Shi, K. Pettet, E. Kita, et al., Phys. Rev. B 54, 15273 ∂P P------is more complex. As already mentioned, this (1996). ∂E 4. O. Levy and D. Bergman, Mater. Res. Soc. Symp. Proc. quantity must be negative for nearly free electrons 195, 206 (1990).  ∂P  5. R. O. Barnard, Termoelectricity in Metals and Alloys -P > 0, ------< 0- . If we take, for an estimate, the total  ∂E  (Taylor and Francis, London, 1972), p. 259. 6. E. Z. Meœlikhov, Zh. Éksp. Teor. Fiz. 115, 1484 (1999) ∂P density of states for Co, we will obtain P < 0, ------< 0, [JETP 88, 819 (1999)]. ∂E 7. J. Inoue and S. Maekawa, Phys. Rev. B 53, R11927 ∂P (1996). and, in accordance with experiment, P------> 0. It is usu- ∂E 8. P. Sheng, B. Abeles, and Y. Arie, Phys. Rev. Lett. 31, 44 ally accepted, however, that d-like electrons are not (1973). involved in tunneling and, therefore, P > 0 for all sys- 9. R. Meservey and P. M. Tedrow, Phys. Rep. 283, 173 ∂P (1994). tems, while for spÐd hybridized states, the sign of ∂------10. H. Sato, Y. Aoki, S. Mitany, H. Fujimory, and K. Taka- E nashi, private communication. depends substantially on the subband filling. Moreover, the local density of states near interfaces may differ considerably from that of the grain material. Using Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

MAGNETISM AND FERROELECTRICITY

Unidirectional Anisotropy in a FerromagnetÐAntiferromagnet System A. I. Morosov and A. S. Sigov Moscow State Institute of Radioengineering, Electronics, and Automation, pr. Vernadskogo 78, Moscow, 119454 Russia e-mail: [email protected] Received December 25, 2001

Abstract—The unidirectional anisotropy arising in a ferromagnetic film on an antiferromagnetic substrate due to the proximity effect is investigated. Consideration is given to the smooth and rough film–substrate interfaces. The conditions of the formation of a domain wall in the film and the passage of the wall into the substrate in the course of magnetization reversal are determined. The parameters of the static spin vortices formed in the vicinity of the rough interface are studied as functions of the magnetic field strength. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION mean-field approximation, the film–substrate interac- Unidirectional anisotropy (the exchange bias) man- tion can be described by the relationship ifests itself in a shift of the magnetization curve of a fer- J faf, S f Saf ()θ θ σ romagnetic film deposited on the surface of an antifer- W faf, = Ð------2 -∫ cos f Ð af d , (1) romagnet, for example, NiFe/FeMn, NiFe/CoO, b Ni/NiO, Fe/FeF , or Fe/Cr. The effect of unidirectional 2 where Sf and Saf are the mean spins of the film and the anisotropy has been investigated in many works (see, substrate, respectively; Jf, af is the exchange integral for example, review [1]). (the integration is performed over the interface); and b In this work, we examined the case when, in the is a lattice constant that is nearly identical for both antiferromagnet, the atomic plane aligned parallel to materials. the ferromagnetÐantiferromagnet interface involves Irrespective of the sign of the integral Jf, af, interac- uncompensated spins. The origin of unidirectional tion (1) leads to a shift of the magnetization curve of the anisotropy in the case of a compensated surface of anti- film from the position symmetric with respect to the ferromagnets was explained by Koon [2]. sign of the external field. In order to estimate this shift, Earlier [3], we considered the limiting case when it is necessary to determine the equilibrium magnetiza- the exchange stiffness of the film is appreciably larger tion curve from the condition for the minimum of the than that of the substrate. In the present work, we ana- total energy W of the system; that is, lyzed the situation when the exchange stiffness of the 3 ()f f f film is substantially smaller than that of the substrate. WW= faf, + ∫d r wex ++wan wZ The smooth film–substrate interface is considered in (2) 3 ()af af Section 2. The “thickness–roughness” magnetic phase + ∫d r wex + wan . diagram in the absence of an external magnetic field is presented in Section 3. The magnetization reversal in Here, the first and second integrals are taken over the single-domain and polydomain phases is treated in Sec- volumes of the film and the substrate, respectively; tions 4 and 5, respectively. 2()∇θ 2 wex = AiSi i (3) 2. AN ATOMICALLY SMOOTH INTERFACE is the excess exchange energy due to the order parameter inhomogeneity (where A is the exchange stiffness); For an atomically smooth interface, the exchange i θ interaction between spins of the upper atomic layer of wan = ÐK cos4 i (4) the substrate and the lower atomic layer of the film has the same sign over the entire interface. It is assumed is the anisotropy energy in the layer plane; and that the magnetization and antiferromagnetic vectors w f = ÐMB cos()θ Ð ϕ (5) lie in the plane parallel to the interface and are charac- Z 0 f θ terized by the angle i (i = f and af) formed by the order is the Zeeman energy (where M is the film magnetiza- parameter with the preferred axis in this plane. In the tion and B0 is the induction of an external magnetic field

UNIDIRECTIONAL ANISOTROPY IN A FERROMAGNETÐANTIFERROMAGNET SYSTEM 2099 which is aligned parallel to the film plane and forms the (a) angle ϕ with the preferred axis). In this case, we disregard the sublattice angularity f (by virtue of the smallness of B0) and the corresponding Zeeman energy of the antiferromagnet. For definiteness, we assume that J > 0 and θ = 0 deep into the f, af af af substrate. As a result, the magnetization curve at ϕ = 0 is shifted toward the negative-field range. (b) Malozemoff [4] and Mauri et al. [5] obtained an estimate of the corresponding shift in the form f

()A K 1/2S2 B0 ∼ ------af af af , (6) af Ma af where a is the film thickness. This result was obtained from simple considerations: the magnetization reversal Fig. 1. Domain walls (gray regions) in (a) the film and (b) of the film brings about the formation of a domain wall the substrate in the vicinity of the interface. Arrows indicate the directions of the magnetization and antiferromagnetic 1/2 2 vectors. with the surface energy (Aaf Kaf) Saf in the substrate, and the Zeeman energy gain should compensate for the expenditure of energy to form the wall. gain. Taking into account that the domain wall can arise ∆ 0 in the film only at f(B0) < a, we obtain This estimate for Baf is valid for the large exchange stiffness of the film when the energy (A K )1/2 S2 of the ()A K 1/2S2 /Ma,ata ӷ ∆ ()0 f f f  f f f f domain wall in the film is higher than that of the domain B ≈  2 wall in the substrate. Otherwise, the above estimate f A f S f Ӷ ∆ () ------,ata f 0 , turns out to be erroneous.  Ma2 (8) Before preceding to the case of a lower energy of the 2 ≈ A f S f domain wall in the ferromagnet, it should be noted that, ------(), ∆ (). ∆ Mamin a f 0 in a magnetic field, the characteristic width f (B0) of ° the 180 domain wall is governed not by the competi- As the magnetic field increases to B ≈ B , the tion between the exchange interaction and anisotropy 0 af domain wall becomes thinner; i.e., it is “pressed” to the energies but by the competition between the exchange interface. The value of Baf can be found from the con- interaction energy and the sum of the anisotropy and dition for the equality of the domain wall energies in the Zeeman energies [6]; that is, ferromagnet and the antiferromagnet; that is,

1/2 A S2 4 ∆ () π f f Aaf Kaf Saf f B0 = ------. (7) B ≈ ------. (9) 2 af 2 K f S f + 2MB0 A f S f M ≈ It is easy to see that the Zeeman energy in the vicin- At B0 Baf, the domain wall begins to move into the ity of the Curie point of the ferromagnet makes the substrate. At B ӷ B , the film becomes homoge- dominant contribution (M ∝ S ). 0 af f neously magnetized and a 180° domain wall arises in Let the external magnetic field be applied antiparal- the antiferromagnet in the vicinity of the film boundary lel to the direction of the film magnetization in the (Fig. 1b). Certainly, this state is metastable, because, absence of the field. Now, we determine the magnetic for the domain wall, it is likely to disappear after pass- ° ing across the whole width of the substrate. induction Bf at which the 180 domain wall aligned parallel to the film–substrate interface is formed in the 0 For very thin films (when B > B ), the 180° film in the vicinity of the interface (Fig. 1a) under the f af domain wall is formed immediately in the antiferro- Ӷ 0 assumption that Bf Baf , i.e., that the substrate magnetic substrate. The dependences of the magnetic remains virtually homogeneous. For this purpose, we fields Bf and Baf on the film thickness are plotted in equate the domain wall energy and the Zeeman energy Fig. 2.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2100 MOROSOV, SIGOV

0 1.5 B DW in AF Baf 1.0 0.5 Baf

M 0 DW in F No domain wall –0.5 –1.0 Bf –1.5 ∆–1(0) a–1 –10 –8 –6 –4 –2 0 2 4 B0

Fig. 3. Magnetization curve in the case of a smooth film– Fig. 2. “Thickness–field” phase diagram in the case of a substrate interface. The magnetic field strength is given in smooth interface. The solid line shows the dependence of arbitrary units, and the magnetic moment is expressed in the bias field on the film thickness. terms of the saturation moment.

80 70 60 50 f

δ 40 30 20 10

07010 20 30 40 50 60 z

Fig. 5. Dependence of the thickness of the “unusual” Fig. 4. A transverse domain wall due to frustration in the domain wall on the distance to the interface. The distances ﬁlm. are expressed in terms of a lattice constant (a = 64).

Thus, the magnetization curve is shifted by min(Bf, spins of the film and the substrate, the presence of steps B0 ) from the symmetric position toward the negative- at the interface leads to frustrations. In our previous af work [7], the phase diagram of the magnetic film on the field range. The corresponding magnetization curve is antiferromagnetic substrate with a large exchange stiff- depicted in Fig. 3. The above analysis can easily be ness was considered in the framework of the continuum extended to the case of magnetic fields applied at arbitrary angles ϕ to the film magnetization. model. In the absence of a magnetic field, an individual step ° 3. PHASE DIAGRAM IN THE CASE is responsible for the formation of a 180 domain wall OF A ROUGH INTERFACE of a new type—a domain wall induced by the frustration. This wall coincides with the step edge and pene- Apparently, the real interface is not perfectly trates throughout the film; i.e., the wall plane is perpen- smooth and involves atomic steps that change the sub- dicular to the film surface (Fig. 4). On different sides of strate thickness by one atomic layer. On different sides the step, spins of the ferromagnet are parallel to spins of the step, spins of the upper atomic layer of the anti- δ f ferromagnet have opposite orientations. As a result, of the upper layer of the antiferromagnet. The width 0 regardless of the sign of the exchange integral Jf, af for of this domain wall in the vicinity of the film–substrate

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 UNIDIRECTIONAL ANISOTROPY IN A FERROMAGNETÐANTIFERROMAGNET SYSTEM 2101 interface depends on the parameter η deﬁned by the formula 60 η = J faf, Saf a/J f S f b. (10) 50 If η ӷ 1, the width δ f is given by the expression 0 40 δ f ≈ ()

0 bJfaf, Saf + J f S f /J faf, Saf . (11) z 30 0.2 An increase in the distance from the interface results in 0.5 an almost proportional increase in the domain wall 20 0.6 width (Fig. 5). The characteristic wall width is esti- 0.3 0.7 mated as δ f ≈ a. For a typical film thickness a ~10 Å, 0.4 0.8 this width is considerably less than the width of conven- 10 0.9 tional domain walls [see expression (7)]. At η Ӷ 1, the 0.1 increase in the width of the domain wall can be ignored 0 and the wall width can be represented as –30 –20–10 0 10 20 30 x δ f ≈ a/ η. (12) Fig. 6. Static spin vortex in the film in the vicinity of the The above relationships hold at a < ∆ (0). At a > film–substrate interface (z = 0) at a ӷ R. Numbers near the f θ θ π ∆ (0), the domain wall width increases to ∆ (0) and then isolines of f are the values of f in units of . Steps are f f located at the points x = ±8 and z = 0. The distances are remains unchanged. expressed in terms of a lattice constant. When the distance R between the step edges is larger δ f than max(0 , a), the film breaks down into domains where with opposite magnetization directions. Distortions of the order parameter in the substrate appear to be insig- J S2 σ C ≡ Cσ ≈ ------f f j (14) nificant. j j Rb ӷ ӷ δ f For a relatively thick film (a R 0 ), the film in ∆ 1/2 2 σ ∆ at f(B0) > R and Cj = (AfKf) S f j at f(B0) < R. If the bulk is in a single-domain state and static spin vor- π tices arise in the film in the vicinity of the interface σ = σ , the angle ψ = --- corresponds to the minimum (Fig. 6). If the exchange stiffness of the film is relatively 1 2 2 large, the vortices are formed in the substrate. Each vor- energy of the system in a zero magnetic field. tex is bounded by the step edges and diverges in the Ӷ δ f ∆ At R 0 , the system is characterized by weak dis- direction away from the edges. At f(B0) > R, the vortex size in the direction perpendicular to the interface is tortions of the order parameter. Below, we will consider ∆ the behavior of each phase in an external magnetic equal to R in order of magnitude. At f(B0) < R, a domain wall aligned parallel to the interface is formed field. in the film and the transverse size of the vortex is equal ∆ to f(B0). 4. THE SINGLE-DOMAIN PHASE As was noted above, the steps separate the interface IN A MAGNETIC FIELD σ into regions of two types with the total surface areas 1 The application of the magnetic field results in rota- σ θ θ and 2. The interfacial energy is minimum at f = af tion of the film magnetization and in the formation of θ π θ in regions of the former type and at f = Ð af in the domain wall first in the film and then in the sub- regions of the latter type. Now, we assume that the mag- strate. The magnetization reversal and unidirectional netization of the ferromagnetic film makes an angle ψ anisotropy in the single-domain phase are similar to with the antiferromagnetic order parameter deep into those observed in the case of the smooth interface. The θ the substrate. In this case, the angle f varies from zero differences are as follows. to ψ in vertices in the former regions and from ψ to π in (i) In the absence of the field, the film magnetization vertices in the latter regions. is perpendicular rather than parallel to the antiferro- As in the case of a large exchange stiffness of the magnetic order parameter. ∆ film [1, 8], it can be shown, by analogy with the “mag- (ii) In the field range corresponding to f(B0) < R, netic proximity” model proposed by Slonczewski [9], the vortex shape changes: the vortex size in the direc- that the energy of the system can be represented by the tion perpendicular to the interface becomes equal to ∆ relationship f(B0) in order of magnitude and decreases with an increase in the field. The vortices take an elongated ψ2 ()πψ2 WC= 1 + C2 Ð , (13) shape (Fig. 7a) and are then displaced into the substrate

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2102 MOROSOV, SIGOV

(a) fore, the shift of the magnetization curve can be determined as 1/2 2 f ()A K S ∆n B* ≈ ------f f -f ------, (16) MR n where ∆n is the excess concentration of particular walls af and n is the total wall concentration. When the field is applied parallel to the domain magnetization, the magnetization curve is not shifted. (b) Let us now analyze the magnetization reversal. If ∆ the domain size is larger than af, the surface can be f treated as virtually smooth; i.e., we can consider the magnetization reversal of an individual domain in the same manner as for the smooth interface. In this case, the region in the vicinity of the domain walls makes an insignificant contribution. The characteristic magneti- af 0 zation field is of the order of min(Bf, Baf ). Ӷ ∆ Fig. 7. Vortices in the film–substrate system in the magnetic In the case when R af and the magnetic field is ˜ ˜ applied normally to the domain magnetization, the fields (a) B0 < Baf and (b) B0 > Baf . The isolines of the ˜ order parameter are shown. magnetization reversal of domains in the field B0 < Baf occurs in much the same manner as in the case of the ˜ by the magnetic field (Fig. 7b). This field is defined by smooth interface. At Bf < Baf , the magnetization vector the expression in the domain in the magnetic field B0 ~ Bf begins to rotate and a spin vortex arises in the film in the vicinity 2 4 of the interface. The transverse size of this vortex A S B˜ ≈ ------af af , (15) ˜ af 2 [], ∆ 2 decreases with an increase in the field. At B0 > Baf , a set MAf S f min()R af of static vortices (similar to that in the single-domain phase) appears in the antiferromagnetic substrate. ∆ π 1/2 where af = (Aaf /Kaf) . Since the film magnetization in a strong external field is orthogonal to the order parameter deep into the antifer- ˜ However, even at B0 > Baf , in the region of radius romagnetic substrate, no domain walls are formed. At ∆ ˜ f(B0) which is adjacent to the step edge, the distortions Bf > Baf , the vortices appear immediately in the sub- of the order parameter occur in the film rather than in strate. The characteristic magnetization field is equal to the substrate (Fig. 7b). Qualitatively, we can make the min(B , B˜ ). inference that the magnetic field displaces the distor- f af tions from the region in which the Zeeman energy The magnetic field applied parallel to the magnetization in ferromagnetic domains leads to a magnetiza- exceeds the difference between the energies of inhomo- tion similar to that considered above, except that the geneities in the antiferromagnet and the ferromagnet. 90° domain wall is formed in the substrate in the magnetic field defined as 5. THE POLYDOMAIN PHASE B ,atB < B ≈ af f af IN A MAGNETIC FIELD Bdw  (17) (), 0 ∆ > min B f Baf af /R ,atB f Baf . In the polydomain phase, the unidirectional anisotropy can be observed in the case when the magnetic The reason for the formation of this wall is the same as in the case when the film magnetization in the single- field is applied perpendicularly to the magnetization domain phase in the absence of the magnetic field is direction in domains and the domain walls with a par- orthogonal to the order parameter deep into the sub- ticular sense of rotation of magnetization are predomi- strate: the formation of the wall results in a decrease in nant [3]. These 180° domain walls disappear with an the energy of the system of vortices. increase in the magnetic field strength for one of the Actually, in the absence of the domain wall, no vor- two directions of the magnetic field and transform into tices are formed in domains whose magnetization is 360° domain walls for the reverse direction of the field. parallel to the external field. At the same time, a 180° ≈ The characteristic change in the wall energy in the vortex arises in the magnetic field B0 min(Bf, 0 ∆ saturation field is of the order of the energy. There- Baf af/R) in domains with initial antiparallel orienta-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 UNIDIRECTIONAL ANISOTROPY IN A FERROMAGNETÐANTIFERROMAGNET SYSTEM 2103 tions of the magnetization and the field. In the presence field increases, the wall becomes sharper and then of the domain wall, 90° vortices appear in domains of moves into the antiferromagnetic substrate. both types. Note that, in different-type domains, the (4) Similar behavior is observed for the static spin magnetization (or the antiferromagnetic order parame- vortices arising in polydomain films during their mag- ter) has an opposite sense of rotation. Since the vortex netization reversal. energy is proportional to the rotation angle squared, the energy of vortices decreases in the presence of the domain wall. The difference between the energies of ACKNOWLEDGMENTS the states with a domain wall and without it should This work was supported by the Russian Foundation exceed the energy of formation of the domain wall. for Basic Research, project no. 00-02-17162.

6. CONCLUSIONS REFERENCES Thus, the main inferences made in the present work 1. J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192 can be summarized as follows. (2), 203 (1999). (1) The unidirectional anisotropy and the character- 2. N. C. Koon, Phys. Rev. Lett. 78 (25), 4865 (1997). istic magnetization field of the ferromagnetic film on 3. V. D. Levchenko, A. I. Morozov, and A. S. Sigov, Fiz. the antiferromagnetic substrate depend on the film Tverd. Tela (St. Petersburg) 44 (1), 128 (2002) [Phys. thickness and the degree of roughness of the film–sub- Solid State 44, 133 (2002)]. strate interface. 4. A. P. Malozemoff, Phys. Rev. B 35 (7), 3679 (1987). (2) The presence of atomic steps (changing the anti- 5. D. Mauri, H. C. Siegmann, P. S. Bagus, and E. Kag, ferromagnet thickness by one monoatomic layer) in the J. Appl. Phys. 62 (7), 3047 (1987). interface leads to frustrations in the ferromagnetÐanti- ferromagnet system. In thin films, the frustrations are 6. S. Mangin, G. Marchal, and B. Barbara, Phys. Rev. Lett. 82 (21), 4336 (1999). responsible for the formation of domain walls of the new type that separate the film surface into domains. In 7. V. D. Levchenko, A. I. Morozov, A. S. Sigov, and Yu. S. Sigov, Zh. Éksp. Teor. Fiz. 114 (11), 1817 (1998) thicker films, static spin vortices arise in the vicinity of [JETP 87, 985 (1998)]. the interface, whereas the magnetization in the bulk of the film remains unchanged. 8. V. D. Levchenko, A. I. Morosov, and A. S. Sigov, Pis’ma Zh. Éksp. Teor. Fiz. 71 (9), 544 (2000) [JETP Lett. 71, (3) In single-domain films, the formation of domain 373 (2000)]. walls in the course of magnetization reversal depends 9. J. C. Slonczewski, J. Magn. Magn. Mater. 150 (1), 13 on the film thickness. In thin films, the domain wall is (1995). immediately formed in the antiferromagnetic substrate. In thick films, the domain wall initially arises in the film in the vicinity of the interface with the substrate. As the Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2104Ð2106. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2010Ð2012. Original Russian Text Copyright © 2002 by Platonov, Tatsenko, Plis, Popov, Zvezdin, Barbara.

MAGNETISM AND FERROELECTRICITY

Magnetic Susceptibility of the V15 Nanocluster in Megagauss Magnetic Fields V. V. Platonov*, O. M. Tatsenko*, V. I. Plis**, A. I. Popov**, A. K. Zvezdin***, and B. Barbara**** * Russian Federal Nuclear Center, Institute of Experimental Physics, Sarov, Nizhni Novgorod oblast, 607188 Russia ** Moscow State Institute of Electronic Engineering Technical University, Zelenograd, Moscow oblast, 103498 Russia e-mail: [email protected] *** Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 117942 Russia **** Laboratoire de Magn. Lois Néel, CNRS, Grenoble, 38042 France Received January 8, 2002

Abstract—The low-temperature behavior of the magnetic susceptibility of the V15 low-spin cluster in ultrastrong magnetic fields of up to 550 T was studied. Ultrastrong magnetic fields were generated by an MK-1 magnetic explosion generator. Anomalies in the susceptibility were found to exist in fields B1 = 200 T and B2 = 350 T. It is concluded that these anomalies indicate the initial phase of a field-induced transformation of the cluster magnetic structure from quasi-ferrimagnetic to ferromagnetic. This transformation occurs by discrete quantum jumps at low temperatures. The experimental data are compared with theory. © 2002 MAIK “Nauka/Interpe- riodica”.

1. INTRODUCTION 2. EXPERIMENT Considerable interest is presently focused on mag- We measured the magnetic susceptibility of a poly- crystal formed of polyoxyvanadate V molecules netic nanoclusters containing d or f ions (Mn12Ac, Fe6, 15 IV Fe8, Fe10, Fe17, V12, etc.) [1Ð11], most of which are (K6[AsV15 6O42(H2O)] á 8H2O). These crystals possess giant-spin organometallic molecules. These clusters are trigonal symmetry (space group R3 with a = 14.029 Å, mesoscopic objects whose behavior reveals, in addition α = 79.26°, V = 2632 Å3 [11, 14]). The unit cell con- to specific quantum features typical of individual IV tains two V15 clusters. The V15 cluster consists of 15 V atoms, classical characteristics inherent in bulk single IV crystals. The clusters form molecular crystals in which ions, each having a spin S = 1/2. The V ions occupy the vertices of two plane hexagons and of a triangle they retain their individuality, because the coupling located between them. The cluster structure is shown between the clusters is fairly weak. The unique proper- schematically in Fig. 1. Each hexagon of the cluster ties of such systems are molecular bistability [12] and macroscopic quantum tunneling of magnetization [5, 8, 12, 13]. These phenomena are of undeniable interest for 5' understanding the fundamental problems of magne- 4' 6' tism. Until recently, only integer-spin clusters, such as 1' Mn12Ac (S = 10 in the ground state), Fe8 (S = 10 in the 3' ground state), and Mn R (S = 12 in the ground state), 6 6 ' were studied. A deeper insight into the physics of mag- 2 c J1 J2 netic nanoclusters can be gained, however, by studying J0 clusters with a half-integer spin, to which the V15 magnetic cluster belongs. b a 5 6 Moreover, one should know the exchange interac- J2 tions between the magnetic ions contained in a cluster. J1 The most direct method of studying these interactions 4 J" is based on measurements of the total magnetization 1 curve in megagauss magnetic fields. This is the object J 3 J' 2 of the present study, which deals with the behavior of the magnetic susceptibility of the V15 cluster in ultras- Fig. 1. Schematic structure of the V15 cluster and exchange trong magnetic fields. interactions between vanadium ions.

MAGNETIC SUSCEPTIBILITY OF THE V15 NANOCLUSTER 2105 contains three pairs of strongly coupled spins (J ≈ U, V −800 K); the spin of each ion belonging to the triangle 6 is coupled to two pairs of spins, with one pair belonging to the upper hexagon and the other to the lower hexagon 5 ≈ ú ≈ ≈ ≈ (J' J1 Ð150 K, J'' J2 Ð300 K). The exchange interaction between the spins located at the vertices of 4 ≈ the triangle is very weak (J0 Ð2.5 K [15]). The magnetic (dipole) interaction between the spins of the neighboring clusters is negligible (a few millikelvins). 3 The ground-state cluster spin is S = 1/2. This value of the spin results from antiferromagnetic interactions 2 between the VIV ions [14]; therefore, such a cluster can be treated on the molecular level as a multisublattice 1 ferrimagnet. An external magnetic field induces transformation of the magnetic structure from quasi-ferri- 0 magnetic with spin S = 3/2 to ferromagnetic with S = 0 100 200 300 400 500 600 15/2. According to [16], this transformation should take , T place in ultrastrong magnetic fields and proceed via B three quantum jumps at low temperatures. Fig. 2. Inductive sensor plotted signal of differential mag- The magnetic susceptibility measurements were netic susceptibility dM/dt vs. magnetic field. conducted at liquid- helium temperature in fields of up to 550 T. Magnetic fields were produced by an MK-1 magnetic explosion generator [17]. The generator was are compensated. An ideal coil compensation is impos- employed in its single-stage version (without interme- sible to achieve, particularly if the rate of magnetic field diate internal stages) to produce a smooth magnetic- variation is as high as dB/dt ≈ 106 T/s. However, if the field pulse and to increase the usable volume. Several rate of magnetic field variation dB/dt is a smooth func- samples (from four to eight) could be studied in one tion of time, the jumps observed on the U(B) curve can experiment. The original magnetic field (B ≈ 16 T) was originate only from a change in the sample magnetic generated in a thin-walled multilayered multiple-coil moment. solenoid by a capacitor battery discharge (W = 2 MJ). The magnetic flux trapped inside the conducting cylin- Figure 2 presents an oscillogram of the signal der was compressed by the products of explosion to a obtained from the inductive sensor of the differential diameter of 20 mm. The magnetic-field compression magnetic susceptibility dM/dt. At fields of ~200 and µ time was about 16 s. At the maximum magnetic field, 350 T, one can see signal jumps, which may be due to the usable volume was a cylinder of approximately changes in the sample magnetic moment produced by 20 mm in diameter and about 100 mm in length. The samples and magnetic-field sensors were disposed on a the V15 cluster undergoing discrete quantum transitions plate of fiberglass laminate and immersed in liquid in the spin structure. helium in a continuous-flow cryostat. The magnetic field was measured with a set of single-turn induction sensors varying from 0.6 to 14.0 mm in diameter 3. DISCUSSION OF RESULTS wound with PÉTV-2 wire. The signals were fed to Tek- tronix 784 and Tektronix 744 four-channel oscillo- The theoretical study of the spin structure rearrange- graphs having a resolution of 2 ns/T. ment in a V15 cluster conducted in [16] yielded analyti- The magnetic susceptibility was measured with a cal relations describing the behavior of the cluster mag- compensation pickup, which represented two oppo- netization upon variation of the magnetic-field induc- sitely connected, well-compensated induction coils. tion B and of temperature T. The expression given in The PÉTV-2 wire, 71 µm in diameter, was laid in spiral [16] for the magnetic moment of a cluster can be cast in grooves 2 mm in diameter cut in two caprolon blocks, the form with 9 turns in each. The degree of coil compensation was checked in a high-frequency magnet. The total coil MB() areas NS (where S is the area of one turn) differed by no 3 (), more than 2%. A hole 1.6 mm in diameter and intended ()2 ESB ∑ S 3SSÐ2+ expÐ------for placing the sample to be studied was drilled in one kT (1) of the blocks. When one of the coils contains a sample, µ S = 0 µ = 4 B------3 + 3 B, the signal is proportional to the derivative of the sample 2 ESB(), magnetic moment, U(B) ∝ dM/dt + KdB/dt. The coeffi- ∑ ()3SSÐ2+ exp Ð------kT cient K depends on the accuracy with which the coils S = 0

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2106 PLATONOV et al. where 4. CONCLUSION 2 Ð S Thus, we have studied, both experimentally (in ESB(), = ÐSJ˜ Ð2------()J'2+ J'' +,µ B 4 B fields of up to 550 T) and theoretically, the spin struc- (2) ture rearrangement induced by megagauss-scale mag- 1 netic fields in the V magnetic nanocluster. It was J˜ = J Ð ---()J + J , 15 4 1 2 shown that the transition from the quasi-ferrimagnetic to the quasi-ferromagnetic structure of the VIV spins with J, J1, J', J2, and J" being exchange integrals IV occurs at low temperatures through three quantum between the V ions (Fig. 1). jumps. The experimental data were compared with the- Equation (1) allows one to derive the following rela- ory. It was established that the values of the exchange tion for the magnetic susceptibility: integrals given in [14] do not allow quantitative χ()∂, ()(), ∂ description of the experimentally observed initial stage BT = MBT/ B T of the spin structure rearrangement of the V15 cluster. A  3 set of exchange parameters was proposed that permits 2 2 ESB(),  S ()3SSÐ2+ exp Ð------one to reconcile quantitatively the theoretical results 2 ∑  8µ  kT with experiment. Observation of a complete picture of = ------B-------S = 0 (3) kT 3 quantum jumps in the V15 spin structure would require  2 ESB(),  ∑ ()3SSÐ2+ exp Ð------fields of up to 103 T. kT  S = 0  REFERENCES  1. D. Gattecshi, A. Canecshi, L. Pardi, and R. Sessoli, Sci- MBT(), 3 2  ence 265, 1054 (1994). -- Ð ------µ Ð --- . 2 B 4  2. R. Sessoli, D. Gattecshi, A. Canecshi, and H. A. Novak,  Nature 356, 141 (1993).  3. J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, It follows from Eq. (2) that in the fields Phys. Rev. Lett. 76, 3830 (1996). 4. L. Thomas, F. Lionti, R. Ballou, et al., Nature 383, 145 1 J'2+ J'' (1996). B = ------ÐJ˜ + ------, 1 2µ 4 5. V. V. Dobrovitski and A. K. Zvezdin, Europhys. Lett. 38, B 377 (1997). 1 J'2+ J'' 6. D. Gattecshi, L. Pardi, A. L. Barra, et al., Nature 354, B = Ð------J˜ + ------, (4) 2 2µ 4 465 (1991). B 7. A. Canecshi, D. Gattecshi, and R. Sessoli, J. Am. Chem. 1 3 Soc. 113, 5872 (1991). B = Ð------J˜ + ---()J'2+ J'' , 8. L. Gunther, Europhys. Lett. 39, 1 (1997). 3 2µ 4 B 9. A. K. Zvezdin and A. I. Popov, Zh. Éksp. Teor. Fiz. 109, the crossing of the lower energy levels of the cluster 2115 (1996) [JETP 82, 1140 (1996)]. occurs. This should initiate sharp anomalies at B1, B2, 10. A. L. Barra, P. Debrunner, D. Gattecshi, et al., Europhys. Lett. 35, 133 (1996). and B3 (at low temperatures) in the magnetization (1) and in the magnetic susceptibility χ(B, T) (3). Accept- 11. A. Muller and J. Doring, Angew. Chem. Int. Ed. Engl. ≈ ≈ 27, 1721 (1991). ing, as a rough estimate, the values J = Ð800 K, J' J1 ≈ ≈ Ð150 K, and J'' J2 Ð300 K for the exchange integrals 12. O. Kahn and C. Jay Martinez, Science 279, 44 (1998). [14], we obtain from Eq. (4) for B1, B2, and B3 the val- 13. A. K. Zvezdin, V. V. Dobrovitski, B. N. Harmon, and ues 371, 650, and 929 T, respectively. The anomalies M. I. Katsnelson, Phys. Rev. B 58, 14733 (1998). revealed experimentally in the differential magnetic 14. D. Gattecshi, L. Pardi, A. L. Barra, and A. Muller, Mol. Eng. 3, 157 (1993). susceptibility occur in fields B1 = 200 T and B2 = 350 T (Fig. 2), which is in obvious disagreement with the the- 15. I. Chiorescu, W. Wernsdorfer, A. Muller, et al., Phys. oretical figures. The observed anomalies in magnetic Rev. Lett. 84, 3454 (2000). susceptibility correspond to the following values of the 16. A. K. Zvezdin, V. I. Plis, A. I. Popov, and B. Barbara, Fiz. = = Tverd. Tela (St. Petersburg) 43 (1), 177 (2001) [Phys. exchange integrals: J = Ð490 K, J' J1 Ð80 K, and = = Solid State 43, 185 (2001)]. J'' J2 Ð161 K. It should be pointed out that the procedure used in comparing the theoretical data with 17. A. I. Pavlovskiœ and R. Z. Lyudaev, in Problems of Mod- experimental results does not allow an unambiguous ern Experimental Science and Technology, Ed. by choice of the exchange parameters. To determine the A. P. Aleksandrov (Nauka, Leningrad, 1984), p. 206. ≈ values of J, J', J1, J", and J2, we used the relations J' ≈ J1 and J'' J2 [14]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2107Ð2111. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2013Ð2017. Original Russian Text Copyright © 2002 by Levitin, Zvezdin, Ortenberg, Platonov, Plis, Popov, Puhlmann, Tatsenko.

MAGNETISM AND FERROELECTRICITY

Faraday Effect in Tb3Ga5O12 in a Rapidly Increasing Ultrastrong Magnetic Field R. Z. Levitin*, A. K. Zvezdin**, M. von Ortenberg***, V. V. Platonov****, V. I. Plis*****, A. I. Popov*****, N. Puhlmann***, and O. M. Tatsenko**** * Moscow State University, Vorob’evy gory, Moscow, 119899 Russia ** Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 117942 Russia *** Institute of Physics, Humboldt University, Berlin, D-10115 Germany **** Russian Federal Nuclear Center, Arzamas-16, Sarov, Nizhni Novgorod oblast, 607189 Russia ***** Moscow State Institute of Electronic Engineering (Technical University), Zelenograd, Moscow oblast, 103498 Russia Received January 8, 2002

Abstract—The Faraday effect is measured in paramagnetic terbium gallate garnet Tb3Ga5O12 at a wavelength λ = 0.63 µm at 6 K in pulsed magnetic fields up to 75 T increasing at a rate of 107 T/s for field orientation along the crystallographic direction 〈110〉. The experimental data are compared with the results of theoretical calculations taking into account the crystal fields acting on the Tb3+ ion and various contributions to the Faraday rotation. Since the measurements in pulsed fields are carried out in the adiabatic regime, the dependence of the sample temperature on the magnetic field acting during a current pulse is obtained from the comparison of the experimental dependence of Faraday rotation with the theoretically calculated dependences of the Faraday effect under isothermal conditions at various temperatures. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION carried out using the Faraday effect, whose features in At present, the magnitude of experimentally attain- rare-earth compounds with the garnet structure have able static magnetic fields reaches 50 T, while fields been studied extensively [2Ð4]. An important argument with a higher induction (which are usually referred to as in favor of the Faraday effect is that the measuring unit ultrastrong) are generated only in the pulse regime [1]. is outside the magnetic field in this method, which makes it possible to minimize electric stray currents Experiments in ultrastrong magnetic fields attract induced in ultrastrong pulsed magnetic fields. considerable attention, since the substances investigated in them are in extreme conditions (huge Zeeman Crystals with garnet structure possess cubic symme- splittings for the spin and orbital degrees of freedom, try. Their crystallographic structure is rather complex quantum limit in semiconductors and semimetals, “rup- 10 and is described by the space group Oh [5Ð7]. The unit ture” of exchange coupling in magnetically ordered cell of the garnet contains 160 atoms. Rare-earth ions in materials and nanoclusters). In such experiments, mag- the garnet structure are located in six inequivalent netic transformations that cannot occur in other condi- dodecahedral positions with orthorhombic symmetry tions take place. of their surroundings (point group D2) with different Although the methods of generation of pulsed mag- orientations of local axes. The orientation of local axes netic fields in the range 50–100 T have been developed for all six inequivalent sites can be obtained by rotating to perfection, the measuring technique for such fields is the crystallographic system of coordinates through an far from perfect. In particular, the entropy redistribution angle ±π/4 relative to the axes [100], [010], and [001] among the magnetic subsystem, crystal lattice, and (Table 1). thermostat and, hence, the temperature conditions for pulsed magnetization remain unclear, since the rate of The low symmetry of the crystal surroundings of the variation of the field in experiments with ultrastrong rare-earth ion R3+ in the garnet structure leads to the fields is very high (107Ð108 T/s). This work is devoted maximum possible removal of degeneracy of its ground to this methodologically important and physically multiplet. For the non-Kramers ion Tb3+, the low- interesting problem. It is expedient to choose for such energy part of the spectrum consists of quasi-doublets: experiments a material whose properties and energy the ground state is a quasi-doublet with a gap of 2.5 cmÐ1, spectrum of the magnetic subsystem are well known. the first excited quasi-doublet lies 34 cmÐ1 above the We chose terbium gallate Tb3Ga5O12 with the garnet ground one, the third is separated from the latter by structure for our investigation. The “tracing” of the 43 cmÐ1, and so on [8]. The exchange interaction energy magnetic subsystem (and of the sample temperature) is in rare-earth gallate garnets is lower than the energy of

2108 LEVITIN et al.

Table 1. Orientation of local symmetry axes for six inequivalent dodecahedral positions of rare-earth ions in rare-earth garnets Unit vector 123456

ex [001] [001] [100] [100] [010] [010]

ey [11 0] [110] [011 ] [011] [1 01] [101]

ez [110] [1 10] [011] [01 1] [101] [101 ]

Table 2. Crystal-ﬁeld parameters for terbium gallate garnet [10] (in cmÐ1)

B20 B22 B40 B42 B44 B60 B62 B64 B66 Ð81 169 Ð2163 249 945 677 Ð155 1045 Ð4 the crystal field (the Néel temperature of terbium gal- where α(Tb) is the contribution from paramagnetic 3+ α late is TN = 0.25 K [9], which corresponds to approxi- Tb ions and D is the contribution from the matrix mately 0.17 cmÐ1); consequently, we can disregard the formed by diamagnetic gallium and oxygen ions. The α exchange interaction in an analysis of experimental term D is independent of temperature and proportional results at liquid-helium temperatures. to the magnetic field B, 3+ The parameters of the crystal field acting on a Tb α = VB. (2) ion in gallate garnet have been determined by various D methods [10, 11]. In our opinion, the set of parameters As the Verdet constant V of the matrix, in the first given in [10] (Table 2) is most adequate. This set of approximation, we can take the Verdet constant of parameters makes it possible to describe quantitatively yttrium gallate garnet V(YGG) = 0.043 min/(cm Oe) the crossing of energy levels in the spectrum of the Tb3+ [14]. ion, which is observed experimentally in Tb3Ga5O12, in In the visible and ultraviolet regions, the contribu- a field parallel to direction 〈110〉, as well as magnetization of rare-earth ions to Faraday rotation is mainly tion curves along this direction in fields up to 15 T [10]. determined by the allowed electric-dipole fÐd transi- One of our tasks was to verify the applicability of the tions, except in narrow spectral regions in the vicinity above crystal-field parameters for describing the prop- of the resonance frequencies of forbidden fÐf transi- erties of the terbium gallate garnet in stronger fields (up tions. In general, the quantity α(Tb) contains three to 75 T). terms: the paramagnetic contribution, the contribution It should be noted that magnetization was measured of mixing, and the diamagnetic contribution [2Ð4, 15]. in [10] in static fields and, hence, under isothermal con- The diamagnetic contribution is linear in the magnetic ditions. In the pulsed magnetic fields used in our exper- field B, is significant only in a narrow spectral region in iments, the measurements were made in the adiabatic the vicinity of resonance frequencies of optical transi- regime. It was demonstrated for the first time in [12] tions, and is reduced to a renormalization of constant V that, by comparing the experimental results obtained in Eq. (2). According to [3], the paramagnetic contribu- under adiabatic conditions with the (experimental and tion and the contribution of mixing have the form theoretical) results obtained in the isothermal regime, it g is possible to determine the magnetocaloric effect. This α() J0 Tb = AM0 Ð ------MVV , (3) was a second problem solved in this study. It should be 2 Ð gJ noted that, apart from its scientific value, an analysis of 0 the magnetocaloric effect is important for applications, where M0 is the magnetization associated with the split- 7 since rare-earth paramagnetic garnets are regarded as ting of the energy levels of the ground multiplet F6, promising materials for low-temperature magnetic MVV is the Van Vleck correction to magnetization [3, 4] refrigerators [13]. associated with admixing of the first excited multiplet 7F of the Tb3+ ion to the ground multiplet, g = 3/2 is 5 J0 2. THEORY the Landé factor of the ground multiplet of this ion, and A is a constant depending on the frequency of incident 2.1. Faraday Effect radiation and the oscillator strength for allowed fÐd The angle of rotation in the light-polarization plane transitions. In order to calculate M0(B) and MVV(B) and α in Tb3Ga5O12 contains two contributions, thus calculate F(B), we must determine the electron structure of Tb3+ ions formed under the action of the α α()α F = Tb + D, (1) ionic surroundings (described by the crystal-field

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 FARADAY EFFECT 2109

Hamiltonian) and the magnetic ﬁeld. The actual Hamil- approximated at low temperatures by the expression tonian of the problem is [6] 12 π4 ()Θ 3 CV = ------R nT/ D , (8) Hˆ = Hˆ cr + Hˆ Z, (4) 5

where n = 20 is the number of atoms in a Tb3Ga5O12 where Θ molecule; R is the universal gas constant; and D is the Debye temperature, whose exact value for Tb Ga O is ˆ ()k () k () 3 5 12 Hcr = ∑Bkq Cq i + CÐq i ; (5) not known to us at the moment; however, a comparison kqi of the available data on the Debye temperature for alu- minate garnets [18] makes it possible to estimate it as k ≤ Θ ≈ Cq are irreducible tensor operators [16]; k = 2, 4, 6; q D 500 K. It should be noted that the approximation k; and summation over i is carried out over all 4f elec- of the heat capacity by Eq. (8) is rather rough, espe- 3+ trons of the Tb ion. In Eq. (4), HZ is the interaction cially for crystals with a complex crystallographic Hamiltonian of the ion with the magnetic field: structure as in rare-earth garnets. This follows, for example, from the results obtained in [19], where it is shown that the Debye temperature of gadolinium gal- Hˆ Z = µ ()L + 2S B. (6) B late garnet depends strongly on the temperature at It was mentioned above that in our calculations we which it is measured. used the set of crystal field parameters from [10] Entropy SM (per Tb3Ga5O12 molecule) in Eq. (7) is (Table 2). given by Using this algorithm, we calculated the energy lev- 6 () 3+ 1 Ð1 En m els En and determined the eigenfunctions of Tb ions in S = Ð--- ∑ ∑Z exp Ð------〈 〉 M 2 m T a wide range of magnetic fields parallel to the 110 m = 1 n axis, taking into account inequivalent positions occu- (9) 3+ Ð1 E ()m pied by Tb ions in the garnet structure (Table 1). It × ln Z expÐ------n - , was found that, as indicated in [10], the lower energy m T levels of the Tb3+ ion located in position 1 cross in a () field B ≈ 9.5 T. Then, we calculated the magnetizations En m where Zm = ∑n expÐ------is the magnetic contri- M0(B) and MVV(B) in Tb3Ga5O12 as a function of the T magnetic field at various temperatures (6 ≤ T ≤ 41 K). bution to entropy, which is averaged over inequivalent It should be noted that, according to calculations, the positions (index m). The solution of Eq. (7) gives the value of MVV(B) turned out to be quite small (it amounts magnetocaloric effect, viz., the field dependence of to 1% of M0(B) in fields on the order of 100 T) and can T(B). be disregarded in the first approximation. The contribu- The theoretical formulas derived in this section were tion of terbium to the Faraday effect is calculated using used for an analysis of experimental data (see below). Eq. (3) (in relative units) for various temperatures under isothermal conditions. 3. EXPERIMENTAL TECHNIQUE AND SAMPLES 2.2. Magnetocaloric Effect We measured the Faraday effect in Tb3Ga5O12 for Since we measured Faraday rotation under adiabatic the initial sample temperature 6 K. Measurements were conditions, it is necessary to take into account in calcu- made at a wavelength of 0.63 µm lying in the transpar- lations the magnetocaloric effect, i.e., the change in the ency window, far away from the absorption lines of sample temperature as a result of magnetization. The rare-earth ions, and were carried out according to the change in temperature upon adiabatic magnetization conventional intensity scheme using a gas laser. A can be calculated by solving the equation of the adiabat pulsed magnetic field up to 75 T was generated by dis- (see, e.g., [17])1: charging a capacitor bank through a one-turn solenoid. Cooling to low temperatures was carried out in a gas- (), () (), () SM T 0 0 + SP T 0 = SM TB+ SP T , (7) flow helium cryostat, so that the sample was in helium vapor during measurements. The pulse duration was µ T CV dT 6 s. Earlier [12], it was proved theoretically and where SP = ∫ ------is the phonon contribution to the experimentally that, for such a cooling regime, magne- T0 T tization is an adiabatic process if the rate of increase in entropy; CV is the molar heat capacity, which can be the field exceeds about 100 T/s. Since the rate of varia- 1 tion of the magnetic field in our case was much higher Here, we assume that entropy is transferred from the magnetic 7 subsystem to the lattice quite rapidly, so that the magnetic sub- (more than 10 T/s), we can state that measurements system and the lattice can be described by the same temperature. were made under adiabatic conditions.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2110 LEVITIN et al.

1.0 6 K of the Faraday effect in the terbium gallate garnet under isothermal conditions at low temperatures. For plotting 0.8 these curves in absolute units, we made use of the fact 41 K that, according to calculations, the magnetization M0 of the terbium sublattice in strong ﬁelds (B ≥ 50 T) and at

°/cm 0.6

4 low temperatures (T ≤ 50 K) is virtually independent of temperature; i.e., the adiabatic and isothermal magneti- , 10 F 0.4 zations coincide under these conditions. This allowed α

– us to determine constant A in Eq. (3) for the Faraday 0.2 effect from a comparison of theoretical results obtained for isothermal conditions with the experimental data for adiabatic magnetization in the region of strong fields µ 0 10 20 30 40 50 60 70 (B = 70 T); the result is A = 1500 deg/( B cm). B, T It can be seen from Fig. 1 that the experimental adiabat of the Faraday effect intersects the theoretically Fig. 1. Dependence of the Faraday effect in Tb3Ga5O12 on α the magnetic field. Thick curve describes the experimental calculated isothermal F(B) dependences. At the inter- α F(B) dependence under adiabatic conditions at the initial section points, the sample temperature in the adiabatic temperature of 6 K of the sample; thin curves are the exper- regime is equal to the temperature of the corresponding α imental F(B) dependences under isothermal conditions, isothermal dependence. Thus, a comparison of the obtained at temperatures varying from 6 to 41 K with a step experimental adiabatic curve of Faraday rotation of the of 5 K. α polarization plane of incident light with the F(B) iso- therms makes it possible to determine directly the magnetocaloric effect, i.e., the dependence of the sample temperature on the magnetic field under adiabatic con- 35 ditions. 30 Figure 2 shows the T(B) dependence for terbium 25 gallate garnet determined in this way. It can be seen that the increase in temperature as a result of adiabatic mag-

, K 20 netization in a field of 75 T attains a value of 35 K. Fig- T ure 2 also shows a theoretical dependence calculated on 15 Θ the basis of Eqs. (7)Ð(9) using the value D = 500 K for 10 the Debye temperature. It can be seen that the experimental and theoretical T(B) dependences are of the 5 0 10 20 30 40 50 60 70 same type and the calculated and experimental values B, T of magnetocaloric effect are close, although the experimental effect is slightly stronger (by 3Ð4 K). The reason for this discrepancy has not been found as yet. Fig. 2. Magnetocaloric effect T(B) in Tb3Ga5O12. Dark squares correspond to experimental results, and the solid According to estimates, the temperature variation as a curve is calculated theoretically. result of adiabatic magnetization depends on the Debye temperature only slightly; its variation within 20Ð30% does not remove the discrepancy between theory and A terbium gallate garnet Tb3Ga5O12 single crystal experiment. This discrepancy is probably due to the was grown from solution in melt. A plate having a non-Debye character of the phonon spectrum in garnets thickness of 0.64 mm and oriented perpendicular to the (see above). 〈 〉 crystallographic direction 110 was cut from this crys- In Fig. 3, the theoretical adiabat of the Faraday tal. effect calculated by using the magnetocaloric effect determined from the solution to Eq. (7) is compared with the experimental α (B) dependence measured at 4. EXPERIMENTAL RESULTS F AND DISCUSSION the initial temperature of 6 K. It can be seen that the agreement between the experimental and theoretical Figure 1 shows the experimental field dependence curves is quite small, which confirms the correctness of of the Faraday effect in Tb3Ga5O12 in a pulsed field for the model and the adequacy of the values of the param- the initial sample temperature 6 K. It can be seen that eters of the crystal field indicated above. We note that Faraday rotation attains saturation in comparatively other sets of crystal-field parameters available in the lit- weak fields B ≈ 40 T. It should be noted that no features erature [11] provide a poorer description of the experi- associated with crossing of energy levels of the Tb3+ ion mental results. in a magnetic field are observed on the experimental It should be noted that, in all probability, the small curves. Figure 1 also shows the theoretical dependences variations of the crystal-field parameters used by us

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 FARADAY EFFECT 2111

1.0 1 REFERENCES 3 1. F. Herlach, Strong and Ultrastrong Magnetic Fields and 0.8 2 Their Applications (Springer, Berlin, 1985). 2. U. V. Valiev, A. K. Zvezdin, G. S. Krinchik, et al., Zh. 0.6 Éksp. Teor. Fiz. 79, 235 (1980) [Sov. Phys. JETP 52, 119

deg/cm (1980)]. 4 3. U. V. Valiev, A. K. Zvezdin, G. S. Krinchik, et al., Zh. 0.4 , 10 Éksp. Teor. Fiz. 85, 311 (1983) [Sov. Phys. JETP 58, 181 F

α (1983)]. – 0.2 4. A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials (Institute of Physics Publ., Bristol, 1997). 0 10 20 30 40 50 60 70 5. S. Krupicka, Physik der Ferrite und der verwandten B, T magnetischen Oxide (Academia, Prague, 1973; Mir, Moscow, 1976), Vol. 2. α Fig. 3. (1) Calculated F(B) isotherm at 6 K and (2, 3) 6. A. K. Zvezdin, V. M. Matveev, A. A. Mukhin, and α F(B) adiabat for the initial sample temperature of 6 K A. I. Popov, Rare-Earth Ions in Magnetic-Ordered Crys- (curve 2 corresponds to experiment, and curve 3 is calcu- tals (Nauka, Moscow, 1985). lated theoretically). 7. Yu. A. Izyumov, V. E. Naœsh, and R. P. Ozerov, Neutron Diffraction Analysis of Magnetics (Atomizdat, Moscow, 1981). here will provide an even better agreement between the experimental and theoretical results. 8. J. A. Koningstein and C. J. Kaue-Maguire, Can. J. Chem. 52, 3445 (1974). Figure 3 also shows the theoretically calculated iso- 9. F. Harbus and H. E. Stanly, Phys. Rev. 88, 1156 (1973). therm of the Faraday effect at 6 K. It clearly displays a 10. M. Guillot, A. Marchand, V. Nekvasil, and F. Tcheou, feature near 9.5 T associated with a crossing of energy J. Phys. C 18, 3547 (1985). levels. The absence of this feature in adiabatic measure- 11. V. Nekvasil and I. Veltruski, J. Magn. Magn. Mater. 86, ments can be explained by the elevation of temperature 315 (1990). due to the magnetocaloric effect, leading to a blurring 12. R. Z. Levitin, V. V. Snegirev, A. V. Kopylov, et al., of this feature. J. Magn. Magn. Mater. 170, 223 (1997). 13. M. D. Kuz’min and A. M. Tishin, Cryogenics 32, 545 5. CONCLUSION (1992). 14. U. V. Valiev, A. A. Klochkov, A. I. Popov, and Thus, using the parameters of the crystal field acting B. Yu. Sokolov, Opt. Spektrosk. 66, 613 (1989) [Opt. 3+ Spectrosc. 66, 359 (1989)]. on Tb ions in Tb3Ga5O12 taken from [10] (Table 2), we have described both the behavior of magnetization 15. R. Serber, Phys. Rev. 41, 489 (1932). of terbium gallate garnet in weak magnetic fields under 16. B. G. Wybourne, Spectroscopic Properties of Rare Earth isothermal conditions and the field dependence of the (Wiley, New York, 1965). Faraday effect in strong magnetic fields under adiabatic 17. K. P. Belov, A. K. Zvezdin, A. M. Kadomtseva, and conditions, as well as the dependence of the sample R. Z. Levitin, Orientational Phase Transitions (Nauka, temperature on the magnetic field during the action of a Moscow, 1979). pulse. 18. B. Nagaian, M. Pam Babu, and D. B. Sirdeshmukh, Indian J. Pure Appl. Phys. 17, 838 (1979). 19. Wen Dai, E. Gmelin, and R. Kremer, J. Phys. D 21, 628 ACKNOWLEDGMENTS (1988). This study was supported by the Russian Founda- tion for Basic Research, project no. 00-02-17240. Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2112Ð2115. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2018Ð2021. Original Russian Text Copyright © 2002 by Bokov, Volkov, Petrichenko.

MAGNETISM AND FERROELECTRICITY

High-Drive-Field Domain Wall Dynamics in Low-Damping Garnet Films V. A. Bokov, V. V. Volkov, and N. L. Petrichenko Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received January 22, 2002

Abstract—Domain wall (DW) dynamics in a low-damping YBiFeGa film with perpendicular magnetic anisotropy was studied under FMR conditions. Measurements were carried out under radial expansion of magnetic bubbles in high pulsed drive fields and in an in-plane dc magnetic field. A high-speed image recording technique was employed. The pattern of the dependence of DW velocity on drive field for the parts of the DW oriented parallel and perpendicular to the in-plane field was established. In all cases, this dependence contains a saturation region in which the DW velocity increases noticeably with increasing in-plane field. The experimental data obtained do not agree with theory. A possible explanation for this discrepancy is proposed. The onset of spatially periodic distortions in a moving DW is discussed. © 2002 MAIK “Nauka/Interperiodica”.

While domain wall (DW) dynamics in ferrite-garnet While there is no lack of conjectures concerning the films with perpendicular magnetic anisotropy has been origin of these effects [1, 7, 11Ð14], the problem extensively studied, the results reported relate primarily remains basically unsolved. In view of the disagree- to weak drive fields Hg not in excess of the saturation ments among the results obtained in various studies and magnetization 4πM. Only a limited number of studies the fairly limited amount of available experimental π have dealt with stronger fields (Hg > 4 M), and the data, it appeared reasonable to study the dynamic results of these studies are contradictory. It is reported behavior of a DW in a low-damping film under high in [1], for instance, that in this region the character of drive fields. It is believed that in such fields, the effects DW motion changes, the saturated velocity behavior specific for this field region should be most clearly being replaced by a regime in which the DW velocity V expressed. increases substantially with the field Hg. In [2, 3], no Our measurements were performed on a (111)-ori- velocity saturation was observed, whereas in [4Ð6] it ented film of the YBiFeGa system with the following was seen. Several communications have discussed the characteristics: the film thickness h = 4.6 µm, 4πM = effect of a dc in-plane field H on DW motion in high 156 G, the field of uniaxial anisotropy was 6200 Oe, the p ∆ × Ð6 drive fields. According to [7], when the field H is weak Bloch wall width parameter = 2 10 cm, the effec- p γ × 7 Ð1 Ð1 (~4πM) and parallel to the film plane, the relation V(H ) tive gyromagnetic ratio = 1.67 10 Oe s , the g α has a region with positive differential mobility followed Gilbert damping parameter = 0.002, and the bubble by another region at higher H that exhibits negative domain static collapse field was 34 Oe. A high-speed g video recording technique was used, with a rhodamine differential mobility. As Hp increases, the mobility in 6G dye laser (pumped by a nitrogen laser) providing the second region reverses sign to become positive. At single light pulses ~5 ns long. The image obtained was the same time, according to [5, 6], as the in-plane field stored in digital video memory and could be displayed increases, the velocity saturation in weak fields Hp dis- on a monitor screen for processing. The spatial resolu- appears and the DW velocity increases monotonically tion was ~0.4 µm. A bubble was stabilized by a dc bias with increasing drive field. The data on the V(H ) p field Hb and expanded by applying a pulsed uniform behavior in high fields H are also contradictory. For g field H directed opposite to Hb. The DW was displaced instance, the findings reported in [7] fit Eq. (12) from a by an effective drive field: theoretical analysis made in [8]. However, the data reported in [5, 6] agree neither with that relation nor Hg = H Ð Hb + He, with the expression derived on the basis of a simple where the last term takes into account the fact that the model in [9]. Domain wall motion is frequently effective field due to the DW curvature and the effective observed to entail the so-called widening effect, i.e., demagnetization field normal to the film plane depend formation of a broad diffuse boundary and generation on the bubble radius. This term in Hg was found using of microdomains in front of the moving DW [4Ð6, 10]. the well-known relations from the theory of bubble sta- These phenomena are sometimes preceded by the onset bility (see, e.g., [15]). The lowest value of the drive field of spatially periodic DW shape distortions [1, 7, 10]. in our measurements was 60 Oe, because below this

HIGH-DRIVE-FIELD DOMAIN WALL DYNAMICS 2113 value of H , the bubble shape underwent irreproducible g 60 distortions. The dc in-plane magnetic field Hp applied 3 to the film could be varied. For Hp < 145 Oe, the maxi- 50 mum value of Hg was 970 Oe. In a stronger field Hp, the a walls of the expanding bubble approached the neigh- 40 b boring domains; to avoid their interaction, measurements under these conditions were conducted in lower , m/s 30 drive fields. The translation of parts of the bubble wall V parallel and perpendicular to the in-plane field was 20 2 measured experimentally as a function of time during the drive field pulse. For this purpose, the laser illumi- 10 1 nation pulse was delayed with respect to the instant of the field pulse application by an amount varied from 0.3 µ 0 200 400 600 800 1000 to 0.7 s. The measurements were made repeatedly for H , Oe each delay time. To exclude the effect of the initial DW g displacement phase, the minimum delay was always µ Fig. 1. Domain wall velocity plotted vs. drive field for dif- 0.3 s. The data thus obtained were used to find the DW ferent values of the in-plane field Hp: (1) 0, (2) 200, and (3) velocity and its dependence on the drive field Hg for 360 Oe. (a, b) The wall is parallel and perpendicular to the field H , respectively. various values of Hp. p Figure 1 exemplifies typical V(Hg) dependences obtained for the in-plane field Hp equal to 0, 200, and Hp [23]. The HBLs in such DWs will apparently have 360 Oe. At Hp = 0, the DW velocity is seen to reach sat- different twist angles and move differently; this is what uration at Vs = 4.5 m/s. This velocity agrees well with initiates an anisotropy in the DW velocity. the value of 4.2 m/s calculated for the film under study ∆γ α from the empirical relation Vs = M (1 + 7.5 ) [16, The V(Hg) relations obtained by us differ from those 17]; this agreement lends additional support to the the- observed in [7]. In our case, regions with negative dif- oretical model developed in [18, 19], which is in accord ferential mobility that transform into those exhibiting with this relation and according to which the saturation positive mobility were not seen. The reason for these regime corresponds to the state of chaos. The analysis discrepancies is unclear; the films studied had similar performed in [18, 19] was applicable to low fields Hg < parameters, except the uniaxial-anisotropy field (in our 4πM. As follows from our data, the state of chaos can sample, this field was ~60% higher). The difference also set in in high drive fields. from the results reported in [5] is as difficult to explain. ≤ For Hp 110 Oe, the experiment reveals only the It is possible that the measurements in [5] were carried velocity saturation region throughout the Hg variation out in fields Hg that were not high enough, and thus the range studied. When Hp > 145 Oe and is oriented per- velocity saturation region was not reached. pendicular to the DW plane, one can see an interval of V(H ), preceding the velocity saturation region, where The data plotted in Fig. 1 were used to derive the g dependence of the saturation velocity on the in-plane the DW velocity increases with the field Hg (Fig. 1). If the field H is parallel to the DW plane, such an interval field for DWs oriented differently relative to this field p (Fig. 2). The saturation regime is known to set in when appears for Hp > 250 Oe. Thus, the initial part of the ≤ the average DW velocity reaches a certain critical value V(Hg) relation (Hg 220 Oe) is characterized by veloc- and does not vary thereafter. The V (H ) relation is ity anisotropy. As already mentioned, the measure- s p ments were conducted in drive fields H ≥ 60 Oe; for likely to reflect the dependence of this critical velocity g on the in-plane field. This field should stabilize the DW this reason, the region of linear DW translation and a spin structure. Therefore, Fig. 2 suggests that, starting maximum in the DW velocity are not observed, because, even in high H fields, the maximum occurs with a certain value of Hp, the DW structure with a p Néel-type region observed in walls oriented perpendic- for Hg < 5 Oe in low-damping films [20]. It is believed that, after the steady-state motion breaks down, the DW ular to the in-plane field is more stable. In this wall, structure contains a horizontal Bloch line (HBL) and chaos arises at a higher DW velocity. The theory devel- undergoes periodic transformations, with the wall oped in [8] permits the conclusion that the velocity saturation effect is caused by a change in the relaxation velocity increasing with the field Hg (see [21, Fig. 1] and [22, Fig. 1]). It is apparently this region of increase mechanisms with increasing DW velocity and yields in velocity that becomes manifest in the corresponding the following expression relating the saturation velocity curves in Fig. 1. The parts of the DW oriented perpen- and the in-plane field: dicular and parallel to the in-plane field should differ in π structure. In the former case, the DW should contain a V = ---∆γH . (1) region of the Néel wall type whose size increases with s 2 p

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2114 BOKOV et al. our data, the formation of spatially periodic distortions 60 a is not related to a crossover to the velocity saturation 1 b regime. 50 Thus, our analysis of the behavior of a low-damping 40 ferrite-garnet film under FMR conditions in high drive fields has established the character of the dependence 2 , m/s of the DW velocity on the drive field for domain walls s 30 V oriented parallel and perpendicular to the in-plane field. It was shown that velocity saturation is reached in all 20 cases, and the dependence of the saturation velocity on the in-plane field was found. On the whole, the results 10 obtained find interpretation in terms of a concept by which the DW velocity saturation corresponds to the 0 1.0 2.0 3.0 4.0 state of chaos. When the in-plane field is perpendicular H /8M to the wall, spatially periodic distortions arise on the p DW; this phenomenon is not related in any way to the velocity saturation. Fig. 2. Domain wall saturation velocity plotted vs. in-plane field. (a, b) The wall is parallel and perpendicular to the field Hp, respectively. Curve 1 is a plot of Eq. (1), and curve 2 is a plot of Eq. (2). ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 00-02-16945. Another relation was obtained in [9] from an analysis of a simple model of wall motion and has the form ∆γ REFERENCES V = ------H . (2) s 4 p 1. V. G. Kleparski, I. Pinter, and G. J. Zimmer, IEEE Trans. Magn. 17 (6), 2775 (1981). Equations (1) and (2) are plotted in Fig. 2. The obvious 2. R. V. Telesnin, S. M. Zimacheva, and V. V. Randoshkin, disagreement with the experimental data is accounted Fiz. Tverd. Tela (Leningrad) 19 (3), 907 (1977) [Sov. for by the fact that the velocity saturation regime is due Phys. Solid State 19, 528 (1977)]. to a mechanism different from the ones treated in the communications cited above. 3. V. V. Randoshkin, Tr. Inst. Obshch. Fiz. Ross. Akad. Nauk 35, 49 (1992). Throughout the ranges of the fields H and H stud- g p 4. L. P. Ivanov, A. S. Logginov, and G. A. Nepokoœchitskiœ, ied here, the wall parallel to the in-plane field did not Zh. Éksp. Teor. Fiz. 84 (3), 1006 (1983) [Sov. Phys. exhibit the spatially periodic distortions reported in [7, JETP 57, 583 (1983)]. 10] for this DW orientation. At the same time, as in [7, 10], such distortions formed on the wall perpendicular 5. K. Vural and F. B. Humphrey, J. Appl. Phys. 50 (5), 3583 to the in-plane field. When this field was 110 Oe, the (1979). distortions appeared when the wall moved in a drive 6. T. Suzuki, L. Gal, and S. Maekawa, Jpn. J. Appl. Phys. ≥ 19 (4), 627 (1980). field Hg 300 Oe. As Hp increases, the value of the drive field above which DW distortions arise decreases; 7. V. V. Randoshkin and M. V. Logunov, Fiz. Tverd. Tela for Hp = 360 Oe, the corresponding value of Hg is (St. Petersburg) 36 (12), 3498 (1994) [Phys. Solid State ~100 Oe. Distortions arise at drive field values below 36, 1858 (1994)]. the velocity saturation region, at least for Hp > 240 Oe. 8. B. A. Ivanov and N. E. Kulagin, Zh. Éksp. Teor. Fiz. 112 The width of the DW region where spatially periodic (3), 953 (1997) [JETP 85, 516 (1997)]. distortions are observed is comparatively small. For µ µ 9. F. H. de Leeuw, IEEE Trans. Magn. 9 (4), 609 (1973). Hp = 360 Oe, this width was ~3 m 0.7 s after application of a field H = 590 Oe; at this instant, the domain 10. M. V. Logunov and V. V. Randoshkin, Tr. Inst. Obshch. g Fiz. Ross. Akad. Nauk 35, 107 (1992). size in the direction collinear with the in-plane field was 115 µm. The spatially periodic DW distortions may be 11. V. V. Randoshkin, Fiz. Tverd. Tela (St. Petersburg) 37 one more reason for the above-mentioned anisotropy in (3), 652 (1995) [Phys. Solid State 37, 355 (1995)]. the wall velocity. 12. A. S. Logginov, G. A. Nepokoœchitskiœ, and T. B. Roza- Formation of a broad diffuse DW in a field H nor- nova, Zh. Tekh. Fiz. 60 (7), 186 (1990) [Sov. Phys. Tech. p Phys. 35, 861 (1990)]. mal to its plane was reported in [5]; this observation found theoretical interpretation [13] within a concept 13. R. A. Kosinski, Phys. Rev. B 50 (10), 6751 (1994). assuming random wall motion in high drive fields. We 14. P. A. Polyakov, Pis’ma Zh. Éksp. Teor. Fiz. 60 (5), 336 did not observe such a DW widening, and, as shown by (1994) [JETP Lett. 60, 343 (1994)].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 HIGH-DRIVE-FIELD DOMAIN WALL DYNAMICS 2115

15. V. V. Randoshkin, Prib. Tekh. Éksp., No. 2, 155 (1995). 20. V. A. Bokov, V. V. Volkov, N. L. Petrichenko, and M. Maryshko, Fiz. Tverd. Tela (St. Petersburg) 39 (7), 16. V. V. Volkov, V. A. Bokov, and V. I. Karpovich, Fiz. 1253 (1997) [Phys. Solid State 39, 1112 (1997)]. Tverd. Tela (Leningrad) 24 (8), 2318 (1982) [Sov. Phys. Solid State 24, 1315 (1982)]. 21. G. N. Patterson, R. C. Giles, and F. B. Humphrey, IEEE Trans. Magn. 27, 5498 (1991). 17. V. A. Bokov, V. V. Volkov, and N. L. Petrichenko, Phys. 22. V. A. Bokov and V. V. Volkov, Fiz. Tverd. Tela Met. Metallogr. 92 (S1), 1 (2001). (St. Petersburg) 39 (4), 660 (1997) [Phys. Solid State 39, 18. E. E. Kotova and V. M. Chetverikov, Fiz. Tverd. Tela 577 (1997)]. (Leningrad) 32 (4), 1269 (1990) [Sov. Phys. Solid State 23. R. A. Kosinski and J. Engemann, J. Magn. Magn. Mater. 32, 748 (1990)]. 50, 229 (1985). 19. A. Sukiennicki and R. A. Kosinski, J. Magn. Magn. Mater. 129, 213 (1994). Translated by G. Skrebtsov

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Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2116Ð2123. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2022Ð2028. Original Russian Text Copyright © 2002 by Buchel’nikov, Bychkov, Nikishin, Palmer, Lim, Edwards.

MAGNETISM AND FERROELECTRICITY

ElectromagneticÐAcoustic Transformation in an Erbium Single Crystal V. D. Buchel’nikov*, I. V. Bychkov*, Yu. A. Nikishin*, S. B. Palmer**, C. M. Lim**, and C. Edwards** * Chelyabinsk State University, Chelyabinsk, 454021 Russia ** University of Warwick, Coventry CV4 7AL, United Kingdom e-mail: [email protected] e-mail: [email protected] Received February 1, 2002

Abstract—The temperature dependences of the electromagnetic–acoustic transformation (EMAT) efficiency and the velocity of transverse sound for the erbium rare-earth metal with three complex magnetic structures are experimentally investigated at different external constant magnetic fields. An intensive generation and anomalies in the velocity of transverse sound are revealed in the temperature range of the magnetic phase transitions. It is found that an increase in the magnetic field leads to an increase in the sound generation efficiency and a decrease in the anomalies in the velocity of sound. The relationships for the efficiency of transverse-sound generation through the magnetoelastic mechanism are theoretically derived for two magnetic structures of erbium. It is demonstrated that an increase in the EMAT efficiency in the phase transition range is associated with the specific features in the static and dynamic magnetic susceptibilities of erbium. © 2002 MAIK “Nauka/Interpe- riodica”.

1. INTRODUCTION spiral phase and a decrease in the temperature ranges of Magnetic ordering of some magnets, among them existence of the longitudinal spin wave and cycloidal rare-earth metals, exhibit a complex nature. As a rule, phases. An increase in the magnetic field in the cycloi- all rare-earth metals undergo a number of phase transi- dal phase is attended by stabilization of the commensu- tions between different magnetic structures with a rate structures, which possess a net magnetization decrease in the temperature. In particular, the erbium along the hexagonal axis [3]. At H > 16 kOe, the cyc- rare-earth metal in the absence of magnetic fields loidal phase is characterized by only one commensu- undergoes the following sequence of spontaneous rate structure with a wave number of 2/7 (in terms of a phase transitions [1Ð4]. At temperatures T > T = 87 K, reciprocal lattice constant along the c axis) [3]. By con- N1 trast, the number of commensurate and incommensu- erbium is a paramagnet. In the temperature range TN1 > rate structures increases in the ferromagnetic spiral T > TN2 = 54 K, erbium has a magnetic structure which phase [3, 4]. is referred to as the longitudinal spin wave structure. This structure is characterized by oscillations of the The occurrence of several magnetic phase transi- longitudinal magnetization projection onto the anisot- tions and different long-period magnetic structures in ropy axis. In this case, the transverse magnetization erbium and other rare-earth metals leads to the fact that components are equal to zero. The transverse and lon- the behavior of different physical characteristics of gitudinal magnetization projections onto the hexagonal these metals can essentially differ from the behavior of axis c oscillate in the structure observed in the temper- similar characteristics in magnets with a simpler mag- ature range TN2 > T > TC = 18 K. This magnetic struc- netic structure. In particular, it is of interest to investi- ture is termed the complex spiral (or cycloidal) struc- gate experimentally and theoretically the electromag- ture. With a decrease in the temperature in this phase, neticÐacoustic transformation (EMAT) in rare-earth the wave vector of a cycloid decreases and passes metals, because all changes in the magnetic structure of through a series of commensurate and incommensurate these materials should affect the sound generation efficiency [6]. values [15]. At T < TC, erbium has a structure of the ferromagnetic spiral type. The electromagneticÐacoustic transformation in The complex magnetic structure of erbium is rare-earth metals with a modulated magnetic structure retained in the external magnetic field H aligned along has been investigated in a number of works (see, for the hexagonal axis up to H = 26Ð28 kOe [3, 4]. An example, review [6]). Andrianov et al. [7, 8] experi- increase in the magnetic field results in an increase in mentally studied the electromagnetic excitation of the temperature range of existence of the ferromagnetic sound in ferromagnetic and modulated phases of ter-

ELECTROMAGNETICÐACOUSTIC TRANSFORMATION 2117 bium and dysprosium. The electromagneticÐacoustic transformation in ferromagnetic phases of modulated magnets was theoretically analyzed in [6, 8]. However, 1 the efficiency of the electromagneticÐacoustic transformation in modulated phases of rare-earth metals was theoretically treated in only one work [9], which dealt with this process in a modulated phase of the simple spiral type. In the present work, we experimentally investigated the electromagneticÐacoustic transformation in an 0.1 erbium single crystal. The efficiency of the electromag- neticÐacoustic transformation through the magne- 2 toelastic mechanism was theoretically calculated for η the longitudinal spin wave and ferromagnetic spiral modulated phases. The results of theoretical calcula- 2 tions were compared with experimental data. 0.01 2. EXPERIMENTAL TECHNIQUE The EMAT efficiency and the velocity of generated sound were investigated using a Matec DSP-8000 automated ultrasonic measuring system [10, 11], which was adapted for measurements according to the standard 1 echo pulse noncontact sound generation technique described, for example, in review [6]. A cylindrical sin- 0.001 gle-crystal sample 2.8 mm long and 4.5 mm in diameter 0 20 40 60 80 100 was placed in a pancake coil. The hexagonal axis c of T, K the sample coincided with the cylinder axis. The coil Fig. 1. Experimental temperature dependences of the with the sample was placed in a constant magnetic field EMAT signal amplitude for the erbium single crystal in the whose vector H was perpendicular to the coil plane and magnetic field H || c at H = (1) 10 and (2) 20 kOe. The signal parallel to the hexagonal axis c of the sample. In this amplitude is given in arbitrary units. geometry, the transverse sound with the wave vector k aligned parallel to the c axis was generated in the sample [6]. Electromagnetic field pulses with an amplitude The temperature dependences of the EMAT signal of 200 V, a width of 0.6Ð0.8 µs, and a carrier frequency amplitude at two constant magnetic fields are displayed of 10 MHz were applied to the coil. These pulses in Fig. 1. It can be seen that, at H = 10 kOe, the depen- excited a sequence of decaying acoustic signals in the dence of the EMAT signal amplitude in the temperature sample. Since the electromagnetic pulse-repetition fre- ranges of the paramagnet Ðlongitudinal spin wave, lon- quency was equal to 1 kHz, the whole sequence of gen- gitudinal spin waveÐcycloid, and cycloidÐferromag- erated acoustic signals fell in this range. The ultrasound netic spiral phase transitions exhibits pronounced peaks pulses at the frequency of the incident electromagnetic in the sound generation. At H = 20 kOe, the generation wave were generated at both cylinder ends, propagated efficiency also increases in these temperature ranges. along the cylinder axis, and recorded at the opposite Note that the generation efficiency in the cycloidal cylinder end by the same coil. Echo signals following phase is characterized by a plateau and exceeds the gen- at intervals t = h/S (where h is the cylinder length and S eration efficiency in other phases. The generation effi- is the velocity of ultrasound) arrived from the coil at the ciency in all the phases increases with an increase in the Matec DSP-8000 precision instrument. The propaga- magnetic field. The temperature dependences of the tion time of the signal through the sample was deter- velocity of generated transverse sound at three mag- mined from the third and fourth echo signals, and the netic fields are plotted in Fig. 2. As is seen from this fig- amplitude of the generated signal was measured from ure, the anomalies in the velocity of sound are observed the third echo signal [10]. The error in the measurement in the above temperature ranges. The anomalies are of the velocity of sound with the use of the Matec DSP- smoothed as the magnetic field increases. The velocity 8000 instrument did not exceed 0.1%. Analysis of the of transverse sound most strongly decreases in the cyc- echo signal shape showed that the ultrasonic attenua- loidal phase at the magnetic field H = 10 kOe. In this tion in the sample was relatively small and had no case, the velocity of sound changes by 15Ð20%. It noticeable effect on the EMAT efficiency [10]. In this should be noted that such a considerable change in the case, the recorded signal amplitude under the above velocity of ultrasound in the range of magnetic phase experimental conditions was proportional to the EMAT transitions has never been observed in other rare-earth efficiency squared [6, 10]. metals.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

2118 BUCHEL’NIKOV et al.

1900 exchange constants; β stands for the anisotropy constants, H and h are the strengths of constant and alter- γ γ nating magnetic ﬁelds, respectively; ik and iklm are the exchange and relativistic magnetostriction tensors, respectively; Uij is the strain tensor, and ciklm is the elas- 3 tic constant tensor. Let us determine the ground state of a crystal with- β 1800 out regard for the anisotropy in the basal plane ( 6 = 0) provided the external magnetic ﬁeld is aligned along the hexagonal axis: H || c || z. For this purpose, it is necessary to solve a set of equations that involves the Euler 2 equations for the magnetic subsystem and the equilib- 1 , m/s rium equations for the elastic subsystem under the Saint S Venant strain compatibility conditions [12]. 1700 We will separately consider the ground states for the longitudinal spin wave and ferromagnetic spiral phases. In the case of a longitudinal spin wave structure, the equilibrium components of the magnetization vector can be written in the form [1]

n = +∞ z iqnz M± ==0, M ∑ M e , (2) 1600 z n n = Ð∞ where 2π/q is the period of the structure along the z 0 20 40 60 80 100 T, K axis. The equilibrium strain tensor, which is obtained by Fig. 2. Experimental temperature dependences of the veloc- solving the equilibrium equations for the elastic ity of transverse sound for the erbium single crystal in the || medium under the Saint Venant compatibility condi- magnetic field H c at H = (1) 10, (2) 15, and (3) 20 kOe. tions with due regard for relationships (2) in the approximation qd ӷ 1 (i.e., in the case when the period of the structure is considerably less than the magnet thickness d), is given in the Appendix [formulas (A1)]. 3. THEORY z 3.1. The free energy and the ground state. Rare- In order to determine the equilibrium values of Mn earth metals have a hexagonal crystal structure. The and q, formulas (2) and (A1) should be substituted into free energy of such a magnet can be described by the relationship (1) for the free energy, which should then phenomenological relationship [1, 9, 12] z be minimized with respect to Mn and q. Analysis of the 1 W = --- FVd , set of equations derived by minimizing the free energy V∫ z z of the system demonstrates that Mn = MÐn . Hence, where z hereafter, we will treat only the components Mn with 1 2 1 4 1 2 1 4 F = ---aM + ---bM Ð ---β M Ð ---β M n = 0, 1, 2, 3, …. We will also restrict our consideration 2 4 2 1 z 4 2 z to weak magnetic fields. In this case, it is sufficient to ӷ retain the terms with n = 0, 1, 2, and 3 (M1 M0, M2, ∂ 2 ∂ 2 1 6 6 1 M M and M3) in expression (2) [1]. To a first approximation + ---β ()M + M + ---α⊥ ------+ ------2 6 + Ð 2 ∂x ∂y ˜ in the small parameter bi /L(q), we obtain the following 2 (1) relationships for the components M , M , M , and M : 1 ∂M 2 1 ∂2M 0 1 2 3 + ---α|| ------+ ---γ ------Ð MH()+ h 2 ∂z 2 ∂ 2 z z () ˜ xi M0 ==H/L(),0 M1 ÐLq/3b, ˜ z 2 3b1M 2 + γ M U ++γ M M U c U U . Mz = Ð------0()Mz , ik ik iklm i k lm iklm ik lm 2 L()2q 1 (3) ± Here, M is the magnetization; M± = Mx iMy are its cir- ˜ z b z 3 cular components; a and b are the homogeneous M ==Ð------1 -()M , q q . exchange constants; α and γ are the inhomogeneous 3 L()3q 1 0

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ELECTROMAGNETICÐACOUSTIC TRANSFORMATION 2119

β α 2 γ 4 Here, L(q) = a Ð 1 + ||q + q is the eigenvalue of the equations with the standard boundary conditions for the differential operator electric and magnetic field vectors, the stress tensor, and the magnetization [6], that is, 2 4 ∂ ∂ 2 α|| Lˆ ==a Ð β Ð α||------+ γ ------, q Ð------, ∂σ 1 ∂ 2 ∂ 4 0 2γ ú [], eff ρ ik z z M ==g MH , uúúi ------∂ -, xk 2 2 2 π ∂ (6) 2γ˜ 8()c γ˜ Ð c γ˜ 2γ˜ 4 σ 1 B ˜ 3 33 1 13 3 ˜ 3 curlH ==------E, curlE Ð------∂ -, b ==b Ð ------+ ------∆ -, b1 b Ð ------. c c t c33 3c33 c33 divB ==0, divE 0. The ground state of the magnet in the ferromagnetic spiral phase can be determined under the assumption Here, B = H + 4πM is the magnetic induction, E is the ∂ ∂∂ that the energy W is minimum when the magnetization eff F F has the form electric field strength, H = Ð∂------+ ∂ ∂∂------()∂ is M x M/ xi the effective magnetic field, ρ is the metal density, u is + iqz θ Ð Ðiqz θ M0 ==M0e sin , M0 M0e sin , the displacement vector, g is the gyromagnetic ratio, c (4) σ z θ is the velocity of light in free space, and is the electri- M0 = M0 cos , cal conductivity. The terms responsible only for the magnetoelastic mechanism of sound generation are θ where is the angle between the magnetization vector retained in Eqs. (6). This mechanism of electromag- M and the symmetry axis z. The solution of the elastic- neticÐacoustic transformation dominates in magnetic ity equations with allowance made for the Saint Venant fields as strong as 100 kOe [6]. compatibility conditions for the ferromagnetic spiral phase and relationships (4) makes it possible to obtain We will analyze small deviations of the magnetiza- the equilibrium strain tensor U0 . The components of tion, elastic displacements, and electromagnetic field ik from the equilibrium values determined by relation- this tensor are given in the Appendix [formulas (A2)]. ships (2) and (3). For this purpose, all the variables are Substitution of expressions (4) and (A2) into relation- represented in the form ship (1) for the free energy, averaging of the energy at ӷ θ qd 1, and minimization with respect to and q give n = +∞ the following equations for the equilibrium magnetiza- ikz iqnz tion and the wave number of a spiral in the ferromag- FF==0()z + f , f e ∑ f ne , (7) netic spiral phase: n = Ð∞

()α γ 1/2 where F0 stands for the equilibrium values and f are qq==0 Ð ⊥/2 , small deviations from the equilibrium values. By sub- M cosθβ[ ˜ ++h ()β˜ Ð h /M2 M2 cos 2θ (5) stituting formulas (7) into the system of coupled equa- 0 1 me 2 me 0 0 tions (6) and eliminating the electric field strength E, + αq2 + ∆˜ ] +0,H = we obtain an infinite set of equations (linearized in the vicinity of the equilibrium values) for the Fourier components f of coupled waves in the longitudinal spin where ∆˜ = γq4, h = (γ Ð γ )2 M2 /(c Ð c ), and β˜ n me 11 12 0 11 12 1 wave phase. Within the approximation that the zeroth ˜ and β2 are the anisotropy constants renormalized by harmonics f0 in the derived infinite set of equations have the magnetostriction [9, 13]. the largest amplitudes [9], the linearized system of equations (6) takes the form 3.2. The ultrasonic generation in the longitudinal spin wave phase. Now, we consider a magnet occupy- γ z ()ω2 ω2 ± ik 44M0 ± ing the half-space z > 0 (this approximation is valid in Ð t u0 +0,------m0 = the case when the acoustic wavelength is considerably ρ ± ± (8) less than the sample thickness and, in particular, under ()k2 Ð k2 h Ð0,4πiδ2m = the aforementioned experimental conditions). We e 0 0 ± 2 ± ± assume that a plane electromagnetic wave is incident ()ωω+− m Ð0.2igkγ ()Mz u + Mz gh = normal to the magnet surface. Let the external magnetic 1 0 44 1 0 0 0 field vector H be aligned along the hexagonal axis c of ± ± ± ω the sample (H || k || c || z). Here, m , h , and u are the circular components; is the frequency of the electromagnetic wave incident on The amplitude of the acoustic wave excited in the the metal; ω = S k and S = c /ρ are the frequency magnet can be determined by solving the system of t t t 44 2 elasticity, LandauÐLifshitz, and Maxwell coupled and velocity of the transverse sound, respectively; ke =

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2120 BUCHEL’NIKOV et al.

δ2 π σω 2 δ ω 2i/ = 4 i /c ; is the skin depth in the metal; 1 = solution of the systems of equations (8) and (11) β ˜ 2 enables us to derive the amplitude of the excited sound: gM0(L(k) + 1 + 6(b + bme)M1 ); and 1/2 ± c 2 γ M χµ ± 2γ˜ 2()c γ˜ Ð c γ˜ u = ------44 0 h . (12) b = ------3()γ Ð γ + ------33 1 13 3- 0 S πσρ ()µζ3/2 0 me 33 3 ∆ t 2 St Ð t c33 3c33 c The coefficient of the transformation of electromag- × γ γ γ γ 13γ γ˜ c333 11 + 12 Ð 4 13 + 1 Ð 3------3 +.4c13 3 netic waves into acoustic waves (the EMAT efficiency) c33 η is defined as the ratio between the acoustic and electromagnetic energy fluxes at the magnet boundary [6]. Note that only the transverse sound can be excited in In the case under consideration, this coefficient is given the geometry under consideration [6]. by the formula It is assumed that the frequency of excited waves is 3 γ 2 2ω2χ2µ considerably less than the frequency of uniform magne- c 44M0 η = ------. (13) tization precession (ω Ӷ ω ) and the spatial dispersion  2 2 3 1 St 2πρσ S ()µζÐ of spin waves is insignificant. Consequently, from Eqs. t t (8), we obtain the dispersion relation for coupled elec- 3.3. The sound generation in the ferromagnetic tromagnetic, spin, and elastic waves: spiral phase. Let us examine small deviations of the displacement, magnetization, and electric and magnetic ()ζ 4 []()µζ 2 2 2 µ 2 2 1 Ð t k ==Ð t ke + kt k +0,ke kt (9) field vectors from the equilibrium values [relationships (4) and (5)]. In the case when the waves propagate where k2 = ω2/S2 , χ = gM /ω is the dynamic magnetic along the z (spiral) axis and the fundamental harmonics t t 0 1 have the largest amplitudes, the linearized system of µ πχ ζ χγ 2 2 ρ 2 susceptibility, = 1 + 4 , and t = 44M1 / St is the equations (6) for the Fourier components of the above dynamic parameter of magnetoelastic interaction. vectors takes the form [13] The dispersion equation (9) has the following solu- ± θω1ω 2 θ − ω tions: m ()k cos 2()k + --- me4 sin + 2 µ µζ 2 2 2 Ð t 2 ӷ k1 ==------kt , k2 ------ke , ke kt, 2 2 1 2 ± µζÐ 1 Ð ζ + igγ M k cos θ Ð --- sin θ u ()k t 1 (10) 44 0 2 2 1 2 2 2 k ==------k , k µk , k Ӷ k . 1 ζ t 2 e e t θ ± 1 Ð t Ð gM0 cosh ()k = 0, (14) Here, k and k are the wave numbers of the quasi-elas- 1 2 ()ω2 2 2 ± ikγ θ ± tic and quasi-electromagnetic waves, respectively. Ð St k u ()k Ð0,----ρ 44M0 cos m ()k =

Under the assumption that the zeroth harmonics f0 2 2 2 ± 2 ± have the largest amplitudes, the linearized system of ()ω Ð ν k h ()k +0. 4πω m ()k = boundary conditions can be written as follows: Here, we introduced the following designations: h±(k), ± ± ± ± ± ± iγ M ()m + m Ð0,c ()k u + k u = m (k), and u (k) are the circular components of the 44 0 1 2 44 1 1 2 2 Fourier vectors h, m, and u, respectively; ν = ick ± ick ± ± (11) ω π σ ω ω ω ω 2θ 1 2 c /4 i ; 2(k) = 20 + gM0L⊥(k); 20 = me4cos ; 1 Ð ------πσ h1 +21 Ð ------πσ h2 = h0 , 4 4 ω γ 2 3 me4 = g 44M0/c44 ; L⊥(k) is the eigenvalue of the dif- ± ∂2 ∂4 where h are the circular amplitudes of the incident 2 4 0 ferential operator Lˆ ⊥ = Ðα⊥q Ð γq Ð α⊥------+ γ ------; electromagnetic wave. Subscripts 1 and 2 refer to the ∂z2 ∂z4 2 waves propagating in the metal and correspond to those ω (k) = ω Ð ω + gM sin θL⊥(k); ω = gM[h Ð in the solutions of the dispersion equation (10). It 1 10 20 0 10 me4 2θ β˜ β˜ β 2 2θ 2θ should be noted that the boundary condition for magne- sin (1 + (2 + 2 2)M0 cos + hmesin )]; and hme4 = ω tization in the system of equations (11) is absent, me4/gM0. because the dynamics of the magnet is considered The dispersion relation of system (14) is deﬁned by within the approximation according to which the inho- expression (9). The solution of this dispersion equation mogeneous exchange is disregarded in the energy den- within the approximation ω ӷ ω (the given approxi- sity (1) [6, 9]. 2 mation, as a rule, is valid in the frequency range used in ӷ In the approximation ke kt, which is usually satis- experimental investigations of the electromagneticÐ ﬁed under experimental conditions, the simultaneous acoustic transformation in metals) is represented by

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ELECTROMAGNETICÐACOUSTIC TRANSFORMATION 2121

ζ γ 2 2 χ θ 2θ expressions (10), where t = 44 M0 cos (sin /2 Ð This maximum, as a rule, is observed in magnetically ordered crystals upon transition from the paramagnetic 2θ ρ 2 µ πχ θ χ ω cos )/ St , = 1 + 4 cos , and = gM0/[( 2(k) + state to the ordered state [14]. The small height of the ω 2θ θ me4sin /2)cos ]. By ignoring the spatial dispersion generation efficiency peak in the vicinity of TN1 can be of spin waves, the linearized system of boundary con- due to the small anisotropic magnetostriction constant γ ditions is determined by formulas (11) in which the 44 in this temperature range [6]. θ component M0 should be replaced by M0cos . The transition from the longitudinal spin wave The simultaneous solution of the systems of equa- phase to the cycloidal phase at T = TN2 = 54 K is πσ Ӷ tions (14) and (11) at cki/4 1 permits us to deter- attended by a sharp increase in the dynamic susceptibil- mine the amplitude of the generated elastic transverse ity (19). This is caused by the fact that the frequency of wave in the case when the skin depth in the metal is ω χÐ1 the quasi-spin mode 1 = gM0 d decreases upon the substantially less than the incident electromagnetic longitudinal spin wave cycloidal phase transition. wavelength; that is, According to relationship (19) (see also [1]), at this 1/2 2 ω 2 γ χµ θ phase transition point, the frequency 1 takes the mini- ± c 44M0 cos ± u = ------h . (15) ˜ 2 2  3/2 0 mum value (ω = 6gM bmeM ) determined by the mag- St 2πσρS ()µζÐ 1 0 1 t t netoelastic coupling and the dynamic susceptibility has From this formula, we obtain the following relationship a maximum. As follows from expression (17), this can for the EMAT efficiency in the ferromagnetic spiral be responsible for the second peak in the experimental phase: dependence of the EMAT efficiency for erbium at T = TN2. Moreover, in accordance with expression (17), 3γ 2 ω2 2χ2µθ4 η c 44 M0 cos the EMAT efficiency depends on the external magnetic = ------2 2 3-. (16) St 2πσ ρS ()µζÐ field and should increase with an increase in its t t strength. This is actually observed in the experimental dependence (Fig. 1). 4. RESULTS AND DISCUSSION It should be noted that the temperatures of the para- Let us dwell once again on the expression for the magnetÐlongitudinal spin wave (TN1) and longitudinal EMAT efficiency in the case of sound generation in the spin waveÐcycloidal phase (TN2) transitions only longitudinal spin wave phase. Taking into account that slightly depend on the magnetic field (see the HÐT µ ӷ ζ the inequality t is almost universally satisfied [6] phase diagrams in [3, 4]). Consequently, the EMAT z efficiency peaks at H = 10 and 20 kOe in Fig. 1 are and using expression (3) for M0 , formula (13) for the observed virtually at the same temperatures. EMAT efficiency in the longitudinal spin wave phase The next intense EMAT efficiency peak at T ≈ 47 K can be rewritten in the form in the cycloidal phase in the field H = 10 kOe, accord- 2 2 2 2 2 ing to the HÐT phase diagrams obtained in [3, 4], can be c 3 γ H ω χ χ η = ------44 s d , (17) explained by the specific features of the static and S πρσ2 2()πχ 2 t 2 St 14+ d dynamic susceptibilities in the range of transition either χ χ from the commensurate phase with a wave number of where s and d are the static and dynamic susceptibil- 2/7 to the commensurate phase with a wave number of ities of the ferromagnetic metal, respectively. The static 3/11 [3] or from the commensurate phase with a wave susceptibility is determined from expression (3) as the number of 2/7 to the incommensurate phase [4]. z proportionality coefficient between M0 and H; that is, In the ferromagnetic spiral phase at T < TC, the mag- χ ()β γ 4 Ð1 netization M0 entering into formula (16) for the EMAT s = a Ð21 Ð q0 . (18) efficiency can be treated as nearly constant (equal to the The dynamic susceptibility relates the alternating mag- saturation magnetization at T 0) and independent netization and the alternating magnetic field and, of the external magnetic field. In this case, the specific according to relationships (3), (8), and (9), can be features in the EMAT efficiency in the ferromagnetic expressed by the formula spiral phase are likely associated with the characteristic properties of the dynamic susceptibility and the equi- χ ()β γ 4 ˜ 2 Ð1 θ d = Ða ++2 1 2 q0 +6bmeM1 . (19) librium angle between the resultant vector of the magnetization along the hexagonal axis and the exter- The experimental dependence of the EMAT effi- nal magnetic field vector. ciency in the longitudinal spin wave phase of erbium Formula (16) for the EMAT efficiency at µ ӷ ζ can (Fig. 1) can be explained in terms of the theoretical rela- t tionship (17) and expressions (18) and (19). As follows be represented in the form from relationship (17), the EMAT efficiency peak at the 2 2 2 2 2 c 3 γ ω M χ cos θ Néel point T = 87 K can be associated with the maxi- η  44 0 d N1 = ------2 2 2. (20) χ St πρσ ()πχ mum of the static susceptibility s [expression (18)]. 2 St 14+ d

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2122 BUCHEL’NIKOV et al. χ Here, the dynamic magnetic susceptibility d at efficiency in the longitudinal spin wave and ferromag- ω ӷ ω 2(k) (as was noted above, this inequality is well netic spiral phases [relationships (17) and (20)] allowed satisfied in the range of ultrasonic frequencies), accord- us to make the following inferences. ing to relationships (5) and (14), can be written as follows: Analysis of relationships (17) and (20) provides a qualitative explanation of the EMAT efficiency peaks Ð1 1 2 4 observed in the experiment (Fig. 1) in the range of para- χ = h 1 Ð --- sin θ + 2γ q . (21) d me42 0 magnetÐlongitudinal spin wave, longitudinal spin waveÐcycloid, and cycloidÐferromagnetic spiral phase The transition from the ferromagnetic spiral phase transitions. These peaks are associated with the specific to the cycloidal phase at T = TC leads to a decrease in features in the static and dynamic susceptibilities of the wave number q0 [1]. As a result, the susceptibility erbium in the vicinity of the above transitions. Unfortu- χ d sharply increases at the transition point, which man- nately, a quantitative comparison of the experimental ifests itself in an increase in the EMAT efficiency in the and theoretical results cannot be performed because of experimental dependence (Fig. 1). According to the the large number of unknown parameters entering into H−T phase diagrams [3, 4], the transition from the fer- relationships (17) and (20). A quantitative comparison romagnetic spiral phase to the cycloidal phase is calls for complex experimental measurements of these accompanied by the appearance of a small peak in the parameters at different temperatures and magnetic EMAT efficiency at T ≈ 27 K in the field H = 10 kOe fields. To the best of our knowledge, similar experi- and a peak in the EMAT efficiency at T ≈ 45 K in the ments with erbium crystals have not been performed to field H = 20 kOe. date. Moreover, quantitative comparison is complicated It can be seen from Fig. 1 that the ferromagnetic spi- by the fact that finite samples are usually used in exper- ral phase is characterized by one more EMAT effi- iments, whereas the theory is constructed for semi-infi- ciency peak in the field H = 10 kOe and, at the mini- nite single crystals. However, as was shown in [6], the mum, two more EMAT efficiency peaks in the field H = theory developed for semi-infinite crystals offers a 20 kOe. The peak in the field H = 10 kOe is observed at qualitative explanation for all the main regularities of the temperature T = 20 K. Analysis of the phase dia- the electromagneticÐacoustic transformation in ferro- grams obtained in [3, 4] (despite their certain disagree- magnetic metals. As follows from analyzing relation- ment) suggests that this peak can be associated with the ships (17) and (20) and Fig. 1, a similar statement is specific features in the characteristics of erbium in the true for the electromagneticÐacoustic transformation in range of the transition between the commensurate erbium single crystals. phase with a wave number of 5/21 and the incommensurate phase within the ferromagnetic spiral phase. In The other EMAT efficiency peaks observed in the the field H = 20 kOe, the first peak at the temperature T experiment can be associated with the specific features ≈ 38 K is very weakly pronounced and can be explained in the characteristics of erbium in the range of transi- by the specific feature in the susceptibility upon the tions either between two commensurate phases or transition from the incommensurate phase to the com- between commensurate and incommensurate phases mensurate phase with a wave number of 1/4 within the within the cycloidal and ferromagnetic spiral phases. In ferromagnetic spiral phase [3, 4]. Similarly, the second the present work, the phenomenological theory of the peak at the temperature T ≈ 27 K can be due to the spe- electromagneticÐacoustic transformation was devel- cific feature in the susceptibility upon the transition oped in the continuum approximation. This theory can- from the commensurate phase with a wave number of not describe the commensurability effects and, corre- 1/4 to the commensurate phase with a wave number of spondingly, the phase transitions between different 5/21 within the ferromagnetic spiral phase [3, 4]. commensurate and incommensurate phases in the cyc- Note that an increase in the external magnetic field loidal and ferromagnetic spiral phases [1]. In order to results in a decrease in the equilibrium angle θ between describe these transitions and the electromagneticÐ the magnetization and the field. Consequently, an acoustic transformation in the presence of commensu- increase in the magnetic field strength can lead to an rability effects, it is necessary to construct the micro- increase in the EMAT efficiency over the entire temper- scopic theory of electromagnetic generation of ultra- ≤ ature range T TC by virtue of the presence of the mul- sound in rare-earth metals, which is a special problem. tiplier cos2θ in relationship (20). This is also observed This problem remains unsolved because of its complex- in the experimental dependence (Fig. 1). ity. However, the comparison between the temperature dependences of the EMAT efficiency at different magnetic fields (Fig. 1) and the HÐT phase diagram of 5. CONCLUSIONS erbium [3, 4] allowed us to draw the conclusion that the Thus, the comparison of the experimental data on commensurability effects in the cycloidal and ferro- the EMAT efficiency for the Er rare-earth metal (Fig. 1) magnetic spiral phases can actually be responsible for and the theoretical expressions describing the EMAT the other EMAT efficiency peaks.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ELECTROMAGNETICÐACOUSTIC TRANSFORMATION 2123

ACKNOWLEDGMENTS REFERENCES This work was supported in part by the Ministry of 1. Yu. A. Izyumov, Neutron Diffraction by Long-Period Education of the Russian Federation, project no. E00- Structures (Énergoatomizdat, Moscow, 1987). 3.4-536. 2. R. A. Cowley and J. Jensen, J. Phys.: Condens. Matter 4, 9673 (1992). APPENDIX 3. D. F. McMorrow, D. A. Jahan, R. A. Cowley, et al., J. The equilibrium strain tensor in the longitudinal Phys.: Condens. Matter 4, 8599 (1992). spin wave phase has the form 4. H. Lin and M. F. Collins, Phys. Rev. B 45, 12873 (1992). U0 = 0 ()ik≠ , 5. D. Gibbs, J. Bohr, J. D. Axe, et al., Phys. Rev. B 34, 8182 ik (1986). γ Ð γ 2c γ U0 = Ð------33 31 M2 Ð ------13-U0 Ð ------1 -M2, 6. A. N. Vasil’ev and V. D. Buchel’nikov, Usp. Fiz. Nauk zz z xx z 162 (3), 89 (1992) [Sov. Phys. Usp. 35, 192 (1992)]. c33 c33 2c33 (A1) 7. A. V. Andrianov, A. N. Vasil’ev, Yu. P. Gaœdukov, et al., c c 0 0 33γ˜ 13γ˜ z z Fiz. Met. Metalloved. 64, 1036 (1987). Uxx ==Uyy Ð------∆ 1 Ð ------3 ∑MnMÐn, c33 n 8. A. V. Andrianov, V. D. Buchel’nikov, A. N. Vasil’ev, et al., Zh. Éksp. Teor. Fiz. 97, 1674 (1990) [Sov. Phys. γ γ γ γ γ γ γ γ ∆ where ˜ 1 = 13 Ð 12 + 1/2, ˜ 3 = 33 Ð 31 + 3/2, and = JETP 70, 944 (1990)]. c (c + c ) Ð 2c2 . 9. V. D. Buchel’nikov, I. V. Bychkov, and V. G. Shavrov, 33 11 12 13 Zh. Éksp. Teor. Fiz. 105, 739 (1994) [JETP 78, 396 The equilibrium strain tensor in the ferromagnetic (1994)]. ӷ spiral phase at qd 1 can be represented as follows: 10. C. M. Lim, S. Dixon, C. Edwards, and S. B. Palmer, J. c Phys. D 31, 1362 (1998). U0 ==U0 Ð------33-()γ Ð γ M2 sin2 θ xx yy 2∆ 11 12 0 11. C. M. Lim, C. Edwards, S. Dixon, and S. B. Palmer, J. Magn. Magn. Mater. 234, 387 (2001). 1 []()γ γ ()γ γ 2 2 θ Ð --- c33 13 Ð 12 Ð c13 33 Ð 31 M0 cos , 12. V. D. Buchel’nikov and V. G. Shavrov, Fiz. Tverd. Tela ∆ (Leningrad) 31 (5), 81 (1989) [Sov. Phys. Solid State 31, 765 (1989)]. 0 2c13 0 1 ()γ γ 2 2 θ Uzz = Ð------Uxx Ð ------33 Ð 31 M0 cos , (A2) 13. V. D. Buchelnikov, I. V. Bychkov, and V. G. Shavrov, J. c33 c33 Magn. Magn. Mater. 118, 169 (1993). γ 14. A. S. Borovik-Romanov, in Advances of Science: Anti- 0 44 2 θ Uxz = Ð------M0 sin2 cosqz, ferromagnetism and Ferrites (Akad. Nauk SSSR, Mos- 4c44 cow, 1962). γ 0 44 2 θ 0 Uyz ==Ð0.------M0 sin2 sinqz, Uxy 4c44 Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2124Ð2129. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2029Ð2034. Original Russian Text Copyright © 2002 by Shvachko, Starichenko, Shmatov, Gobov.

MAGNETISM AND FERROELECTRICITY

Width of the Ferromagnetic Resonance Line in an Anisotropic Magnet in Misoriented Resonance and Scanning Magnetic Fields Yu. N. Shvachko, D. V. Starichenko, G. A. Shmatov, and Yu. V. Gobov Institute of Metal Physics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoœ 18, Yekaterinburg, 620219 Russia e-mail: [email protected] Received February 11, 2002

Abstract—The dependence of the width ∆Hβ of the ferromagnetic resonance (FMR) line in anisotropic magnets on the angle of misorientation β between the static (resonance) and scanning magnetic fields was investigated both experimentally and theoretically. The change in the linewidth is due to the dependence of the equilibrium orientation of the magnetization vector on the direction of the magnetic field upon the passage through the resonance. Using iron garnet films as an example, it was shown that under such a misorientation (a nonzero angle β) the linewidth is smallest. It was also shown that a two-dimensional representation of FMR spectra, in contrast to one-dimensional angular dependences of the resonance parameters, contains full information on the spectral characteristics of the film, including noncollinear field configurations. © 2002 MAIK “Nauka/Interpe- riodica”.

1. INTRODUCTION The physical cause of the change in the linewidth ∆H in films of anisotropic ferromagnets is the change The width of the line of ferromagnetic resonance pp in the orientation of the equilibrium magnetization M (FMR) measured in the process of scanning of an external magnetic field H is determined as a range of fields during the registration of the absorption line, as was shown in [2] and [3]. The calculations performed in ∆H = H Ð H (1) these works showed that the width of the FMR line pp p+ pÐ ∆ Hpp increased with the deviation of the magnetic field from the anisotropy axis (directed along the normal to whose initial and final points, H p+ and H pÐ , respectively, correspond to the positions of the maximum and the film). The linewidth reaches maximum values in the vicinity of angles ϕ corresponding to the ranges of the minimum of the first derivative dP/dH of the signal of ϕ absorption of the high-frequency field P(H) (the point fastest variation of the function HR( ). The increase in of the maximum slope). In this case, the magnitude of the linewidth obtained in the above-cited works is related to the fact that they were concerned with a spe- the resonance field HR, determined as the point of passage of the dP/dH signal through zero, always lies in cial case of orientation of the magnetic and scanning fields, namely, the fields were collinear. We may sup- the range of values H p+ < HR < H pÐ . In the traditional pose that in the case of misorientated resonance and process of measuring an FMR linewidth, the directions scanning fields the above mechanism of the linewidth of the scanning field hsw and resonance field HR are variation will lead not only to a broadening but also, at coincident. In other words, the signal is detected due to certain angles, to a narrowing of the FMR line. a change in only the amplitude of the external field in the vicinity of the resonance value. In this method of In this paper, we analyze the width of the FMR line ∆ β measurement of the linewidth, the orientation of the Hβ measured at a certain angle to the static field HR magnetic field with respect to the anisotropy axes rather than along this field. In this case, the amplitude remains unaltered in the process of measurement. of the scanning field changes in the range [–hsw, +hsw] ∆ along a new direction and the resulting field H = H + It is known that the linewidth Hpp determined by R this method is independent of the angle ϕ that specifies hsw changes in both magnitude and direction. It follows the orientation of the resonance magnetic field relative from an analysis of the experimental dependence of the ∆ ϕ FMR linewidth ∆Hβ in the mixed iron garnet on the to the anisotropy axes. This function Hpp( ) is nonmonotonic; it has local maxima and minima whose misorientation angle β and on the direction of the mag- number and positions are determined by the type of netic field (ϕ) that the line width ∆Hβ proves to be ∆ anisotropy and the magnitude of the corresponding smaller than the conventionally measured value Hpp. constants [1]. It was shown experimentally and substantiated theoret-

WIDTH OF THE FERROMAGNETIC RESONANCE LINE 2125 ically that a two-dimensional representation of the signal P(H, ϕ) of FMR absorption contains full informa- OZ OY tion on all possible values of ∆Hβ and, thus, permits one h to determine an optimum configuration of the fields and sw the smallest value of the linewidth. Ms +hsw ϕ 2. EXPERIMENTAL β θ As an object of investigation, we used a film HR of mixed iron garnet of composition µ (Y1.15Eu0.5Bi0.5Ca0.85)(Fe4.15Si0.85)(O12) 2.92 m thick grown by liquid-phase epitaxy. The values of the uniaxial anisotropy constant and quality factor were K = 1.696 × 104 erg/cm3 and Q = 4.3, respectively. The film composition was controlled by the weight ratio of the –hsw components in the initial charge. As a substrate, we used a gallium gadolinium garnet sample 0.5 mm thick with a surface (111) on which a thin film of yttrium iron Fig. 1. Geometry of the experiment (for details, see the µ main text). The rotation of the film occurs about the OX garnet (YIG) 0.2 m thick was applied. The narrow line axis. The vector h directed at an angle β to the field H = of YIG was not observed in the FMR spectra of the sw HR corresponds to the effective scanning field under misori- sample. The application of an YIG layer ensures a entation conditions (in the figure, corresponding parallel higher time and temperature stability of the two-layer and perpendicular projections of hsw are indicated). system and suppresses the formation of hard bubble domains. As is known, it is precisely those films with an easily modified composition that are the basis of mag- In the process of measurements, each spectrum was netic electronics [4]. Therefore, obtaining additional integrated (dP/dH P(H)) and was plotted in the information on the resonance characteristics under the form of a set of experimental points in a cylindrical conditions of noncollinear magnetic fields is of great coordinate system (ρ, ϕ, Z) (H, ϕ, P). Then, for importance. The problems of the interaction of modes each set of points with a fixed parameter H = const, we in magnetically coupled two-layered systems and the performed a procedure of smoothing of the dependence techniques of measuring magnetic parameters from of P on the angle ϕ and subsequent approximation of FMR data have been discussed in detail earlier in [5, 6]. the entire set of data in the form of a single surface ϕ A sample (1 × 0.5 mm) was mounted on a vertical P(H, ). Based on realistic spectra, we used a Lorentz- stock at the antinodal point of the magnetic component ian shape of the line when approximating the field depen- of the microwave field of a TE resonator. The FMR dence. As a result, the experimental set of 75000 points 102 was described by a single surface in a relatively com- spectra were recorded using a standard CW homodyne pact form. The general appearance of the surface for EPR spectrometer in the X range (9.45 GHz). The (Y Eu Bi Ca )(Fe Si )(O ) is represented derivative of the FMR signal dP/dH was recorded at 1.15 0.5 0.5 0.85 4.15 0.85 12 power levels p < 2 mW every 5° in the range of ϕ = 0°Ð in Fig. 2. 360°. All measurements were performed at room tem- Note that in the general case (complex spectra, perature. anisotropies of various types, multilayered objects), the The geometry of the experiment is shown in Fig. 1. procedure of the choice of a single surface may prove The axis of the sample rotation perpendicular to the fig- to be inefficient because of a significant complexity. In ure plane coincides with the OX axis, and the direction this case, it appears to be more expedient to deal with of the normal to the film coincides with the OZ axis. As an original set of data in the form of a 3D grid (a three- the film is rotated, the angle ϕ between the field direc- dimensional representation of a discrete set of data) and tion H and the OZ axis changes. Upon analyzing the to perform the approximation using a smooth surface in R the vicinities of singular points. FMR spectra under misorientation conditions, a configuration is assumed in which the direction of scanning of Note also that the method of recording in which a β the field hsw makes an angle with the direction HR. double scanning of the external field is performed for The magnetization vector lags behind the direction of each fixed value of the azimuthal angle may also prove the external field, which is characterized by the angle θ. inefficient for the 3D representation of angular depen- It is shown in the work that in the two-dimensional rep- dences. A fundamental advantage may be given by a resentation of the spectra, we can always determine a method in which the sample rotates about the OX axis variety of points corresponding to the FMR signal in such a way that the sample is rotated by 360° in a sin- under conditions of misoriented fields with an arbitrary gle elementary interval (step) of sweeping of the mag- angle β. netic field. In this case, a large number of points at var-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2126 SHVACHKO et al. the sample and the plane YOZ, this procedure yields full information on the resonance properties of a magnet in spherical coordinates in the experiment with a single scanning of the magnetic field. In turn, the passage from the 1D spectra to a three-dimensional “portrait” Y permits one to pass to a qualitatively new analysis of ϕ the resonance properties of magnets from the viewpoint HR hsw Z β of the topological features of the corresponding equipotential surfaces and their sections. From the methodological viewpoint, the P(H, ϕ) P(H, ϕ) surface shown in Fig. 2 represents a special case (section) of a 3D portrait. In turn, the case of a misoriented resonance and scanning fields, as can easily be shown, represents the intersection of this surface with a vertical β ϕ plane that passes at an angle to the plane (HR( ), OX). In the general case, this plane can be inclined to the YOZ plane. In this case, the resultant curve represents an FMR spectrum under conditions of a noncollinear ϕ configuration. At fixed values of (HR, ), the points of the new spectrum P(h, β) can be determined numerically by changing the coordinates into P(H', ϕ') according to the following rule (Fig. 2): Hy 2 2 2 β H ' = HR ++hsw 2HRhswcos , (3)

Hz ϕ ϕ ()β ' = Ð arcsin hswsin . (4) It is seen that for anisotropic magnets, the dependence Fig. 2. Representation of the experimental angular ∆ β dependence of the FMR spectrum P(H, ϕ) for of the linewidth Hβ on the angle is nonmonotonic, (Y1.15Eu0.5Bi0.5Ca0.85)(Fe4.15Si0.85)(O12) in the form of an exhibiting a local minimum with smallest values at cer- approximated surface in cylindrical coordinates (H, ϕ, P). tain angles ϕ that are different for different magnets. The conventional absorption signal is a section of the surface along the radius vector HR. The FMR signal under misoriented conditions corresponds to a section in the vicinity 3. PRINCIPAL EQUATIONS of H = H along the field h . R sw In this section, we present a model that permits one to calculate the ∆Hβ dependences and compare the experimental data with the results of calculation. It is ious angles could be recorded in the course of a single known that in films of mixed iron garnets, the contribu- act of sweeping with a field. The duration of the sweep- tion to the angular dependence of HR comes from both ing, which is determined by the inertia properties of the uniaxial (growth-related) and cubic (crystallographic) magnetic system, is usually T = 3Ð10 min, which means magnetic anisotropies. The applicability of the model that the duration of an elementary step is ∆T ≈ 0.2 s. At of a uniaxial ferromagnet to samples with a cubic the same time, it can easily be ensured that the sample anisotropy is based on the fact that the constant of cubic is rotated at rates Ω ≤ 1 sÐ1, so that during a rotation of anisotropy in the sample at hand was smaller by an the sample by 360° only a single scanning with the field order of magnitude than that of uniaxial anisotropy. is required at each sweeping step. If the condition The dependence of the resonance field HR on the angle ϕ that characterizes its orientation relative to the Ω Ӷ ()τ Ð1 Ӷ π 2 f mod (2) anisotropy axis is known to be determined from the τ equation for the natural frequency of uniform vibra- is fulfilled, where is the time of reading (with an ana- tions of magnetization [7]. In the model of a uniaxial logÐdigital converter) and fmod is the frequency of ferromagnet, the magnitude of the resonance field is modulation of the field, the admissible number of val- found from the equation ues (angles) is N ~ 1/Ωτ. As a result, the total time of Ð1 2 the experiment at Ω ≤ ∆T does not exceed the duration ω 2 ------R = []H cos()θϕÐ + H cos θ of the recording of a single spectrum. At ΩÐ1 = 1 s, the γ R A (5) time of an experiment increases by a factor of Ω∆T = 5 × []()θϕ ()θ to be 15 min. HR cos Ð + H A cos 2 , ω γ It can easily be shown that, upon successive varia- where R is the resonance frequency, is the gyromag- tion of the angle between the normal to the surface of netic ratio, θ is the angle of the equilibrium orientation

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 WIDTH OF THE FERROMAGNETIC RESONANCE LINE 2127 of the magnetization vector M in the field HR reckoned 5000 π H (ϕ) from the anisotropy axis, HA = 2K/M Ð 4 M is the effec- p+ tive field of anisotropy, K is the uniaxial anisotropy H (ϕ) 4000 R constant, and M is the saturation magnetization. The ϕ angle θ is found by minimizing the free energy of the Hp–( ) system, which, in the case of a uniaxial ferromagnet, leads to the relations 3000 , Oe θ ()θϕ y H A sin2 +0, 2HR sin Ð = H (6) 2000 θ ()θϕ> H A cos2 + 2HR cos Ð 0. Thus, at a given angle ϕ of the orientation of the 1000 magnetic field and the parameters of the problem ω γ θ ϕ ( R, K, M, ), we can find HR and can be found from Eqs. (5) and (6). –2500 –1500 –500 500 1500 2500 The method of the calculation of the linewidth ∆Hβ Hz, Oe is described in [8], where, using the example of a uniaxial ferromagnet, the following expression was obtained Fig. 3. Experimental angular dependences of three for the width of the FMR line measured at an arbitrary parameters of the FMR spectrum for angle β to the direction of the resonance field: (Y1.15Eu0.5Bi0.5Ca0.85)(Fe4.15Si0.85)(O12) in polar coordi- ϕ ()ϕ ()ϕ nates (H, ): maximum H p+ , minimum H pÐ , and ∆ ∆ ϕ Hβ = Fβ H pp, (7) the values of the resonance field HR( ) corresponding to the first derivative of the absorption signal dP/dH in the angular where range 0° ≤ ϕ ≤ 180° (the angle ϕ is shown in Fig. 1).

Ð1 Fβ = ()cosβ + Gsinβ , (8) the linewidth ∆H . Namely, at ϕ = 0 (the resonance 1 dHR ()ϕθ { ()θϕ pp G ==------ϕ tan Ð + sin Ð field parallel to the anisotropy axis), we have HR d (9) ω θ[]ϕ()ϕ 3 θ Ð1 } ∆ ≡ ∆ α R + hsin2 1 + sin 2sin Ð hsin H|| H pp()0 ==H pÐ Ð2H p+ ------γ . (11) ×θϕ[]() θ Ð1 cos Ð + hcos2 If ϕ = π/2 (the resonance field perpendicular to the ∆ anisotropy axis), we obtain (h = HA/HR). Thus, the linewidth Hβ is expressed through the known quantity ∆H and the function Fβ ∆ ≡ ∆ π pp H⊥ H pp()/2 depending on the angles β and ϕ and on the ratio H /H . A R ω 2 Ð1/2 (12) In this case, the equilibrium angle of the magnetization α RhA θ = H pÐ Ð2H p+ = ------1 + ----- . and the magnitude of the resonance field HR entering γ 4 into Eq. (8) can be found from the set of equations (5) ω γ and (6). Here, hA = HA/( R/ ) is the normalized anisotropy field. ∆ It follows from Eq. (12) that if the magnetic field is per- The method of calculating Hpp was described in pendicular to the anisotropy axis, then, irrespective of ϕ [3], where it was shown that the fields H p+() and the sign of the effective field HA, the linewidth ϕ H pÐ() entering into the definition of the linewidth decreases as compared to the isotropic case. ∆ ϕ Hpp( ) (Eq. (1)) can be found from the set of equations (5) and (6), in which one should substitute the fre- 4. DISCUSSION quencies In the case of a single FMR line of a Lorentzian ω ()ωα ω ()ωα p+ ==1 Ð R, pÐ 1 + R (10) shape, the analysis of the P(H, ϕ) dependences is significantly simplified. Figure 3 displays experimentally ω α for the resonance frequency R ( is the damping obtained angular dependences of the three fields parameter). In other words, the linewidth ∆H is found ϕ ϕ ϕ ° ≤ ϕ ≤ ° pp H p+(), HR( ), and H pÐ() in the range 0 180 . by doubly solving the set of equations (5), (6) with In this figure, the angle ϕ is a parameter and the coordi- respect to the fields H p+ and H pÐ . For the angles 0 < nate axes correspond to the projections of the external ϕ < π/2, this set of equations can be solved only numer- field onto the axes OZ and OY, respectively. These ically. However, for two directions (ϕ = 0, π/2), as was curves also contain information on the angular depen- ∆ ϕ shown in [8], we can obtain analytical expressions for dence of the linewidth Hpp( ); namely, the linewidth

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2128 SHVACHKO et al.

5500 Figure 4 compares the experimental data with the ϕ HR( ) dependence calculated according to Eqs. (5) and 5000 (6). As the parameters of the theory, we also used experimentally measured quantities (for the numerical values of the parameters, see the caption to Fig. 4). As might 4500 be expected, the model of the uniaxial ferromagnet sat-

, Oe isfactorily describes experimental data in the case of a R

H 4000 small constant of cubic anisotropy. Figure 5 displays the experimentally obtained data (points) and the results calculated in accordance with 3500 the above model (lines) for the dependences of the linewidth on the misorientation angle β between the reso- 3000 nance and scanning fields. When comparing them, we 0 60 120 180 240 300 360 ϕ should take into account that the error of the experi- Angle mental measurement of the width of the FMR line is about 5%. In the case of a single symmetrical line, there Fig. 4. Experimental (points) and theoretical (line) depen- ϕ is no need to construct sections of the surface for each dences of the resonance field HR( ). The magnitudes of the ϕ β ∆ ω π ( , ) set (Fig. 2), since the magnitudes of Hβ coincide parameters of the model are as follows: fR = R/2 = ω γ with the distance [H ()ϕ + H ()ϕ ] intercepted by 9.5 GHz, R/ = 4.31 kOe, M = 25 G, and HA = 1.04 kOe. pÐ 1 p+ 2 ϕ a straight line passing through the point HR( ) and the origin at an angle β to the radius vector (Fig. 3). For 1000 more complex spectra, this simplification is invalid and 165 we should perform the procedure of intersecting the surface described above. The width of the FMR line 160 observed experimentally along the resonance field var- 155 ied from 133 to 152 Oe. 150 In Fig. 5, both the theoretical and experimental 145 curves are given for two directions of the magnetic field with respect to the anisotropy axis. Cases 1 and 2 cor- 140 respond to the angles ϕ = 60° and 90°. It is seen from 135 the figure that if ϕ = 90° (the resonance field lies in the 130 film plane), then the minimum width of the FMR line is –40–30–20–10 0 10 20 observed at β = 0; i.e., the minimum width corresponds to the linewidth traditionally measured along the reso- Line width, Oe 400 nance field. However, as follows from the experiment 300 and calculations, when the resonance field deviates by 30° from the film plane (ϕ = 60°), the minimum line- β ° 200 width is observed at the misorientation angle = 20 . Note that although the minimum linewidth for the 100 (Y1.15Eu0.5Bi0.5Ca0.85)(Fe4.15Si0.85)(O12) film at hand decreases insignificantly (by 10%), a significant fact is that the minimum linewidth is observed upon misorien- –80 –60 –40 –20 0 20 40 60 80 tations between the resonance and scanning fields as Angle β small as 20°. The latter means that we can select such a geometry of the resonance and scanning fields for the Fig. 5. Variation of the FMR linewidth as a function of the fields to be misoriented up to angles close to 90° but the misorientation angle between the resonance and scanning linewidth to increase only insignificantly. This fact is, fields at α = 0.018. Other parameters: ϕ = 60° (squares, in particular, important from the applied viewpoint, experiment); ϕ = 90° (rhombi, experiment); lines show the corresponding calculated dependences. since it offers additional opportunities, e.g., for the development of devices of microwave electronics based on FMR of spin waves. at a fixed ϕ is numerically equal to the distance between the first and third curves along the position vector 5. CONCLUSION ϕ HR( ). The slight asymmetry of the curves in quadrants Thus, in this paper, we established, using the exam- I and II is due to the deviation of the easy axis from the ple of an iron garnet film, that the minimum width of normal to the surface and to the contribution from cubic the FMR line in anisotropic magnets is realized in the anisotropy. case of a noncollinear configuration of the resonance

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 WIDTH OF THE FERROMAGNETIC RESONANCE LINE 2129 and scanning magnetic fields. Information on the opti- additionally miniaturize the devices of microwave elec- mum configuration (on the angles ϕ and β) is contained tronics. in the two-dimensional representation of the FMR ϕ spectra in the form of a P(H, ) surface. The determi- ACKNOWLEDGMENTS nation of this configuration reduces to seeking the section of this surface that yields a line with the smallest We are grateful to A.V. Kobelev for fruitful discus- width. Such an analysis can be useful for the investiga- sions. The work was supported in part by the Russian tion of the interaction of modes in two-layer and multi- Foundation for Basic Research, project no. 00-15- layer magnetic films. Note that the passage from 1D 96745. spectra to a three-dimensional portrait permits us to go to a qualitatively new analysis of resonance properties REFERENCES of magnets from the viewpoint of topological features 1. G. V. Skrotskiœ and L. V. Kurbatov, in Ferromagnetic of the corresponding equipotential surfaces and their Resonance (Fizmatgiz, Moscow, 1961), p. 25. sections. 2. F. G. Bar’yakhtar, V. L. Dorman, and N. M. Kovtun, Fiz. Tverd. Tela (Leningrad) 26 (12), 3646 (1984) [Sov. In addition, the results obtained permit us to suggest Phys. Solid State 26, 2192 (1984)]. a new method of controlling the width of the FMR line 3. A. M. Zyuzin, Fiz. Tverd. Tela (Leningrad) 31 (7), 109 in anisotropic magnets, which consists in variation of (1989) [Sov. Phys. Solid State 31, 1161 (1989)]. the orientation of the scanning field with respect to the 4. A. H. Eschenfelder, Magnetic Bubble Technology resonance field. Such a variation can be sufficiently (Springer, Berlin, 1980; Mir, Moscow, 1983). simply realized by applying, in the presence of a con- 5. A. V. Kobelev, V. P. Gogin, V. A. Matveev, et al., Zh. stant resonance field, an additional (scanning) magnetic Tekh. Fiz. 59 (2), 95 (1989) [Sov. Phys. Tech. Phys. 34, field at a certain angle with respect to the resonance 185 (1989)]. field. This method of controlling the width of the FMR 6. A. V. Kobelev, M. V. Perepelkina, A. A. Romanyukha, line is of practical importance. For example, in the et al., Zh. Tekh. Fiz. 60 (5), 117 (1990) [Sov. Phys. Tech. devices of microwave electronics, the resonance field is Phys. 35, 605 (1990)]. most frequently created with dc magnets, whereas the 7. A. G. Gurevich and G. A. Melkov, Magnetic Oscillations controlling (scanning) field is produced with coils. In and Waves (Nauka, Moscow, 1994), p. 55. such a situation, it is sometimes technologically conve- 8. G. A. Shmatov and Yu. L. Gobov, Fiz. Met. Metalloved. nient if the resonance and controlling fields are misori- 88 (4), 16 (1999). ented, but in such a way that the linewidth does not increase significantly. In particular, this permits one to Translated by S. Gorin

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2130Ð2144. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2035Ð2048. Original Russian Text Copyright © 2002 by Golovenchits, Sanina, Levin, Shepelev, Smolin.

MAGNETISM AND FERROELECTRICITY

Thermal Vibrations and the Structure of Quasi-Two-Dimensional R2CuO4 Crystals (R = La, Pr, Nd, Sm, Eu, and Gd) E. I. Golovenchits1, V. A. Sanina1, A. A. Levin2, 3, Yu. F. Shepelev2, and Yu. I. Smolin2 1 Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] 2 Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia 3 Institute of Crystallography and Solid-State Physics, Technical University, Dresden, D-1062 Germany Received January 18, 2002

Abstract—Thermal vibrations of ions in R2CuO4 crystals (R = La, Pr, Nd, Sm, Eu, Gd) were studied by x-ray diffractometry. A comparative analysis of thermal displacements of the copper and rare-earth ions permitted a conclusion as to the main interactions responsible for the structural state of the CuO2 sheets and of a crystal as a whole. The structural properties were found to correlate with the magnitude of the ionic radius and with the ground state of the rare-earth ions. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION La2CuO4 and Gd2CuO4 undergo phase transitions from Thermal vibrations of ions in a crystal are deter- the high-temperature tetragonal to the low-temperature mined by the symmetry and strength of the local crys- orthorhombic phase (see [3] and [4], respectively). A tal-field potential. This permits one to obtain informa- common feature in the structure of the La2CuO4 and tion on the local potential by studying the spatial distri- R2CuO4 crystals with RE ions is its quasi-2D character. bution of thermal ion displacements using diffraction The ions are arranged in layers perpendicular to the c methods. Our previous communications [1, 2] reported axis. Layers of R3+ and O2Ð ions are sandwiched 2+ on an investigation of thermal vibrations of the Cu between the CuO2 planes. The distances between the and rare-earth (RE) ions in R2CuO4 crystals (R = La, CuO2 sheets and the layers containing copper and RE Eu, Pr, Gd). Main attention was focused on the temperature dependence of the probability density function (PDF), which characterizes the spatial distribution of equally probable displacements of ions relative to their equilibrium positions. We succeeded in following the R3+ temperature-induced local structural distortions and 2– structural phase transitions in the crystals under study. O The purpose of this study was to compare the PDFs Cu2+ of the Cu2+ and RE ions at a fixed temperature for a broader class of R2CuO4 crystals (R = La, Pr, Nd, Sm, Eu, Gd) and to follow the influence of the RE ions on the properties of the CuO2 sheets and on the crystal structure as a whole. To do this, we studied, in addition to data obtained earlier, thermal vibrations in Nd2CuO4 and Sm2CuO4 crystals (at temperatures of 296 and 373 K). The analysis of the PDF for the Cu2+ and RE ions was carried out for the whole class of R2CuO4 crystals (R = La, Pr, Nd, Sm, Eu, Gd) at room temperature, taking into account the earlier data reported in [1, 2]. TT' The structure of the La2CuO4 and R2CuO4 crystals (a) (b) with RE ions (R = Pr, Nd, Sm, Eu, Gd) is presented in Fig. 1. The R CuO crystals (R = Pr, Nd, Sm, Eu) have 2 4 Fig. 1. Crystal structure of (a) La2CuO4 and (b) R2CuO4 T' structure (space group I4/mmm) at all temperatures. (R = Pr, Nd, Sm, Eu, Gd).

THERMAL VIBRATIONS AND THE STRUCTURE 2131 ions exceed by far those between the in-plane ions. We A study of the spin and lattice dynamics in R2CuO4 are interested primarily in the properties of CuO2 layers crystals (Pr, Sm, Eu), carried out in the millimeter range separated at a distance of c/2. The nearest environment at low temperatures, revealed jumps in the microwave 2+ of Cu ions in La2CuO4 differs from that in R2CuO4 power absorption which were assigned to structural with RE ions. In La CuO , we have octahedra of O2Ð phase transitions [9]. A structural phase transition in 2 4 ≈ Eu2CuO4 at T 150 K was also observed to occur using ions (coordination number Z = 6), and in R2CuO4 with RE ions, we have squares of oxygen ions (Z = 4). The other techniques [10Ð12]. The properties of the low- structural properties of the crystals under study are temperature phases and the transition points in R2CuO4 (Pr, Sm, Eu) were different for different crystals and are determined by in-plane interactions in the CuO2 sheets and the layers containing RE ions, as well as by inter- most likely determined by the RE ions. layer coupling (primarily between the Cu2+ and R3+ Our present data show that the structural properties ions). We study here the main interactions responsible of the CuO2 layers and of the crystal as a whole for for the structural properties of the crystals and their R2CuO4 crystals with RE ions depend on the RE ions dependence on the properties of the RE ions. present in the crystals. The cubic crystal-field potential is usually ~1 eV. The tetragonal distortions of the cubic lattice are related 2. EXPERIMENTAL RESULTS to a change in this potential by about ~0.1 eV. The anomalously strong 2D Heisenberg exchange interac- We carried out a complete x-ray diffraction charac- tion between the Cu2+ ion spins (exchange constant J ~ terization of the Sm2CuO4 and Nd2CuO4 crystals at 0.13Ð0.15 eV [5]) and the spinÐorbit (SO) coupling of temperatures of 296 and 373 K. The measurements Cu2+ ions with a constant λ ~ 0.06Ð0.08 eV [6] are prac- were performed on single crystals grown by the sponta- tically of the same magnitude in the crystals under neous-crystallization method described in [2]. The 2+ crystals were stoichiometric in composition and study. As a result, Cu ions in the R2CuO4 crystals with RE ions are involved in a number of interactions of sim- dielectric. ilar magnitude, which determine various, including The samples used for the diffractometric studies structural, properties of the crystals. were rectangular prisms measuring 0.17 × 0.18 × × × As shown in [1, 2], the character of the PDF and, 0.11 mm (Sm2CuO4) and 0.12 0.11 0.11 mm hence, the symmetries of the local crystal-field poten- (Nd2CuO4). All measurements were taken on the same 2+ samples and in the same setup. tial of the Cu and RE ions in R2CuO4 crystals (R = La, Pr, Eu, Gd) are different. However, local distortions in A preliminary diffractometric investigation showed CuO2 layers were found to be dominant in all these the Sm2CuO4 and Nd2CuO4 crystals to have tetragonal crystals in the temperature interval 140Ð400 K. As fol- symmetry (space group I4/mmm). The x-ray reflection lows from the present study, distortions in these layers intensities were measured up to the value sinΘ/λ = Ð1 are also dominant in Nd2CuO4. Only in Sm2CuO4 at 1.075 Å on an automated three-circle, single-crystal room temperature do JahnÐTeller distortions for the diffractometer in the normal-inclination, layer-by-layer Sm3+ ions dominate and is the character of the copper ω scanning mode by properly rotating the crystal about λ ion PDF determined primarily by the Sm3+ÐCu2+ inter- the a axis. The MoKα radiation used ( = 0.71069 Å) action. was passed through a graphite monochromator. The crystal structures were refined by least-squares Thermal vibrations in Pr2CuO4 at room temperature were probed using neutron scattering [7]. A comparison techniques in the block-matrix approximation, basically, by the scheme employed in [1, 2]. The parameters of the measurements carried out on Pr2CuO4 and CuO crystals suggests [7] that the dynamic behavior of cop- of the cation temperature factors were refined in the per ions in Pr CuO is a result of interaction mainly fourth-order anharmonic approximation. The data thus 2 4 obtained were used to construct the ion probability den- with the nearest neighbor oxygen atoms. At the same sity functions, which are Fourier transforms of the cor- time, Raman scattering measurements performed on a responding temperature factors. To reduce the regions number of R CuO crystals (R = Pr, Nd, Sm, Gd), also 2 4 of negative values of the PDF which form when refin- at room temperature, suggest that the RE ions have a ing anharmonic parameters by least-squares tech- substantial effect on phonon spectra. It has been niques, we employed the GramÐCharlier model of the pointed out that the part played by the CuÐR coupling temperature factor [13]: increases with decreasing RE ionic radius [8]. Gd CuO occupies a special place among the [ ()π 3 2 4 T()h = T harm()1h + 2i /3!cpqrh phqhr R2CuO4 crystals with RE ions. This crystal undergoes a (1) ()π 4 ] structural phase transition to the low-temperature +2i /4!d pqrsh phqhrhs . orthorhombic phase (T < 650 K) [4]. The orthorhombic β β distortions in the CuO2 layers originate from the oxy- Here, Tharm(h) = exp(Ð pqhphq); pq are anisotropic har- gen OI ions rotating about the copper ions [4]. monic temperature parameters; cpqr and dpqrs are the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2132 GOLOVENCHITS et al.

Table 1. Details of the experiment and structural parameters of R2CuO4 single crystals at room temperature a Parameter La2CuO4 Pr2CuO4 Nd2CuO4 Sm2CuO4 Eu2CuO4 Gd2CuO4 Space group Abma I4/mmm I4/mmm I4/mmm I4/mmm I4/mmm a, Å 5.397(3) 3.953(2) 3.944(1) 3.921(1) 3.897(3) 3.895(1) b, Å 5.365(3) aaaaa c, Å 13.165(2) 12.232(3) 12.147(2) 11.994(2) 11.905(2) 11.8952(6) σ b Nrefl (I > 3 (I)) 640 250 198 170 285 250 R, %c 3.23 2.79 1.79 2.09 4.00 2.79 d RW, % 3.34 2.24 2.37 2.27 4.19 2.95 µ, cmÐ1e 275.3(3) 304.3(1) 324.9(3) 370.6(3) 400.8(4) 423.3(1) f rext , Å 1835(14) 3288(15) 2082(17) 5564(44) 7868(49) 2897(35) a Note: La2CuO4 is twinned pseudomerohedrally along the (110) planes. Extinctions of the twin-domain reflections satisfy the Abma and Bmab space groups. Least-squares refinement yields for the twinning coefficient Ktw = 0.68(2), which corresponds to the 2 : 1 ratio for the components described by lattices A and B; b Nrefl is the number of independent nonzero reflections; c R is the structural divergence factor; d RW is the weighting factor of divergence according to the Cruckshank scheme [14]; e µ is the absorption coefficient (for MoKα). The absorption was taken into account by numerical integration over the crystal volume; f r is the least-squares value of the extinction parameter. The extinction was taken into account using the method of Becker and ext Ӷ λ Coppens [15] in isotropic approximation for crystals with rext g and a Lorentzian mosaic-block distribution.

2+ Table 2. Reﬁned room-temperature parameters for Cu (0, 0, 0) ions in R2CuO4 crystals (R = La, Pr, Nd, Sm, Eu, Gd) Parameter La Pr Nd Sm Eu Gd 2 Beq, Å 0.67(1) 0.439(6) 0.43(2) 0.48(1) 0.42(1) 0.45(1) U11, Å 0.085(1) 0.0861(3) 0.055(1) 0.075(1) 0.072(2) 0.072(1) U22, Å 0.088(2) U11 U11 U11 U11 U11 U33, Å 0.101(1) 0.079(1) 0.100(4) 0.084(4) 0.075(2) 0.075(2) U13, Å 0.006(30) 00000 × 9 d1111 10 154(49) 130(148) Ð172(141) 1065(794) Ð19(305) 305(262) × 9 d2222 10 Ð686(69) d1111 d1111 d1111 d1111 d1111 × 9 d3333 10 Ð0.4(7) 0.3(10) 39(14) Ð5(2) Ð5(2) Ð7(21) × 9 d1113 10 2(8) 00000 × 9 d1333 10 1(1) 00000 × 9 d1122 10 73(18) 325(71) 82(63) Ð424(129) Ð321(163) 649(132) × 9 d1133 10 13(3) Ð8(8) Ð41(17) Ð17(19) 69(16) 58(16) × 9 d2233 10 Ð23(4) d1133 d1133 d1133 d1133 d1133 × 9 d1223 10 Ð1(7) 00000 β β π 2 Note: Uii are RMS temperature-induced atomic displacements (harmonic part of the temperature factor); Uij = sgn( ij)sqrt( ij/(2 ) aiaj); Σβ dpqrs are fourth-order anharmonic temperature parameters; Beq is the equivalent isotropic temperature factor; Beq = 4/3 ijaiaj, β where ij are anisotropic temperature factors. anharmonic temperature parameters of the third and temperature factors [13]. The cation PDFs obtained in fourth order, respectively; and h is the scattering vector both models of the anharmonic temperature factor are with components (h1, h2, h3) = (h, k, l). qualitatively similar. The same method was applied to reﬁne the room- The experimental and main structural parameters of temperature structural parameters of the La2CuO4 and the crystals studied are listed in Tables 1Ð5. Figures 2 2+ Eu2CuO4 crystals studied by us earlier [1] by using the and 3 display the PDFs of the Cu ions in R2CuO4 crys- Edgeworth anharmonic approximation of the cation tals (R = La, Pr, Nd, Sm, Eu, Gd) calculated for room

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3+ Table 3. Reﬁned room-temperature parameters for R ions (x/a, 0, z/c) in R2CuO4 crystals (R = La, Pr, Nd, Sm, Eu, Gd) Parameter La Pr Nd Sm Eu Gd x/a 0.00227(4) 00000 z/c 0.36136(1) 0.35132(2) 0.35149(4) 0.35057(3) 0.35009(3) 0.34922(3) 2 Beq, Å 0.718(4) 0.522(2) 0.556(7) 0.494(3) 0.389(4) 0.478(3) U11, Å 0.0938(5) 0.0861(3) 0.0578(4) 0.0850(4) 0.0758(6) 0.0832(4) U22, Å 0.1126(6) U11 U11 U11 U11 U11 U33, Å 0.0761(6) 0.0707(4) 0.1201(13) 0.0658(7) 0.0575(10) 0.0655(7) U13, Å 0.057(13) 00000 × 8 c111 10 14(13) 00000 × 8 c333 10 Ð2(5) Ð0.7(8) Ð20(6) Ð2.4(15) Ð2(1) Ð1.4(14) × 8 c122 10 Ð5(10) 00000 × 8 c113 10 8(4) Ð19(10) 6(12) Ð1(160) Ð19(16) Ð25(16) × 8 c133 10 1(1) 00000 × 8 c223 10 1(5) c113 c113 c113 c113 c113 × 9 d1111 10 Ð309(7) 786(49) Ð139(47) 887(242) 491(95) 765(78) × 9 d2222 10 Ð1410(26) d1111 d1111 d1111 d1111 d1111 × 9 d3333 10 Ð0.2(2) 1.5(3) 135(7) Ð3.6(5) 1.6(6) Ð3.4(5) × 9 d1113 10 Ð2(3) 00000 × 9 d1333 10 0.5(5) 00000 × 9 d1122 10 363(6) Ð513(24) 67(22) Ð363(62) Ð632(47) Ð673(36) × 9 d1133 10 16(1) Ð12(2) Ð47(7) Ð14(5) 13(5) 36(4) × 9 d2233 10 Ð15(1) d1133 d1133 d1133 d1133 d1133 × 9 d1223 10 1(3) 00000

Note: Notation for the quantities Beq, Uii, Uij, and dpqrs is the same as in Table 2; cpqr are third-order anharmonic temperature parameters.

Table 4. Refined cation parameters for Nd2CuO4 (space group I4/mmm, a = 3.950(1) Å, c = 12.153(1) Å, Nrefl = 178, R = 1.87%, RW = 2.82%) and Sm2CuO4 (space group I4/mmm, a = 3.926(1) Å, c = 12.014(1) Å, Nrefl = 168, R = 2.37%, RW = 2.44%) at 373 K

Nd2CuO4 Sm2CuO4 Parameter Nd Cu Sm Cu z/c 0.35079(7) 0 0.35055(2) 0 2 Beq, Å 2.33(2) 2.31(5) 0.666(3) 0.65(1) U11, Å 0.0588(6) 0.047(2) 0.0969(4) 0.087(1) U22, Å U11 U11 U11 U11 U33, Å 0.2854(12) 0.289(3) 0.0807(6) 0.097(2) × 8 c333 10 Ð25(19) 0 38(14) 79(359) × 8 c113 10 31(25) 0 Ð124(166) Ð1(160) × 8 c223 10 c113 0 c113 0 × 9 d1111 10 Ð132(79) Ð86(210) 564(105) 79(359) × 9 d2222 10 d1111 d1111 d1111 d1111 × 9 d3333 10 49(17) Ð34(48) Ð0.6(5) Ð10(2) × 9 d1122 10 69(33) 44(97) Ð74(28) Ð49(97) × 9 d1133 10 Ð43(15) Ð22(47) Ð12(4) 19(16) × 9 d2233 10 d1133 d1133 d1133 d1133

Note: Notation for the quantities Beq, Uii, Uij, cpqr, and dpqrs is the same as in Tables 2 and 3.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2134 GOLOVENCHITS et al.

Table 5. Room-temperature distances between ions (Å) in R2CuO4 crystals (R = La, Nd, Sm, Eu)

Parameter La2CuO4 Nd2CuO4 Sm2CuO4 Eu2CuO4 RÐO2 2.323(4) 2.3256(5) × 4 2.3019(5) × 4 2.284(1) × 4 RÐO1 2.620(3) × 2 2.6726(5) × 4 2.6563(8) × 4 2.642(1) × 4 RÐO1' 2.654(3) × 2ÐÐÐ RÐO2' 2.684(6) Ð Ð Ð RÐO2'' 2.752(2) × 2ÐÐÐ RÐO2''' 2.850(6) Ð Ð Ð CuÐO1 1.9028(8) × 4 1.9720(5) × 4 1.9605(5) × 4 1.948(1) × 4 CuÐO2 2.437(4) × 2ÐÐÐ temperature. Figures 4 and 5 show the room-tempera- The density of the spatial distribution of PDF lines ture PDFs of the R3+ ions calculated for the same crys- relative to its maximum value is a measure of the steep- tals. Figures 6 and 7 present the PDFs of the Cu2+ and ness of the local potential. For multiply connected 3+ PDFs, the value of the PDF at the center of a panel R ions calculated for Nd2CuO4 and Sm2CuO4, respectively, for T = 373 K. We note that the ion PDFs in the (field) characterizes the depth of the wells produced by the CJTE of the corresponding ions. Angular symmetry (100) and (010) planes of R2CuO4 crystals (R = Pr, Nd, Sm, Eu, Gd), whose structure is described by the space in the PDF distribution provides information on the symmetry group I4/mmm, are equivalent. At the same directions of extremal ion displacements in thermal vibrations. time, for La2CuO4 (space group Abma), the PDFs in the (100) and (010) planes are different. Because the (100) PDFs are constructed taking into account both the cut of the PDF is in this case more informative, it is the harmonic and anharmonic contributions to thermal ion (100) PDFs that are presented in the figures. vibrations (Tables 2Ð4). Large anharmonic contributions are indicated by the flat regions in the PDFs [13]. As seen from Figs. 2Ð7, while PDFs for R2CuO4 To find the effect of the RE properties, as well as of crystals with different RE ions differ substantially, we interactions between the copper and RE ions, on the observe, however, a certain pattern in the PDF variation structural states of a crystal, a comparative analysis is which correlates with the increase in atomic number carried out of the PDF patterns for the copper and RE and decrease in radius of the RE ion. ions. As an illustration of two limiting cases, let us con- Consider the main features of the room-temperature 2+ sider the PDFs of Cu ions for La2CuO4 and Eu2CuO4. Cu2+ PDF in the series of crystals studied by us. In La2CuO4, one observes a two-well potential with In Eu2CuO4 and Gd2CuO4, one observes singly con- splitting along the a and b axes in the CuO2 layer and a nected PDFs with the maximum value at the field cen- minimum of the PDF at the special ion position. We ter, which implies the absence of vibronic CJTE. By believe the JahnÐTeller vibronic interaction, which contrast, R2CuO4 crystals (R = La, Nd, Pr, Sm) exhibit gives rise to the cooperative JahnÐTeller effect (CJTE) multiply connected PDFs, which indicates the presence of copper ions [1], to dominate here. Eu CuO exhibits 2 4 of vibronic CJTE for the Cu2+ ions. a singly connected PDF with the maximum value at the special position. In this case, the orbitÐorbit interaction In La2CuO4, Pr2CuO4, and Nd2CuO4, the crystal- mediated by 2D spin fluctuations is dominant [1]. field potential in the presence of vibronic CJTE splits in the (001) plane along the [100], [010], and [110] direc- Let us formulate the main criteria underlying the tions, respectively. In Sm CuO , this splitting occurs analysis of the data presented here for all R CuO crys- 2 4 2 4 along the c axis ([001] direction). tals. In crystals with multiply connected PDFs, the deep- First of all, the actual type of the PDF is important, est wells are observed in La CuO and Nd CuO , with namely, whether the PDF is multiply or singly con- 2 4 2 4 nected. We believe that multiply connected PDFs, cor- close-to-zero PDF values at the field centers. In the responding to double- or multiwell crystal-field poten- Pr2CuO4 and Sm2CuO4 crystals, the well depth is con- tials, form as a result of vibronic CJTE for the corre- siderably smaller. In the latter crystal, a practically flat sponding ions. In the case of a singly connected PDF crystal-field potential is observed near the field center with the maximum value (accepted as 100%) at the (Figs. 2, 3). equilibrium ion position, there is no vibronic CJTE. As also can be seen from Figs. 2Ð7, the symmetry of The equilibrium position of an ion in the lattice corre- the PDFs in the (001) planes, corresponding to the sponds to the center of the square panels in Figs. 2Ð7. CuO2 sheets, and in the (100) and (010) planes, charac-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 THERMAL VIBRATIONS AND THE STRUCTURE 2135

[010] [001]

[100] [100]

2+ Fig. 2. Room-temperature PDF of Cu ions for La2CuO4 (La), Pr2CuO4 (Pr), and Nd2CuO4 (Nd) calculated for the (001) and (010) planes. Field span 0.48 × 0.48 Å. PDF isolines drawn in steps of 10% of the maximum value. Dot speciﬁes 100% PDF. Dot-and- dash line corresponds to PDF = 0. Dashed lines identify negative values of PDF. Field centers correspond to equilibrium lattice ion positions in the lattice.

terizing copper ions off the CuO2 sheets, is substan- vibronic CJTE in the (001) plane, while the square tially different. anisotropy in the PDF distribution is retained. The PDF patterns in the (100) and (010) planes for R2CuO4 crys- In R2CuO4 crystals (R = Sm, Eu, Gd), the probability tals with different RE ions differ more strongly than in density function in the (001) planes for small ion ther- the (001) plane. Copper ions in La CuO , Nd CuO , mal-vibration amplitudes (PDF > 50%) has a practi- 2 4 2 4 cally isotropic pattern, which implies a harmonic char- and Sm2CuO4 exhibit the largest thermal displacement acter of the thermal vibrations. In the case of large along the c axis. In Sm2CuO4, the 100% value point of PDF > 50% (large thermal vibration amplitudes), ﬂat the PDF is shifted along the c axis from the equilibrium PDF regions characteristic of an anharmonic compo- ion position. In Nd2CuO4, one sees only residual indi- nent of thermal vibrations appear and the PDF distribu- cations of a vibronic CJTE along the c axis, which pos- tion acquires a square anisotropy. In Nd2CuO4 and sibly occurs at a lower temperature, as is the case with Pr2CuO4 crystals, one can see manifestations of a Pr2CuO4 (see [2]).

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2136 GOLOVENCHITS et al.

[010] [001]

[100] [100]

2+ Fig. 3. Room-temperature PDF of Cu ions for Sm2CuO4 (Sm), Eu2CuO4 (Eu), and Gd2CuO4 (Gd). For other explanatory notes, see Fig. 2.

Almost all the crystals studied exhibit ﬂat PDF Let us turn now to the experimental features of the regions, indicating a large anharmonic component of PDF distribution of RE ions. As seen from Figs. 4 and thermal vibrations of the copper ions (see also Table 2). 5, crystals with the Pr3+, Eu3+, and Gd3+ ions at room Note that the anharmonic contribution to thermal cop- temperature exhibit singly connected PDFs and those per-ion vibrations is most prominent in crystals with with La3+, Nd3+, and Sm3+ ions exhibit multiply con- weak CJTE manifestations. Also, anharmonic contribu- nected PDFs. Note that, in contrast to the copper ions, tions manifest themselves more strongly in the (100) the PDF of the RE ions is singly connected but asym- and (010) than the (001) plane. metric relative to the [001] direction (see Table 5). This deviation from symmetry is also most likely a manifes- As follows from the pattern of the angular PDF dis- tation of the CJTE. tribution, the nonsplit, centered part of the local poten- A comparison of the copper and RE PDFs shows tial has cubic symmetry (Z = 8) in all R2CuO4 crystals them to have the same pattern for almost all crystals, with RE ions. with the exception of Pr2CuO4.

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[010] [001]

[100] [100]

3+ Fig. 4. Room-temperature PDF of R ions for La2CuO4 (La), Pr2CuO4 (Pr), and Nd2CuO4 (Nd). For other explanatory notes, see Fig. 2.

As seen from Fig. 4, Nd3+ ions are characterized by shifted along the [001] direction with respect to the flat potentials in the (001) plane with weakly pro- equilibrium ion positions. The isoline that envelopes nounced shallow wells which are shifted along the both wells and is closest to the equilibrium position [110] diagonals. Comparison of the PDFs of the copper corresponds to a PDF value of 40%. Thus, unlike 3+ and Nd ions in Nd2CuO4 makes it possible to con- Nd2CuO4, the degree of local crystal-field distortion for 3+ clude that the CJTE-induced distortion of the local the Sm ions in Sm2CuO4 is larger than that for the crystal-field potential for the copper ions is consider- copper ions. We note that only in Sm2CuO4 is there a ably larger than that for Nd3+. double-well potential with splitting along the c axis of the crystal for both the copper and RE ions. The Sm3+ ions in the (100) plane [as well as in the equivalent (010) plane] in Sm2CuO4 exhibit a vibronic In crystals with dominant vibronic JahnÐTeller dis- CJTE, with wells that are asymmetric in depth and tortion of the Cu2+ ions, this distortion is observed in

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2138 GOLOVENCHITS et al.

[010] [001]

[100] [100]

3+ Fig. 5. Room-temperature PDF of R ions for Sm2CuO4 (Sm), Eu2CuO4 (Eu), and Gd2CuO4 (Gd). For other explanatory notes, see Fig. 2.

the (001) planes. At the same time, in Sm2CuO4, the large third- and fourth-order anharmonicity parameters dominant vibronic distortion of the Sm3+ ions occurs (see Table 3). along the [001] direction, with no vibronic CJTE present for the Cu2+ ions in the (001) plane. 3. ANALYSIS OF EXPERIMENTAL DATA 3+ In Gd2CuO4, there is no CJTE for the Gd ions and One can thus conclude that the structural properties the PDF distributions for the copper and RE ions are of the CuO2 layers in R2CuO4 crystals with RE ions dif- 3+ fer substantially, depending on the actual type of the RE similar. In Eu2CuO4, the PDF of Eu ions is singly connected but asymmetric relative to the [001] direction; ion, more speciﬁcally, on its ionic radius and the character of the ground state. this pattern results from the CJTE. No indication of the CJTE for the copper ions is seen, however. The absence of apical oxygen ions in the nearest environment of the copper ions in R2CuO4 crystals with For the RE ions, as well as for Cu2+, there are large RE ions should enhance the effect of the latter on the anharmonic contributions, i.e., ﬂat PDF regions and properties of the CuO2 sheets and the crystal structure

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[010] [001]

[100] [100]

2+ 3+ Fig. 6. PDF of Cu and Nd ions for Nd2CuO4 calculated for 373 K. For other explanatory notes, see Fig. 2.

[010] [001]

[100] [100]

2+ 3+ Fig. 7. PDF of Cu and Sm ions for Sm2CuO4 calculated for 373 K. For other explanatory notes, see Fig. 2.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2140 GOLOVENCHITS et al. ∆ as a whole compared with La2CuO4 (in which the apical interactions comparable in strength to the splitting = oxygen ions in the CuÐO layers screen the effect of the (eg Ð t2g) ~ 0.1 eV in the cubic lattice are observed, RE ions on them). As a result, in La2CuO4, the local which makes admixture of excited states to the ground crystal-field potential of the copper ions is determined state possible. The t2g triplet in the tetragonal lattice by the nearest neighbors in the 2D CuO2 sheet, whereas splits into a dxy singlet and a dxz, yz tetragonal doublet. in R2CuO4 crystals with RE ions, the effective local The character of this splitting depends on the relative potential of the Cu2+ ions does not derive from them magnitude of the vibronic and SO couplings. If the alone. To the field of the 2D plane square lattice in the former coupling dominates, the lower state is the dxy CuO sheet (the first coordination shell) is admixed to 2 singlet, and if the latter interaction is stronger, the dxz, yz the crystal field of the next coordination shells and, pri- tetragonal doublet is in a lower position [6, 16]. marily, of the RE ions, because the R3+ ions are closest to the Cu2+ ions in the c direction (see Fig. 1). If the SO interaction in R2CuO4 crystals with RE Let us turn now to an analysis of the intralayer inter- ions is strong, the tetragonal doublet is admixed to the orbital ground singlet d 2 2 for the copper ions. actions, which determine the structural properties of ()x Ð y crystals. These are, first of all, the Jahn–Teller interac- Under these conditions, the orbitÐorbit coupling tions in the CuO2 sheets and in layers with RE ions. through 2D spin fluctuations can give rise to a CJTE of R2CuO4 crystals with RE ions contain JahnÐTeller ions spin origin. A CJTE of this type was assumed to be of two types, namely, Cu2+ and R3+ [16, 17]. As follows responsible for the 2D orbital glass state revealed in from our results, the dominant lattice distortions exist- Eu2CuO4 for T > 150 K [11]. No strong lattice distor- ing in the crystals under study, which actually deter- tions occurred, but correlated spin and orbital states mine the crystal structure as a whole, are those caused formed. We believe that the existence of a singly con- by the JahnÐTeller effect of either the copper or RE nected PDF in Eu2CuO4 for T > 150 K is due to JahnÐ ions. In Sm2CuO4, the JahnÐTeller RE ion interactions Teller stabilization through a CJTE of spin origin. A are dominant, whereas in the Nd2CuO4 and Pr2CuO4 singly connected PDF is observed in Gd CuO , while crystals, the JahnÐTeller interaction of copper ions pre- 2 4 2+ 3+ the Pr2CuO4 and Nd2CuO4 crystals exhibit an admix- dominates. In this case, the intralayer Cu ÐR cou- ture of centered PDF to the split PDF induced by pling plays, on the whole, an important role in govern- vibronic CJTE. It appears only natural to assign the ing the crystal structure. nonsplit, centered PDF of the copper ions to a contribu- Note that the vibronic CJTE for Cu2+ ions is tion of orbit–orbit coupling through spin fluctuations. observed in R2CuO4 crystals with the maximum possible RE ionic radii (R = Pr, Nd). By contrast, in crystals The multiply connected copper-ion PDFs observed with smaller ionic radii (R = Sm, Gd, Eu), the vibronic in R2CuO4 crystals (R = Nd, Pr) are evidence of a dom- CJTE for the Cu2+ ions is substantially weaker or is inant contribution of vibronic JahnÐTeller interactions 2+ altogether absent. for the Cu ions in CuO2 sheets.

Let us consider the situation in the CuO2 sheets in 2+ Thus, in the CuO2 layers of R2CuO4 crystals with more detail. The character of the Cu orbital ground RE ions, there are two different interactions bringing state and, accordingly, the character of the JahnÐTeller about JahnÐTeller lattice stabilization, namely, vibronic effect are known to be determined by the type of nearest (phonon-mediated orbitÐorbit coupling) and orbitÐ environment in the cubic-lattice approximation [6, 16]. In R CuO crystals with RE ions, the nearest environ- orbit coupling mediated by 2D spin fluctuations. As 2 4 already mentioned, the character of the JahnÐTeller sta- ment of the copper ions derives from the contribution of two crystal-field components with different symme- bilization is different for these two interactions. Their coexistence may produce nonuniform structural states, tries, namely, of the oxygen ion squares in the CuO2 sheets (CuÐ4OI) and of the cubic RE environment (CuÐ and variation of the relative strength of the interactions 2+ with temperature may give rise to structural phase tran- 8R). As is well known, the orbital ground state of Cu sitions. ions in an oxygen ion square is a d 2 2 singlet. The ()x Ð y cubic arrangement of R3+ ions around the Cu2+ ion cor- In the cases where the JahnÐTeller RE interaction in R CuO crystals is dominant, a distortion of the cubic responds to an orbital ground state of Cu2+ in the form 2 4 of an e doublet [18]. symmetry in the RE environment plays a primary role. g Interaction between the copper and RE ions should also Splitting of the eg doublet in the tetragonal lattice change the structural state of the Cu2+ ions. It is this sit- produces a vibronic CJTE for copper ions [6, 16]. We uation that is apparently realized in Sm2CuO4. believe, however, that the structural states of the CuO2 sheets and of the crystals as a whole are also deter- Let us consider some characteristic features of the mined by the excited orbital states of the t2g triplet. As structural states of specific R2CuO4 crystals with RE already mentioned, in R2CuO4 crystals, a number of ions in more detail.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 THERMAL VIBRATIONS AND THE STRUCTURE 2141

3.1. La2CuO4 The first excited orbital doublet is spaced energywise from the ground state by 16 meV [20]. Atomic number Z(La) = 57; ionic radius r(La3+) = 1.032 Å [19]. The Nd2CuO4 crystal at room temperature exhibits a 3+ dynamic vibronic CJTE for both the copper and neody- The La ion is nonmagnetic. As mentioned above, mium ions; however, the distortion of the local crystal for temperatures T < 650 K, La CuO has a T-type 2 4 field for copper ions is larger than that for Nd3+. In other orthorhombically distorted structure, with the nearest 2+ 2+ words, vibronic interactions for the Cu ions dominate environment of the Cu ions being oxygen octahedra in Nd CuO . One readily sees a correlation between the in the CuO sheets. 2 4 2 copper ion and Nd3+ states. We believe this correlation As seen from Fig. 2, the copper ion PDFs have a pat- to be due to Cu2+ÐNd3+ coupling giving rise to corre- tern typical of crystals with a vibronic CJTE. The crys- lated ion displacements. The ionic radius of Nd3+ is tal field has a double-well potential with a splitting in 3+ the (001) plane (in the CuO layer) and a close-to-zero smaller than that of Pr ; this accounts for the higher 2 mobility of the former ion in the lattice. field-center PDF. The copper ion displacement in thermal vibrations is maximum along the c axis, which The (001) plane displays a centered part of the cop- indicates that the nearest environment of the ion is an per ion PDF near the equilibrium position, which we octahedron extended along the c axis. assign to a contribution of the orbitÐorbit interaction through spin fluctuations to the Jahn–Teller stabilization. Note that the relative contribution of the centered 3.2. Pr2CuO4 nonsplit potential in Nd2CuO4 is larger than that in Z(Pr) = 59; r(Pr3+) = 0.99 Å [19]. Pr2CuO4. 3+ 3 Thus, the crystal structure of Nd2CuO4 is a result of The ground state of the Pr ion is H4 (S = 1, L = 5, J = 4). The orbital and spin ground singlet is separated three types of interactions, namely, the intralayer vibronic JahnÐTeller and SO coupling of copper ions energywise from the excited orbital doublet by 2+ 3+ ~18 meV [20]. and the Cu ÐNd interlayer interaction. At room temperature, the JahnÐTeller vibronic interaction for the Vibronic JahnÐTeller distortions in the CuO2 layers copper ions is dominant. are dominant for copper ions at room temperature. 3+ As seen from Figs. 2 and 6, an increase in tempera- Such distortions are not observed for the Pr ions. As ture enhances copper-ion displacements along the already mentioned, Pr2CuO4 is the only R2CuO4 crystal [001] directions and decreases the contribution of sin- with RE ions in which there are no obvious correlations gly connected PDFs in the (001) plane; i.e., it enhances in PDF shape between the copper and RE ions. the role of the in-plane vibronic JahnÐTeller coupling. The Pr3+ ion has the maximum ionic radius in The symmetry of the effective nearest environment of R2CuO4 crystals with tetragonal T' structure. The lattice copper ions in Nd2CuO4 approaches ever more that of accommodating the slightly larger La3+ ion has T struc- an octahedron extended along the [001] direction, mak- ture. The displacements of the Pr3+ ions in a lattice of T' ing the PDF distribution pattern similar to that of symmetry are apparently constrained. As a result, the La2CuO4. Coulomb repulsion between the Pr3+ and Cu2+ ions gives rise to a relatively larger copper-ion displacement 3.4. Sm2CuO4 in the CuO2 layer; i.e., the Coulomb repulsion efficiently enhances the vibronic JahnÐTeller interaction Z(Sm) = 62; r(Sm3+) = 0.958 Å [19]. between the copper ions. Nevertheless, near the field As already mentioned, the structure of Sm2CuO4 is center, there is also a centered part of the copper ion dominated by the CJTE for the RE ions and the interac- PDF in the (001) plane; we associate this part with the tion between the copper and RE ions. The ionic radius contribution of the orbitÐorbit coupling through spin of Sm3+ is slightly smaller than that of Nd3+ but is larger fluctuations to the Jahn–Teller stabilization. than those of europium and gadolinium. Thus, although the vibronic interaction of copper 3+ 6 The ground multiplet of the Sm ion is H5/2 (S = ions in the CuO2 sheets is dominant in Pr2CuO4, the 5/2, L = 5, J = 5/2). Crystal field splitting makes the properties of the Pr3+ ion (in this case, its ionic radius) orbital singlet and Kramers doublet the ground state. and the R3+ÐCu2+ interaction noticeably affect the crys- The distance to the nearest excited orbital doublet is tal structure. ~18 meV [21]. At room temperature, the Sm3+ ion is seen to 3.3. Nd CuO undergo equilibrium-position displacement along the 2 4 [001] direction from the field center (Fig. 5), which is a Z(Nd) = 60; r(Nd) = 0.983 Å [19]. consequence of static CJTE for the Sm3+ ions. The local 3+ 4 3+ The Nd ground state is I9/2 (S = 3/2, L = 2). The JahnÐTeller distortions for the Sm ions are larger than orbital ground doublet is crystal-field split by 0.5 meV. those for the copper ions. The PDFs for the copper and

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2142 GOLOVENCHITS et al. samarium ions are similar. The off-plane displacement ground state by ~2 eV. Gd3+ ions have the smallest ionic 2+ of the Cu ions from the CuO2 sheets along the c axis radius among the RE ions in R2CuO4 crystals. A T'-type is most likely due to the Sm3+ÐCu2+ interaction. Note lattice does not form for RE ions with smaller ionic 2+ that the pattern of the Cu displacement along the c radii. For temperatures T < 600 K, Gd2CuO4 exhibits 2+ axis corresponds to the case of dynamic CJTE (Cu orthorhombic distortions in the CuO2 sheets caused by displacements up and down with respect to the layer are the O2Ð (OI) ions rotating about the undisplaced Cu2+ equally probable). At the same time, copper ions do not ions [4, 8]. exhibit any manifestation of the vibronic CJTE in the (001) plane at room temperature. One sees, however, The shape of the copper-ion PDF suggests that the PDFs extended along the a and b axes, which can give vibronic JT interaction is weak and does not affect the rise to switching, at a low temperature, of the dominant structural state of the crystal significantly. There is a type of interaction from the vibronic for RE ions to the similarity in the spatial PDF distribution between the copper and gadolinium ions. The PDF symmetry in the vibronic for copper ions in the CuO2 layer and, thus, can lead to a structural phase transition. (100) and (010) planes reflects a Z = 8 coordination of the nearest environment of ions, which is in agreement, As seen from Fig. 7, the pattern of PDF distribution for the Gd3+ ions, with the actual nearest environment of copper and samarium ions in Sm CuO obtained at 2 4 of OI and OII ions and indicates, for the Cu2+ ions, that T = 373 K changes qualitatively compared to that for 3+ 2+ room temperature. At T = 373 K, no vibronic CJTE is the Gd ÐCu interaction plays a dominant role in the 3+ 2+ formation of the crystal-field potential. There is a sub- observed either for the Sm or the Cu ions. In the stantial anharmonic contribution to the PDF in the same interval 300Ð373 K, Sm2CuO4 apparently undergoes a planes, which usually reflects structural instability of structural phase transition manifest in a transformation the lattice, i.e., closeness to a displacive phase transi- of the PDF from the doubly to singly connected pattern tion [24]. (compare Figs. 3, 4, 7). The situation becomes similar to that observed in Eu2CuO4 for T > 150 K (in the tet- In the (001) plane, a square PDF symmetry is ragonal phase) [1]. It may be assumed that orbitÐorbit observed, which is a sign of a predominant contribution coupling through 2D spin fluctuations in the CuO2 lay- of the CuÐOI coupling to the potential formation in the ers becomes dominant in Sm2CuO4 for T > 296 K. CuO2 sheets. As already pointed out, in this layer, one observes orthorhombic distortions associated with the Note that below the Néel temperature (TN ~ 250Ð 290 K), all R CuO crystals with RE ions support long- OI ions turning with respect to the undisplaced copper 2 4 ions. range 3D antiferromagnetic order. If for T < TN the crystal is in a uniform 3D antiferromagnetic state, then One may thus assume that the Gd3+ÐCu2+ Coulomb 2D spin fluctuations with large correlation lengths are repulsion and the CuÐOI coupling in the CuO2 sheets formed in CuO sheets for temperatures T > T . If, how- 2 N are of the same order of magnitude in Gd2CuO4. Also, ever, the 3D antiferromagnetic state is nonuniform, 2D the Gd3+ÐCu2+ interaction in the (100) and (010) planes, spin fluctuations also persist for T < TN. The existence combined with a strong anharmonicity, results in the of well-developed 2D spin fluctuations with large cor- RE and copper ion thermal displacements being corre- relation lengths can enhance the part played by a CJTE lated; this, possibly, gives rise to stabilization of the of spin origin. It is possible that Sm2CuO4, in contrast nearly unstable lattice. to Eu2CuO4, supports a uniform magnetic and a structural state for T < TN. At the same time, in Pr2CuO4 and As mentioned above, we associate the presence of 2+ Nd2CuO4, which do not suffer noticeable changes in nonsplit centered PDFs of the Cu JahnÐTeller ions their structural properties (in the shape of the PDF) near with a CJTE of spin origin. In Gd2CuO4, however, the Néel temperature, a nonuniform 3D antiferro- orthorhombic distortions in the CuO2 sheets should magnetic state most likely exists for T < TN. The possi- bring about splitting of the dxz, yz tetragonal doublet and bility of existence of well-developed 2D spin fluctua- predominance of the vibronic JahnÐTeller interaction tions in Pr2CuO4 for T < TN is also corroborated by spin for the copper ions. If the structural state of the crystals dynamics studies [22, 23] and by the presence of cen- were to be uniform and have orthorhombic symmetry, tered, nonsplit PDFs of copper ions in CuO2 sheets for the PDFs would be split, reflecting the predominantly T < T [2]. vibronic character of the JahnÐTeller lattice stabiliza- N tion, which is at odds with the observed PDF distribution pattern. It may be conjectured that Gd2CuO4 sup- 3.5. Gd2CuO4 ports a nonuniform structural state with a strong anhar- Z(Gd) = 60; r(Gd3+) = 0.938 Å [19]. monicity and that effects characteristic of both the tetragonal [23] and orthorhombic symmetry [4, 8, 22] Unlike other RE ions, Gd3+ is not a JahnÐTeller ion, are observed. Possible coexistence of the tetragonal and 8 because its ground state is S7/2. A nonzero orbital orthorhombic phases was reported recently in a study moment appears in the next term spaced from the of weak ferromagnetism and Raman spectra [25].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 THERMAL VIBRATIONS AND THE STRUCTURE 2143

3.6. Eu2CuO4 the main factor accounting for the structural features is the absence of RE ion screening for the copper ions. Z(Eu) = 63; r(Eu3+) = 0.947 Å [19]. The larger the RE ionic radius, the more significant 3+ 7 The ground state of the Eu ion F0 (S = 3, L = 3, the part played by the vibronic JahnÐTeller interactions J = 0) is singlet and nonmagnetic. However, the first for the copper ions in crystal structure formation. The 3+ 7 copper ion displacements resulting from the CJTE in excited multiplet of the Eu ion ( F1), which is magnetic and degenerate (triplet), lies low in energy, the CuO2 layers efficiently increase the distance to the 0.03 eV away from the ground-state multiplet [1]. nearest positively charged RE ions. Correlated dis- 2+ 3+ Structural studies of Eu2CuO4 showed static CJTE of placement of the Cu ÐR ions brings about stabiliza- the Eu3+ ions (7F ) to form for temperatures T > 350 K tion of a T'-type tetragonal structure. In the cases where 1 3+ 7 when there is a noticeable thermal population of the 7F the RE ions are not of the JahnÐTeller type [Eu ( F1) 1 3+ excited triplet state [1]. and Gd ] and at the same time have minimum ionic radii, the Cu2+ÐR3+ interaction gives rise to a strong Eu2CuO4 is reported [9Ð11] to undergo a structural anharmonicity characteristic of lattice states close to phase transition at T ≈ 150 K. For T < 150 K, orthor- instability. hombic distortions form in the CuO2 layers that are similar to those observed in Gd2CuO4. For T > 150 K, these distortions disappear, restoring the square in- ACKNOWLEDGMENTS plane symmetry and tetragonal symmetry throughout This study was supported by the Russian Founda- the crystal [9]. However, the structural state is nonuni- tion for Basic Research (project no. 02-02-16140a) form and an orbital glass state is observed [11]. and, partially, by the “Basic Research” Foundation, We believe that the phase transition at T ≈ 150 K Presidium RAS (project “Quantum Macrophysics”), may be driven by a change in the character of the Cu2+Ð and RBRF “Scientific Schools” grant (project no. 00- 3+ 7 2+ 3+ 7 15-96757). Eu ( F1) interaction compared with Cu ÐEu ( F0). For T ≥ 150 K, the displacement of the JahnÐTeller ions 3+ 7 Eu ( F1) from the central positions may be assumed to REFERENCES be more probable than rotation of the oxygen ions OI in 1. E. I. Golovenchits, V. A. Sanina, A. A. Levin, et al., Fiz. the CuO2 layers. As a result, for T < 150 K, Eu2CuO4 Tverd. Tela (St. Petersburg) 39 (9), 1600 (1997) [Phys. acquires a structural state similar to that in Gd2CuO4. Solid State 39, 1425 (1997)]. The difference is that Eu2CuO4 retains tetragonal sym- 2. A. A. Levin, Yu. I. Smolin, Yu. F. Shepelev, et al., Fiz. metry throughout the crystal, with orthorhombic distor- Tverd. Tela (St. Petersburg) 42 (1), 147 (2000) [Phys. tions present only in local regions. The state persisting Solid State 42, 153 (2000)]. in the temperature interval 150 < T < 350 K is charac- 3. L. A. Muradyan, R. A. Tamazyan, A. M. Kevorkov, et al., terized by a singly connected, centered PDF for the Kristallografiya 35, 861 (1990) [Sov. Phys. Crystallogr. copper ions, and we believe that the interactions in the 35, 506 (1990)]. CuO2 sheets are dominated by orbitÐorbit coupling 4. M. Braden, W. Paulus, A. Cousson, et al., Europhys. through 2D antiferromagnetic spin fluctuations with Lett. 25, 625 (1994). large correlation lengths [11]. Nonuniform magnetic 5. S. Chakravarty, B. Halperin, and D. Nelson, Phys. Rev. and structural states allowing the existence of 2D anti- B 39, 2344 (1989). ferromagnetic fluctuations form in Eu2CuO4 for T > 6. A. Abragam and B. Bleaney, Electron Paramagnetic 150 K as a result of simultaneously occurring structural Resonance of Transition Ions (Clarendon, Oxford, 1970; and magnetic phase transitions (i.e., considerably Mir, Moscow, 1972), Vol. 2. ≈ below TN 250 K) [9, 12]. In this case, the role of the 7. P. P. Parshin, M. G. Zemlyanov, A. S. Ivanov, et al., Fiz. 2+ 3+ 7 Cu ÐEu ( F1) interaction can be enhanced by its long- Tverd. Tela (St. Petersburg) 41 (7), 1149 (1999) [Phys. range character if it is mediated by 2D spin correlations Solid State 41, 1046 (1999)]. in the CuO2 layer. 8. M. Udagawa, Y. Nagaoka, N. Ogita, et al., Phys. Rev. B 49, 585 (1994). 9. E. I. Golovenchits and V. A. Sanina, Pis’ma Zh. Éksp. 4. CONCLUSION Teor. Fiz. 74 (1), 22 (2001) [JETP Lett. 74, 20 (2001)]. 10. V. P. Plakhty, A. B. Stratilatov, and S. Beloglazov, Solid Thus, our analysis of experimental data shows that State Commun. 103 (12), 683 (1997). the character of structural distortions occurring at room 11. A. V. Babinskiœ, S. L. Ginzburg, E. I. Golovenchits, and temperature in the CuO2 sheets of R2CuO4 crystals with V. A. Sanina, Pis’ma Zh. Éksp. Teor. Fiz. 57 (5), 289 RE ions and the type of the dominant interaction are (1993) [JETP Lett. 57, 299 (1993)]. determined by the RE properties (ionic radius and the 12. E. I. Golovenchits, S. L. Ginzburg, V. A. Sanina, and ground state). In all R2CuO4 crystals with RE ions, the A. V. Babinskiœ, Zh. Éksp. Teor. Fiz. 107, 1641 (1995) part played by the Cu2+ÐR3+ coupling is important and [JETP 80, 915 (1995)].

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13. V. G. Tsirel’son, Itogi Nauki Tekh., Ser.: Kristallokhim. 21. R. Sachidanandam, T. Yidirim, A. B. Harris, et al., Phys. 27, 3 (1993). Rev. B 56 (1), 260 (1997). 14. D. W. J. Cruckshank, in Computing Methods in Crystal- 22. E. I. Golovenchits and V. A. Sanina, Fiz. Tverd. Tela lography, Ed. by J. S. Rollett (Pergamon, Oxford, 1965), (St. Petersburg) 41, 1437 (1999) [Phys. Solid State 41, p. 112. 1315 (1999)]. 15. P. J. C. Becker and P. Coppens, Acta Crystallogr., Sect. A 30, 129 (1974). 23. E. I. Golovenchits and V. A. Sanina, Physica B (Amster- dam) 284Ð288, 1369 (2000). 16. K. I. Kugel’ and D. I. Khomskiœ, Usp. Fiz. Nauk 136 (4), 621 (1982) [Sov. Phys. Usp. 25, 231 (1982)]. 24. B. A. Strukov and A. P. Levanyuk, Physical Principles of 17. G. A. Gehring and K. A. Gehring, Rep. Prog. Phys. 38, 1 Ferroelectric Phenomena in Crystals (Nauka, Moscow, (1975). 1995). 18. I. B. Bersuker, Electronic Structure and Properties of 25. Z. Fisk, P. G. Pagliuso, J. A. Sanjurjo, et al., Physica B Coordination Compounds (Khimiya, Leningrad, 1986). (Amsterdam) 305 (1), 48 (2001). 19. R. D. Shannon, Acta Crystallogr. A 32, 751 (1976). 20. A. T. Boothroyd, S. M. Doyle, D. M. K. Paul, and R. Osborn, Phys. Rev. B 45 (17), 10075 (1992). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2145Ð2150. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2049Ð2054. Original Russian Text Copyright © 2002 by Shur, Rumyantsev, Nikolaeva, Shishkin, Baturin.

MAGNETISM AND FERROELECTRICITY

Kinetic Approach for Describing the Fatigue Effect in Ferroelectrics V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, E. I. Shishkin, and I. S. Baturin Institute of Physics and Applied Mathematics, Ural State University, Yekaterinburg, 620083 Russia e-mail: [email protected] Received February 15, 2002

Abstract—A new kinetic approach is proposed for explaining the fatigue effect in ferroelectrics. A self-consistent variation in the area and geometry of the switching region of a sample upon a cyclic switching accompanied by the formation and growth of kinetically frozen domains is considered. It is assumed that fatigue is due to self-organized formation of a spatially inhomogeneous internal bias field due to retardation of bulk screening of the depolarization field. Variations in the switching charge and in the amplitude of switching current, which are calculated with the help of computer simulation of domain kinetics upon cyclic switching, are in good agreement with experimental data obtained for thin lead zirconate–titanate (PZT) thin films. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In this study, we propose an approach to describing the evolution of switched regions during cycling A decrease in the switching charge as a result of switching, which is based on the decisive role of retar- long-time cyclic switching of polarization in thin ferro- dation in the bulk screening of the depolarization field. electric films, which is known as the fatigue effect, For cyclic switching, this retardation leads to the self- remains a key problem, whose solution is essential for organized formation of a spatially nonuniform internal wide practical application of nonvolatile storage bias field suppressing the formation of new domains devices based on such films [1–16]. Recently, several and slowing down the switching. As a result, the alternative mechanisms have been proposed for switching cannot be completed in the regions with the explaining the fatigue effect in films. In accordance maximum value of the internal bias field, which corre- with the bulk-locking mechanism, charged domain sponds to the formation of kinetically frozen domains. walls in a film subjected to cyclic switching are fixed The main predictions of the proposed model (the exist- (pinning effect) by charge carriers localized at traps [9, ence of an additional stage of formation (of a rejuvena- 10]. It should be noted that this mechanism was pro- tion stage) preceding the fatigue stage and a change in posed for the first time by Kudzin et al. [16] for explain- the geometry of the switched region upon cycling ing fatigue effects in barium titanate single crystals. switching) are in good agreement with the results of experimental investigations in thin films. According to another mechanism (interface scenario), fatigue is associated with the slowing down of domain growth, which is due to suppression of nucleation (seed 2. MODEL termination) at the film–electrode interface [5, 6]. The role of oxygen vacancy redistribution in an electric field It is well known that, after switching of polarization has also been discussed extensively [14]. In [12], it is in a ferroelectric capacitor, external screening (charge proposed that fatigue can be explained by the growth of redistribution on the electrodes accompanied by a nonferroelectric passive surface layer upon cyclic switching current) rapidly compensates the depolariza- switching. It should be noted that, in all contemporary tion field Edep. However, there is no complete field com- models, the relation between the fatigue effect and the pensation in the bulk, since a nonpolar surface layer domain-structure kinetics upon cyclic switching is dis- (dielectric gap) exists in ferroelectrics [18, 19]. After regarded while considering fatigue in thin films. At the termination of external screening, the field Edep is com- same time, recent direct observations of the domain pensated only partly and a residual depolarization field structure of thin films with the help of scanning-probe Erd = Edep Ð Eex.scr is preserved in the bulk [19, 20]. The microscopy proved that the fatigue effect is accompa- compensation of Erd due to slow processes of bulk nied by the formation of a complex-shaped region screening leads to the formation of an internal bias field being switched and by the appearance and growth of Eb. This field is responsible for the unipolarity of the regions with a frozen orientation of polarization (“fro- switching process, which is manifested in a shift of a zen domains”) [8, 17]. hysteresis loop upon a quite rapid (as compared to the

2146 SHUR et al.

(‡) (b) growth of unswitched regions (kinetically frozen domains) [29, 30]. 2 T+

3 s

P 0 1 – 3. COMPUTER SIMULATION –2 T The domain kinetics for cyclic polarization switch- 1 2 ing in a thin plate (film) of a uniaxial ferroelectric under 0 the action of rectangular field pulses were simulated on Field –1 T a two-dimensional (2D) matrix. We used the classical 0 1 2 3 model of polarization switching, in which the domain Time structure kinetics involve the formation of new domains and their subsequent growth (nucleation and growth Fig. 1. (a) Three types of nuclei considered in simulation: model) [31, 32]. We assumed that both processes are (1) a nucleus at the end of a step, (2) a nucleus on a domain controlled by nucleation. The formation of nuclei of wall, and (3) an isolated nucleus. (b) Diagram showing vari- three different types was considered (Fig. 1a). ations in the relative local value of spontaneous polarization and in the relative value of the external field during a simu- (i) Formation of nuclei at the ends of the existing lated switching cycle. steps leads to step growth. (ii) Nucleation at a domain wall leads to the formation of new steps. bulk screening) cyclic switching in an alternating field [20, 21]. It has been shown, using lead germanite single (iii) The appearance of isolated nuclei leads to the formation of new domains. crystals as an example, that the field Eb is spatially nonuniform [22] and varies upon a long-time cyclic switch- The probability pk of formation of nuclei of a spe- ing [19Ð23]. Bulk screening can be regarded as a result cific type in a given element of the matrix (i, j) during of competition between three groups of mechanisms: the Nth switching cycle is determined by the local field (1) orientation of defect dipoles [24Ð26], (2) redistribu- Eloc(i, j, N): tion of bulk charge carriers [18, 19], and (3) injection of p ()ijN,, ∼ exp{}ÐE /[]E ()ijN,, Ð E , (1) charge carriers from electrodes through a dielectric gap k ac, k loc th,k [6, 27]. where Eac, k and Eth,k are the activation field and the We take into account the fact that the domain kinet- threshold field, respectively [19]. ics during cyclic switching are a self-organized process, It is well known that the nucleation probability at a since the spatial distribution of Eb is determined by the domain wall is much higher than that at a distance from preceding evolution of domains and determines, in the wall, since Eac,1 < Eac,2 < Eac,3 [19]. Conse- turn, their subsequent kinetics. Earlier, we demon- quently, an increase in the length of a domain wall facil- strated experimentally that the field Eb is virtually uni- itates polarization switching. form over the entire switched region in the case of long- The local field E (i, j, N) is the sum of the uniform time cyclic switching with asymmetric (T ≠ T ) loc pos neg external field E , the residual depolarization field E , rectangular pulses if the switching time is much shorter ex rd and the spatial nonuniform internal bias field E (i, j, N) than the period of the switching field (t Ӷ T). In this b s formed by the end of the preceding switching cycle: case, the mean value of Eb is determined by the asym- Ð1 (),, (),, metry of pulses c ~ (Tpos Ð Tneg)(Tpos + Tneg) [28]. It fol- Eloc ijN = Eex ++Erd Eb ijN. (2) ≤ lows that in the case of slow switching (ts T), the In experiments on fatigue, bulk screening proceeds, internal bias field E becomes spatially nonuniform τ ӷ b as a rule, quite slowly, and T. In this case, Eb(i, j, N) even when symmetric pulses are used, since different tends to compensate the sum of the external and the regions in the sample are in states with opposite direc- residual depolarization field averaged over a switching 〈 〉 tions of polarization for different periods of time. Dur- cycle, Eex + Erd . For symmetric pulses of rectangular ing long-time switching, the local value of E relaxes to 〈 〉 〈 〉 b shape, we have Eex = 0 at any point, while Erd is spa- a value determined by the relative difference in the res- 〈 〉 + Ð + tially nonuniform: Erd = Erd(T Ð T )/T, where T and idence times of a given part of the sample in the states TÐ are the times for which the spontaneous polarization + Ð with opposite directions of polarization, (T Ð T )/T at a given point is directed along one of the two oppo- (Fig. 1b). In this case, the magnitude and sign of the site directions, respectively, during a switching cycle internal bias field Eb(x, y) are spatially inhomogeneous (Fig. 1b). It should be noted that the local values of T+ and the distribution function f(Eb) varies upon cyclic and TÐ for each cycle depend on the domain kinetics in switching. The computer simulation of the domain the entire sample (since the probability of switching at kinetics carried out by us revealed that the variation of a given point is determined by the state of its surround- f(Eb) upon cyclic switching leads to the appearance and ings) and, hence, can be determined only as a result of

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 KINETIC APPROACH FOR DESCRIBING THE FATIGUE EFFECT 2147 simulation of the domain kinetics in the entire matrix. In our analysis, we did not aim at choosing a preferred screening mechanism from those described above and characterized the kinetics of screening only by a time (a) constant τ. The domain structure kinetics in each switching cycle were simulated by taking into account the field 0.05T 0.2T 0.4T Eb(i, j, N) formed by the end of the preceding cycle. The calculated local values of T+ and TÐ were used for recal- culating the spatial distribution of Eb for the next switching cycle in accordance with the relation (b) (),, (),, ()τ Eb ijN = Eb ijNÐ 1 exp ÐT/ (3) []∆()τ (),, + Erd 1 Ð exp ÐT/ TijN/T. Assuming the condition T Ӷ τ to be satisfied, we used the simplified relation Fig. 2. Sequence of instantaneous domain configurations (),, (),, ()τ forming during a switching cycle, found with the help of Eb ijN = Eb ijNÐ 1 1 Ð T/ computer simulation (a) during the first switching and (b) (4) 〈〉(),, τ after a long-time cyclic switching. Switched domains of Ð Erd ijNÐ 1 T/ . opposite polarities are shown by black and white, while frozen domains of various polarity are shown by light-gray or Obviously, the results of simulation must depend to dark-gray color. The initial state corresponds to zero inter- a considerable extent on the spatial distribution of Eb nal bias field. prior to the first switching cycle. In the course of simulation, we considered two versions of the initial state: an idealized version with a uniform zero internal bias field (without bulk screening) Eb(i, j, 1) = 0 and a realistic version with a completely screened polydomain 0.2 (‡) (b) 6 1.0 4 initial state Eb(i, j, 1) = ÐErd(i, j). The latter version cor- max N = 5 j 2 responds to the spatial distribution observed after pro- 0.1 ∆

), % / : E w 0 longed holding of the domain structure in a static state. N 0 0 1 2 3 ,

b ∆ In this case, the ﬁelds Eb in domains with opposite ori- 0.5 1.6 E/w

E 0.05

( 3.2 , arb. units f N = 100 entations are equal in magnitude but opposite in sign. j

0 3.1. Initial State with Zero Internal Bias Field –1 0 1 0 0.5 1.0 1.5 The evolution of the domain structure (a set of con- Eb/Erd Field, arb. units secutive domain configurations each of which corresponds to a single switching cycle) at various stages of Fig. 3. Computer simulation (initial state with zero internal fatigue process, which is obtained for the initial state bias field): (a) distribution functions of the internal bias field for various values of N approximated by a Gaussian; and (b) with zero internal bias field, is presented in Fig. 2. It can switching current caused by a triangular pulse for N = 50: be seen that complete switching occurs during the first ∆E/w = 0 corresponds to the Preisach theory, and ∆E/w ≠ 0, cycle (Fig. 2a), while after a long-time cyclic switch- to a modified approach [25], where ∆E is the difference ing, polarization reversal occurs predominantly in nar- between the threshold fields required for the formation of an row regions separating kinetically frozen domains of isolated seed and for domain growth, respectively. The inset different polarities (Fig. 2b). Such a change in the shows the dependence of jmax on 1/w fitted by formula (5). geometry of the switching region leads to a qualitative change in the type of domain kinetics. In the initial state, switching is mainly due to a 2D growth of indi- The spatially nonuniform field Eb formed after the vidual domains (Fig. 2a), while the appearance of fro- completion of the Nth cycle was characterized by the zen domains leads to an increase in the contribution instantaneous value of the distribution function f(Eb, N) from 1D motion of domain walls. After a long-time of the internal bias field: cyclic switching, the domain kinetics are mainly determined by a reversible parallel displacement of domain (), Ð2 δ[](),, walls (Fig. 2b). It is important to note that a similar fEb N = L ∑ Eb Ð Eb ijN . (5) change in the geometry of the switching region was also detected by directly observing domain kinetics in The distribution functions f(Eb, N) obtained as a PZT films using scanning-probe microscopy [8]. result of simulation are closely fitted by a Gaussian

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2148 SHUR et al.

(‡) (b) 0.8 (‡) 2 (b) 1.0 1.0 15 s s C/cm Q max 10 Q j 0.5 0.5 0.4 µ , r 5 P 0 0 0 0 1 10 100 1 10 100 10 100 100 102 104 106 108 Cycles Cycles Cycles Cycles

Fig. 4. Decrease (a) in the relative values of switched charge and (b) in the maximum value of switching current as a Fig. 5. Rejuvenation and fatigue stages: (a) simulation of result of the cyclic switching process obtained by simula- switching from a completely screened polydomain initial tion. The initial state corresponds to zero internal bias ﬁeld. state; and (b) experimental results for thin PZT ﬁlms.

( , ), % f Eb N f(Eb, N), % (‡) 8 (b) 8 6 6 4 4 2 2 40 30 100 20 200 N N 10 300 1 400 N = 2 Rejuvenation N = 50 Fatigue N = 450 –1 0 1 –1 0 1 Eb, arb. units Eb, arb. units

Fig. 6. Variation of switching region upon cyclic switching obtained as a result of simulation. Switching from a completely screened polydomain initial state. Dark ﬁeld corre- Fig. 7. Evolution of the distribution function of Eb obtained sponds to the switching region, while light-gray and as a result of simulation in (a) the rejuvenation and (b) dark-gray ﬁelds correspond to frozen domains of different fatigue stages. The switching starts from a completely polarities. screened polydomain initial state.

(Fig. 3a). The initially narrow distribution function where J and a are constants. spreads in the course of cyclic switching (variance w It is important to note that the maximum value of the increases significantly). The formation and growth of switching current, j , decreases in the fatigue process ± max two peaks of the distribution function at Eb = Erd cor- to a considerably greater extent than the switched respond to the formation and increase in the area of charge does (Fig. 4). kinetically frozen domains (Fig. 3a). According to the Preisach theory, the distribution function for the internal bias field determines the 3.2. Completely Screened Polydomain Initial State dependence of switching charge and current on the A simulation of cyclic switching from a polydomain applied voltage during testing with triangular pulses completely screened state revealed an initial increase in [33, 34]. The spread of f(Eb, N) upon cyclic switching the switching charge. This feature enabled us to single decreases the switching charge, since polarization out an additional rejuvenation stage preceding the switching is terminated in the regions in which the local fatigue stage (Fig. 5a). It can be seen that the geome- field Eloc becomes smaller than the threshold field Eth. tries of the switching region in the rejuvenation and In our modification [35] of the Preisach approach, we fatigue stages are qualitatively different (Fig. 6). In the took into account the above-mentioned fact that the course of rejuvenation, the width of the switching threshold fields required for the formation of a single region increases considerably and its connectivity nucleus and for domain growth (nucleation at a domain changes. After the completion of rejuvenation, this wall) differ significantly. It is shown that the depen- region has the form of a connected labyrinth structure. dence of the maximum value of switching current jmax At the fatigue stage, a self-consistent smoothing and on the variance w of the distribution function, which is simplification of the labyrinth structure is observed. found in our model with the help of computer simula- The stimulation revealed that frozen domains of the tion, can be fitted by the formula (Fig. 3) [35] same polarity are predominantly formed and grow (uni- polar fatigue). The sign of unipolarity is determined by () () []() jmax 1/w = jmax 0 + Jaexp/w Ð 1 , (6) the geometry of the initial domain structure. This ten-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 KINETIC APPROACH FOR DESCRIBING THE FATIGUE EFFECT 2149

(‡) (b) (a) (b) 4 N = 20 N = 40 N = 60 Initial 7 60 0.4 A

40 10 3 10 µ , × × 40

, arb. units 0.2 A max = 2 = 3

2 j

µ 20 N N , max

j 20 j 0 0

, arb. units 1 0 1 2 3 j 50 100 200 400 10 10 10 10 Cycles Cycles 0 –0.5 0.5 –1 1 –1 1 0–40 4–4 0 4 –4 0 4 Field, arb. units Field, V

Fig. 8. Variation of the shape of switching current at the Fig. 9. Increase in the maximum value of switching current rejuvenation and fatigue stages: (a) simulation of switching at the rejuvenation stage (a) obtained as a result of simula- from a completely screened polydomain initial state and (b) tion, and (b) experimental data on cyclic switching in thin experimental results for thin PZT films. PZT films. dency is in good agreement with the reported experi- discrepancy between the theoretical and experimental mental results [13]. values of N, for which the magnitude of the switched The evolution of the internal-bias-field distribution charge decreases to half the initial value (endurance). function in this case differs qualitatively from that in the case of a zero initial internal bias field considered 5. CONCLUSION above. Two peaks corresponding to initial domains of opposite signs spread and merge into a single broad Thus, the proposed model of self-consistent evolu- peak during rejuvenation (Fig. 7a). The subsequent tion of the local internal bias field enabled us to describe the variations in the geometry of the switching behavior of this peak repeats the variation of f(Eb, N) described earlier (Fig. 7b). At the rejuvenation stage, region during cyclic switching and to predict the exist- the switching current is the sum of two contributions ence of the rejuvenation stage. With the help of com- corresponding to switching in the regions differing in puter simulation, we observed a correlation between the sign of the internal bias field (Fig. 8a). the variations in the internal-bias-field distribution function and in the shape of switching current upon cyclic switching. The established correlation makes it 4. EXPERIMENT possible to extract important information concerning The rejuvenation and fatigue were studied in lead the fatigue kinetics from measurements of the switching current. The good agreement between the results of zirconateÐtitanate Pb(Zr0.2Ti0.8)O3 films 100–200 nm thick. The films were deposited by sol–gel method on simulation and the experimental data obtained for thin Pt/Ti/SiO /Si/ substrates [11]. The upper platinum elec- PZT films confirm the applicability of the proposed 2 approach. trode was used for applying rectangular bipolar pulses with 100% filling (without a pause between the pulses). The amplitude of pulses Ucyc was 5Ð8 V, and the fre- ACKNOWLEDGMENTS quency fcyc was chosen in the interval from 10 Hz to The authors are grateful to T. Schneller and R. Ger- 1 MHz. Hysteresis loops and switching currents were hardt for providing thin-film samples and to O. Lohse measured by applying triangular pulses (fm = 10Ð for consultations on measurements. 100 Hz, Um = 5Ð7 V) following a certain number of This study was partially supported by the Russian switching cycles with rectangular pulses. Cyclic switch- Foundation for Basic Research (grant no. 01-02- ing and measurements were carried out at room temper- 17443), the program “Russian Universities: Fundamen- ature using an automated measuring complex [29]. tal Studies,” the program “University Studies in Priority The measured variations in switching charge Trends of Science and Technology: Electronics” (grant (Fig. 5b) and in current (Fig. 8b) upon cyclic switching no. 03-03-29), and the American Foundation for Civil in PZT films confirm the existence of the rejuvenation Studies and Development of Independent States of the stage, which is manifested most clearly in the increase Former Soviet Union (grant no. REC-005). in jmax (Fig. 9). The experimental data on the evolution of the shape of the current during cyclic switching is in qualitative agreement with the predictions of computer REFERENCES simulation (Figs. 5a, 8a). It should be noted that, in 1. H. M. Duiker, P. D. Beale, J. F. Scott, et al., J. Appl. order to reduce the computer time required for simula- Phys. 68, 5783 (1990). tion, the value of the time constant τ characterizing the 2. C. J. Brennan, Integr. Ferroelectr. 2, 73 (1992). kinetics of screening was chosen to be much smaller 3. J. Lee, S. Esayan, A. Safari, and R. Ramesh, Appl. Phys. than its experimental value, which leads to the observed Lett. 65, 254 (1994).

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4. G. Arlt and U. Robels, Integr. Ferroelectr. 3, 343 (1993). 20. V. Ya. Shur, Yu. A. Popov, and N. V. Korovina, Fiz. 5. E. Colla, D. Taylor, A. Tagantsev, and N. Setter, Appl. Tverd. Tela (Leningrad) 26, 781 (1984) [Sov. Phys. Solid Phys. Lett. 72, 2478 (1998). State 26, 471 (1984)]. 6. I. Stolichnov, A. Tagantsev, E. Colla, and N. Setter, Appl. 21. V. Ya. Shur, Phase Transit. 65, 49 (1998). Phys. Lett. 73, 1361 (1998). 22. V. Ya. Shur, Yu. A. Popov, and G. B. Soldatov, Fiz. Tverd. 7. V. Shur, S. Makarov, N. Ponomarev, et al., J. Korean Tela (Leningrad) 25, 265 (1983) [Sov. Phys. Solid State Phys. Soc. 32, S1714 (1998). 25, 148 (1983)]. 8. E. Colla, S. Hong, D. Taylor, et al., Appl. Phys. Lett. 72, 23. V. Ya. Shur and E. L. Rumyantsev, Ferroelectrics 191, 2763 (1998). 319 (1997). 9. W. Warren, D. Dimos, B. Tutler, et al., Appl. Phys. Lett. 24. V. A. Yurin, Izv. Akad. Nauk SSSR, Ser. Fiz. 24, 1329 65, 1018 (1994). (1960). 10. V. V. Lemanov and V. K. Yarmarkin, Fiz. Tverd. Tela 25. U. Robels and G. Arlt, J. Appl. Phys. 73, 3454 (1993). (St. Petersburg) 38, 2482 (1996) [Phys. Solid State 38, 26. P. Lambeck and G. Jonker, J. Phys. Chem. Solids 47, 453 1363 (1996)]. (1986). 11. M. Grossmann, D. Boten, O. Lohse, et al., Appl. Phys. 27. I. Stolichnov, A. Tagantsev, N. Setter, et al., Appl. Phys. Lett. 77, 1894 (2000). Lett. 74, 3552 (1999). 12. A. M. Bratkovsky and A. P. Levanyuk, Phys. Rev. Lett. 28. V. Ya. Shur, Author’s Abstract of Doctoral Dissertation 84, 3177 (2000). (UPI, Sverdlovsk, 1990). 13. A. Kholkin, E. Colla, A. Tagantsev, et al., Appl. Phys. 29. V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, et al., Lett. 68, 2577 (1996). Integr. Ferroelectr. 33, 117 (2001). 14. J. F. Scott and M. Dawber, Appl. Phys. Lett. 76, 3801 30. V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, et al., J. (2000). Appl. Phys. 90, 6312 (2001). 15. R. Ramesh, W. K. Chan, B. Wilkens, et al., Integr. Ferro- 31. R. C. Miller and G. Weinreich, Phys. Rev. 117, 1460 electr. 1, 1 (1992). (1960). 16. A. Yu. Kudzin, T. V. Panchenko, and S. P. Yudin, Fiz. 32. M. Hayashi, J. Phys. Soc. Jpn. 33, 616 (1972). Tverd. Tela (Leningrad) 16, 2437 (1974) [Sov. Phys. 33. G. Robert, D. Damjanovic, and N. Setter, Appl. Phys. Solid State 16, 1589 (1974)]. Lett. 77, 4413 (2000). 17. A. Gruverman, O. Auciello, and H. Tokumoto, Appl. 34. A. Bartic, D. Wouters, H. Maes, et al., J. Appl. Phys. 89, Phys. Lett. 69, 3191 (1996). 3420 (2001). 18. V. M. Fridkin, Photoferroelectrics (Nauka, Moscow, 35. V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, et al., J. 1976; Springer, Berlin, 1979). Appl. Phys. (in press). 19. V. Ya. Shur, Ferroelectric Thin Films: Synthesis and Basic Properties (Gordon and Breach, New York, 1996), Vol. 10, Chap. 6. Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2151Ð2156. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2055Ð2060. Original Russian Text Copyright © 2002 by Shur, Nikolaeva, Shishkin, Kozhevnikov, Chernykh.

MAGNETISM AND FERROELECTRICITY

Kinetics of Domain Structure and Switching Currents in Single Crystals of Congruent and Stoichiometric Lithium Tantalate V. Ya. Shur, E. V. Nikolaeva, E. I. Shishkin, V. L. Kozhevnikov, and A. P. Chernykh Institute of Physics and Applied Mathematics, Ural State University, Yekaterinburg, 620083 Russia e-mail: [email protected] Received February 15, 2002

Abstract—A comparative analysis of polarization-switching kinetics in single crystals of congruent and stoichiometric lithium tantalate LiTaO3 is carried out by recording a sequence of instantaneous domain conﬁgura- tions (optical visualization of the evolution of the domain-structure) and the switching current simultaneously. A new mechanism of fast kinetics of domains in congruent lithium titanate due to the growth of steps formed during domain coalescence is discovered experimentally and studied with the help of computer simulation. Additional information on the domain kinetics in stoichiometric lithium tantalate is obtained on the basis of statistical analysis of the noise component of a switching current. A model is proposed for description of the jerky motion of domain walls. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. EXPERIMENT Recently, a new class of periodically polarized non- The experiments were performed on monocrystal- linear optical materials has been used extensively for line samples of commercial CLT (Crystal Technology, creating coherent radiation sources and frequency con- CA) and on lithium tantalate samples with a close-to- verters using the quasi-phase matching effect [1, 2]. stoichiometric composition (SLT), which were grown by using the Czochralski method with a double crucible Lithium tantalate LiTaO3 is one of the most important representatives of this new class, since it is character- at the National Institute of Materials Science, Tsukuba, ized by high values of electrooptical and nonlinear opti- Japan [8]. The concentration of lithium vacancies was about 2% in CLT and 0.3% in SLT samples. The sam- cal coefficients. A periodic domain structure can be cre- ples under investigation were in the form of mon- ated by applying an electric field to a system of periodic odomain plates having a size of 6 × 5 mm and cut per- electrodes [3Ð5] deposited by photolithography. In pendicularly to the polar axis. The thickness was recent years, attempts have been made to use stoichio- 0.2 mm for CLT samples and 0.9 mm for SLT samples. metric lithium tantalate (SLT) with a much lower con- Switching was carried out by rectangular field pulses centration of defects instead of congruent lithium tanta- using transparent electrodes having a diameter of 1 mm late (CLT) for manufacturing nonlinear optics devices. and made of liquid electrolyte (aqueous solution of It was shown that making use of SLT not only consid- LiCl). In addition, SLT samples were investigated using erably reduces photorefraction and qualitatively In2O3:Sn (ITO) transparent electrodes of diameter changes the domain shape but also decreases the coer- 2.5 mm, deposited by reactive sputtering. All measure- cive field by an order of magnitude [6]. However, the ments were made at room temperature. kinetics of domain structure in the case of polarization switching in SLT have not been studied comprehen- The evolution of the domain structure during switching in the entire switching region of a sample (for sively as yet. An integrated study of switching pro- electrodes of diameter 1 mm) was observed directly cesses, including a comparison of the results obtained with the help of a polarizing microscope in transmitted simultaneously by using direct and indirect methods, is light. The sequence of instantaneous images was of special interest. It has been proved recently for CLT recorded by a video camera with a frequency of that the domain kinetics can be observed with the help 25 frames per second. The switching current was of an optical microscope directly during switching [7]. recorded simultaneously by using traditional methods. In this study, we report on the results of measurements In this way, a unique possibility of establishing a corre- of switching current and direct observation of evolution spondence between the switching current and domain of the domain structure in CLT and SLT and analyze the kinetics was realized. Subsequent computer processing mechanisms determining the domain kinetics. of images was used for a quantitative analysis. Our

2152 SHUR et al.

(a) (b) (c) 3. DOMAIN KINETICS IN CONGRUENT LITHIUM TANTALATE The domain-structure switching in CLT has made it possible for the first time to experimentally observe a new domain wall motion mechanism, which was proposed by us earlier for explaining the domain kinetics in lead germanate [9, 10]. The switching from the single-domain state in CLT Fig. 1. Optical observation of domain-structure evolution upon the application of a field (in the form of a rectan- during switching under the action of a rectangular field pulse in CLT (a) 0.3, (b) 0.9, and (c) 1.5 s from the instant gular pulse) begins with the almost instantaneous (as of field application. The diameter of the switching region is compared to the switching time) formation of a large 1 mm, E = 190 kV/cm. number of small domains with a density attaining 1000 mmÐ2 (Fig. 1a). Subsequently, domain formation ceases completely and switching is carried out only due (a) (b) (c) (d) to domain growth. The following two mechanisms of 50 µm domain growth can be singled out: (i) extremely slow growth of individual domains (at a rate vs) and (ii) fast motion of domain walls (DWs) after coalescence of domains with considerable anisotropy (a difference in the velocities in the Y+ and YÐ directions) (Figs. 1b, 1c). (e) The fast motion of a DW (at a velocity vf) in the three + Y + vf Y directions is the result of formation of steps on the Y– DW due to the coalescence of the moving wall with sin- v gle isolated domains, followed by the growth of steps sf (Fig. 2). In addition, superfast zigzag DWs with a high concentration of steps are formed as a result of coalescence of large-size domains (Figs. 2c, 2d), which leads + Fig. 2. Fast motion of a domain wall in the Y direction and to an anomalously fast motion (at velocity vsf) of DWs the formation of a superfast zigzag domain wall moving in in the YÐ directions. Typical values of velocity of DWs, the YÐ directions in CLT (a) 0.7, (b) 0.8, (c) 0.9, and (d) 1 s obtained as a result of statistical analysis of a sequence from the instant of field application. E = 190 kV/cm. (e) The of instantaneous images for switching in a field of results of simulation of domain kinetics. µ µ 190 kV/cm, were vs ~ 1 m/s, vf ~ 20Ð60 m/s, and µ vsf ~ 130 m/s. It can be seen that for such a relation of velocities, the switching process is determined not 0.9 (b) by the growth of individual domains due to nucleation (a) at the walls [10Ð12] but by the growth of steps formed 0.6 as a result of coalescence of domains. It should be noted 0.3 that the mechanism under consideration differs qualita- 0 tively from the KolmogorovÐAvrami model [13, 14] used conventionally for analysis of switching currents

, arb. units 1

j and based on the study of the growth of individual 0 domains. –1 Computer simulation of the switching process in 0 200 400 600 CLT was carried out in accordance with the proposed Time, arb. units mechanism. The only parameter of the model is the density of randomly distributed seeds formed on the Fig. 3. (a) Diagram of domain growth (dashed lines indicate the final position of the domain wall). (b) Switching current first simulation step. Only the switching due to the and its noise component obtained as a result of simulation growth of the steps formed during domain coalescence [fitting by Eq. (1)]. was simulated (Fig. 3a). The growth of individual domains was disregarded. The obtained domain configurations and the form of the switching current (Fig. 3b) qualitatively agree with the experimental results measurements proved that CLT and SLT differ quanti- (Figs. 2, 4), which confirms the applicability of the pro- tatively, not only in the shape of domains, but also in the posed mechanism of domain evolution. A detailed dis- domain-structure kinetics and, hence, in the shape of cussion of the results of simulation will be given in a the switching current. separate publication.

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KINETICS OF DOMAIN STRUCTURE AND SWITCHING CURRENTS 2153

4. SWITCHING CURRENT IN CONGRUENT j, arb. units ∆S, arb. units LITHIUM TANTALATE (a) (b) In the switching mechanism under investigation, an 0.9 0.9 elementary act of domain-structure rearrangement 0.6 0.6 (coalescence of domains or rapid growth of steps) is accompanied by the generation of an elementary pulse 0.3 0.3 of switching current. In CLT, the high initial concentra- 0 0 tion of isolated domains leads to a large number of 0 0.1 simultaneous events. While measuring the switching –0.2 0 –0.4 –0.1 current, elementary pulses over the entire switching 0 1 2 3 0 1 2 3 region are added; for this reason, a smooth current Time, s pulse with an insignificant noise component is usually observed (Fig. 4a). Fig. 4. (a) Traditional and (b) optical switching currents and In order to quantitatively compare the switching their noise components in CLT [fitting by Eq. (1)]. E = 190 current and the domain-structure kinetics in the course kV/cm. of computer processing of the domain-structure images, we calculated the “optical switching current” ∆ proportional to the change S(t) of the switching- qt()= k exp[]Ð()ttÐ /t , t > t , (3) region area between two consecutive instantaneous 2 cat 02 cat domain configurations (Fig. 4b). It can be seen from where k1 and k2 are the fractions of area, t01 and t02 are Fig. 4 that the optical current obtained for the entire time constants, and tcat is the catastrophe time. switching region is similar to the switching current measured simultaneously according to the conventional Such an analysis makes it possible to determine the method. The time resolution for the optical current is effective velocity of lateral motion and the nucleation relatively small; however, the application of this rate. Alternative methods of the analysis of the current method makes it possible to determine the optical shape for the switching mechanism under investigation switching current for any part of the switching region, will be discussed in a separate publication. which provides a unique possibility to attribute the fea- Analysis of an image clearly shows that the forma- tures of the switching current to specific processes of tion and growth of domains [α(2D) process] dominates domain-structure rearrangement. at the first stage (Fig. 1b), while coalescence of domains due to annihilation of domain walls [β(1D) Our detailed analysis of instantaneous domain con- process] mainly occurs at the second stage (Fig. 1c). figurations made it possible to separate three mechanisms of domain evolution, which we used for choosing It should be noted that the method of analysis an adequate mathematical description of the form of the described above can be applied to extract information switching current. The switching current was approxi- on the domain kinetics from the results of measurement mated by the sum of two current components: of the conventional switching current, without direct optical observations of the domain kinetics in the () () () Ðγ jt ==j1 t + j2 t At Ð 2PSdq/dt. (1) course of switching. The jerky rearrangement of the domain structure The first current component, j1(t), corresponds to leads to a considerably nonmonotonic switching cur- coalescence of the initial small individual domains. For rent and to the emergence of a noise component a low initial concentration of domains (in a weak field), (Barkhausen noise). This component can be found by only partial switching is observed, which terminates subtracting the approximating curve from the switching with the formation of individual domains with a consid- current (Fig. 4). Elementary acts of domain-structure erable size dispersion. Traditionally, the first compo- rearrangement leading to individual jumps were sin- nent of the switching current is not considered, since it gled out from an analysis of instantaneous domain con- is attributed to the “dielectric contribution” associated figurations. with the recharging of the ferroelectric capacitor.

The second current component, j2(t), is observed only for a high density of nuclei (in a strong field). Its 5. DOMAIN-STRUCTURE KINETICS time dependence has two stages, which can be approx- IN STOICHIOMETRIC LITHIUM TANTALATE imately described by the KolmogorovÐAvrami formula Domain kinetics in SLT were studied in the case of modified by us [15] (this formula is normally used for switching under the action of a field and in the case of analysis of experimental currents) with a transition of spontaneous reversal of polarization after the removal growth dimensionality from 2D to 1D (geometrical of the field [4, 16]. It can be seen from Figs. 1 and 5 that catastrophe), α(2D) β(1D): the shapes of domains in CLT and SLT differ qualitatively. Single domains in SLT have the shape of hexag- () []()3 < qt = k1 exp Ð t/t01 , t tcat, (2) onal prisms and pyramids (for nonthrough domains), as

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2154 SHUR et al.

(a) (b) (c) nal domains are formed in SLT under the edges of an electrode (Fig. 5a). The subsequent kinetics of the domain structure are determined by jerky motion of a small number of domain walls oriented strictly along preferred crystallographic directions (Figs. 5b, 5c). The DWs stop and, after a certain “rest time,” rapidly jump to a new static position. As a result, the switching current is a sequence of individual short current pulses (Fig. 6) corresponding to the jumps of DWs. The aver- Fig. 5. Optical observation of domain-structure evolution age time between pulses (rest time) decreases with during switching in a dc field in SLT with liquid electrodes increasing applied field (Figs. 6a, 6b). In contrast to (a) 2.4, (b) 4.44, and (c) 10.32 s from the instant of field application. The diameter of the switching region is 1 mm, congruent lithium niobate [19], the evolution of E = 32 kV/cm. domains in SLT is independent of the type of electrodes used (cf. Figs. 6a, 6c).

6. ANALYSIS OF BARKHAUSEN NOISE 2 2 150 (a) 150 (c) IN STOICHIOMETRIC LITHIUM TANTALATE 100 100 It is obvious that classical methods cannot be used A/mm A/mm µ µ for an analysis of switching currents in SLT (as in the

, 50 , 50 j j case of lithium niobate) [9, 15, 20Ð22]; for this reason, 0 20 60 1000 100 200 we applied statistical methods developed for the analy- Time, ms Time, ms sis of noise.

2 (b) (d) An analysis of the frequency spectrum proved that 150 the Fourier spectrum of switching current in SLT 100 (Fig. 6d) is continuous and contains no well-deﬁned

A/mm 0.1 µ

, arb. units harmonics. The spectrum can be approximated by a

, 50 2 j j power-law dependence with an exponent close to 0.1. 0 10 20 30 110 100 A statistical analysis of switching currents by using Time, ms Frequency, Hz a modiﬁed Korcak method [23] made it possible to determine the cumulative distribution functions for the Fig. 6. Switching currents in SLT with (a, b) liquid and (c, duration of individual current pulses (Fig. 7a) and for d) ITO electrodes. E, kV/cm: (a, c) 20 and (b) 33. (d) The the rest time (Fig. 7b). The distribution function for cur- Fourier spectrum of switching current (E = 25 kV/cm). rent-pulse duration demonstrates the power-law-dependence with exponents 2.8 and 2.0 for polarization switching with the ﬁeld switched on and off, respectively. It should be noted that such behavior, which is ) t (a) (b) invariant to scaling, is typical of self-organized pro- δ 101

> cesses [24].

t 0 ∆ The power-law dependence for the rest time, corre-

( 10

N sponding to a self-organized behavior, is observed only 10–1 100 100 101 in a narrow time interval. For this reason, we approxi- δt, ms mated the experimental data by a formula taking into account the finite scaling range (upper cutoff) [25]: α Fig. 7. Analysis of switching currents in SLT with liquid N()δt = BδtÐ exp()Ðδt/ξ , (4) electrodes: cumulative distribution functions of (a) current pulse duration and (b) rest time. Fitting (a) by a power-law where δt is a time interval, α is the scaling exponent, dependence and (b) by Eq. (4). Squares and circles corre- and ξ is the fractal correlation time. spond to polarization switching with the field switched on and off, respectively. E = 20 kV/cm, liquid electrodes. As a result of the approximation, we determined the scaling exponent α = 0.33 and the fractal correlation time ξ = 5 ms. With the help of the R/S analysis of switching cur- is the case with switching in lithium niobate with the rent [24], we determined the value of the Hurst expo- help of liquid electrodes [17, 18], in contrast to the tri- nent H = 0.70 ± 0.05 (Fig. 8a), which is direct evidence angular prisms and pyramids observed in CLT [17, 18]. of the persistence of domain-wall motion with a long correlation time. It is well known that persistent sto- Domain kinetics in CLT and SLT also differ consid- chastic processes are characterized by the preservation erably. After the application of a field, several hexago- of tendencies with low noise, as compared to Brownian

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 KINETICS OF DOMAIN STRUCTURE AND SWITCHING CURRENTS 2155

R/S N 7. CONCLUSION 2 (a) 10 (b) Thus, we have carried out a detailed analysis of the 101 switching kinetics by comparing the sequences of 1 10–1 instantaneous domain configurations with switching 10 current, which allowed us to find for CLT a new mech- 10–3 101 101 102 anism of fast domain kinetics associated with the coa- Time, ms j, arb. units lescence of domains in ferroelectrics. It is shown that important information on the domain kinetics in SLT Fig. 8. Analysis of switching currents in SLT: (a) R/S anal- can be extracted from a statistical analysis of the noise ysis and (b) distribution function for current-pulse ampli- component of the switching current. The proposed tudes. E, kV/cm: (a) 20 and (b) 24. Fitting by a power-law model of jerky motion of a domain wall is also applica- dependence. ble to lithium niobate. The mechanisms responsible for the observed differences will be considered in a sepa- (random) processes [24]. The distribution function for rate publication. the current-pulse amplitudes demonstrates a power-law dependence with an exponent 2.3 over the entire range ACKNOWLEDGMENTS investigated (Fig. 8b). The authors are grateful to Prof. K. Kitamura and In order to explain the obtained results, a mecha- Dr. K. Terabe for providing the samples of stoichiomet- nism of jerky motion of a DW was proposed. It is well ric lithium tantalate. known [26, 27] that the velocity of a planar DW This study was partially supported by the Russian decreases after its displacement from the initial posi- Foundation for Basic Research (grant no. 01-02- tion. This slowing down is associated with a decrease in 17443), the program “Russian Universities: Fundamen- the local field at the wall under the action of a residual tal Studies” (grant no. 5563), the program “University depolarization field (partly compensated by a fast exter- Studies in Priority Trends of Science and Technology: nal screening) created by bound charges in the region Electronics” (grant no. 03-02-29), and the American behind the moving wall [10]. For switching in fields Foundation of Civil Studies and Development of Inde- exceeding the threshold value insignificantly, this effect pendent States of the Former Soviet Union (grant leads to a halt of the DW after its displacement [27, 28]. no. REC-005). Obviously, defects (which locally enhance the threshold field) may play the role of pinning centers in this REFERENCES case. 1. R. L. Byer, J. Nonlinear Opt. Phys. Mater. 6, 549 (1997). The pinning force and the distance between the cen- 2. G. Rosenman, A. Skliar, and A. Arie, Ferroelectr. Rev. 1, ters determine the power law of the distribution func- 263 (1999). tions for the duration of current pulses and their ampli- 3. M. Yamada, M. Saitoh, and H. Ooki, Appl. Phys. Lett. tudes with close values of exponents. A similar power- 69, 3659 (1996). law dependence was also obtained from an analysis of 4. R. G. Batchko, V. Ya. Shur, M. M. Fejer, and R. L. Byer, Barkhausen pulses emerging during the motion of a Appl. Phys. Lett. 75, 1673 (1999). DW in magnets [29]. 5. V. Ya. Shur, E. L. Rumyantsev, R. G. Bachko, et al., Fiz. Tverd. Tela (St. Petersburg) 41, 1831 (1999) [Phys. Solid A domain wall can continue its motion only if the State 41, 1681 (1999)]. local field exceeds its local threshold value. This can be 6. K. Kitamura, Y. Furukawa, K. Niwa, et al., Appl. Phys. realized during DW rest due to bulk screening of the Lett. 73, 3073 (1998). residual depolarization field as a result of simultaneous 7. V. Gopalan and T. E. Mitchell, J. Appl. Phys. 85, 2304 operation of the following three competing mecha- (1999). nisms: (i) redistribution of volume charges [10, 30], 8. Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, (ii) reorientation of dipole defects [31, 32], and J. Cryst. Growth 197, 889 (1999). (iii) injection of a charge from electrodes through the 9. V. Ya. Shur, A. L. Gruverman, V. V. Letuchev, et al., Fer- dielectric gap [33]. According to estimates, the rest roelectrics 98, 29 (1989). time for a single planar DW must be considerably 10. V. Ya. Shur, in Ferroelectrics Thin Films: Synthesis and shorter than the time constant of bulk screening. In Basic Properties (Gordon and Breach, New York, 1996), addition, the experimentally determined rest time must Vol. 10, Chap. 6. decrease due to the fact that several DWs move simul- 11. R. C. Miller and G. Weinreich, Phys. Rev. 117, 1460 taneously and independently in the sample. This con- (1960). clusion is confirmed by the relation between the value 12. E. Fatuzzo and W. J. Merz, Ferroelectricity (North-Hol- of the fractal correlation time obtained by us (5 ms) and land, Amsterdam, 1967). the experimentally determined time constant of bulk 13. A. N. Kolmogorov, Izv. Akad. Nauk SSSR, Ser Mat. 3, screening in SLT (40 ms) [34]. 355 (1937).

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14. M. Avrami, J. Chem. Phys. 7, 1103 (1939); 8, 212 25. A. Hasmy, M. Foret, J. Pelous, and R. Jullien, Phys. Rev. (1940); 9, 177 (1941). B 48, 9345 (1993). 15. V. Ya. Shur, E. L. Rumyantsev, and S. D. Makarov, J. 26. M. Drougard and R. Landauer, J. Appl. Phys. 30, 1663 Appl. Phys. 84, 445 (1998). (1959). 16. V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, et al., 27. V. Ya. Shur, E. L. Rumyantsev, V. P. Kuminov, et al., Fiz. Appl. Phys. Lett. 76, 143 (2000). Tverd. Tela (St. Petersburg) 41, 126 (1999) [Phys. Solid 17. S. Kim, V. Gopalan, and B. Steiner, Appl. Phys. Lett. 77, State 41, 112 (1999)]. 2051 (2000). 28. V. Ya. Shur, A. L. Gruverman, V. P. Kuminov, and 18. V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, et al., N. A. Tonkachyova, Ferroelectrics 111, 197 (1990). Proc. SPIE 3992, 143 (2000). 29. B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, 19. V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, and J. Appl. Phys. 68, 2908 (1990). E. I. Shishkin, Appl. Phys. Lett. 77, 3636 (2000). 30. V. M. Fridkin, Photoferroelectrics (Nauka, Moscow, 20. V. Ya. Shur, E. L. Rumyantsev, and S. D. Makarov, Fiz. 1976; Springer, Berlin, 1979). Tverd. Tela (St. Petersburg) 37, 1687 (1995) [Phys. Solid 31. U. Robels and G. Arlt, J. Appl. Phys. 73, 3454 (1993). State 37, 917 (1995)]. 32. P. Lambeck and G. Jonker, J. Phys. Chem. Solids 47, 453 21. V. Ya. Shur, S. D. Makarov, N. Yu. Ponomarev, et al., Fiz. (1986). Tverd. Tela (St. Petersburg) 38, 1889 (1996) [Phys. Solid 33. I. Stolichnov, A. Tagantsev, N. Setter, et al., Appl. Phys. State 38, 1044 (1996)]. Lett. 74, 3552 (1999). 22. V. Ya. Shur, E. L. Rumyantsev, S. A. Makarov, et al., 34. V. Ya. Shur, E. V. Nikolaeva, E. I. Shishkin, et al., Appl. Integr. Ferroelectr. 27, 179 (1999). Phys. Lett. (in press). 23. J. Russ, Fractal Surfaces (Plenum, New York, 1994). 24. J. Feder, Fractals (Plenum, New York, 1988; Mir, Mos- cow, 1991). Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2157Ð2164. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2061Ð2068. Original Russian Text Copyright © 2002 by Yu. Boœkov, Erts, Claeson, A. Boœkov. MAGNETISM AND FERROELECTRICITY

Dielectric Response of Ba0.75Sr0.25TiO3 Epitaxial Films to Electric Field and Temperature Yu. A. Boœkov*, D. Erts**, T. Claeson**, and A. Yu. Boœkov*** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia ** Chalmers Technical University, Göteborg, S-41296 Sweden *** St. Petersburg State University, Universitetskiœ pr. 2, St. Petersburg, Petrodvorets, 198094 Russia Received January 15, 2002; in ﬁnal form, February 18, 2002

Abstract—The structure and dielectric parameters of the intermediate ferroelectric layer in the (001)SrRuO || || 3 (100)Ba0.75Sr0.25TiO3 (001)SrRO3 heterostructure grown by laser ablation on (001)La0.294Sr0.706Al0.647Ta0.353O3 were studied. Tensile mechanical stresses accounted for the polar axis in the ferroelectric, being oriented predominantly parallel to the substrate plane. The remanent polarization in the Ba0.75Sr0.25TiO3 layer increased approximately linearly with decreasing temperature in the interval 320Ð200 K. The real part of the dielectric ε ε ≈ permittivity of the intermediate ferroelectric layer reached a maximum '/ 0 = 4400 at TM 285 K (f = 100 kHz). The narrow peak in the temperature dependence of the dielectric loss tangent for the Ba0.75Sr0.25TiO3 ferroelectric layer, observed for T < TM, shifted toward lower temperatures with decreasing frequency and increasing bias voltage applied to the electrodes. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The mechanisms responsible for the substantial differences in the temperature and field dependences of For the device potential of ferroelectric films to be both the real and imaginary (ε'') parts of the dielectric realizable in microelectronics and microwave technol- permittivity ε = ε' Ð iε'' between ferroelectric films and ogy, they should be employed as an integral part of a single crystals remain unclear. Our knowledge of the multilayer heterostructure grown on the corresponding temperature dependences of the remanent polarization substrate and including interlayers of metals, conduct- and coercive field for c- and a-oriented BaxSr1 Ð xTiO3 ing oxides, or superconductors. films (with the polar axis perpendicular and parallel to the substrate plane, respectively) is far from complete. In the large family of perovskite-like ferroelectrics, the BaxSr1 Ð xTiO3 solid solutions are among the most This communication reports on a study of the likely candidates for application in storage cells [1], response of ε' and dielectric loss tangent tanδ of tunable microwave devices [2], IR detector sensors [3], (100)Ba0.75Sr0.25TiO3 (BSTO) epitaxial films to varia- etc. The possibility of using physical techniques of film tions in temperature, electric field, and frequency. Esti- preparation (laser ablation, magnetron sputtering, mates of the activation energy and of the characteristic molecular-beam epitaxy, etc.) to grow multilayer het- relaxation time of the process governing the value of erostructures, including films of ferroelectrics and met- tanδ at temperatures below the phase transition point als (superconductors), has been demonstrated in a num- are presented. ber of publications [4, 5].

In the case of metallic electrodes (Pt, Au, etc.), the 2. EXPERIMENT Shottky barrier forming near the electrode/ferroelectric interface favors suppression of leakage currents in a The laser ablation method (COMPex 100, KrF, λ = plane-parallel capacitor structure [5]. However, the real 248 nm, τ = 30 ns) was employed to grow part of the dielectric permittivity ε' of the intermediate SRO/BSTO/SRO trilayer heterostructures on a ferroelectric layer becomes noticeably reduced [6] and (001)LSATO substrate [(LaAlO3)0.3 + (Sr2AlTaO6)0.7]. its temperature dependence cannot be ﬁtted to the Consecutive SRO BSTO SRO ablation of the CurieÐWeiss relation. While replacing the metal elec- original ceramic targets was conducted at a laser radia- trodes with electrodes made from an oxide having tion density on their surface of 1.5 J/cm2 and an oxygen metallic conductivity [SrRuO3 (SRO), La0.5Sr0.5CoO3, pressure of 0.4 mbar. The deposited layers of the metal etc.) strongly enhances the response of ε' to tempera- oxide and the intermediate ferroelectric layer were sat- ture and electric ﬁeld, dielectric losses are also urated by oxygen in the course of cooling of the increased in this case. SRO/BSTO/SRO heterostructure (760°C 20°C,

2158 BOŒKOV et al. 3. EXPERIMENTAL RESULTS LCR AND DISCUSSION meter An SRO layer grown epitaxially at condensation ° SRO temperatures of 700Ð800 C on the LSATO surface has a smooth surface [9] with rare growth steps whose BSTO height is a multiple of the metal-oxide unit cell param- E PS eter. Because the temperature coefficients of linear β × SRO thermal expansion of strontium ruthenate ( SRO = 11 10Ð6 KÐ1 for the orthorhombic phase at T = 300Ð650 K β × Ð6 Ð1 LSATO [11]) and of the substrate ( LSATO = 10 10 K according to the specification of Crystec, which provided the substrates) are similar, the character of Fig. 1. Sketch of a plane-parallel capacitor mechanical strains in a SRO film grown on LSATO is SRO/BSTO/SRO heterostructure formed by photolithogra- determined by the lattice misfit of the two crystals [m = phy and ion milling. The spontaneous polarization vector P ≈ S (aL Ð aS)/aS +2%, where aL is the lattice parameter of in the BSTO layer is oriented predominantly parallel to the the film and a is that of the substrate]. An SRO layer substrate plane. S grown on (001)LSATO is acted upon by compressive mechanical stresses in the substrate plane. ° The temperature coefficient of linear expansion of 20 C/min) in an oxygen environment (1 atm). The tech- BSTO (β ≈ 12.5 × 10Ð6 KÐ1) is considerably larger than nology for growing the SRO/(Ba,Sr)TiO /SRO hetero- 0 3 those of LSATO and SRO. As a result, at temperatures structures is described in more detail in [7, 8]. close to that of the phase transition (TC = 340 K for bulk The microstructure and phase composition of the samples [12]), a BSTO layer grown on SRO/LSATO is heterostructures thus grown were studied by x-ray dif- subject to tensile mechanical stresses in the substrate fraction (CuKα radiation, ω/2θ and φ scans, rocking plane. The mechanical strains generated in the hetero- curves). To determine the lattice unit cell parameters of structure due to the BSTO and SRO lattice misfit the BSTO layer in the direction parallel to the substrate become partially relieved during the formation of the plane (a||) and along the normal to its surface (a⊥), dif- ferroelectric layer. The tensile mechanical stresses act- fractograms were measured in the geometry where the ing in the substrate plane favor the in-plane orientation plane containing the incident and reflected X-ray of the polar axis of the BSTO ferroelectric layer (paral- beams was orthogonal to either (001) or (101)LSATO. lel to the substrate plane). To estimate the effective grain size and strain distribu- Let us analyze first the x-ray diffractograms tion in the ferroelectric layer, the first four peaks in a obtained for the grown SRO/BSTO/SRO heterostruc- (00n)ω/2θ x-ray scan were measured using high-preci- tures and the data on the free-surface morphology of the sion x-ray optics [7]. BSTO layer and, after this, the dielectric parameters of this layer. The surface morphology of the BSTO layer grown on SRO/LSATO was studied with an atomic-force microscope (Nanoscope-IIIa, tapping mode). The data 3.1. The Structure of the BSTO Layer on the surface morphology of the SRO layer grown on and the Morphology of Its Free Surface LSATO can be found in [9]. The x-ray diffractograms obtained did not contain The preparation of contact pads on the top SRO peaks indicating the presence of macroinclusions of layer and of auxiliary holes in the intermediate ferro- secondary phases in the grown SRO/BSTO/SRO het- electric layer is outlined in [10]. erostructures (Fig. 2 and inset to Fig. 3). The preferred orientation of the ferroelectric layer and the oxide elec- The capacitance C and tanδ of the plane-parallel trodes in the SRO/BSTO/SRO heterostructure was SRO/BSTO/SRO capacitors formed (Fig. 1) were mea- (001)[010]SRO || (100)[010]BSTO || (001)[010]SRO || sured with an hp 4263A LCR meter in the frequency (001)[010]LSATO. ± interval 1Ð100 kHz, both with a bias voltage Vb = 2.5 V The parameter a⊥ = 3.981 Å, calculated for the applied to the electrodes and without it. The real and BSTO layer from the x-ray data (T = 300 K), was sub- imaginary parts of the dielectric permittivity of the stantially smaller than a|| = 3.992 Å. This suggests that ε BSTO layer were calculated from the relations ' = the polar axis in the grown ferroelectric layer is ori- Cd/S and ε'' = ε'tanδ , where d = 700 nm is the ferro- ented predominantly parallel to the substrate plane (a- electric layer thickness and S = 200 × 200 µm2 is the oriented layer). The FWHM of the rocking curve for the area of the upper contact in the capacitor heterostruc- (200)BSTO x-ray peak of the SRO/BSTO/SRO hetero- ture; the bottom electrode was common for all the structure was 0.23° [inset (a) in Fig. 3]. The FWHM capacitors grown on the chip. obtained for the BSTO layer is in accord with the typi-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 DIELECTRIC RESPONSE OF Ba0.75Sr0.25TiO3 EPITAXIAL FILMS 2159

5 cal values reported in the literature for (Ba,Sr)TiO3 10 epitaxial layers with the polar axis parallel to the substrate plane. The slight increase in the FWHM of the ferroelectric layer in the SRO/BSTO/SRO com-

pared to that of the rocking curve measured for a (202)BSTO (202)SRO , arb. units (100)BSTO I (202)LSATO (002)LSATO

3 (004)LSATO precisely c-oriented Ba0.25Sr0.75TiO3 ﬁlm in the 10 (200)BSTO θ SRO/Ba0.25Sr0.75TiO3/SRO heterostructure [7] should 65 2 , deg 70 1 , counts/s be attributed to the presence of grains whose polar axis I 2 is orthogonal to the substrate plane in the bulk of the 3 BSTO layer. The effective lattice parameter for the (001)LSATO

1 (300)BSTO BSTO layer in the SRO/BSTO/SRO heterostructure, 10 (003)LSATO 2 1/3 ≈ aeff = (a⊥a||) 3.98 Å, was larger than the corresponding parameter for stoichiometric bulk samples (a ≈ 3.97 Å) [13]. This signals a high density of oxygen 20 40 60 80 100 120 vacancies in the intermediate ferroelectric layer. 2θ, deg Figure 3 presents the FWHM ¯ of the (n00) peaks plotted vs. θ for an x-ray (001)ω/2θ scan. The ¯ vs. θ Fig. 2. Diffractogram (CuKα, ω/2θ) obtained for a dependence can be written as [14] SRO/BSTO/SRO heterostructure in the geometry where the incident and reﬂected x-ray beams were in a plane normal ∅ λ ()θ θ∂ to (001)LSATO. (1, 2) CuKβ x-ray peaks from the ferroelec- = 0.9 0 d0 cos + 2tan a/a, (1) tric layer and the substrate, respectively, and (3) (400)BSTO. Inset shows a fragment of the diffractogram λ where 0 = 1.54056 Å is the x-ray wavelength, d0 is the measured on the same heterostructure in the geometry ∂ where the incident and reﬂected x-ray beams were in a sample grain size, a is the lattice parameter, and a is plane normal to (101)LSATO. the lattice parameter distortion associated with nonuniform lattice strains. The slope of the straight line in Fig. 3 was used to determine 2∂a/a ≈ 1.2 × 10Ð3, the value characterizing the average strain in the BSTO layer. The intercept of the same line on the ordinate axis 5 provided an estimate of the grain size in the BSTO layer, d0 = 35 nm. The values obtained in this way for ∂ 4 a/a and d0 for the BSTO layer in the SRO/BSTO/SRO

heterostructure agree well, on the whole, with the cor- , rad

θ 3 2 responding literature ﬁgures for the SrTiO3 [15] and (a) (b) Ba Sr TiO [7] epitaxial ﬁlms. The lattice parame-

0.25 0.75 3 cos ter distortion in the BSTO layer may be caused by off-

2 counts/s × ∅ ,

3 stoichiometry, the presence of structural defects (grain 3 , arb. units I 10 boundaries, stacking faults, dislocations, etc.) in its vol- 10 ume, and mechanical stresses. 1 –0.40 0.4 ω, deg 0 360 Data on the morphology of the free surface of a film φ, deg are useful when analyzing the mechanisms governing 0 0.2 0.4 0.6 0.8 its growth. These data are particularly important in the sinθ case of thin multicomponent oxide layers grown on the surface of auxiliary (buffer or electrode) layers with a high density of structural defects. Figure 4a presents an Fig. 3. FWHM of x-ray peaks ¯ obtained in a (00n)ω/2θ atomic-force microscope (AFM) image of the scan plotted as a function of θ for the BSTO layer in the (100)BSTO || (001)SRO || (001)LSATO free surface SRO/BSTO/SRO heterostructure. Inset a shows a rocking obtained in height mode. One can clearly see grains of curve for the (200)BSTO peak, and inset b, a φ scan for the (111)BSTO x-ray peak from the same heterostructure. size d1 = 150Ð200 nm (dimensions in the substrate plane) whose interfaces are decorated by characteristic grooves. The formation of grooves on the free film surface at the points where it crosses the grain boundaries excess free energy compared to the ions in the grain results from free-energy minimization in the growing bulk). As follows from the x-ray data [φ scan of the film–substrate system. The character of the grooves is (111)BSTO peak], the azimuthal misorientation of determined to a considerable extent by the relative grains in the ferroelectric layer did not exceed 0.4°. magnitude of the free energy of the free surface and of Because grain boundaries can substantially affect the that of the grain boundaries (the ions present in the parameters of films of the perovskite-like oxides, it region of grain boundaries and in the surface layer have appears appropriate to mention at least the main mech-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2160 BOŒKOV et al.

(a) (b)

50 deg

0 deg

0 200 400 600 0 200 400 600 nm nm

0 200

nm 400 600

Fig. 4. (a) AFM image of the free surface of a 700-nm-thick BSTO layer obtained in height mode (top view). The ferroelectric layer was grown on (001)SRO || (001)LSATO. (b) AFM image of the same part of the ferroelectric layer obtained in phase mode (top || view). (c) Image of the free surface of a 700-nm-thick SrTiO3 layer grown on (001)SRO (001)LSATO obtained in height mode (viewed at 45°). anisms responsible for their formation. One of the rea- material are deposited on the substrate in a coherent sons for grain formation in a BSTO layer is lattice mis- manner. Grain boundaries can originate here due to the fit with the substrate [in our case, with (001)SRO || phase adsorbed on the surface of the growing layer (001)LSATO]. Minimization of the energy of elastic being off-stoichiometric or as a result of the low mobil- mechanical stresses in the nucleusÐsubstrate system ity of the particles of this phase. The excess compo- can produce a certain azimuthal misorientation of sta- nents segregate at the outer boundaries of the growing ble islands forming in the initial growth stage of a fer- islands. As a result, the grown film is made up of grains roelectric layer. This accounts for the formation of a of the given composition, with interlayers of an off-sto- polycrystalline layer with grains separated by small- ichiometric phase. The density of such interlayers in the angle boundaries. The small-angle grain boundaries are layer volume can be reduced by appropriate heat treat- seen clearly in high-resolution electron-microscopy ment in an oxygen environment [16]. images of cross sections of multilayer heterostructures The pronounced difference between the effective containing layers of perovskite-like oxides [4]. grain sizes derived from x-ray measurements and from Grain boundaries can also form in a ferroelectric images of the BSTO layer free surface can be film in the case where the nuclei of the condensing accounted for by the grains growing with increasing

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 DIELECTRIC RESPONSE OF Ba0.75Sr0.25TiO3 EPITAXIAL FILMS 2161 layer thickness or by the presence of a ﬁne structure of (a) 6000 block interfaces within a grain, which is not resolved 0

ε 5

'/ 1

when viewing the ﬁlm free surface in height mode. ε

In contrast to the height mode, the AFM images × 2 obtained in the phase mode contain information on 4000 –3 1 10 –30 3 variations in the nanoscale mechanical characteristics 0 6 ε

E, 10 V/m of the sample surface layer. The image obtained in the '/ 3 phase mode (Fig. 4b, T = 300 K) of the same part of the ε BSTO layer that is shown in Fig. 4a resolves clearly not 2000 4 only the small-angle crystallographic boundaries but also characteristic features within the grains in the form 5 6 of sharp boundaries where the signal changes its phase. 0 Azimuthal orientation of these intragrain boundaries is (b) predominantly along the directions making an angle of 0.8 45° with [001] and [010]BSTO. At room temperature, 14 the grown BSTO layer (or at least a sizable part of its 0.6 ) [Hz] 2 volume) was in the ferroelectric phase. One of the rea- 1 –1 sons for the appearance of the above features in the 2 τ 1 ln (

δ 3 8 phase-mode images of the free surface of the BSTO 0.4 410 film can be the formation of 90° walls between ferro- tan 103/T, K–1 electric domains in its volume. Phase-mode AFM 4 images of Ba0.25Sr0.75TiO3 epitaxial films [7], which 0.2 were in paraelectric phase at 300 K, did not reveal any features indicating a complex structure of crystallo- 5 graphic grains. The atomic-force microscope was used 0 100 200 300 400 successfully in the height mode to study the domain T, K structure of PZT films [17]. To find the extent to which the granular character of Fig. 5. (a) Temperature dependences of the real part of a grown BSTO layer can be accounted for by off-sto- ε ε ∆ε ε dielectric permittivity '/ 0 (curves 1Ð4) and of '/ 0 ichiometry of the adsorbed phase and by the lattice mis- (curves 5, 6) obtained for the BSTO layer in the fit between BSTO and SRO/LSATO, we grew a 700-nm SRO/BSTO/SRO heterostructure. f (kHz): (1) 10 and (2Ð4) || thick SrTiO3 epitaxial layer on the (001)SRO 100; Vb (V): (1, 2) 0, (3) +1, and (4) +2.5; (5, 6) experiment (001)LSATO surface under the same technological and calculation made using Eq. (7), respectively. Inset ε ε conditions. Strontium titanate is a better lattice match shows the '/ 0(E) dependence measured for a BSTO layer to SRO and LSATO (m < 0.7%). The mechanisms that at T = 295 K and f = 100 kHz. (b) Temperature dependences δ can give rise to a BSTO layer becoming off-stoichio- of tan measured on the BSTO layer at different frequen- metric but that do not operate in STO were considered cies and bias voltages applied to the electrodes. f (kHz): (1, 4, 5) 100, (2) 10, and (3) 1. Vb (V): (1Ð3) 0, (4) +1, and in [18]. The FWHM of the rocking curve of the (5) +2.5. Inset shows Arrhenius plots for the relaxation time (200)STO reflection from the SRO/STO/SRO hetero- τ−1 (1/T) in the BSTO layer obtained at Vb (V): (1) 0 and structure is smaller by about a factor of three [9] than (2) +2.5. the corresponding figures for the BSTO layer. Figure 4c shows an image of the free surface of a 700-nm thick STO layer grown on SRO/LSATO. On the free STO surface, we clearly see resolved growth steps whose As in the case of bulk BSTO crystals, the maximum in height is a multiple of the strontium titanate lattice the ε'(T) curve shifted toward higher temperatures parameter. No grooves decorating grain boundaries on when a bias voltage was applied to the electrodes the free STO layer surface were found. This, however, ε (curves 3, 4 in Fig. 5a). At temperatures below TM, ' cannot be considered proof of the absence of grain exhibited a noticeable frequency dispersion. As f boundaries in the strontium titanate layer but rather decreased, the real part of the dielectric permittivity indicates that the azimuthal misorientation of grains increased, with the difference ∆ε' = ε'(10 kHz) Ð and the density of grain boundaries in its volume are ε'(100 kHz) passing through a sharp peak at T ≈ 185 K smaller than those in BSTO/SRO/LSATO. ∆ε (Fig. 5a). For T > TM, the difference ' decreased monotonically with increasing temperature. When f ∆ε 3.2. Dielectric Parameters of the BSTO Layer was lowered from 10 to 1 kHz, the peak in the '(T) curve shifted toward lower temperatures by about 30 K. The temperature dependence of ε' measured at Taking into account the fact that the real part of dielec- 100 kHz for the BSTO layer had a clearly pronounced tric permittivity of epitaxial layers of the (Ba,Sr)TiO3 ≈ maximum at a temperature TM 280Ð290 K (Fig. 5a). solid solutions in the paraelectric phase is virtually fre-

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2162 BOŒKOV et al. quency-independent [7, 19], it seems natural to dielectric permittivity, we used ε'(T) and tanδ()T attribute the presence of the peak in the ∆ε(T) relation dependences measured at different frequencies. It to dielectric relaxation of the ferroelectric phase. At seems natural to assign the sharp peak in the tempera- temperatures below TC, a BSTO film is divided into fer- ture dependence of tanδ to resonance in the interaction roelectric domains. Ferroelectric domains in an a-ori- of domain walls with the electromagnetic wave, when ° ented BSTO layer are separated by 90 domain walls. its frequency approaches τÐ1. When the frequency was The displacement of ferroelectric domain walls by an δ electromagnetic wave propagating in a BSTO layer is lowered, the peak in the tan ()T dependence for the accompanied by redistribution of the electrons, cancel- BSTO layer shifted toward lower temperatures (Fig. 5b), which is in full agreement with the tempera- ing the polarization charge. Incomplete compensation τ of the polarization charge at the points where domain ture dependence of [see Eq. (4)]. walls cross the grains accounts for the generation of an Within the Debye model, the contribution of the ε internal electric depolarization field, which acts relaxation process to dielectric permittivity r can be directly on ε' and tanδ [19]. The dielectric relaxation written in the form [23] time τ associated with charge redistribution in the D ε ε ()ε ε ()ωτ BSTO layer can be written as [20] r = ∞ + S Ð ∞ /1Ð i , (5) τ ε ρ ε ε D = ' , (2) where S and ∞ are the relaxation contributions to the dielectric permittivity at low and high frequencies, where ρ is the electrical resistivity. ω π ε ӷ ε respectively, and = 2 f. For BSTO, we have S ∞ The electrical resistivity of the (Ba,Sr)TiO3 interme- [24]. It follows from Eq. (5) that diate layer in an epitaxial heterostructure with electrodes of conducting oxides is, as a rule, substantially ε'' ∼ ωτ/1()+ ω2τ2 . (6) lower than that of the corresponding single crystals. One of the main reasons accounting for the substantial According to Eq. (6), ε'' should be maximum at ωτ = 1. τ ϕ conductivity of epitaxial ferroelectric layers is the high To estimate 0 and 0, we made use of the temperature concentration of oxygen vacancies in their volume. The dependence of the reciprocal relaxation time. The val- conductivity of oxygen-deficient BSTO films is deter- ues of τÐ1 at different temperatures were derived from mined to a considerable extent by the intensity of elec- the position of the maximum in the measured ε''(T) tron promotion from donor centers into the conduction ()τÐ1 band [18]. Under these conditions, the carrier concen- plots. The ln ~ 1/T dependences for the BSTO tration in the ferroelectric layer depends exponentially layer at E = 0 are plotted in the inset to Fig. 5b (curve 1). τ on the temperature and electric field (the Poole–Frenkel The Arrhenius plot yielded the following values: 0(E = × Ð9 ϕ emission [21]) and the temperature dependence of ρ 0) = 5 10 s and 0(E = 0) = 105 meV. follows the relation The maximum in the tanδ()T curve for the BSTO ρϕ∼ exp[]()Ð βE1/2 /k T , (3) layer shifted toward lower temperatures when a bias L B voltage was applied to the metal-oxide electrodes where ϕ is the energy gap between the donor level and (curves 4, 5 in Fig. 5b). In accordance with Eq. (3), the L height of the barrier governing the intensity of electron the conduction band edge, kB is the Boltzmann con- β 3 πε ε 1/2 ε promotion into the conduction band from donor centers stant, = (e / hf 0) , e is the electronic charge, hf is ε decreased in an electric field, which was accompanied the high-frequency dielectric permittivity, and 0 is the by a decrease in τ. The relation ln(τÐ1) ~ 1/T obtained at dielectric permittivity of the vacuum. From the temper- E = 3.5 × 106 V/m for the BSTO layer in the ature dependence of the conductivity determined for SRO/BSTO/SRO heterostructure is plotted in the inset oxygen-deficient STO single crystals and BSTO epi- ϕ to Fig. 5b (curve 2). When biased at E = 3.5 × 106 V/m, taxial films, the value of L was found to be ~0.1 eV [9, ϕ × 7 the activation energy 0 decreased by about 12% to 18]. For E = Vb/d > 2 10 V/m, ions can also contribute 93 meV, while the attempt time did not change: τ (E = to the conductivity of the ferroelectric layer [22]. 0 3.5 × 106 V/m) = τ (E = 0) = 5 × 10Ð9 s. Recalling Eqs. (2) and (3), the dielectric relaxation 0 time associated with electron redistribution in the According to [21], β = 5.1 × 10Ð24 J m1/2 VÐ1/2 for domain wall region in the BSTO layer can be written as bulk STO crystals. With such a coefficient β, the activation energy should have decreased in a field E = 3.5 × ττ ()ϕ 6 = 0 exp 0/kBT , (4) 10 V/m by about 50%. Since there are no grounds to suspect a sharp increase in the high-frequency dielec- τ ϕ ϕ β 1/2 where 0 is the attempt time and 0 ~ ( L Ð E ) is the tric permittivity of the BSTO layer compared with the activation energy. corresponding figures available for bulk single crystals, ϕ To make a comprehensive analysis of the extent to the relatively weak response of 0 to an external electric which the relaxation process affects the dynamics of field in the BSTO layer should be apparently attributed

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 DIELECTRIC RESPONSE OF Ba0.75Sr0.25TiO3 EPITAXIAL FILMS 2163

4 temperature. For T < 200 K, the growth of Pr with 2 1 decreasing temperature became less steep. The coer- 10 cive ﬁeld increased with decreasing temperature in the C/m 0 –1 interval 320Ð200 K without a visible sign of saturation. 2 3 8 –1 200 K , 10 Thus, the polar axis orientation in the BSTO layer is P

C/m 1 V/m

6 determined to a considerable extent by the actual type –2 6 –10 1 E, 107 V/m of mechanical strains due to the lattice misfit and a dif- 2 , 10 , 10 r C ference in the thermal expansion coefficient between P 4 E 2 2 the layer and the substrate. For T < TC, relaxation of ferroelectric domain walls in the BSTO layer accounts 2 f = 50 kHz 1 2 for the substantial frequency dependence of dielectric τ 0 permittivity. For E = 0, the characteristic time 0 and the ϕ 200 240 280 320 activation energy 0 describing domain wall relaxation T, K in the BSTO layer are 10Ð9 s and 105 meV, respectively, ϕ so that 0 decreases by about 12% when a bias voltage Fig. 6. Remanent polarization Pr and coercive field EC plot- of ±2.5 V is applied to the electrodes. ted as a function of temperature for the BSTO layer in the SRO/BSTO/SRO heterostructure. Inset shows the field dependence of the polarization P(E) measured on the same layer at T = 200 K. ACKNOWLEDGMENTS This study was carried out under the aegis of scientific cooperation between the Russian and the Swedish to the electric fields in the bulk of crystal grains and at Royal Academies of Sciences and was partially sup- grain boundaries being substantially different. ported by the Ministry of Science of the Russian Feder- τ ϕ We used the values of 0 and 0 thus obtained to ation, project no. 4B19. approximate the measured temperature dependence of ∆ε'(T) = ε'(10 kHz) Ð ε'(100 kHz) for the BSTO layer (Fig. 5a). It follows from Eq. (5) that REFERENCES ∆ε ()ε ε 1. C. S. Hwang, Mater. Sci. Eng. B 56, 178 (1998). '()T = S Ð ∞ (7) 2. J. P. Hong and J. S. Lee, Appl. Phys. Lett. 68, 3034 ×ω()τ2 ω2 2 ()ω2τ2 ()ω2τ2 2 Ð 1 /1+ 1 1 + 2 , (1996). 3. J.-G. Cheng, X.-J. Meng, J. Tang, et al., Appl. Phys. Lett. ω π ω π where 1/2 = 10 kHz and 2/2 = 100 kHz. The 75, 3402 (1999). experimental ∆ε'(T) relation can be fitted quite well by ε ε ε ≈ 4. Yu. A. Boikov, Z. G. Ivanov, A. N. Kiselev, et al., J. Appl. Eq. (7) for ( S Ð ∞)/ 0 3300. Phys. 78, 4591 (1995). ε The '(T = 295 K) curve (inset to Fig. 5a) exhibits a 5. B. Nagaraj, T. Sawhney, S. Perusse, et al., Appl. Phys. weak hysteresis, signaling the presence of a ferroelec- Lett. 74, 3194 (1999). tric phase in the bulk of the BSTO layer. At T = 295 K, the dependence of ε' of the BSTO layer on electric field 6. L. J. Sinnamon, R. M. Bowman, and J. M. Gregg, Appl. was virtually symmetrical with respect to the E = 0 Phys. Lett. 78, 1724 (2001). point. In contrast to a c-oriented Ba0.25Sr0.75TiO3 film 7. Yu. A. Boœkov and T. Claeson, Fiz. Tverd. Tela [7], the electronic processes occurring at the (St. Petersburg) 43, 2170 (2001) [Phys. Solid State 43, SRO/BSTO interface do not noticeably affect the 2267 (2001)]. dynamics of BSTO dielectric permittivity. At T = 8. B. M. Gol’tsman, Yu. A. Boœkov, and V. A. Danilov, Fiz. 300 K, ε' of the ferroelectric layer in the Tverd. Tela (St. Petersburg) 43, 874 (2001) [Phys. Solid SRO/BSTO/SRO heterostructure decreased by about a State 43, 908 (2001)]. ± factor of three when a bias voltage 2.5 V was applied 9. Yu. A. Boikov and T. Claeson, Physica C (Amsterdam) to the electrodes. 336, 300 (2000). The field dependence of polarization P(E) measured 10. Yu. A. Boœkov and T. Claeson, Fiz. Tverd. Tela at T < 320 K exhibited distinct hysteresis loops. For (St. Petersburg) 43, 323 (2001) [Phys. Solid State 43, T < 300 K, the hysteresis loops were practically sym- 337 (2001)]. metrical with respect to the E = 0 point (inset to Fig. 6) and remained unchanged with the frequency varied 11. J.-P. Maria, H. L. McKinstry, and S. Trolier-McKinstry, from 10 to 50 kHz. The temperature dependences of the Appl. Phys. Lett. 76, 3382 (2000). remanent polarization Pr and coercive field EC for the 12. A. D. Hilton and B. W. Rickkets, J. Phys. D 29, 1321 BSTO layer are displayed in Fig. 6. For 200 < T < 320 K, (1996). Pr increased approximately linearly with decreasing 13. A. von Hippel, Rev. Mod. Phys. 22, 221 (1950).

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2164 BOŒKOV et al.

14. E. D. Specht, R. E. Clausing, and L. Heatherly, J. Mater. 20. N. P. Bogoroditskiœ, Yu. M. Volokobinskiœ, A. A. Vorob’ev, Res. 5, 3251 (1999). and B. M. Tareev, Theory of Dielectrics (Moscow, 1965). 15. D. Fuchs, M. Adam, P. Schweiss, et al., J. Appl. Phys. 88, 21. J. R. Yeargan and H. L. Taylor, J. Appl. Phys. 39, 5600 1844 (2000). (1968). 16. Yu. A. Boœkov, T. Claeson, and A. Yu. Boœkov, Zh. Tekh. 22. J.-H. Lee, R. Mohammedali, J. H. Han, et al., Appl. Fiz. 71 (10), 54 (2001) [Tech. Phys. 46, 1260 (2001)]. Phys. Lett. 75, 1455 (1999). 17. A. Seifert, F. F. Lange, and L. S. Speck, J. Mater. Res. 16, 23. C. Kittel, Introduction to Solid State Physics (Wiley, 680 (1995). New York, 1966, 3rd ed.), p. 392. 24. W. Kleemann and H. Schremmer, Phys. Rev. B 40, 7428 18. Yu. A. Boikov and T. Claeson, Supercond. Sci. Technol. (1989). 12, 654 (1999). 19. Yu. A. Boikov and T. Claeson, Appl. Phys. Lett. 79, 2052 (2001). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2165Ð2170. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2069Ð2074. Original Russian Text Copyright © 2002 by Shmyt’ko, Afonikova, Torgashev.

MAGNETISM AND FERROELECTRICITY

Anomalous States of the Crystal Structure of (Rb0.1(NH4)0.9)2SO4 Solid Solutions in the Temperature Range 4.2Ð300 K I. M. Shmyt’ko*, N. S. Afonikova*, and V. I. Torgashev** * Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia ** Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344090 Russia Received February 25, 2002

Abstract—Crystals of (Rb0.1(NH4)0.9)2SO4 solid solutions are studied using x-ray diffractometry. It is revealed that the temperature dependence of the lattice parameter a exhibits an anomalous behavior, namely, the “invar effect” at temperatures above the ferroelectric phase transition point Tc and an anomalous increase in the temperature range from Tc to the liquid-helium temperature. An anomalous increase in the lattice parameter a and an increase in the intensity of Bragg reﬂections with a decrease in the temperature are interpreted within the model of the coexistence of two sublattices hypothetically responsible for the ferroelectric phase transition. A series of superstructure reﬂections observed along the basis axes corresponds to a sublattice formed in the matrix of the host structure. Analysis of the ratio between the lattice parameters of these structures allows the inference that, in the temperature range 4.2Ð300 K, the structure of the crystal under investigation can be considered an incommensurate single-crystal composite. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the temperature behavior of the physical properties of these compounds and elucidation of the nature and It is known that crystals belonging to the structural mechanisms of the phase transitions involved. family of rubidium ammonium sulfates of the general formula (Rbx(NH4)(1 Ð x))2SO4 form a continuous series of solid solutions [1]. Crystals of the rubidium-free 2. SAMPLE PREPARATION AND EXPERIMENTAL TECHNIQUE ammonium sulfate (NH4)2SO4 undergo a ﬁrst-order ferroelectric phase transition with a change in the symme- The experiments were carried out using single-crys- try from Pnam to Pna21 at a temperature of 223 K. A tal samples with a rubidium content x = 0.1. According decrease in the temperature brings about reversal of the to the phase diagram, these crystals at temperatures spontaneous polarization of these crystals [2]. In the below 223 K belong to ferroelectrics [1]. The choice of (Rbx(NH4)(1 Ð x))2SO4 system, the ferroelectric phases this composition was made for the following reasons: are retained up to compositions with x = 0.6Ð0.65 [1]. (i) the low content of rubidium in the solid solution and, consequently, the possible manifestation of the proper- The origin of the ferroelectric phase transition in ties of the rubidium-free ammonium sulfate crystals of this family has been discussed in the frame- (NH4)2SO4, on the one hand, and (ii) the presence of work of different models, such as the orderÐdisorder substitutional atoms that would provide a manifestation model [3], improper ferroelectrics [4], coupled oscilla- of the structural features typical of the entire series of tors [5], coupled oscillatorsÐrelaxors [6], and the model compositions 0 < x < 0.6 containing low-temperature of two ferroelectric nonequivalent sublattices [7]. As is ferroelectric phases, on the other hand. evident from the aforementioned approaches, the mechanism of the ferroelectric phase transition under The quality of the studied samples was controlled investigation is rather complicated and, until presently, using traditional rolling-crystal and Laue methods. The has not been clearly understood [8]. determination of the crystal symmetry and precision measurements of the unit cell parameters were per- The compounds under consideration belong to a formed on a Rigaku AFC6S four-circle diffractometer very interesting class, namely, the class of orientational (MoKα radiation) at temperatures above and below Tc glasses, in which the subsystem of multipole moments (300 and 203 K, respectively). The low-temperature is efﬁciently frozen with a decrease in the temperature. measurements were carried out using a Strimer cryostat These compounds exhibit a disorder intermediate (Rigaku). For x-ray diffraction analysis, samples were between the disorders observed in crystals and conven- prepared in the form of balls ~0.25Ð0.35 mm in diame- tional “canonical” glasses [9–12]. ter. The space groups were determined prior to and after In the present work, we performed an x-ray diffrac- the transformation and coincided with those available in tion investigation of the real structure of the literature, namely, Pnam and Pna21, respectively. (Rb0.1(NH4)0.9)2SO4 solid solutions. In our opinion, this The results of the solution of the atomic structure at investigation is necessary for correct interpretation of these temperatures will be published in a separate paper.

2166 SHMYT’KO et al.

Lattice parameters of (Rb0.1(NH4)0.9)2SO4 crystals T, K Space group a, Å (∆a) b, Å (∆b) c, Å (∆c) α, deg (∆α) β, deg (∆β) γ, deg (∆γ) 300 Pnma 7.8081 5.9862 10.5525 90.003 90.038 89.994 (0.0010) (0.0008) (0.0016) (0.012) (0.010) (0.011)

203 Pna21 7.8161 5.9564 10.5206 90.024 89.985 89.978 (0.0018) (0.0012) (0.0012) (0.012) (0.013) (0.018)

The temperature measurements of the lattice param- Reasoning from the results obtained, we can pro- eters were performed on a Siemens D500 x-ray diffrac- pose three models of the crystal structure under investi- tometer with the use of a helium cryostat designed at gation. According to the first model, the crystal struc- the Institute of Solid-State Physics, Russian Academy ture is formed by layers of new phase precipitates of Sciences. The diffractometer was modified using whose lattice spacing differs from the lattice spacing of special programs with the aim of recording reciprocal the matrix along the b* direction. It should be noted lattice maps of the single-crystal sections. that these phase precipitates are unrelated to enrichment or depletion of particular regions of the crystal with rubidium atoms in the case when these atoms are 3. RESULTS AND DISCUSSION statistically distributed over the (NH4) positions. As follows from analyzing the reciprocal lattice maps The measured characteristics of the crystal structure depicted in Fig. 1, the lattice parameter bph of the pos- of (Rb0.1(NH4)0.9)2SO4 solid solutions are presented in sible phase precipitates is less than the lattice parameter the table. The angular distribution of substructure ele- b of the matrix ()b* > b* . This ratio between the lat- ments of the samples (ω scan mode) did not exceed m ph m 0.3°. Analysis of the rocking x-ray patterns revealed tice parameters of the new phase precipitates and the additional (superstructure) reflections lying off the matrix should correspond to an enrichment of the new- main layer lines. The superstructure reflections phase regions with rubidium atoms in accordance with the concentration dependence of the lattice parameter observed were identified using the reciprocal lattice b(x) obtained in [1]. However, the lattice parameter b of cross sections along the basis axes. Figure 1 shows the the possible phase precipitates proves to be equal to characteristic (b*Ða*) plane sections of the reciprocal 9.2593 Å, whereas this parameter even for pure Rb SO space with the aforementioned superstructure reflec- 2 4 tions at room temperature. It can be seen that the super- must be no less than 10.44 Å. Moreover, within the structure reflections are located along the b* direction framework of the concentration model, a change in the not only for the (0 k 0) reflections but for the (h k 0) lattice parameter b should be attended by variations in reflections as well. The distance from these reflections the lattice parameters a and c, which is in contradiction to the main reflection changes with variations in the with the experiment. The lattice parameters a and c of order of the reflection in accordance with the coexist- the new phase precipitates observed are assumed to be ence of two lattices with the following parameters: nearly identical to those of the matrix. This inference 10.5525 Å for Bragg reflections and 9.2593 Å for satel- stems from the fact that, in the case of the (h k 0) reflec- lite reflections. Similar sections were obtained for the tions (see, for example, the reciprocal lattice map in the vicinity of the (1 6 0) point in Fig. 1d), the satellite (b*Ðc*) plane of the reciprocal space. An examination reflections (–h k 0), (0 k l), and (0 k Ðl) are located, of these sections revealed that the (0 k l) superstructure accurate to within the scan step, along the a* (c*) direc- reflections are also located along the b* direction. In tion at the same distance in the reciprocal space as the the reciprocal space, the obtained sections correspond main reflection. Furthermore, the absence of the satel- to two interpenetrating lattices in which the projections lite reflection for the Bragg reflection (0 4 0) and the of all the points onto the (a*Ðc*) plane virtually coin- presence of the satellite reflections for the Bragg reflec- cide, thus retaining the motif of the initial structure, tions (0 2 0) and (0 6 0) suggest a structural dissimilar- whereas the points along the b* direction are spaced at ity between the possible phase precipitates and the progressively larger intervals with distance away from 1 matrix. For a statistical distribution of rubidium atoms, the origin of the coordinates. each Bragg reflection should be accompanied by the 1 It is quite possible that the points of both sublattices also do not satellite reflections. coincide in the (a*Ðc*) plane. This assumption is based on the reciprocal lattice map in the vicinity of the (1 0 0 0) point The model of phase precipitates is consistent with (Fig. 1e), which contains a very weak diffuse reflection in addi- the experiment under the assumption that, in particular tion to the strong Bragg reflection. Similar superstructure reflec- regions of the crystal, the rubidium atoms occupy regu- tions are also observed for higher orders of reflection along the c lar positions in the unit cell (for example, a certain layer direction. The difference between the parameters a and c of the satellite lattices and the relevant Bragg lattices is so small that along the b direction) instead of being statistically dis- they can be considered to be nearly identical in the (a*Ðc*) plane. tributed over the (NH4) positions. This structure can be

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ANOMALOUS STATES OF THE CRYSTAL STRUCTURE 2167

(a) (b) 2.09 4.09

2.04 4.03

2.00 3.97

1.95 3.91 (0 2 0) (0 4 0) 1.90 3.85 –0.06 –0.02 0.02 0.06 –0.15 –0.05 0.05 0.15 b* 6.14 6.14 (c) (d)

6.09 6.09

6.04 6.04

6.00 6.00

5.95 5.95

(0 6 0) (1 6 0) 5.90 5.90 –0.06 –0.02 0.02 0.06 0.94 0.98 1.02 1.06 a* a* (e) 10.12

10.08

10.03

9.99

9.94 (1 0 0 0) 9.90 –0.06 –0.02 0.02 0.06 c*

Fig. 1. Reciprocal lattice maps in the vicinity of the (a) (0 2 0), (b) (0 4 0), (c) (0 6 0), (d) (1 6 0), and (e) (1 0 0 0) points for (Rb0.1(NH4)0.9)2SO4 crystals.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2168 SHMYT’KO et al. considered a polytype structure with a layer packing of tice must be a multiple of the superstructure lattice in Rb and (NH4). In this case, the absence of certain super- both the polytype model and the model of substitutional structure reflections and a considerable change in the modulations. For an incommensurate single-crystal lattice parameter along the direction of layer packing composite, the ratio between the lattice parameters of become possible. For this distribution of rubidium the substructures must have an irrational value. In this atoms, the lattice parameters in the (aÐc) plane remain case, the multiplicity or the irrationality can be esti- unchanged. The verification of the above model calls mated using the ratio asat/(aBragg Ð asat), which follows for investigation of (Rbx(NH4)(1 Ð x))2SO4 compounds of from the coincidence (or noncoincidence) of the sites in other compositions for which, in the case under consid- both lattices within N periods: asat(N + 1) = aBraggN. eration, the positions of the satellite reflections will This ratio must have an integral value for the first two change with variations in the rubidium content. models and an irrational value for incommensurate The second model describes a crystal structure with structures. From the aforementioned parameters, we substitutional modulations such that, in the reciprocal obtain N = 7.36877513…. Therefore, the most appro- space, the distance from the satellite reflection to the priate model is an incommensurate single-crystal com- main reflection can change with variations in the order posite. of reflection and, moreover, the satellite reflections can In order to obtain additional information on the arise on only one side of the main reflection [13]. In this crystal structure of (Rb0.1(NH4)0.9)2SO4 solid solutions, model, the (NH4) tetrahedron changing its orientation we measured the temperature dependences of the lat- with respect to the (SO4) tetrahedron can serve as a sub- tice parameters (Fig. 2). As can be seen from Fig. 2a, stitutional element and the rubidium atoms can be sta- the lattice parameter a exhibits an anomalous behavior, tistically distributed over the (NH4) positions. The namely, a decrease in the temperature from Tc to 4.2 K choice of the (NH4) tetrahedron as an ordering element is accompanied by an increase in the lattice parameter corresponds to an extremely low intensity of super- a. To account for the anomalous behavior of the lattice structure reflections. In order to verify the validity of parameter a, it is necessary to elucidate the physical the second model, it is necessary to examine rubidium- nature of the changes observed in the lattice parameters free ammonium sulfate crystals of the composition with variations in the temperature. It should be noted (NH4)2SO4, which, in this case, are also characterized that the changes in the lattice parameters are associated by substitutional modulations. primarily with the anharmonicity of atomic vibrations in local potential wells. It is evident that, in the general According to the third model, the structure of the case, changes in the temperature (or, what amounts to crystal under investigation is treated as an incommen- the same, changes in the energy of atomic vibrations) surate single-crystal composite in which two weakly should be attended by variations in the amplitude of interacting and interpenetrating nonequivalent (host atomic vibrations and, consequently, in the mean inter- and guest) substructures coexist in such a way that their atomic distances and lattice parameters. Within this lattice parameters coincide in one of the basal planes interpretation, there must be no expansion of the crystal and are incommensurate along the direction perpendic- lattice with a decrease in the temperature. ular to this plane [14Ð20]. In this case, it would appear reasonable that the guest substructure is formed by A different situation can arise when atoms are not (NH4) groups, which, owing to the low reflection independent of one another but form coupled nonequiv- power, are responsible for lower intensities of the satel- alent sublattices. As the temperature decreases, these lite reflections from the guest lattice sites as compared sublattices can expand not through anharmonicity of atomic vibrations but through other interactions, for to those of the host substructure containing (SO4) groups. Therefore, the structures described above example, due to a dipoleÐdipole interaction of their become possible for rubidium-free ammonium sulfate, constituent structural elements (in our case, for exam- as is the case with substitutional modulations. ple, due to an interaction of dipole moments of (SO4) and (NH4) distorted tetrahedra). In the case under con- Finally, the assumption that the additional reflec- sideration, the mechanism of the anomalous increase in tions are associated with the degradation of the surface the lattice parameter a can be determined from the layer of the sample should be rejected. Judging from µ changes observed in the intensity of reflections due to the depth of the diffracting layer (~5Ð50 m) and the variations in the temperature. For anharmonicity of intensity ratio of the additional and main reflections atomic vibrations, the increase in the lattice parameters (1/500), the disturbed surface layer does not exceed with variations in the temperature should lead to a 100–1000 Å. This layer should be easily polished. decrease in the diffracted intensity at the expense of the However, the additional reflections were observed even DebyeÐWaller factor. In the case when the lattice after deep mechanical polishing. This indicates a vol- parameter a increases as a result of displacement of the ume distribution of the phase precipitates over the sublattices, the diffracted intensity either can decrease matrix. with a decrease in the temperature or can increase with Analysis of the results obtained permits us to choose temperature variations in the structure factor of the the most appropriate model. In actual fact, the main lat- reflection involved. In general, upon transformation of

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ANOMALOUS STATES OF THE CRYSTAL STRUCTURE 2169

7.86 (a) 10.54 (b)

7.84 a, Å 10.52 Å

Å 7.815 , , b a 7.82 10.50 7.800

7.80 7.785 10.48 210 220 230 240 250 260 7.78 T, K 10.46 0 50 100 150 200 250 0 50 100 150 200 250 T, K T, K

5.98 (c)

5.96 Å , c 5.94

5.92 0 50 100 150 200 250 T, K

Fig. 2. Temperature dependences of the lattice parameters (a) a, (b) b, and (c) c for (Rb0.1(NH4)0.9)2SO4 crystals. the sublattices, the intensity of reflections increases or cumstance suggests that the modulation, if it exists, decreases depending on the temperature range. We exhibits a complex nature in these crystals. observed only an increase in the integrated intensity of The temperature dependence of the unit cell volume the (h 0 0) reflection with a decrease in the temperature. (Fig. 3) provides additional information on the origin of This is indirect evidence that nonequivalent sublattices the ferroelectric phase transition. It can be seen from coexist in the studied structures. In this respect, the Fig. 3 that, below Tc, the unit cell volume first smoothly observed increase in the lattice parameter a and the increases with a decrease in the temperature and then increase in the integrated intensity of the (h 0 0) reflec- decreases. Note that the bulk thermal expansion coeffi- tion with a decrease in the temperature count in favor of cient of the studied sample substantially changes in the the model of the coexistence of two nonequivalent sub- vicinity of T . An increase in the unit cell volume with lattices, especially from the standpoint of the elucida- c tion of the mechanism of the ferroelectric phase transition [7]. 492.0 Apart from the anomalous increase at temperatures below Tc, the lattice parameter a exhibits an “invar effect” with a change in the temperature above Tc (see 490.5 inset in Fig. 2a). It is known that the invar effect can be 3

observed in the temperature range of the existence of Å , incommensurately modulated phases. However, we V 489.0 failed to reveal superstructure reflections that would be symmetric with respect to the Bragg reflections and, moreover, would be located at regular intervals with 487.5 variations in the order of reflection (wave vectors ±k) along the basis axes (by analogy with the known direc- 0 50 100 150 200 250 tions for isomorphic compounds [21, 22]). No equidis- T, K tant reflections along the basis axes were observed until the intensity ratio of the superstructure reflection to the Fig. 3. Temperature dependence of the unit cell volume for main reflection became equal to 1/1000. The latter cir- (Rb0.1(NH4)0.9)2SO4 crystals.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2170 SHMYT’KO et al. a decrease in the temperature of the crystal indicates a REFERENCES radical transformation of the crystal structure upon the 1. K. Ohi, J. Osaka, and H. Uno, J. Phys. Soc. Jpn. 44, 529 ferroelectric phase transition. A considerable decrease (1978). in the bulk thermal expansion coefficient at tempera- 2. N. G. Unruh, Solid State Commun. 8, 1915 (1970). tures below T suggests that, after the ferroelectric c 3. D. E. O’Reily and T. Tsang, J. Chem. Phys. 46, 1301 phase transition, the lattice atoms reside in potential (1967). wells with a smaller anharmonicity coefficient. 4. T. Ikeda, K. Fudjibayashi, T. Nada, and J. Kobayashi, Phys. Status Solidi A 16, 279 (1973). 4. CONCLUSIONS 5. A. Sawada, Y. Tagagi, and Y. Ishibashi, J. Phys. Soc. Jpn. 34, 748 (1973). Thus, the results obtained in our investigation clearly 6. J. Petzelt, J. Grigas, and I. Myerova, Ferroelectrics 6, demonstrated that crystals of (Rb0.1(NH4)0.9)2SO4 solid 225 (1974). solutions are characterized by structural states of com- 7. A. Sawada, S. Ohya, Y. Ishibashi, and Y. Takagi, J. Phys. plex nature. This manifests itself in the following phe- Soc. Jpn. 38, 1408 (1975). nomena: (i) a decrease in the temperature leads to an 8. G. V. Kozlov, S. P. Lebedev, and A. A. Volkov, J. Phys. C anomalous increase in the lattice parameters and the 21, 4883 (1988). unit cell volume, (ii) the temperature dependence of the 9. E. Courtens, Ferroelectrics 72, 229 (1987). lattice parameters exhibits an invar effect at tempera- 10. U. T. Hochli, K. Knorr, and A. Loidl, Adv. Phys. 39, 405 tures above the ferroelectric phase transition point Tc, (1990). and (iii) a set of superstructure reflections indicates the 11. P. Simon, Ferroelectrics 135, 169 (1992). presence of an incommensurate sublattice in the host 12. D. DeSousa Meneses, G. Hauret, and P. Simon, Phys. structure. The observed increase in the integrated inten- Rev. B 51, 2669 (1995). sity of scattering with a decrease in the temperature and 13. A. Guinier, Theorie et Technique de la Radiocristallog- an anomalous increase in the lattice parameter below Tc raphie (Dunod, Paris, 1956; Fizmatgiz, Moscow, 1961). are indirect evidence in favor of the model of the coex- 14. D. Brown, B. D. Cotforth, C. G. Davides, et al., Can. J. istence of two ferroelectric sublattices. However, the Chem. 52, 791 (1974). available data are still insufficient to interpret defini- 15. J. M. Hastling, J. P. Pouget, G. Shirane, et al., Phys. Rev. tively the structural state of a crystal composed of two Lett. 39, 1484 (1977). nonequivalent incommensurate substructures. In par- 16. A. J. Schultz, J. M. Williams, N. D. Miro, et al., Inorg. ticular, the role played by rubidium atoms in the forma- Chem. 17, 646 (1978). tion of superstructure reflections and, consequently, the 17. Sander van Smaalen, Cryst. Rev. 4, 79 (1995). anomalous structural states of these crystals are not clearly understood. In our opinion, a complete under- 18. R. J. Nelmes, D. R. Allan, M. T. McMahon, and standing of the nature of the observed anomalous states S. A. Belmonte, Phys. Rev. Lett. 83, 4081 (1999). of the crystal structure calls for detailed investigation 19. M. I. McMahon, T. Bovornratanaraks, D. R. Allan, et al., into the structure of crystals of the other compositions, Phys. Rev. B 61 (5), 3135 (2000). specifically of rubidium-free ammonium disulfide crys- 20. V. Heine, Nature 403, 836 (2000). tals. 21. H. Dohata and S. Kawada, J. Phys. Soc. Jpn. 57, 4284 (1988). 22. A. Kunishige and H. Mashiyama, J. Phys. Soc. Jpn. 56, ACKNOWLEDGMENTS 3189 (1988). This work was supported by the Russian Foundation for Basic Research, project no. 99-02-18238. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

LATTICE DYNAMICS AND PHASE TRANSITIONS

Mechanism of Homogenization of the Martensite State of Crystals with a Shape Memory Effect G. A. Malygin Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received February 20, 2002

Abstract—The mechanism of homogenization of the martensite state of crystals with a shape memory effect under mechanical stresses applied to a crystal is theoretically analyzed in the framework of the thermodynamic approach and the theory of smeared martensite transitions. The homogenization of the martensite state of the crystal is considered for two variants of martensite that differ in all parameters (the temperature and the heat of transformation, spontaneous strains, etc.) and for many variants of martensite that differ from one another in the orientation of the habit planes. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION equilibrium in a crystal and the homogenization of the Experimental investigations have revealed several martensite state are considered in the framework of the variants of martensite in the course of martensite trans- theory of smeared martensite transitions as applied to formation of crystals with a shape memory effect [1, 2]. the crystal with two variants of martensite. In Section 3, Different martensites either can have the same crystal a similar analysis is performed for a larger but finite lattice and differ only in the orientation of the habit number of martensite variants. planes or can exhibit different crystal structures. This multivariance takes place when the martensite transfor- 2. TWO VARIANTS OF MARTENSITE mation is initiated by variations in the temperature. Owing to the isotropic nature of the thermodynamic Within the thermodynamic approach, the phase forces involved, the martensite transformation, as a equilibrium in a crystal undergoing a structural transformation is determined by the minimum (minima) of rule, leads to the formation of a system of self-accom- ∆ ∆ ∆ modated (twinned) martensite layers in the crystal. the free energy F = U Ð T S, where T is the temperature, ∆U is the change in the internal energy of the Under external mechanical stresses or internal crystal upon the structural transition, and ∆S is the stresses arising in the crystal, the isotropy of the crystal change in the entropy of the crystal. Let us assume that and the multivariance and self-accommodation of the particles (atoms) in the crystal exist in three states with martensite state disappear. When the stresses are suffi- ϕ ϕ concentrations (relative volume fractions) 1, 2, and ciently strong, there occurs a complete or partial ϕ ϕ 3, where 3 is the volume fraction of particles in the homogenization of the martensite state due to “freez- ground state. Since the number of atoms in the crystal ing-out” of the martensites unfavorably oriented with during the transformation remains unchanged, we can respect to the acting mechanical force [1, 2]. At present, write the condition this phenomenon has been understood qualitatively. ϕ ϕ ϕ However, in a quantitative sense, the physical mecha- 1 ++2 3 = 1. (1) nism of homogenization remains poorly investigated. For the given system of particles, the free energy can be In this work, the homogenization mechanism was represented in the form analyzed in terms of the recently developed phenome- ∆ ϕ ϕ ϕ ()ϕ ,,ϕ ϕ nological theory of smeared martensite transitions [3, FU= 1 1 +++U2 2 U3 3 Uint 1 2 3 (2) 4], which is based on a thermodynamic approach to + kT[]ϕ lnϕ ++ϕ lnϕ ϕ lnϕ . first-order phase transitions, among them martensite 1 1 2 2 3 3 transformations in alloys with a shape memory effect. Here, U1, U2, and U3 are the internal energies of parti- ϕ This theory describes the phase equilibrium in a crystal cles in states 1, 2, and 3, respectively, and Uint( ) is the as a function of temperature and different forces acting energy of interaction between particles in different on the crystal, including mechanical forces. states. The last term in the right-hand side of Eq. (2) is Earlier [3, 4], I developed the theory describing the the change in the entropy of the crystal due to mixing phase equilibrium in a crystal with one variant of mar- of the particle states, and k is the Boltzmann constant. tensite. In Section 2 of the present paper, the phase Since our concern here is with the martensite embryos

2172 MALYGIN

1.5 ∆u is the change in the density of the internal energy of (a) the crystal upon the structural transition:

∆ TTÐ c12, ξ()12, τ ξ()12, 1.0 u12, = q12, ------Ð ik ik Ð 0 P, (5) T c12, ) 1 T (

ϕ where q is the heat of transformation; Tc is the critical ξ (characteristic) temperature of transformation; ik and 0.5 2 ξ 0 are the shear strain and the dilatation of the lattice τ upon its structural transformation, respectively; ik is 0 the shear stress applied to the crystal; and P is the uni- 1.5 form pressure. (b) It is assumed that the two variants of martensite dif- ω ξ ξ fer from each other in all parameters ( , q, Tc, ik, 0). For simplicity, we assume that the shear strain tensor 1.0 and stress tensor have only one nonzero component, )

T ()12, ( 1 namely, ξ ≡ ξ and τ ≡ τ. As a result, in expres- ϕ ik 1, 2 ik sions (4), we obtain 0.5 2 ∆ τ U1 σ τ ± f 1 ------= B1T/T c1 Ð 1 Ð m1 / 1 Ð P/P1 ------τ - , (6a) kT 1 0 0.5 1.0 1.5 2.0 ∆U τ T/Tc1 2 σ τ ± f 2 ------= B2T/T c2 Ð 1 Ð m2 / 2 Ð P/P2 ------τ - , (6b) kT 2 Fig. 1. Temperature dependences of the volume fraction of ω ω (1) martensite 1 and (2) martensite 2 according to formulas B1 ==1q1/kT, B2 2q2/kT, (4) and (6a)Ð(6c): (a) in the absence of stresses and (b) in (6c) τ ξ ξ()12, the presence of stresses. The dotted lines correspond to the 12, ==q12, / 12, , P12, q12, / 0 , total amount of martensite in the crystal. where m1 and m2 are the orientational factors of the habit planes during the structural transformation of the whose sizes exceed the critical embryo size, in what lattice with respect to the mechanical stress σ applied to follows, we will disregard the interaction energy of par- the crystal and ±τ is the stress of dry friction in the ϕ f ticles in different states Uint( ), i.e., the surface energy course of the interaction between the interphase bound- [3, 4]. aries and defects in the crystal during the direct and According to condition (1), two concentrations are reverse martensite transitions. ϕ ϕ independent, namely, 1 and 2. In order to determine Figure 1a displays the temperature dependences of their thermodynamically equilibrium values, we have the volume fractions of martensites 1 and 2 according two conditions, to expressions (4) and (6a)Ð(6c) in the absence of the stress (σ = 0) and the uniform pressure (P = 0) at the ∂∆ ∂∆ F F following parameters: B2 = 0.8B1, B1 = 100, Tc2 = ------∂ϕ-0,==------∂ϕ- 0. (3) τ τ 1 2 1.25Tc1, and f 1 = f 2 = 0. It can be seen that, at these parameters, there exist two temperature regions of the Substituting potential (2) into these equations and tak- homogenization of the martensite state of the crystal: ing into account the particle balance equation (1), we (i) T < 0.5Tc1, in which martensite 1 dominates, and (ii) obtain the following expressions for equilibrium con- 0.6Tc1 < T < 1.3Tc1, in which martensite 2 dominates. In centrations: the intermediate temperature range 0.5Tc1 < T < 0.6Tc1, ϕ = ϕ exp()Ð∆U /kT , ϕ = ϕ exp()Ð∆U /kT , there exists a mixed martensite state. The dotted line in 1 3 1 2 3 2 Fig. 1a corresponds to the total amount of martensite in (4) ϕ 1 the crystal. 3 = ------()∆ ()∆ -, 1 ++exp Ð U1/kT exp Ð U2/kT Let us now consider the change in the martensite state of the crystal under mechanical stresses. Figure 1b ∆ ∆ where U1 = U1 Ð U3 and U2 = U2 Ð U3 are the changes shows the volume fractions of martensites 1 and 2 in the particle energies upon the transitions from the according to formulas (4) and (6a)Ð(6c) at the external σ τ τ τ ground state to new structural states 1 and 2, respec- stress = 0.5 1 and m1 = 0.5, m2 1/ 2 = 0.2, and P = 0. tively. Since the transition to a new state occurs in por- It can be seen that, under the applied stress, the temper- tions of volume ω (where ω is the elementary volume ature range of existence of martensite 2 becomes nar- ∆ ω ∆ of transformation), we have U1, 2 = 1, 2 u1, 2. Here, rower, whereas the temperature range of existence of

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 MECHANISM OF HOMOGENIZATION OF THE MARTENSITE STATE 2173 martensite 1 broadens. This results in the homogeniza- 1.5 tion of the martensite in the temperature range 0.5Tc1 < (a) T < 0.6Tc1, in which a mixed state is observed in the σ τ absence of stresses. An increase in the stress to = 1 1.0

leads to the complete disappearance of martensite 2. It 1 ε

is evident that the uniform pressure P, when applied to )/ T the crystal in the presence of nonzero lattice dilatations ( 1 ε ξ ()12, 0 , should also affect the ratios of the martensite 0.5 volume fractions in certain temperature ranges. 2 The martensite transformation in the crystal is accompanied by its macrodeformation (transformation 0 plasticity), depending on the amount of martensite in 1.5 ξ (b) the crystal, the spontaneous strains ik, and the orientational factors m. In the example considered above, the strain arising in the crystal due to the formation of the 1.0 two types of martensite (martensites 1 and 2) is deter- 1 ε mined as ε(T, σ) = ε(1)(T, σ) + ε(3)(T, σ). Here, )/ T ( ε ε()1 (), σ ξ []ϕ ()ϕ, σ (), σ T = m1 1 1 Tm1 Ð 1 TmÐ 1 , (7a) 0.5

ε()2 (), σ ξ []ϕ ()ϕ, σ (), σ T = m2 2 2 Tm2 Ð 2 TmÐ 2 . (7b) In the absence of external stresses, martensites 1 and 2 0 0.5 1.0 1.5 2.0 reside in a twinned state. The martensite variants in a T/Tc1 twinned pair differ from each other in that their orien- ± ± tational factors are opposite in sign ( m1 and m2, Fig. 2. Temperature dependences of (a) the partial strains respectively). Consequently, we obtain ε(1)(T) = upon transformation of (1) martensite 1 and (2) martensite ε(2)(T) = 0; i.e., the strain of the crystal is absent. The 2 (the solid line indicates the total strain of transformation) and (b) the total strain of the direct and reverse martensite application of stresses leads to partial or complete transformations in the absence (dotted line) and in the pres- detwinning of the martensites and a nonzero strain of ence (solid lines) of the hysteresis of the transformation. the crystal. Figure 2a shows the temperature dependences of ε(1) and ε(2) according to expressions (7a) and (7b) at the σ τ ε ξ ε above parameters, = 0.5 1, and 2 = m2 2 = 0.5 1, ε ξ 1.5 where 1 = m1 1. The solid curve indicates the temperature dependence of the total strain of transformation ε. It can be seen that this curve has a two-step shape due to the formation of two variants of martensite with a decrease in the temperature. Similar step curves of 1.0

1 2 transformation are observed, for example, in crystals of ε )/

titanium nickelide TiNi [2]. The steps correspond to the σ ( formation of martensite of types R and B19, respec- ε tively. The reverse martensite transition is characterized 0.5 by a hysteresis of the transformation due to the dry fric- τ tion stress f. Figure 2b depicts the curves of the direct and reverse martensite transformations at the friction 1 τ τ ± τ τ ± × Ð2 stresses f1/ 1 = 0.1 and f 2/ 2 = 5 10 . 0 0.5 1.0 1.5 The existence of two variants of martensite is σ/τ responsible for the step shape of hyperelastic strain 1 curves of the crystal [2]. Figure 3 depicts the hyperelastic strain curves according to expressions (7a) and (7b) at the temperature T = 1.3T for loading and unloading c1 Fig. 3. Hyperelastic strain curves for two variants of mar- of the crystal. The dotted line corresponds to the hyper- tensite in the absence (dotted line) and in the presence (solid elastic strain in the absence of a hysteresis of the trans- lines) of the hysteresis of the transformations: (1) loading formation. and (2) unloading.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2174 MALYGIN

3. MANY VARIANTS OF MARTENSITE It follows from the above expressions that, in the case when the stress σ is absent, the volume fractions of The obvious generalization of expressions (4) to the ϕ case of N variants of martensite (instead of two vari- martensites n are identical and the temperature depen- ants) can be represented by the following relationships: dence of the sum of fractions of all the martensite variants can be represented by the relationship ϕ = ϕ exp()Ð∆U /kT , n N n []() ϕ() NBTexp Ð /T c Ð 1 ϕ 1 T = ------[]()-. (10) N = ------N , (8a) 1 + NBTexp Ð /Tc Ð 1 ()∆ 1 + ∑ exp Ð Un/kT In the presence of external stresses, the volume frac- n = 1 tion of each martensite depends on the particular orientational factor m, i.e., on the orientation of the crystal N with respect to the applied stress. For example, under ()∆ N ∑ exp Ð Un/kT uniaxial tension or compression of the crystal in the ϕϕ==∑ ------n = 1 , [111] direction, the shear stress in the (111) octahedral n N (8b) plane is equal to zero, whereas the orientational factors n = 1 ()∆ 1 + ∑ exp Ð Un/kT in the other three octahedral planes are nonzero. Note n = 1 that, in each of these planes, there exists one variant ∆U = U Ð U . with the maximum orientational factor m = 0.416. n n N According to formulas (9a) and (9b), this means that As an illustration, we consider a conventional structural application of a sufﬁciently strong stress to the crystal transformation—a transition from a face-centered should suppress all martensite variants, except for the cubic lattice to a hexagonal close-packed lattice. Theo- three variants with maximum orientational factors. For retically, there can exist 24 variants of martensite upon less symmetric directions of loading of the crystal, a this transition. These variants differ from each other in variant whose orientational factor is maximum in this both the orientation of the habit planes (four octahedral scheme of loading has a good chance of surviving. planes) and the direction of their displacements (six directions in each octahedral plane, including displacements of the opposite sign). All the parameters are iden- 4. CONCLUSION tical except for the orientational factors m. Conse- Thus, the theory of smeared martensite transitions quently, taking into account the designations of formu- offers an adequate quantitative description of the las (6a)Ð(6c), using expressions (8a) and (8b), and homogenization of the martensite state of a crystal sub- assuming that P = 0, we obtain jected to mechanical stresses over the entire tempera- ϕ (), σ ture range of the transformation. n T []()σ τ exp ÐBT/T c Ð 1 Ð mn / 1 = ------N , (9a) REFERENCES 1 + exp[]ÐBT()/T Ð 1 ∑ exp()m Bσ/τ 1. T. Saburi and C. M. Waymen, Acta Metall. 28 (1), 1 c n 1 (1980). n = 1 2. V. N. Khachin, V. G. Pushin, and V. V. Kondrat’ev, Tita- ϕ()T, σ nium Nickelide: The Structure and Properties (Nauka, N Moscow, 1992). exp[]ÐBT()/T Ð 1 exp()m Bσ/τ 3. G. A. Malygin, Fiz. Tverd. Tela (St. Petersburg) 36 (5), c ∑ n 1 1489 (1994) [Phys. Solid State 36, 815 (1994)]. n = 1 (9b) = ------N . 4. G. A. Malygin, Usp. Fiz. Nauk 171 (2), 187 (2001). []()()σ τ 1 + exp ÐBT/T c Ð 1 ∑ exp mn B / 1 n = 1 Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2175Ð2180. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2079Ð2083. Original Russian Text Copyright © 2002 by Gordon, Kukushkin, Osipov.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Perturbation Methods in the Kinetics of Nanocluster Growth P. V. Gordon, S. A. Kukushkin, and A. V. Osipov Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, Bol’shoœ pr. 61, Vasil’evskiœ ostrov, St. Petersburg, 199178 Russia e-mail: [email protected] Received January 21, 2002

Abstract—The rigorous perturbation theory of the evolution of a small-sized cluster is developed in the framework of the density functional method. The solution of the general equation for relaxation of the order parameter ﬁeld is derived in the form of a power series of the metastability parameter (an analog of supersaturation or supercooling) and the curvature. The proﬁle of the cluster density and the cluster growth rate are determined in an analytical form. The surface tension and the Tolman parameter are calculated. The results obtained are applied to a van der Waals three-dimensional gas and a two-dimensional lattice gas. It is shown that the theoretical results are in good agreement with experimental data. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION a grand thermodynamic potential of the general form that would allow first-order phase transitions. This Over the last six decades, first-order phase transi- solution was represented in the form of a convergent tions have attracted the particular attention of many series in terms of the ε difference in the chemical poten- researchers [1Ð3]. At present, there exist two principal tials of the initial and equilibrium phases, which was approaches to the description of the kinetics of first- used as the small parameter of the theory. In other order phase transitions. According to the first (classical) words, we developed a rigorous theory of perturbations approach, a phase transition is treated as a fluctuation of the general equation for relaxation of the order nucleation and the growth of a new phase in the size parameter. Within this approach, we uniquely deter- space [1]. Within this approach, the growth of individ- mined the surface tension and the Tolman parameter ual nuclei is described by an appropriate microscopic (the correction to the surface tension for the curvature equation, for example, the diffusion equation or the in the case of small sizes). The results obtained were heat conduction equation [3]. The second approach is applied to the van der Waals three-dimensional gas and based on the so-called density functional theory. In the the two-dimensional lattice gas (which provides the framework of the second approach, the growth of indi- growth of a wetting layer for quantum dots [10]). It was vidual nuclei is regarded as a relaxation of the order shown that the proposed model is in good agreement parameter field [4–6]. In the case of first-order phase with the experimental data. transitions, the order parameter is a local density of the material and the effective Hamiltonian represents a grand thermodynamic potential. The second approach 2. THE EQUATION FOR RELAXATION is considered to be more general because it can offer an OF THE ORDER PARAMETER adequate description of the nucleation and growth of a new phase even when the first approach is inapplicable, According to the general kinetic theory, the most for example, due to a small size of nuclei. In this probable path of the evolution of a thermodynamic sys- respect, the density functional method becomes espe- tem can be described by the Markov master equation cially efficient when applied to the description of the [5]. For a slowly varying order parameter, this equation evolution of nanometer-sized clusters and islands, i.e., can be simplified. Specifically, if the system is the so-called quantum dots, which are widely used in described by only one order parameter ρ, this equation modern optoelectronics. The disadvantage of the den- takes the form sity functional method is that, within this approach, no ∂ρ(), δ analytical solution describing either the cluster growth r t 1 H ˜ ------∂ - = Ð------δρ + h. (1) or the evaporation of clusters has hitherto been obtained t tρ for real thermodynamic potentials. There are only a few results obtained from numerical simulations [7Ð9]. Here, H is the dimensionless effective Hamiltonian of the system, t is the time, r specifies the coordinates of a The purpose of the present work was to derive a gen- point of the medium, the dimensionless field ρ is the eral analytical solution to the relaxation equation with density of the material at each point (ρ is conveniently

2176 GORDON et al. ρ expressed in terms of the critical density c), tρ is the referred to as the metastability parameter. This param- characteristic time of variation in ρ, and h˜ is the exter- eter is an analog of supersaturation or supercooling. In order for a liquid drop to nucleate from a supersaturated nal force simulating a thermal ensemble. Generally vapor, the metastability parameter must be equal to speaking, this equation describes the relaxation of the (T/T )ln(p/p ) [1], where p is the supersaturation vapor field of the nonpersistent order parameter ρ [4]. c e pressure and pe is the equilibrium pressure. Within this In the case of first-order phase transitions, the effec- theory, ε is considered to be so small that the system tive Hamiltonian is equal to a grand thermodynamic resides in a metastable state (well away from the spin- potential (because the variable of the description is the odal). It is assumed that the external forces maintain a chemical potential rather than the number of particles) constant value of ε. Moreover, we will deal only with a expressed in terms of kBTc, where kB is the Boltzmann temperature range in which thermal fluctuations of the constant and Tc is the critical temperature. The inhomo- order parameter are negligible compared to the density geneous part of the grand thermodynamic potential rep- of the denser phase. For a van der Waals system, this resents a complex nonlocal density functional. How- range satisfies the condition 0 < T/Tc < 0.8. According ever, when the density field slowly varies in the space to the classification of metastable states [11], we con- (as is the case with first-order phase transitions), the sider only a nonfluctuating region. As a result, the equa- inhomogeneous part is significantly simplified and tion of the evolution without regard for thermal fluctu- becomes local. In this approximation, we obtain ations h˜ can be written in the form 2 λ 2 µρ ()∇ρ ∂ρ()r, t 2 HF= ∫ Ð + ----- dr + H0, (2) λ ∆ρ[] µ()µ ρ, () ε 2 tρ------∂ - = Ð T Ð e T + , (5) V t where F is the free energy density of the homogeneous where µ(ρ, T) = ∂F/∂ρ is the chemical potential per par- medium (expressed in terms of k T ), dr is the elemen- ticle of the material with density ρ and temperature T. B c Let us assume that, in this case, the time is measured in tary volume, H0 stands for the contributions of all the λ remaining degrees of freedom, λ is the typical scale of terms of tρ and the length is expressed in terms of . For spatial variation in ρ, and µ is the chemical potential. spherical symmetry, we obtain the relationship λ 2 The quantities and tρ can be determined by com- ∂ρ()rt, ∂ ρ()rt, ------= ------paring the results obtained in the framework of the clas- ∂t ∂ 2 sical theory and the density functional theory. Specifi- r (6) cally, in the case when ρ(r, t) is represented by the d Ð 1∂ρ()rt, + ------Ð Ω' ()ρ + ε. expression derived below, an unambiguous correspon- r ∂r e dence between these theories can be achieved only under the condition Here, d is the space dimension (d = 3 for a drop and d = 2 for a disk-shaped island in the growth of thin films) Ω λ m and e is the grand thermodynamic potential, which is = 3 ρ-----, (3) µ µ c defined at the equilibrium chemical potential e ( e can be found from phase equilibrium equations) as follows: where m is the mass of one molecule and ρ is the crit- c ρ ical density. The characteristic time tρ can be deter- Ω []ξΩµξ()µ, () mined by making the cluster growth rate obtained in e = ∫ T Ð e t d + 0 (7) terms of the density functional theory equal to the cor- 0 responding result of the classical microscopic theory. In ()µρ, ρ Ω particular, when applied to the free-molecular growth = F T Ð e + 0. of a spherical drop, the perturbation theory developed At temperatures below the critical point, the depen- below gives Ω ρ ρ dence of e on exhibits two isolated minima at G and ρ , which correspond to the gaseous and liquid states. 23k T 2/3 L B mm The constant of integration Ω is chosen in such a way tρ = ------σ -----ρ- , (4) 0 0V c that these minima become zero. σ Specifically, in the case of a van der Waals three- where V is the volume per molecule in a liquid and 0 is the surface tension coefficient for a planar interface dimensional gas, which is an important example of sys- between the vapor and the liquid. tems undergoing a first-order phase transition, we have The chemical potential µ can be represented as the ρ 9 2 µ ε µ F()ρ, T = Tρln------Ð ---ρ , (8) sum e + , where e is the equilibrium chemical poten- 1 Ð ρ/3 8 ε µ µ tial and = Ð e is the difference in chemical poten- ρ T 9 tials (expressed in terms of kBTc) between the initial and µρ(), T = Tρln------+ ------Ð ---ρ. (9) equilibrium phases. The second term of the sum is often 1 Ð ρ/3 1 Ð ρ/3 4

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 PERTURBATION METHODS IN THE KINETICS 2177

Here, the temperature and the density are expressed in 0.6 critical units. For a two-dimensional lattice gas [6], we can write the following expressions: 0.5 c

()ρ, T 0.4

F T B k c 0.3 (10) ρ

a / = ---{}T[]ρ()2 Ð ρ ln()2 Ð ρ + ρρln + ()2 Ð ρ , e

2 Ω 0.2 a ρ 0.1 µρ(), T = ---T ln------+ a()1 Ð ρ . (11) 2 2 Ð ρ 0 0.5 1.0 1.5 2.0 2.5 3.0 Ω ρ ρ ρ ρ Figure 1 shows the dependence of e on for the van G * L ρ ρ der Waals three-dimensional gas at T = 0.8Tc. It is evi- / c dent that relationship (6) is a typical GinzburgÐLandau equation which is frequently used in the density func- Fig. 1. Dependence of the grand thermodynamic potential tional theory for ﬁrst-order phase transitions. on the density for the van der Waals three-dimensional gas at T = 0.7T . Below, we will derive an analytical solution to the c evolution equation (6) with allowance made for the fact that ε is the small parameter. 2.5 3. THE PERTURBATION THEORY 2.0 Before proceeding to an analytical solution of the evolution equation (6), we consider a simpler equation

c 1.5

Ω ρ ρ for arbitrary grand thermodynamic potentials e( ) / ρ ρ Ω ρ ρ with two minima at L and G [in this case, e( L) = 1.0 Ω (ρ ) = 0], that is, e G 0.5 ∂ρ ∂2ρ Ω ()ρ ε ------= ------Ð e' + . (12) ∂t ∂r2 0 5 10 15 20 25 r/λ We seek an analytical solution to this equation in the following form: Fig. 2. Density profile for a liquid cluster in the van der ∞ Waals three-dimensional gas at T = 0.7Tc. ρ(), εkρ ()θ rt = ∑ k , (13) k = 0 Equation (15) has an obvious solution, ∞ k θ ==rRÐ ()τε, , R()τε, ∑ ε R ()τ , (14) f k dρ k = 0 ρ ==f ()θ , η ------r, (18) 0 ∫ Ω ()ρ τ ε ρ 2 e where = t is the so-called “slow time.” By substitut- * ing expressions (13) and (14) into Eq. (12) and equating terms with identical exponents εk, we obtain the follow- where ρ∗ is the point at a maximum of the grand ther- Ω ρ ing recurrent system of equations: modynamic potential between the minima at L and ρ (Fig. 1), η = ±1 (sign “–” corresponds to a nucleus ρ'' Ð0Ω' ()ρ = (15) G 0 e 0 of the liquid phase L, and sign “+” corresponds to a for ε0 and nucleus of the gas phase G). Figure 2 depicts the density profile f(r) for a liquid cluster in the van der Waals ρ'' Ð Ω''()ρρ = F ()ρ ,,,ρ …ρ , three-dimensional gas at T = 0.7T . According to the k e 0 k k k Ð 1 k Ð 2 0 (16) c k ≥ 1 rigorous perturbation theory [12], the power series defined by expression (13) converges at any values of r for εk. Here, F are the known functions determined and t (i.e., it is a uniformly convergent series) provided k τ from expressions (12)–(14). Specifically, at k = 1, we the growth law Rk( ) is chosen in such a way that the ρ ρ ∞ ≥ have inequalities k + 1/ k < and k 0 hold at any values of θ and τ. Essentially, this is the renormalization method dR ()ρ , τ 0ρ' in the rigorous perturbation theory [12]. In particular, F1 0 = Ð------τ- 0 Ð 1. (17) ρ d the zeroth-order approximation 0 is uniformly suitable

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2178 GORDON et al. ρ ρ ∞ provided R0 is chosen such that the condition 1/ 0 < Now, we turn to the analysis of the initial equation (6). is universally satisfied. Equation (16) is linear with The term [(d Ð 1)/r](∂ρ/∂r) can be treated as a small respect to ρ (to within the kth correction) and can be perturbation of the antikink-type solution (13) for the solved using the following method. For definiteness, new small parameter 1/R (the curvature of the studied ρ we consider an equation for 1, that is, cluster) because we have ρ'' Ð Ω''()ρρ = F ()ρ . (19) 1ρ' 1 ρ' 1 1 e 0 1 1 0 --- 0 = --- 0 + O----- . (27) r R R2 The solution to this equation will be sought in the form The asymptotic construction of the zeroth-order ρ ()ρθ ()θ 1 = z 0' . (20) approximation with respect to ε has precisely the same form; however, in this case, the function F is more Substitution of expression (20) into Eq. (19) gives 1 complicated; that is, ρ' ρ'' z'' 0 + 2z' 0 = F1. (21) dR d Ð 1 F = Ð------0ρ' Ð1.------ρ' Ð (28) Equation (21) can be simplified by multiplying the 1 dτ 0 εR 0 ρ left-hand and right-hand sides of this equation by 0' ; Substitution of relationship (28) into expression (24) that is, gives the growth law for clusters of the new phase. Con- sequently, to the zeroth order in ε and to the first order ()ρ 2 ' ρ z'' 0' = F1 0' . (22) in 1/R, we have As a consequence, we obtain ρ(), ()() ()ε 1 rt = frÐ Rt ++O O-----2 , (29) θ θ R 0 θ 1 d 1 ρ = ρ' ------F ρ' dθ . (23) dR 1 1 2 1 1 0 ∫ ρ 2 ∫ 1 0 2 ------= ()d Ð 1 ----- Ð --- ++O()ε O ----- , (30) 0'  2 0 Ð∞ dt Rc R R ρ 2 ()σ It follows from relationship (18) that the quantity 1/ 0' d Ð 1 0 Rc = ()ε------ρ ρ . (31) takes on exponentially large values; therefore, the nec- L Ð G essary condition for convergence of the exterior integral in expression (23) has the form It is evident that Rc plays the role of a radius of a critical nucleus. ∞ In general, the method described above can be used ρ θ ∫ F1 0' d = 0. (24) to determine higher orders of the approximation both in ε Ð∞ and in the curvature 1/R. It should be noted that, in this case, the surface tension depends on 1/R. The inclu- This is an important result of the present work. Only ρ ρ ∞ sion of this dependence is particularly important for when condition (24) is met can inequality 1/ 0 < be small-sized nanoclusters. We leave out the mathemati- satisfied and can the zeroth-order approximation in the cal treatment and write the final expressions to the first ρ form of a traveling wave 0(r Ð R(t)) have physical order in ε and to the second order in 1/R: meaning. By substituting relationship (17) into expres- θ θ sion (24), we found the cluster growth law in the fol- θ 1 ρ ()ρ() ()θ d 1 ρ 2()θ lowing form: = frÐ Rt Ð 0' ∫------∫ 0' 2 ρ'2()θ dR ρ Ð ρ 0 0 1 Ð∞ ------0 = ------L G, (25) ρ ρ ()δθ() τ σ L Ð G 1 d Ð 1 + 2 d 0 ×ε------+ ------Ð ------dθ (32) σ ρ' ()θ 2 2 where 0 0 2 R ε ρ 2  1 ∞ L ()ε + O ++O-----2 O-----3 , σ ρ 2 θ Ω ()ρρ R R 0 ==∫ 0' d ∫ 2 e d . (26) ∞ ρ ()δ Ð G dR ()1 1 d Ð 1 ------= d Ð 1 ----- Ð --- + ------2 - Thus, within the framework of the perturbation theory, dt Rc R R (33) the fulfillment of the uniform convergence condition ε ()ε3  1 for the power series (14) of the metastability parameter + O ++O-----2 O-----3 , leads to the cluster growth law (25) and the surface ten- R R sion coefficient (26). It is obvious that formula (26) is ()σ() in complete agreement with the results obtained in d Ð 1 R Rc = ------()ερ ρ , (34) terms of the stationary perturbation theory [13]. L Ð G

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 PERTURBATION METHODS IN THE KINETICS 2179

R 1 3.5 σ()R = σ------+ O ----- , (35) 0 Rd+ ()δÐ 1 2 1 R 3.0 2 3 ∞ ∞ 2/3

) 2.5

c 4

2 ρ θρ()θ θρ θ / Ð 0' d 0' d ∫ ∫ m 2.0 )( Ð∞ Ð∞ c δ = ------Ð ------. (36) T 1.5 ρ ρ σ B L Ð G 0 k / 0 1.0 σ It is obvious that δ plays the role of a correction to the ( surface tension for the curvature; i.e., δ is the Tolman 0.5 parameter. We see little reason for refining expres- 0 sions (32)Ð(36) by calculating new corrections, 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 because the initial GinzburgÐLandau equation is valid T/Tc to the second order in 1/R. In the case of bubbles (gas nuclei in a liquid), the correction to the surface tension Fig. 3. Temperature dependences of the surface tension of for the curvature changes sign: the planar interface for the van der Waals three-dimensional gas (solid line) and the two-dimensional lattice gas (dashed ()δ σ() σRd+ Ð 1 1 line). Points are the experimental data taken from [14]: (1) R = 0------+ O----- . (37) CH , (2) Ne, (3) Ar, and (4) Kr. R R2 4 The important advantage of the above approach is λ that the quantities and tρ can be uniquely determined 0.12 by identifying expressions (32)Ð(36) with the appropriate relationships derived in the framework of the classi- 0.10 σ cal thermodynamic theory; furthermore, 0 can be 0.08 1/3 identified with the surface tension of a planar interface, ) m / δ can be identified with the Tolman parameter, etc. In c 0.06 ρ ( particular, the free energy of formation of a liquid clus- δ 0.04 ter from a supersaturated vapor in the three-dimensional case (d = 3) can be determined from the relation- 0.02 ship 0 0.2 0.4 0.6 0.8 1.0 ∞ T/T ρ 2 c π 2 ' Ω ()ρ ερ() ρ F = ∫4 r ------+ e Ð Ð G dr + const. (38) 2 Fig. 4. Temperature dependence of the Tolman parameter 0 for the van der Waals three-dimensional gas. By using relationship (32) and integrating to a required accuracy, after changing over to a dimensional form with the use of formula (4), we obtain function with respect to ρ Ð ρ∗, the Tolman parameter 3 ρ ρ for this system is equal to zero). It can easily be shown 2 4πR L Ð G FR()= 4πσ()R R Ð ------ε + const. (39) that, in the temperature range 0.5 < T/Tc < 1, the Tolman 3 m δ 3 ρ × −10 parameter is of the order of ~ 0.1c/m ~ 0.5 10 m. Expression (39) completely coincides with the classical This will suffice to have a pronounced effect on the formula for the energy of formation of a nucleus. This nucleation [15]. confirms the validity of formula (4). The temperature dependences of the surface tension (in a dimension form) calculated according to expres- 4. CONCLUSIONS sions (8), (10), and (26) for the van der Waals three- dimensional gas and the two-dimensional lattice gas are Thus, we developed a rigorous perturbation theory displayed in Fig. 3. This figure also shows experimental for the evolution of nanoclusters of a new phase for an values of the surface tension for four gases, namely, arbitrary grand thermodynamic potential that allows methane, neon, argon, and krypton [14]. As is clearly first-order phase transitions. The solution of the general seen, the theoretical results obtained above for the sur- equation for relaxation of the order parameter was face tension of the van der Waals gases are in excellent derived in the form of a uniformly convergent series of agreement with experimental data. Figure 4 depicts the the metastability parameter. It was demonstrated that temperature dependence of the Tolman parameter for the fulfillment of the uniform convergence condition the van der Waals three-dimensional gas (since the ther- for this series uniquely determines the evolution law for modynamic potential of the lattice gas is a symmetric nanoclusters and, consequently, the coefficient of their

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2180 GORDON et al. surface tension. The analytical results obtained are in 7. Yu. E. Kuzovlev, T. K. Soboleva, and A. E. Filippov, Zh. good agreement with experimental data. Éksp. Teor. Fiz. 103 (5), 1742 (1993) [JETP 76, 858 (1993)]. 8. Y. C. Shen and D. W. Oxtoby, J. Phys. Chem. 105 (8), ACKNOWLEDGMENTS 6517 (1996). This work was supported by the Russian Foundation 9. A. Laaksonen, J. Phys. Chem. 106 (10), 7268 (1997). for Basic Research (project nos. 02-02-17216 and 02- 10. N. N. Ledentsov, V. M. Ustinov, V. A. Shchukin, et al., 03-32471), the Russian Federal Center “Integration” Fiz. Tekh. Poluprovodn. (St. Petersburg) 32 (4), 385 (project no. A0151), and the Complex Program (project (1998) [Semiconductors 32, 343 (1998)]. no. 19). 11. V. G. Boœko, Kh.-Œ. Mogel’, V. M. Sysoev, and A. V. Chalyœ, Usp. Fiz. Nauk 161 (2), 77 (1991) [Sov. Phys. Usp. 34, 141 (1991)]. REFERENCES 12. A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1. D. Kashchiev, Nucleation: Basic Theory with Applica- 1973). tions (ButterworthsÐHeinemann, Oxford, 2000). 13. M. P. A. Fisher and M. Wortis, Phys. Rev. B 29 (11), 2. Nucleation Theory and Applications, Ed. by 6252 (1984). J. W. P. Schmelzer, G. Ropke, and V. B. Priezzhev (Joint Inst. for Nuclear Research, Dubna, 1999). 14. B. D. Summ, in Handbook of Physical Quantities, Ed. by I. S. Grigoriev and E. Z. Meilikhov (Énergoatomizdat, 3. S. A. Kukushkin and A. V. Osipov, Usp. Fiz. Nauk 168 Moscow, 1991; CRC Press, Boca Raton, 1997). (10), 1083 (1998) [Phys. Usp. 41, 983 (1998)]. 15. E. N. Brodskaya, J. C. Eriksson, A. Laaksonen, and 4. D. W. Oxtoby and R. J. Evans, Chem. Phys. 89 (10), A I. Rusanov, J. Colloid Interface Sci. 180 (1), 86 7521 (1998). (1996). 5. C. Varea and A. Robledo, J. Chem. Phys. 75 (10), 5080 (1981). 6. A. V. Osipov, J. Phys. D 28 (8), 1670 (1995). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2181Ð2195. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2084Ð2096. Original Russian Text Copyright © 2002 by Lang, Korovin, Contreras-Solorio, Pavlov.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

A Manifestation of the Magnetopolaron Effect in Reflection and Absorption of Light by a Three-Level System in a Quantum Well I. G. Lang1, L. I. Korovin1, D. A. Contreras-Solorio2, and S. T. Pavlov2, 3 1 Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia 2 Escuela de Fisica de la UAZ, Apartado Postal c-580, 98060 Zacatecas, Mexico 3 Lebedev Institute of Physics, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 117924 Russia e-mail: [email protected] e-mail: [email protected] Received February 8, 2002

Abstract—The absorption and reflection of light by a quantum well are investigated in the case of two closely spaced levels of electronic excitations in the well. The dependences of the dimensionless absorptance Ꮽ and ᏾ ω reflectance on the frequency l of the exciting light are calculated. The overall sequence of processes involving absorption and reemission of photons is taken into account. This is beyond the scope of the perturbation theory for the photonÐelectron coupling constant. It is shown that the perturbation theory is inapplicable when the reciprocal radiative lifetimes of excitations are comparable to the reciprocal nonradiative lifetimes. In this Ꮽ ω ᏾ ω case, the nontrivial dependences ( l) and ( l) are obtained. The total reflection and the total transparency points are determined. The relationships derived are used to analyze the special case of two excitation levels that are formed in the quantum well in a strong magnetic field H normal to its plane due to the JohnsonÐLarsen magnetopolaron effect. The reciprocal radiative lifetimes of electronÐhole pairs are calculated far from and in the vicinity of the magnetophonon resonance. It is found that these lifetimes are proportional to H in the range far from the resonance and depend strongly on the difference HÐHres in the vicinity of the resonance. The dependences of the coefficients Ꮽ and ᏾ on the magnetic field H at different frequencies of the exciting light are deduced. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tion of the frequency dependence of magnetooptical The role of the electronÐphonon interaction in the effects, such as reflection, interband absorption, cyclo- formation of a polaron state in magnetic fields sharply tron resonance, and Raman light scattering (see, for enhances when the following resonance condition is example, reviews [2Ð4]). The difference between the satisfied: systems manifests itself in the energy spectrum of an electron (hole), which is represented by Landau one- ω Ω ,,,… LO ==j , j 123 , (1) dimensional bands in the three-dimensional system and ω discrete levels in the quasi-two-dimensional system. where LO is the frequency of longitudinal optical (LO) This difference leads to a different divergence of the phonons, energy levels of the electronÐphonon system: the diver- Ω () α2/3បω = eH/ cmeh() (2) gence is of the order of LO in the three-dimen- α1/2បω sional case [5] and LO in the quasi-two-dimen- is the cyclotron frequency, e is the elementary charge, α and m is the effective electron (hole) mass. In this sional case [6Ð9], where is the dimensionless con- e(h) stant of the electronÐphonon coupling [10]. case, the resonant coupling between electron levels clearly manifests itself. The fulfillment of condition (1) A single quantum well is considered as a quasi-two- leads to the intersection of the magnetic-field depen- dimensional system. The well is assumed to be ideal, dences of the energy terms for the electronÐphonon and the inhomogeneous broadening of levels is system. The formation of the polaron states results in ignored. The dimensionless reflectance ᏾ and absorp- divergence of the energy levels at the intersection tance Ꮽ of the light are calculated as the ratio between points. The divergence of the terms was first revealed the reflected or absorbed light flux and the incident flux. by Johnson and Larsen [1] in the interband magnetoop- The quantum well has a finite depth. Moreover, the tical absorption in massive InSb. quantum well is characterized by the discrete energy The polaron states are formed in three-dimensional levels of electronÐhole pairs and a continuous spec- and quasi-two-dimensional systems. In systems of both trum. We are interested in the optical effects caused by types, these states play an important role in the forma- the resonance between the exciting light of frequency

2182 LANG et al. ω l and the discrete energy levels of electronÐhole pairs perpendicular to the well plane with due regard for the or magnetopolarons. For magnetic fields H and frequen- magnetopolaron effects. ω cies l far from those corresponding to the magnetopo- This paper is organized as follows. In Section 2, we laron resonance, the quantum well is treated as a two- derive the relationships for the electric fields to the right level system (one energy level corresponds to the and the left of the quantum well under normal irradia- ground state of the crystal, and the other energy level is tion by exciting light in the case of a multilevel system. associated with the electronÐhole pair). Under condi- The formulas for the coefficients of reflection and tions close to the magnetopolaron resonance, the well is absorption of light by the quantum well are deduced in regarded as a three-level system (the ground energy and Section 3. The reciprocal radiative lifetimes of the elec- two polaron levels). tronÐhole pair and two magnetopolaron states in the The absorption and reflection of light in quasi-two- quantum well in a strong magnetic field are calculated dimensional systems have been investigated repeatedly. in Sections 4 and 5. In Section 6, we determine the The absorption is traditionally calculated within the broadening associated with the nonradiative lifetimes perturbation theory to the lowest order in the interac- of the magnetopolaron states. The results of numerical tion of light with the electronic system [11Ð13]. The calculations of the reflectance and absorptance are expression for the absorptance involves the multiplier given in Section 7. ∆ បω In Sections 2 and 3, we only disregard the inhomo- γ ( l Ð Eρ), where Eρ is the electronic excitation p geneous broadening of levels, without any other energy reckoned from the ground-state energy, approximations. This broadening can be reduced to a 1 បγ /2 minimum in the quantum well with a high perfection of ∆γ()E = ------(3) π 2 ()បγ 2 boundaries. In these sections, we consider the case of E + /2 two excited discrete levels of the electronic system in is the function transforming into the Dirac delta func- the quantum well. The locations of the levels are tion at γ 0, and γρ is the reciprocal nonradiative assumed to be arbitrary, and the levels are characterized γ γ lifetime of the level ρ. This relationship is rather contra- by the nonradiative [ 1(2)] and radiative [ r1(2)] recipro- dictory. Actually, we obtain ∆γ(E) ∞ at E = 0 and γ cal lifetimes. This pair of levels can exist in an exciton 0, whereas the dimensionless absorptance Ꮽ can- in the absence of a magnetic field or in a magnetopo- not exceed unity. The contradiction can be resolved laron in a quantizing magnetic field in the quantum when going beyond the scope of the perturbation theory well. No assumptions regarding the smallness of the in the photonÐelectron coupling constant, i.e., by sum- exciton effect and a weak influence of longitudinal opti- ming the contributions of all the orders in this constant. cal phonons on the magnetopolaron spectrum are made Summation of the contributions means inclusion of the in Sections 2 and 3. In our earlier work [19], we showed overall sequence of processes involving the absorption that neither exciton effect nor phonon dispersion can បω transform the magnetopolaron levels into energy bands and reemission of a photon l. The new theory if the equality K⊥ = 0 is satisfied (where បK⊥ is the pro- γ Ð1 includes the notion of the radiative lifetime rρ of elec- jection of the quasi-momentum of the magnetopolaron tronic excitation, which was introduced by Andreani onto the xy plane of the quantum well). It is assumed et al. [14] as applied to excitons in quantum wells at that the light creates magnetopolarons with K⊥ = 0, H = 0. because we consider only the normal incidence of light Initially, the new approach was used to describe the on the quantum well plane. At K⊥ = 0, the exciton effect ω light reflection from quantum wells at frequencies l and the phonon dispersion result only in a shift of the close to those corresponding to exciton energies [12, magnetopolaron energy levels and in an additive addi- 14Ð16]. More recently, Kavokin et al. [17, 18] con- tion to the nonradiative broadening of the levels. In Sec- structed the consistent theory for absorption. It was tions 4Ð7, for the purpose of obtaining qualitative demonstrated that, in the case of absorption, the results results and simplifying calculations, we assume that the obtained using the perturbation theory are applicable Coulomb attraction between the electron and the hole, when the following condition is satisfied: the phonon dispersion, and the nonparabolicity of the bands of the crystal have a weak effect. These assump- γ Ӷ γ rρ ρ. (4) tions hold true in the case of strong magnetic fields and narrow quantum wells. Indeed, according to Lerner and The new theory should work well, especially, when Lozovik [20], the exciton effect can be disregarded applied to an analysis of the specific features of reflec- under the conditions tion and absorption of light by ideal quantum wells in a strong magnetic field, because the reciprocal nonradia- a2 ӷ a2 , a ӷ d. (5) γ exc H exc tive lifetimes ρ for discrete levels of electronÐhole Here, pairs in strong magnetic fields are very small and con- ប2ε ()µ 2 dition (4) can be violated. We will calculate the coeffi- aexc = 0/ e (6) cients ᏾ and Ꮽ for the normal incidence of light on the is the WannierÐMott exciton radius in the absence of a ε µ quantum well in the presence of a strong magnetic field magnetic field, 0 is the static permittivity, =

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2183 1 memh/(me + mh) is the reduced effective mass, aH = where n is the refractive index of the barrier. We intro- (cប/|e|H)1/2 is the magnetic length, and d is the quantum duce the designation well width. For GaAs, we have [21] Ᏹ ω π Ᏸ ω 0α()= 2E0elα 0(), (9) res aexc ==146 Å, aH 57.2 Å, (7) where E0 is the scalar amplitude; el is the polarization Ᏸ ω res vector of the exciting light; and 0( ) is the frequency where aH corresponds to the magnetic field Hres function, which, upon excitation by monochromatic ω derived from condition (1) at j = 1. From formulas (7), light with the frequency l, can be represented by the res 2 Ӎ expression we obtain (aH /aexc) 0.154. Therefore, the first inequality (5) is satisfied and the second inequality Ᏸ ()ω = δω()Ð ω . (10) imposes a limitation on the quantum well width. 0 l In Sections 4Ð6, we use the results of our earlier Now, we assume the incident wave to have a circular work [22], in which we developed the theory of magne- polarization; that is, topolarons. As in [22], we assume that the frequencies ()()± of confined and interface phonons [23] involved in the el = 1/ 2 ex iey , (11) ω formation of the magnetopolaron coincide with LO. The approximation according to which the interaction where ex and ey are the unit vectors along the x and y with confined phonons is replaced by the interaction axes, respectively. with bulk phonons and the interaction with interface In the quantum well, the states of the electronic sys- phonons is ignored is used only in Section 6 in order to tem are excited by the incident electromagnetic wave. γ estimate the nonradiative broadening p of the polaron The energy of the excited states will be characterized levels. This approximation is applicable for wide quan- by a set of indices ρ. For example, upon normal inci- tum wells [24]. The nonradiative broadening γ with dence of the wave in a strong magnetic field, the set of p ρ allowance made for the confined and interface phonons indices for electronÐhole pairs involves le, lh, and ne = was calculated in our previous work [25]. Hereafter, we nh, where le (lh) are the quantum numbers of the quan- will use the magnetopolaron classification proposed in tum confinement of electrons (holes) and ne (nh) are the [24]. The specific calculations are performed for the Landau quantum numbers of electrons (holes). In the polaron A, in which the electronÐphonon interaction case of an infinitely deep quantum well, we have le = lh. binds two adjacent Landau levels (with n = 0 and 1) The state with the set ρ is characterized by the energy belonging to the same quantum-well level. The បωρ reckoned from the ground-state energy and the obtained results, as in [22, 24], are valid when the γ γ radiative ( rρ) and nonradiative ( ρ) reciprocal lifetimes. polaron splitting is small compared to the difference Now, we analyze the quantum well with width d Ӷ between the energies of the adjacent quantum-well lev- ω c/ ln. In this case, the electric fields EL(R)(z, t) on the els. Since the level spacing decreases with an increase in left (right) of the quantum well are given by the rela- d, this condition places an upper limit on the permissible tionship [27] well widths (see numerical estimates in [24]). In the present work, we examine the irradiation of the quantum E ()zt, = E ()zt, + ∆E (),zt, (12) well in a strong magnetic field by monochromatic light. LR() 0 LR() Note that, in [26], we proposed a technique for detecting ∆ , EαLR()()zt = E0elα and examining the magnetopolaron effect with the use of ∞ irradiation of the quantum well by light pulses and (13) recording of transmitted and reflected pulses. ×ω∫ d exp[]Ðiω()tzn± /c Ᏸ()ω + c.c., Ð∞ 2. ELECTRIC FIELDS TO THE RIGHT where the upper (lower) sign corresponds to the sub- AND THE LEFT OF A QUANTUM WELL script L (R). The frequency distribution is defined as follows:2 Let us assume that an electromagnetic wave (or a Ᏸ ω πχ ω Ᏸ ω []πχ ω superposition of waves that corresponds to a time-lim- ()= Ð 4 () 0()/1+ 4 (), (14) ited light pulse) is incident on a single quantum well normally to its plane from the left and the electric field 1 It is possible to omit the complex conjugate and to replace the of this wave has the form Ᏹ ω Ᏹ ω Ᏹ ω Ᏹ* ω π ω term 0α( ) by 0α( ) = 0α( ) + 0α (Ð ) = 2 E0[elαD0( ) + , ()π E0, α(zt )= 1/2 * * ω elαD0 (Ð )]. ∞ 2 (8) Expression (15) is written taking into account that either of two circular polarizations corresponds to the excitation of one of two ×ωd exp[]Ðiω()tznÐ /c Ᏹ , α()ω + c.c., ∫ 0 types of electronÐhole pairs with identical energies from the Ð∞ ground state [see formula (78)].

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2184 LANG et al. The three-level system involves the ground level of χω ()γπ () ()= i/4 ∑ rρ/2 the electronic system in the quantum well and two lev- ρ (15) បω els characterized by the energies 1(2) and the radia- Ð1 Ð1 tive [γ ] and nonradiative [γ ] reciprocal lifetimes. ×ωω[]()Ð ρ + iγ ρ/2 + ()ωω++ρ iγ ρ/2 . r1(2) 1(2) The calculation of the electric fields for the three-level Relationships (12)–(15) describe the electric fields on system is somewhat more complicated, because it is both sides of the quantum well and, hence, the energy necessary to solve the quadratic equation [see Eq. (23)]. fluxes of the transmitted and reflected light for the mul- By using relationships (10) and (12)Ð(15) and rejecting tilevel system. The second term in the square brackets the nonresonant term on the right-hand side of expres- on the right-hand side of relationship (15) is nonreso- sion (15), we obtain nant but is required to fulfill the relationship  χ∗()ω = χω().Ð (16) , ()ω () EαL()zt = E0elαexp Ði l tznÐ /c --- In real quantum wells with a finite depth in a strong  magnetic field, there always exist discrete energy levels γ /2 (or at least one discrete level). Moreover, there exists a ()ω () r1 Ð iiexp Ð l tzn+ /c ω------Ω (21) continuous spectrum in addition to the discrete spec- l Ð 1 + iG1/2 trum. Therefore, the index ρ on the right-hand side of γ  relationship (15) takes both discrete and continuous r2/2 values. In a strong magnetic field, proper allowance + ω------Ω  + c.c., l Ð 2 + iG2/2  must sometimes be made for a large number of levels (see [28, 29]). For a two-level system, it is assumed that the influence of all the levels, except for the level that is Eα ()zt, = E e α exp()Ðiω ()tznÐ /c ω R 0 l l at resonance with the frequency l, can be ignored. (22) γ γ The reflectance and absorptance for the two-level × r1/2 r2/2 1 Ð i ω------Ω + ω------Ω + c.c. system corresponding to one exciton level in the quan- l Ð 1 + iG1/2 l Ð 2 + iG2/2 tum well at H = 0 were calculated in [14Ð18]. The following relationships were derived for the electric fields: Ω Ω The quantities O1, 2 = 1, 2 Ð iG1, 2/2 (G and are real quantities by definition) obey the equation EαL = E0elα ()ω γ ()ω γ Ðiω ()tznÐ /c iγ Ðiω ()tzn+ /c (17) O Ð 1 + i 1/2 O Ð 2 + i 2/2 × l r l e Ð ------()ω ω Γ e + c.c., ()γ ()ω γ 2 l Ð + i /2 + i r1/2 O Ð 2 + i 2/2 (23) ()γ ()ω γ + i r2/2 O Ð 1 + i 1/2 = 0 ω () Ði l tznÐ /c EαR = E0elαe and are given by the formula iγ (18) × r ()Ω Ð iG/2 , 1 Ð ------()ω ω Γ + c.c., 12 2 l Ð + i /2 (24) []ω ω ± ()ω ω 2 γ γ = ˜ 1 + ˜ 2 ˜ 1 Ð ˜ 2 Ð r1 r2 /2 Γγγ = + r. (19) (the plus and minus signs refer to subscripts 1 and 2, Formulas (17) and (18) can be obtained from expres- respectively), where sions (10) and (12)Ð(15) by retaining one term in the ω˜ ω Γ Γ γ γ sum over ρ on the right-hand side of relationship (15) 12()==12()Ð i 12()/2, 12() 12()+ r12(). (25) using the designations In general, the relationship under the radicand in for- ωρ ===ω, γ ρ γ , γ ρ γ , (20) mula (24) is a complex-valued expression. Moreover, r r we used the designations and omitting the nonresonant term in the square brack- γ γ ∆γ γ γ ∆γ ets. r1 ==r1 + , r2 r2 Ð , (26)

γ []γΩ ω ()γ []Ω ω ()γ ∆γ r1 2 Ð 2 Ð iG2 Ð 2 /2 + r2 1 Ð 1 Ð iG1 Ð 1 /2 = ------Ω Ω () ------. (27) 1 Ð 2 + iG2 Ð G1 /2

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2185

A comparison of relationships (21) and (22) with the and the transmittance of light is given by the expression corresponding expressions (17) and (18) for the two- ᐀ ᏾Ꮽ level system shows that the summation of the contribu- ==1 Ð Ð SR / S0 . (35) tions from levels 1 and 2 is not a simple summation but is accompanied by the replacement With the use of relationships (17) and (18) for the two- level system, we obtain ω Ω Γ 12() 12(), 12() G12(), (28) ᏾ ()γ 2 []()ω ω 2 ()Γ 2 γ γ = r/2 / l Ð + /2 , (36) r12() r12(). Ꮽ ()γγ []()ω ω 2 ()Γ 2 The calculation of the electric fields for a four-level = r/2 / l Ð + /2 (37) system is even more difficult because it requires the γ solution of a cubic equation, etc. At present, the prob- (see [14Ð18]). From formula (37), it follows that, at = lem for an arbitrary number ρ of levels has defied exact 0, the absorption of light by the electronic system in the quantum well is absent [17, 18]. solution. However, at γ ρ Ӷ γρ, the perturbation theory r γ Ӷ γ γ ӷ for the interaction of light with the electronic system in Let us consider two limiting cases: r and r γ γ Ӷ γ the quantum well is applicable and, hence, the term . At r , the perturbation theory is applicable to the 4πχ(ω) in the denominator on the right-hand side of lowest order in the interaction of light with electrons. relationship (14) can be rejected. As a result, we obtain The absorptance and reflectance appear to be quantities the solution for any number ρ of levels in the form of the second and fourth order in the interaction, respectively. From relationships (36) and (37), we have  ω () ω () M , Ði l tznÐ /c Ði l tzn+ /c 2 EαL()zt = E0elαe Ð ie - ᏾ Ӎ ()γπប/2 ()∆/γ γ []ប()ω Ð ω , (38)  r l (29) Ꮽ Ӎ πបγ ∆γ []ប()ω Ð ω . (39) γ  r l × rρ/2 ∑------ + c.c., Here, ∆γ(E) is the function defined by formula (3). At ω Ð ωρ + iγ ρ/2 ρ l  γ Ӷ γ r , the following inequalities are satisfied: ω () M (), Ði l tznÐ /c Ꮽ Ӷ 1, ᏾ Ӷ Ꮽ. (40) EαR zt = E0elαe γ ӷ γ Γ (30) For r , the quantity in relationships (36) and γ ρ/2 × r (37) is approximately replaced by γ . As a result, at res- 1 Ði∑------ω ω γ - + c.c. r l Ð ρ + i ρ/2 ᏾ ω ω Ӎ ρ onance, we have ( l = ) 1; i.e., the total reflection takes place and Ꮽ Ӷ 1. Consequently, the absorptance is small in both limiting cases. The maximum absorp- Ꮽ ω ω γ γ 3. REFLECTANCE AND ABSORPTANCE tance ( l = ) = 1/2 is observed at r = . OF LIGHT For the three-level system with the use of expres- With knowledge of the relationships for the electric sions (21), (22), (33), and (34), we obtain fields to the left and the right of the quantum well, it is possible to calculate the reflectance and absorptance of  ᏾ 1 []γ ω ω γ ω ω 2 light. Let us introduce the Poynting vector S on the = ------ r1()l Ð 2 + r2()l Ð 1 --- L(R) 4Z left (right) of the quantum well. The vector SL can be written in the form (41)  1()γ γ γ γ 2 S = S + ∆S , (31) + --- r1 2 + r2 1 , L 0 L 4  where ()π 2  γ 2 S0 = c/2 E0ez (32) Ꮽ 1 γ γ ()ω ω 2 2 = ------ r1 1 l Ð 2 + ----- ∆ 2Z 4 is the flux of the exciting light and SL is the flux of the (42) reflected light directed along the vector Ðe . The reflec- z γ 2  tance of light is defined by the ratio + γ γ ()ω Ð ω 2 + -----1 , r2 2 l 1 4 ᏾ ∆  = SL / S0 , (33) the dimensionless absorptance is represented by the where formula 2 2 ()ω Ω 2 G1 ()ω Ω 2 G2 Ꮽ Z = l Ð 1 + ------l Ð 2 +.------(43) = SL Ð/SR S0 , (34) 4 4

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2186 LANG et al. γ γ It follows from formula (42) that the absorptance of Setting 1 = 2 = 0 on the right-hand sides of relation- light for the three-level system is equal to zero under ships (41) and (42), we obtain Ꮽ = 0 and the condition ᏾ γ γ 1 ==2 0. (44) []()γ + γ /2 2()ω Ð ω 2 (51) With due regard for Eq. (23), the quantity Z can be rear- = ------r1 r2 l 0 -, ()ω ω 2()ω ω 2 []()γ γ 2()ω ω 2 ranged to the form l Ð 1 l Ð 2 + r1 + r2 /2 l Ð 0 γ γ γ γ γ γ 2 r1 2 ++r2 1 1 2 ω ()ω γ ω γ ()γ γ Z = ()ωω Ð ω ()Ð ω Ð ------0 = 1 r2 + 2 r1 / r1 + r2 . (52) l 1 l 2 4 (45) As follows from relationship (51), the reflectance for 1[]()Γω ω ()Γω ω 2 + --- l Ð 1 2 + l Ð 2 1 . the three-level system is characterized by the specific 4 properties when the reciprocal radiative times are pre- γ γ dominant. At any constants r1 and r2, the reflectance is Next, we derive the simplified expressions for the ω ω coefficients ᏾ and Ꮽ in different limiting cases. equal to zero at the point l = 0 and unity at the points ω ω ω ω γ γ γ Relationships (41) and (42) are conveniently used l = 1 and l = 2. Assume that r1 = r2 = r. Then, we ᏾ ω ω ω γ ӷ ω ω sometimes with formula (43) and sometimes with have = 0 at l = ( 1 + 2)/2. At r 1 Ð 2, the ω ω ω formula (45). The first limiting case corresponds to the reflectance at the point l = ( 1 + 2)/2 exhibits a nar- inequalities row minimum with ᏾ = 0. The half-width at half-min- ω ω 2 3/2γ γ Ӷ γ imum is equal to ( 1 Ð 2) /(2 r). r12() 12(), (46) Figure 1a shows the functions ᏾(ω ) under the con- when the perturbation theory for the interaction of light l ditions with the electronic system can be applied. Under con- Ω Ӎ ω γ dition (46) and by setting 1(2) 1(2) and G1(2) = 1(2) γ ==γ γ , γ =γ =γ , γ Ӷ γ . (53) in formula (43), we obtain the relationships 1 2 r1 r2 r r

2 Figure 1b depicts the dependences Ꮽ(ω ) under the ()γ /2 l ᏾ Ӎ ------r1 same conditions (53). At ﬁxed values of γ, an increase ()ω ω 2 ()γ 2 γ Ꮽ ω l Ð 1 + 1/2 in r leads to changes in the curves ( l). Curves 1Ð6 បγ 2 are constructed for the following values of r: 0.002, ()γ /2 γ γ 3 + ------r2 + ------r1 r2 (47) 0.005, 0.008, 0.04, 0.125, and 0.5. Two maxima are 2 2 បγ ()ω Ð ω + ()γ /2 2 observed in curve 1 (at r = 0.002). The other curves l 2 2 បγ ()ωω ω ()γω γ have only one maximum. Curve 5 ( r = 0.125) exhibits l Ð 1 l Ð 2 + 1 2/4 Ꮽ × ------, the most pronounced maximum with max = 0.5. The []ω()ω ω 2 ()γ 2 []()ω 2 ()γ 2 Ꮽ l Ð 1 + 1/2 l Ð 2 + 2/2 absorptance 0 that corresponds to the central point ω ω ω Ꮽ l = ( 1 + 2)/2 and coincides with max in curves 2Ð5 γ γ /2 γ γ /2 Ꮽ Ӎ ------r1 1 + ------r2 2 . (48) can be described by the exact formula ()ω Ð ω 2 + ()γ /2 2 ()ω Ð ω 2 + ()γ /2 2 l 1 1 l 2 2 γ γω[]()ω 2 γ 2 Ꮽ 4 r 1 Ð 2 + According to relationship (48), the absorptance Ꮽ is 0 = ------, (54) []()ω ω 2 γ γγ2 2 the sum of contributions (39) from levels 1 and 2, 1 Ð 2 ++2 r because the absorptance under condition (46) is linear in the constants γ and γ . According to formula (47), which can be easily deduced from expression (42) by r1 r2 ω ω ω Ꮽ ᏾ γ γ setting l = ( 1 + 2)/2. The absorptance 0 is maxi- the reﬂectance is quadratic in the constants r1 and r2 and, hence, includes the interference contribution in mum at addition to the contributions from the individual levels. ()ω Ð ω 2 + γ 2 By generalizing relationship (48) to an arbitrary γ 1 2 r0 = ------. (55) number of levels under condition (4), we obtain 2γ ប ω ω បγ γ γ Substitution of ( 1 Ð 2) = 0.005 and = 0.0001 ᏭM Ӎ 1 rρ ρ ---∑------2 -2. (49) (these values correspond to Fig. 1b) into formula (55) 2 ()ω Ð ωρ + ()γ ρ/2 បγ Ꮽ Ꮽ ρ l gives r0 = 0.125 and max = 0 = 0.5 (curve 5 in Fig. 1b). The next limiting case is opposite to that considered above; i.e., it is described by the inequality 3 All the parameters and frequencies in Figs. 1 and 2 are given in γ ӷ γ arbitrary units, because relationships (47) and (48) include only r12() 12(). (50) the ratios of these quantities.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2187

The next limiting case is specified by the smallness 1.0 ᏾ γ γ (a) of the reciprocal lifetimes 1(2) and r1(2) as compared to ω ω 1 Ð 2. Assume that the following conditions are met: γ Ӷ ω ω γ Ӷ ω ω 12() 1 Ð 2, r12() 1 Ð 2. (56) Note that the ratio between the nonradiative and radia- γ γ 3 tive reciprocal lifetimes 1(2) and r1(2) can be arbitrary. 0.5 ω Cosequently, when the frequency l is close to reso- ω Ӎ ω 2 nance with one of the levels, for example, l 1, from relationships (41) and (42), we obtain 1 γ 2/4 ᏾ Ӎ ------1 , ()ω ω 2 ()Γ 2 l Ð 1 + 1/2 (57) –0.005 0.005 γ γ r1 1/2 "[ω – (ω + ω )/2] Ꮽ Ӎ ------. l 1 2 ()ω ω 2 ()Γ 2 l Ð 1 + 1/2 ᏭᏭ These formulas coincide with expressions (36) and (37) (b) for the two-level system. 0.4 0.10 Finally, we examine the case of merging levels. 4 0.2 Under the conditions 5 1 γ γ γ ω ω ω 6 1 ==2 , 1 =2 =, (58) × –4 × –4 γ ==γ γ , –1 10 1 10 r1 r2 r "[ω – (ω + ω )/2] 0.05 l 1 2 2 from relationships (41) and (42), we have γ 2 ᏾ Ӎ ------r , (59) 3 ()ω ω 2 ()γ γ 2 l Ð + r + /2 γ γ –0.01 0 0.01 r ω ω ω Ꮽ Ӎ ------. (60) "[ l – ( 1 + 2)/2] ()ω ω 2 ()γ γ 2 l Ð + r + /2 Fig. 1. Dependences of the dimensionless (a) reflectance ᏾ Ꮽ ω Analysis of these expressions demonstrates that the for- and (b) absorptance of light on the light frequency l for γ Ӷ γ γ γ γ mulas for the two-level system with twice the recipro- a three-level system at r (where r = r1 = r2 is the radi- γ γ γ γ cal lifetime r are valid in the case of the doubly degen- ative broadening of the levels and = 1 = 2 is the nonradi- ប ω ω បγ erate excited level. ative broadening of the levels). ( 1 Ð 2) = 0.005; = បγ 0.0001; and r = (1) 0.002, (2) 0.005, and (3) 0.008. The Ꮽ ω បγ inset shows the dependences ( l) at r = (4) 0.04, (5) 4. CALCULATION OF THE RADIATIVE 0.125, and (6) 0.5. LIFETIME FOR AN ELECTRONÐHOLE PAIR IN A QUANTUM WELL IN A STRONG MAGNETIC FIELD of indices s. The interaction U can be written in the γ form The reciprocal radiative lifetime r of the electronÐ 1 hole pair in a strong magnetic field can be calculated U = Ð--- drA() r jr(). (62) from the following formula of the perturbation theory c∫ in the second order: Here,

2π 2 γ 〈〉η δ()បω πបc Ð1/2 ikr + Ðikr rη = ------∑ sU s Ð Eη . (61) 2 ω () ប Ar()= ------∑ s csese + cs es*e (63) s V 0 n s Here, |η〉 and |s〉 are the wave functions in the second is the vector potential, the set of indices s consists of the |η〉 quantization representation. The function of the ini- wave vector k and the polarization index i (which takes tial state corresponds to the presence of one pair with + the set of indices η, and function |s〉 of the ﬁnal state two values); cs and cs are the creation and annihilation បω corresponds to the presence of one photon with the set operators for the photon s, respectively; es and s are

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2188 LANG et al. the polarization vector and the energy of the photon, used the gauge of the vector potential A = A(0, xH, 0). respectively; and V0 is the normalization volume. The In our earlier work [32], we demonstrated that func- current density operator can be represented by the tions (67) are the eigenfunctions of the momentum expression operator Pˆ ⊥ of the electronÐhole pair in the xy plane and correspond to the eigenvalues of បK⊥. The operator e + jr()= ------∑[]p v Fξ*()r aξ + p*v Fξ()r aξ . (64) ˆ m c c P⊥ was determined in [34]. The energies of the states 0 ξ of functions (67) are as follows: + Here, m0 is the mass of the free electron, aξ and aξ are e h Eξ ==បωξ E ++ε ε the creation and annihilation operators for the electronÐ g le lh (69) បΩ ()បΩ () hole pair in the state ξ; + e ne + 1/2 + h nh + 1/2 , Ψ Fξ()r = ξ(),re ==rh r (65) Ω | | where Eg is the band gap and e(h) = e H/me(h). Now, the Ψ reciprocal radiative lifetime γ ξ of the electronÐhole ξ(re, rh) is the slowly varying part of the electronÐhole r pair wave function (in the effective-mass approxima- pair with the set of indices ξ can be calculated with the use of formulas (61)Ð(69). By assuming that the x axis tion) dependent on the radius vectors re and rh of the is aligned along K⊥, we find (for more details, see [25]) electron and hole, and pcv is the interband matrix element of the momentum operator. 2 2 Ψ γ ξ = []2e Ω /()ប cnm ωξ B , ()K⊥ Let us determine the wave functions ξ(re, rh). The r 0 0 ne nh symbol ξ designates the set of indices characterizing 2 2 2 (70) × R , ()()ωξn/c Ð K⊥ Fξ(),K⊥ the state of the electronÐhole pair, that is, le lh ξ ,, , , ,, < ω γ > ω c v ne nh K⊥ le lh. (66) K⊥ ξn/c, rξ = 0, K⊥ ξn/c, The subscripts c and v indicate the conduction and Ω | | where 0 = e H/m0c is the cyclotron frequency corre- valence bands, respectively (which, in principle, can be sponding to the mass m of the free electron, degenerate). By introducing the coordinates 0 2 () r⊥ (), r⊥ ==re⊥ Ð rh⊥, R⊥ mhrh⊥ + mere⊥ /M ᏷ 0 min n! m! Bnm, ()K⊥ ==nm, Ð------, for the relative and cooperative motion of the electron aH max()n! m! and hole in the xy plane, we obtain [30Ð33] 2 2 2 2 nmÐ 2 2 2 × aH K⊥ aH K⊥ nmÐ aH K⊥ exp Ð------Lmin()nm, ------, (71) Ψ , 1 ()yX 2 2 2 ξ()re rh = ------exp iKx XK+ yY Ð i------2 π 2 a ∞ 2 aH Lx Ly H Ðikz R , ()k = dze ϕ ()z ϕ (),z le lh ∫ cle v lh me Ð mh 2 xy × exp i------()ÐK⊥r⊥ + a K K + ------(67) Ð∞ 2M H x y 2 aH 2 2 ()ωξn/c Ð K⊥ r⊥ Ð r⊥ 2 × ᏷ 0 ϕ ϕ () Fξ()K⊥ = ------pcvx n , n ------()z v z , ω e ha cle e lh h ξn/c H (72) ω 2 2 where បK⊥ is the quasi-momentum of the electronÐhole ξn/c 2 K⊥ pcvz + ------pcvy + ------. pair in the xy plane, M = me + mh, Lx and Ly are the nor- 2 2 2 2 ()ωξn/c Ð K⊥ ()ωωξn/c ()ξn/c Ð K⊥ 2 × malization lengths, and r⊥0 = aH H K⊥/H. The function ᏷ is defined by the relationship [31] A comparison of relationship (70) with the results n, m obtained in [14] for the exciton level in the quantum ᏷ , (), nmÐ nm, ()p = min(n! m! )/max n! m! i well at H = 0 shows that the dependences on pcv for both cases coincide with each other and γ ξ = 0 at K⊥ > 2 nmÐ r × exp()Ðp /4 ()p/2 (68) ωξn/c. × []()φπ ()nmÐ 2 exp i Ð /2 nmÐ Lmin()nm, (),p /2 The right-hand side of relationship (70) involves the Ω multipliers 0 and Bn , n ()K⊥ associated with the mag- 2 2 φ () n e h where p = px + py , = arctan py/ px , and Lm()x is 2 2 2 netic field. The multiplier R , ()()ωξn/c Ð K⊥ is the associated Laguerre polynomial. The form of the le lh ϕ real functions c(v)l(z) corresponding to the quantum absent in the expressions deduced in [14], because number l of quantum confinement for the quantum well these authors considered the quantum wells with d Ӷ ω with a finite depth is given in [22]. In formula (67), we c/n l, whereas we did not impose this limitation in the

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2189 derivation of relationship (70). For narrow wells (d Ӷ pairs of these sorts differ in the interband matrix ele- λ), we have ments pcv, which can be written in the following form:

2 2 I 1 () R , ()()ωξn/c Ð K⊥ Ӎ π , pcv = ------pcv ex Ð iey , le lh le lh 2 ∞ (73) (78) ϕ ϕ II 1 () = ∫ dz cl ()z v l ().z pcv = ------pcv ex + iey . e h 2 Ð∞ Within the proposed model, the wave functions ϕ ()z In the case of infinitely deep quantum wells, we obtain cle and ϕ ()z do not depend on the subscripts c and v. v lh π , = δ , . (74) le lh le lh With the use of the circular polarizations (11), each polarization (the left-hand or right-hand polarization Ӎ ω For the quantum well with d c/n l, the multiplier with respect to the z axis) is rigidly associated with a 2 2 2 particular sort (I or II) of electronÐhole pairs, because R , ()()ωξn/c Ð K⊥ depends on the quantum well le lh the interaction of the pair with light is proportional to width, namely, decreases with an increase in d. As a the quantity espcv. From formula (76) for model (78), γ result, the reciprocal lifetime rξ also decreases. In the we find the reciprocal radiative time of pairs of any sort, ӷ λ γ that is,4 limiting case d , we obtain rξ 0, as should be expected when changing over to the bulk crystal [12]. 2Ω 2 2e 0 2 pcv γ ξ = ------R , ()ωξ n/c ------. (79) For normal incidence of light on the quantum well, r 0 ប le lh 0 បω cn m0 ξ it is sufficient to determine the reciprocal lifetimes 0 described by formula (70) at K⊥ = 0. Since the equality In the case of narrow quantum wells (d Ӷ λ), we use B , ()0 = δ , is satisfied, we calculate the recipro- approximation (73) and have ne nh ne nh 2 2 cal lifetime γ ξ for the set of indices r 0 2e Ω p γ Ӎ 0π2 cv rξ ------l , l ------. (80) 0 បcn e h m បωξ ξ ,, ,, 0 0 0 c v ne ==nh, K⊥ 0 le lh. (75) By using the parameters taken from [21] for GaAs and With the use of expression (70), we find approximation (74) and also setting បωξ Ӎ E , we 0 g obtain the numerical estimate for the reciprocal radia- 2Ω 2 2 2e 0 2 pcvx + pcvy tive time in the form γ ξ = ------R , ()ωξ n/c ------, (76) r 0 ប le lh 0 បω cn m0 ξ 0 បγ Ӎ × Ð5() r 5.35 10 H/Hres eV, (81) បω εc εh បΩ () ξ = Eg +++l l µ ne + 1/2 , where Hres corresponds to polaron A, i.e., is given by the 0 e h ω | | (77) formula Hres = mec LO/ e . eH memh Ωµ ==------, µ ------. µc M 5. RECIPROCAL RADIATIVE LIFETIMES It follows from formula (76) that the reciprocal radi- OF THE MAGNETOPOLARON ative lifetime of the electronÐhole pair is proportional Now, let us calculate the radiative lifetime of the to the magnetic field H when the energy បωξ weakly 0 system composed of a magnetopolaron and a hole. As ˜ ˜ an example, we choose polaron A in combination with depends on the field H (Eξ Ӎ Eg , where Eg = E + 0 g the hole described by the quantum number lh of the εe + εh ). quantum confinement and the Landau quantum number le lh ω Ω nh = 1. In the vicinity of the resonance LO = e, the Now, we calculate the reciprocal lifetime γ ξ for energy level of the electronÐhole pair is split into two r 0 energy terms. Either energy term is characterized by the the band model of GaAs. The conduction band is dou- γ γ reciprocal radiative lifetime ra or rb according to the bly degenerate (with respect to the spin), and the sub- designations proposed in [22] (the subscripts a and b script c can take two values: c = 1 or 2. The valence refer to the higher and lower terms of the magnetopo- band (of heavy holes) is also doubly degenerate: v = 1 laron). The reciprocal radiative lifetimes γ and γ can or 2. There exist pairs of two sorts denoted by the indi- ra rb ces I and II such that c = 1 and v = 1 in pairs of sort I 4 The relationship derived in [28] coincides with expression (79) at and c = 2 and v = 2 in pairs of sort II. The electronÐhole n = 1.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2190 LANG et al.

᏾ where the upper (lower) sign corresponds to the term 0.4 (a) Ω ω p = a (p = b). At resonance, we have e = LO and res ()បω ± 2 E p = 3/2 LO A , (85) ∆ res res 2 EE==a +2Eb A . 0.2 For the reciprocal radiative lifetimes, we obtain 3 γ []2Ω ()ប2 ω 2 rη = 2Q0 pe 0/ cnm0 η B11()K⊥ 2 2 2 × R , ()()ωηn/c Ð K⊥ Fη()K⊥ , 1 le lh K⊥ < ωηn/c, –0.01 0 0.01 ω ω ω γ > ω "[ l – ( 1 + 2)/2] rη = 0, K⊥ ηn/c, (86) ()()± λ λ2 2 Ꮽ Q0 p = 1/2 1 / + 4A . (87) (b) Compared to expression (70), formula (86) includes an additional multiplier Q0p, which strongly depends on λ Ω 0.4 the deviation of the cyclotron frequency e from the ω resonance frequency LO, i.e., on the magnetic field strength H. The reason for this strong dependence is as follows. In [22], we demonstrated that the wave func- 0.2 tion of the magnetopolaron is a linear combination of 3 two functions of the electronÐphonon system, one of 2 which corresponds to the electron with the Landau 1 quantum number n = 1 and phonon vacuum and the other function is associated with the electron with the quantum number n = 0 and one phonon with the fre- –0.02 –0.01 0 0.01 0.02 quency ω . The coefficients Q and Q are the prob- "[ω – (ω + ω )/2] LO 0p 1p l 1 2 abilities of finding the system in these states (the index 0 indicates the absence of phonons, and the index 1 indi- Fig. 2. Dependences of the dimensionless (a) reflectance ᏾ Ꮽ ω cates the presence of one phonon). The coefficient Q0p and (b) absorptance of light on the light frequency l for γ γ ប ω ω បγ is defined by formula (87), and the coefficient Q1p can a three-level system at = r. ( 1 Ð 2) = 0.005 and r = (1) 0.002, (2) 0.005, and (3) 0.008. be represented by the expression ()()− λ λ2 2 Q1 p ==1 Ð1/2Q0 p 1 + / + 4A . (88) be calculated using formula (61). The initial state is Ω ω res res characterized by the set of indices At resonance, we have e = LO and Q0 p = Q1 p = 1/2. The presence of the multiplier Q0p in relationship (86) η c,,,v p K⊥ ,,l l , (82) e h can be explained by the fact that only one of two exci- where the index p is equal to a or b and បK⊥ is the tations (whose superposition comprises the magneto- quasi-momentum (lying in the xy plane) of the system. polaronÐhole system), namely, the excitation with the The wave functions of the system with indices (82) indices ne = nh = 1 and N = 0, can undergo light annihi- were calculated in our previous works [22, 25]. The lation. The light annihilation of the other excitation corresponding energies can be written in the form with the indices ne = 0, nh = 1, and N = 1 is impossible. It follows from expressions (86) and (87) that, at λ = 0 e h Eη ==បωη E +++ε ε ()3/2 បΩ +E , (83) (i.e., at true resonance), the reciprocal radiative life- g le lh h p γ γ times ra and rb are equal to each other and involve an where Ep is the polaron energy measured from the level additional multiplier 1/2 as compared to those of the e electronÐhole pair with the indices n = n = 1. At K⊥ = ε . According to [22], the polaron energy is defined by e h le η 0, we introduce the set of indices 0 c, v, p, K⊥ = the relationship 0, le, and lh. In this case, the formulas derived for the γ 2 2 reciprocal lifetimes η differ from relationships (76), បΩ បω ± ()λ r 0 E p = e + LO/2 /2 +,A ξ (84) (79), and (80) only in the replacement of the indices 0 λ ប()Ω ω η = e Ð LO , by the indices 0 and the presence of the multiplier Q0p.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2191

γ γ γ γ The replacement of ωξ by ωη is of no significance / 0, / 0 0 ra a ra (a) ˜ because we approximately have ωξ Ӎ ωη Ӎ Eg /ប, 0 0 1.2 ˜ where Eg = E + ε + ε . As a result, instead of g le lh expression (80), we obtain 0.8 2Q e2 Ω p2 1 γ Ӎ 0 p 0 cv π2 rη ------l , l . (89) 3 0 បc nm បωη e h 0 0 The strong dependence of the reciprocal radiative life- 0.4 time on the index p and the magnetic field H is deter- λ 2 a3 mined by the multiplier Q0p. The dependences of the λ γ γ a1 reciprocal radiative lifetimes ra and rb on the field H in the vicinity of the resonance are plotted in Fig. 3. –2–1 0 1 2 λ/∆E 6. NONRADIATIVE LIFETIMES OF THE MAGNETOPOLARON 1.4 γ γ γ γ rb/ 0, b/ rb (b) We will not calculate the reciprocal nonradiative lifetimes of the electronÐhole pair in a quantum well in 1 a strong magnetic field far from the magnetopolaron resonance because the nature of the processes responsi- 1.0 ble for these times remains unclear. One-phonon processes are forbidden by the energy conservation law. Possibly, two-phonon processes with the participation 0.6 of acoustic phonons make the main contribution. How- 3 ever, in the vicinity of the magnetophonon resonance, γ 2 there arises a contribution to the reciprocal lifetimes p λ (p = a and b) due to the finite lifetimes of the longitudi- b1 0.2 λ nal optical phonons comprising the magnetopolaron. b3 This contribution will be calculated in this section. In γ –2–1 0 1 2 this way, the lower limit of the quantities p (p = a and λ ∆ b) can be determined in the vicinity of the resonance. / E In the quantum well, the phonon with the set of indi- Fig. 3. Magnetic-field dependences of the radiative and ces ν is characterized by the reciprocal nonradiative nonradiative broadening of the excited level of the system lifetime γν governed by the phononÐphonon interac- consisting of polaron Ꮽ and the hole with the quantum tion, for example, by the decay of one longitudinal opti- numbers nh = 1 and l in the quantum well at p = (a) a and ∆ បω cal phonon into two acoustic phonons. First, we derive (b) b. E/ LO = 0.18 (this is characteristic of GaAs [24]). γ γ the relationship for the reciprocal lifetime γν. This life- Solid lines correspond to rp/ 0, and dashed lines represent γ γ γ γ time can be written in the form p/ rp at LO / 0 = (1) 10, (2) 1, and (3) 0.1. π γ 2 〈〉Ᏼ 2δ() ν = ------ប ∑ f pp i Ei Ð E f , (90) the polaron in the quantum well. In the present work, f we use an even simpler model in which the interaction Ᏼ | 〉 of electrons with confined phonons can be approxi- where pp is the phononÐphonon interaction, i = +|〉 mated by the Fröhlich interaction with bulk phonons bν 0 is the initial state corresponding to one phonon and the interaction with interface phonons is disre- ν បω | 〉 with the set of indices and the energy Ei = LO, f = garded. In [24], we demonstrated that the interaction of + aτ |〉0 is the final state of the phonon system with the set electrons with bulk phonons can be used for sufficiently wide quantum wells. τ + of indices and the energy Ef = Eτ, and aτ is the phonon operator corresponding, for example, to the In the case of bulk longitudinal optical phonons, the creation of two acoustic phonons. In our earlier work set of indices ν reduces to the three-dimensional wave γ γ [25], we calculated the reciprocal lifetimes ν and p for vector q. The final phonon state is characterized by the the model proposed in [22], according to which con- resultant three-dimensional wave vector q' and other fined and interface phonons with the same frequency indices ϕ. For example, when the phonon decays into ω LO without dispersion participate in the formation of two acoustic phonons, we have q' = q1 + q2.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2192 LANG et al. λ ប Ω ω The matrix element in formula (90) can be defined = ( e Ð LO) describing the deviation of the mag- Ω ω by the expression netic field from the resonance e = LO. At resonance, ᏷ + δ (), ϕ we have 0 aτ ppbν 0 = qq, 'W q . (91) γ ()γ Then, formula (90) can be rearranged to give p = 1/2 LO (99) for both terms p = a and b. γ γ π ប , ϕ 2δ()បω ν ==q 2 / ∑ W()q LO Ð Eq, ϕ , (92) Now, we analyze the results obtained with the use of Ω ω ϕ relationship (93) far from the resonance e = LO. It is seen from Fig. 3 that, under the condition |λ| ӷ |A|, the where E ϕ is the energy of the final phonon state repre- q, term a at λ > 0 and the term b at λ < 0 transform into sented, for example, by two bulk acoustic phonons. In the electron level with the indices n = 1 and l. In this the framework of the same model, the reciprocal nonra- Ӎ 2 λ2 diative lifetime of the polaron can be deduced in the fol- case, according to formula (88), we have Q1p A / . lowing form [25]: As a result, from expression (93), we obtain π 2 2 2πQ1 p 2 2 γ Ӎ 2 A U()q , ϕ 2δ()បΩ γ = ------U()q W()q, ϕ p ------∑------W()q e Ð Eq, ϕ . (100) p ប 2 ∑ ប λ2 2 , ϕ A A q, ϕ (93) q 2 2 γ ×δ() λ ± ()λ បω This quantity cannot be expressed through LO but, /2 /2 + A + LO Ð Eq, ϕ . compared to it, includes the small multiplier A2/λ2. It is Here, the upper (lower) sign corresponds to the polaron worth noting that the argument of the δ function in for- state p = a (p = b), the quantity Q1p is given by relation- mula (100) corresponds to the transition from the elec- ship (88), and the function U(q) was derived in [22] in tron level n = 1 to the level n = 0 with the emission, for the form γ example, of two acoustic phonons. The quantity p [formula (100)] is determined to the second order in the ᏹ Ꮿ ᏷* U()q = ()qz q 10(),q⊥ (94) interaction of electrons with longitudinal optical where phonons and to the second order in the phononÐphonon interaction. This quantity is one of the contributions to ∞ the reciprocal nonradiative lifetime of the electronÐ iq z ᏹ()q = dz[]ϕ ()z 2e z , hole pair far from the magnetophonon resonance. Two z ∫ l λ ∞ other branches of the terms, namely, a at < 0 and b at Ð λ > 0 (|λ| ӷ |A|), correspond to the state of an electron 1/2 Ꮿ = Ð()iបω /ql ()4παl3/V , at the level n = 0, l + one longitudinal optical phonon. q LO 0 Ӎ For these branches, we have Q1p 1 and []ប ()ω 1/2 α 2()εÐ1 εÐ1 ()បω l ==/2me LO , e ∞ Ð 0 /2 LOl , γ Ӎ γ p LO, (101) ε ε 0 (∞) is the static (high-frequency) permittivity, ᏷ as should be observed for the state including the longi- 10 (q⊥) is defined by expression (68), and tudinal optical phonon. Therefore, formula (93) cor- |λ| ӷ | | 2 2 rectly describes the limiting cases at A . A = ∑ U()q . (95) γ Thus, the reciprocal nonradiative lifetime p of the q electronÐhole pair sharply increases in the vicinity of In formulas (93)Ð(95), we used the designations given the intersection between the electronic terms and in [22]. From a comparison of relationships (92) and reaches half the reciprocal lifetime of the longitudinal Ω ω (93), we find that, in the vicinity of the polaron reso- optical phonon. At resonance e = LO, the reciprocal nance, i.e., under the condition nonradiative lifetime for each term is no less than γ λ ± ()λ 2 2 Ӷ បω LO /2. /2 /2 + A LO, (96) γ γ the reciprocal time p is expressed through q. As a result, we obtain 7. RESULTS OF NUMERICAL CALCULATIONS γ γ Figures 1 and 2 depict the dependences of the reflec- p = Q1 p LO, (97) ᏾ ω Ꮽ ω tance ( l) and absorptance ( l) for the three-level γ γ γ 2 2 system at different ratios between the parameters r1(2), LO = ∑ q U()q /∑ U()q . (98) γ ω ω 1(2), and 1 Ð 2. These results should be used in the q q case of any two excited, rather closely spaced levels in γ Ӷ ω ω γ Ӷ ω Owing to the multiplier Q1p in expression (97), the the quantum well. At r1(2) 1 Ð 2 and 1(2) 1 Ð γ ω reciprocal lifetime p strongly depends on the quantity 2, the results for two two-level systems are applicable.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2193 ᏾ ω Ꮽ ω The dependences ( l) and ( l) for the three- 0.008 ᏾ (a) γ γ γ γ γ γ γ ӷ γ level system at r1 = r2 = r, 1 = 2 = , and r are γ ω ω depicted in Fig. 1. Curves 1 are obtained at r < 1 Ð 2. γ ω ω Curves 2 are constructed at r = 1 Ð 2. Curve 3 in γ ω 0.006 Fig. 1a and curves 3Ð6 in Fig. 1b are plotted at r > 1 Ð ω 2. Curves 1 correspond to the dependences for two two-level systems. The other curves exhibit a specific ᏾ ω 0.004 behavior. In particular, the reflectance ( 0) at the ω ω point l = 0 in all three curves in Fig. 1a is close to zero. 1 0.002 As for two-level systems, the reflectance and 1 γ Ӷ γ absorptance at r1(2) 1(2) are considerably less than 2 2 ᏾ ω Ӷ Ꮽ ω γ ӷ γ 3 3 unity and ( l) ( l). At r1(2) 1(2), the absorptance reaches values close to unity. ᏾ ω Ꮽ ω Ꮽ (b) Figure 2 shows the dependences ( l) and ( l) γ γ γ γ γ γ 0.15 for the three-level system at r1 = r2 = r, 1 = 2 = , and γ γ γ ω ω r = . Curves 1 are obtained at r < 1 Ð 2, curves 2 γ ω ω are constructed at r = 1 Ð 2, and curves 3 are plotted γ ω ω at r > 1 Ð 2. It can be seen that the absorptance 0.10 reaches a maximum but never exceeds 0.5. 1 Below, the formulas obtained for the coefficients 1 2, 3 Ꮽ ω ᏾ ω ( l) and ( l) in the case of the three-level system in the quantum well will be used for the system comprised 0.05 2 of a magnetopolaron and a hole with the indices nh = 1 3 and lh = l. The energy levels of this system are determined by expressions (83) and (84). If the energy of the system is reckoned from the level –0.5 0 0.5 εe εb ()()បω λ/∆E E0 = Eg +++l l 3/2 1 + me/mh LO, relationship (83) is conveniently rewritten in the form Fig. 4. Dependences of the dimensionless (a) reflectance ᏾ and (b) absorptance Ꮽ on the magnetic field in the vicinity of Ᏹ λ λ 2 Ꮽ បω p 3me ± 1  the resonance for polaron . me/mh = 0.2, LO = 0.036 eV, ∆------= ∆------1 + ------1 +,∆------(102) ∆ × Ð3 បγ × Ð5 γ γ E E 2mh 2 E E = 6.65 10 eV [24], 0 = 5.35 10 eV, LO / 0 = បω បω ∆ បω 10. (1) l = E0, (2) l = E0 + E/2, and (3) l = E0 Ð where λ = ប(Ω Ð ω ), ∆E is the splitting of the terms ∆ e LO E/2. E0 is determined in Section 7. The given parameters at λ = 0, and the upper (lower) sign corresponds to the correspond to GaAs. term p = a (p = b). With the use of relationships (89) and (87), we obtain 0.18.5 According to the formulas derived in Section 4, γ 1λ/∆E λ/∆E ------rp = --- 1 + ------1 ± ------, (103) we obtain the relationship γ 2បω ∆ 2 0 LO/ E 1 + ()λ/∆E 2e2ω m p2 γ γ = ------LO e ------cv π2 , (105) where the constant 0 is the reciprocal radiative lifetime 0 ប ll cnm0 m0Eg of the electronÐhole pair with the indices ne = nh = 1 and λ le = lh = l at = 0 without regard for the polaron effect. which, with the use of the parameters taken from [21], The multiplier in the parentheses describes the depen- gives the estimate Ω dence of the quantity 0 entering into the expression for γ on the magnetic field. By using relationships (89) បγ Ӎ × Ð5 rp 0 5.35 10 eV. (106) and (97), the ratio between the nonradiative and radia- γ tive reciprocal lifetimes can be estimated as Thus, the reciprocal lifetime rp can be estimated (as γ was done above), whereas the reciprocal lifetime p γ γ /γ 1 + ()λ/∆E 2 +− ()λ/∆E ------p = ------LO 0 ------. (104) γ 1 + ()λ/បω 2 5 rp LO 1 + ()λ/∆E ± ()λ/∆E According to [24], the splitting of the terms is estimated to be ∆E Ӎ 6.65 × 10Ð3 eV for polaron A in the quantum well 300 Å បω When plotting Figs. 3 and 4, we used the following thick in GaAs. Since LO = 0.0367 eV, we obtain the ratio ∆ បω ∆ បω Ӎ parameters for GaAs: me/mh = 0.2 and E/ LO = E/ LO 0.181.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2194 LANG et al. 6 γ γ Ӷ remains unknown. For this reason, Figs. 3 and 4 resonance. In this range, we have p/ rp 1 and the γ γ present the results obtained at different ratios LO / 0. reflectance is larger than the absorptance. Figure 3 shows the dependences of the reciprocal In the other limiting case γ /γ = 0.1 (curve 3), the γ γ γ LO 0 radiative lifetime rp and the ratio p/ rp on the magnetic γ γ Ӷ inequality p/ rp 1 is satisfied virtually over the entire field in the vicinity of the magnetophonon resonance. range of the magnetophonon resonance. This range The dependences for the terms a and b are plotted in involves the resonance point λ = 0. The absorptance is Figs. 3a and 3b, respectively. The dependences of the γ γ λ ∆ appreciably less than the reflectance, which is as large ratio p/ rp on / E are constructed for three parameters ᏾ γ γ as = 1 at the points of both maxima in the curve LO / 0. As can be seen from Fig. 3, the reciprocal life- ᏾(ω ). At λ /∆E = Ð1.24 and λ /∆E = 1.68, the radia- γ γ l a3 b3 times rp and p in the range of the magnetophonon res- tive and nonradiative reciprocal lifetimes become equal onance very strongly depend on the magnetic field. to each other, which corresponds to the maximum The magnetopolaron effect is observed in the range absorptance. The term a does not interact with the light λ approximately from λ/∆E = Ð2 to λ/∆E = 2. The term a to the left of the point a3, as judged from the very small λ ∆ λ ∆ γ at / E > 2 and the term b at / E < Ð2 transform into values of ra. The same holds true for the term b to the λ the energy level of the electronÐhole pair with the quan- right of the point b3. Curves 2 in Fig. 3 correspond to tum numbers of the electron and hole ne = nh = 1 and γ γ λ ∆ the intermediate case LO / 0 = 1. The maximum le = lh = l. The term b at / E > 2 and the term a at λ ∆ absorptance is observed at the resonance point λ = 0. / E < Ð2 transform into the energy level of the system γ γ λ For the term a, the condition a/ ra < 1 is fulfilled at > composed of an electronÐhole pair (with the quantum γ γ λ 0 and the condition a/ ra > 1 is met at < 0. The oppo- numbers ne = 0, nh = 1, and le = lh = l) and one phonon. This state involves the longitudinal optical phonon and site situation occurs for the term b. The reflectance is γ γ weakly interacts with the exciting light outside the predominant at p/ rp < 1, and the absorptance is domi- γ γ range of the magnetophonon resonance. nant at p/ rp > 1. The energy splitting ∆E is very large in the case of The calculated dependences ᏾(H) and Ꮽ(H) in the γ Ӷ ω ω polaron A. Therefore, the conditions rp 1 Ð 2 and vicinity of the magnetophonon resonance are displayed γ Ӷ ω ω ᏾ Ꮽ p 1 Ð 2 are satisfied and the coefficients and in Fig. 4. Curves 1Ð3 are constructed at three different ω can be calculated in the approximation of two two-level frequencies l with the use of the parameters for GaAs systems [see formulas (36) and (37) in Section 3]. The at the ratio γ /γ = 10. expressions derived for the three-level system in Sec- LO 0 tion 3 should be used for the small splittings ∆E, which can be observed for weakly coupled polarons [24, 25]. 8. CONCLUSIONS In the range of the magnetopolaron effect, the func- ᏾ ω Ꮽ ω tions ( l) and ( l) should have two maxima at a The main results obtained in the present work can be ω constant magnetic field H. At l = const, two maxima summarized as follows. The reflection and absorption should be observed in the functions ᏾(H) and Ꮽ(H). of light by a quantum well was analyzed in the case of three-level electronic systems. The relationships for the The coefficients ᏾(ω ) and Ꮽ(ω ) depend on the l l dimensionless reflectance and absorptance were γ γ λ ∆ ratio LO / 0. In the range of the ratios / E (i.e., the derived upon normal incidence of light on the quantum γ γ ӷ fields H) at which p/ rp 1, the absorptance is substan- well plane. The calculations were performed with due tially less than unity but exceeds the reflectance. For regard for the radiative lifetimes of two excited levels in γ γ curves 1 plotted in Fig. 3 at the ratio LO / 0 = 10, this the electronic system. The radiative lifetime of the elec- range for both terms (a and b) covers almost the entire tronÐhole pair in the quantum well in a strong magnetic magnetic field range in which the polaron effect is pro- field was calculated far from the magnetophonon reso- γ γ ⊥ nounced. Actually, in this case, we have a/ ra = 1 at nance at an arbitrary wave vector K of the pair in the λ ∆ γ γ λ ∆ a1/ E = 1.24 and b/ rb = 1 at b1/ E = Ð1.82. In the quantum well plane. The radiative lifetime of the sys- λ λ tem composed of a magnetopolaron and a hole in the vicinity of the points a1 and b1, the absorption by the terms a and b reaches a maximum (Ꮽ = 1/2 and ᏾ = valence band was determined. The expressions for the λ ӷ λ λ Ӷ λ contributions associated with finite lifetimes of longitu- 1/4 at a maximum). In the ranges a1 and b1, the polaron effect is of no significance. The second dinal optical phonons to the nonradiative lifetimes of ᏾ ω Ꮽ ω the polaron states were deduced. It was found that the maxima in the curves ( l) and ( l) disappear. The results obtained are similar to those for the pair with the radiative and nonradiative lifetimes of the polarons indices n = n = 1 and l = l = l far from the polaron very strongly depend on the magnetic field H in the e h e h vicinity of the magnetophonon resonance. The func- 6 γ tions Ꮽ(H) and ᏾(H) were determined taking into In principle, the quantity rp can be estimated from the line width of one-phonon scattering in bulk GaAs. account the magnetopolaron effect.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 A MANIFESTATION OF THE MAGNETOPOLARON EFFECT 2195

ACKNOWLEDGMENTS 14. L. C. Andreani, F. Tassone, and F. Bassani, Solid State We are grateful to V.M. Bañuelos for his coopera- Commun. 77 (11), 641 (1991). tion and A. D’Amore for critical remarks. 15. E. L. Ivchenko, Fiz. Tverd. Tela (Leningrad) 33 (8), 2388 (1991) [Sov. Phys. Solid State 33, 1344 (1991)]. This work was supported by the Russian Foundation for Basic Research (project no. 00-02-16904) and the 16. F. Tassone, F. Bassani, and L. C. Andreani, Phys. Rev. B International Scientiﬁc and Technical Program “Phys- 45 (11), 6023 (1992). ics of Solid-State Nanostructures” (project no. 97- 17. A. V. Kavokin, M. R. Vladimirova, L. C. Andreani, and 1099). S.T. Pavlov acknowledges the support and hos- G. Panzarini, in Proceedings of the International Sympo- pitality of Zacatecas University and the Consejo Nacio- sium “Nanostructures: Physics and Technology 97,” Ioffe Physicotechnical Institute, Russian Academy of nal de Ciencia y Tecnologœa (CONACyT). D.A. Contr- Sciences, St. Petersburg, 1997, p. 31. eras-Solorio acknowledges the support of the CONA- CyT (27736-E). 18. L. C. Andreani, G. Panzarini, A. V. Kavokin, and M. R. Vladimirova, Phys. Rev. B 57 (8), 4670 (1998). 19. L. I. Korovin, I. G. Lang, and S. T. Pavlov, Zh. Éksp. REFERENCES Teor. Fiz. 118 (2), 388 (2000) [JETP 91, 338 (2000)]. 1. E. J. Johnson and D. M. Larsen, Phys. Rev. Lett. 16 (15), 20. I. V. Lerner and Yu. E. Lozovik, Zh. Éksp. Teor. Fiz. 78 655 (1966). (3), 1167 (1980) [Sov. Phys. JETP 51, 588 (1980)]. 2. A. Petrou and B. D. McCombe, in Landau Level Spec- 21. A. Garsia-Cristobal, A. Cantarero, and C. Trallero- troscopy, Ed. by G. Landwer and E. I. Rashba (North- Giner, Phys. Rev. B 49, 13430 (1994). Holland, Amsterdam, 1991), Modern Problems in Con- 22. I. G. Lang, V. I. Belitsky, A. Cantarero, et al., Phys. Rev. densed Matter Sciences, Vol. 27.2. B 54 (24), 17768 (1996). 3. R. J. Nicholas, D. J. Barnes, D. R. Leadley, et al., in Spectroscopy of Semiconductors Microstructures, Ed. by 23. N. Mori and T. Ando, Phys. Rev. B 40 (9), 6175 (1989). G. Fasol, A. Fasolino, and P. Lugli (Plenum, New York, 24. L. I. Korovin, I. G. Lang, and S. T. Pavlov, Zh. Éksp. 1990), NATO ASI Ser., Ser. B: Phys., Vol. 206, p. 451. Teor. Fiz. 115 (1), 187 (1999) [JETP 88, 105 (1999)]. 4. P. J. Nicholas, in Handbook of Semiconductors, Ed. by 25. I. G. Lang, L. I. Korovin, A. Cantarero, and S. T. Pavlov, M. Balkanski (North-Holland, Amsterdam, 1994, 2nd cond-mat/0001248. ed.), Vol. 2. 26. D. A. Contreras-Solorio, S. T. Pavlov, L. I. Korovin, and 5. L. I. Korovin and S. T. Pavlov, Zh. Éksp. Teor. Fiz. 53 I. G. Lang, Phys. Rev. B 62 (24), 16815 (2000); cond- (5), 1708 (1967) [Sov. Phys. JETP 26, 979 (1968)]; mat/0002229. Pis’ma Zh. Éksp. Teor. Fiz. 6 (2), 525 (1967) [JETP Lett. 6, 50 (1967)]. 27. I. G. Lang and V. I. Belitsky, Solid State Commun. 107 (10), 577 (1998). 6. L. I. Korovin, S. T. Pavlov, and B. É. Éshpulatov, Fiz. Tverd. Tela (Leningrad) 20 (12), 3594 (1978) [Sov. 28. I. G. Lang and V. I. Belitsky, Phys. Lett. A 245 (3Ð4), 329 Phys. Solid State 20, 2077 (1978)]. (1998). 7. S. Das Sarma and I. Madhukar, Phys. Rev. B 22, 2823 29. I. G. Lang, L. I. Korovin, D. A. Contreras-Solorio, and (1980). S. T. Pavlov, Fiz. Tverd. Tela (St. Petersburg) 43 (6), 8. S. Das Sarma, Phys. Rev. Lett. 53, 859 (1984); 52, 1570 1117 (2001) [Phys. Solid State 43, 1159 (2001)]; cond- (1984). mat/0004178. 9. G. Q. Hai, T. M. Peeters, and J. T. Devreese, Phys. Rev. 30. A. G. Zhilich and S. G. Monozon, Magnetoabsorption B 47 (16), 10358 (1993). and Electroabsorption of Light in Semiconductors (Len- 10. H. Fröhlich, Adv. Phys. 3 (2), 325 (1954). ingr. Gos. Univ., Leningrad, 1984), p. 204. 11. L. I. Korovin and B. É. Éshpulatov, Fiz. Tverd. Tela 31. V. I. Belitsky, M. Cardona, I. G. Lang, and S. T. Pavlov, (Leningrad) 21 (12), 3703 (1979) [Sov. Phys. Solid State Phys. Rev. B 46 (24), 15767 (1992). 21, 2137 (1979)]. 32. I. G. Lang, S. T. Pavlov, and A. V. Prokhorov, Zh. Éksp. 12. L. C. Andreani, in Conﬁned Electrons and Photons, Ed. Teor. Fiz. 106 (1), 244 (1994) [JETP 79, 133 (1994)]. by E. Burstein and C. Weisbuch (Plenum, New York, 33. I. G. Lang, A. V. Prokhorov, V. I. Belitsky, et al., J. Phys.: 1995), p. 57. Condens. Matter 8, 6769 (1996). 13. L. I. Korovin, I. G. Lang, and S. T. Pavlov, Pis’ma Zh. Éksp. Teor. Fiz. 65 (7), 511 (1997) [JETP Lett. 65, 532 (1997)]. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2196Ð2199. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2097Ð2100. Original Russian Text Copyright © 2002 by Meœlanov, Abramova, Gadzhialiev, Dzhabrailov. LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Specific Features of Chemisorption on a Size-Quantized Film in an External Quantizing Magnetic Field R. P. Meœlanov, B. A. Abramova, M. M. Gadzhialiev, and V. V. Dzhabrailov Institute of Geothermal Problems, Dagestan Scientiﬁc Center, Russian Academy of Sciences, pr. Kalinina 39a, Makhachkala, 367030 Russia e-mail: [email protected] Received November 22, 2001; in ﬁnal form, February 18, 2002

Abstract—The energy of an electron of an adatom chemisorbed on a size-quantized film is investigated as a function of the external quantizing magnetic field. Consideration is given to the cases when the external magnetic field is directed parallel and transverse to the film surface. It is demonstrated that, as the magnetic field increases, the energy of chemisorption jumpwise decreases. © 2002 MAIK “Nauka/Interperiodica”.

1. The specific features of chemisorption on size- and the generalization of the AndersonÐNewns model quantized films were investigated in [1Ð3]. It has been within the KadanoffÐBaym formalism are presented demonstrated that the chemisorption energy is an oscil- in [13, 14]. lating function of the film thickness due to specific fea- In the present work, based on the results obtained in tures of the energy spectrum and the density of states of [14], we analyzed the influence of an external magnetic the electrons of a thin film. The chemical bonding field on the energy of an electron of an adatom chemi- between an atom and the surface of a crystal has been sorbed on a thin film. We considered two cases: (i) the studied within the AndersonÐNewns approach with the external quantizing magnetic field is directed trans- use of model Hamiltonians [4Ð6]. According to the versely to the film surface, and (ii) the external quantiz- AndersonÐNewns model, the renormalized energy of ing magnetic field is aligned parallel to the film surface. an electron of an adatom can be represented by the rela- ε 〈 〉 ε 2. The equations of motion for the Green’s functions tionship Ea, s = a + U ns . Here, a is the energy of an of an “adatom + thin film” electronic subsystem in the electron of an isolated atom; U is the potential of the stationary case have the form [14] 〈 〉 intraatomic Coulomb repulsion; and ns is the perturbation of the electron density of the atom due to the []ωεω < ω {σ< ω A ω interaction with the substrate, which is determined by Ð λs()gλs, λ's'()Ð ∑ λs, λ''s'()gλ''s'', λ's'() ∞ 〈 〉 ω π < ()ω < λ''s'' the expression ns = ∫ ∞ d / gs , where gs is the Ð σR ()ω < ()}ω + λs, λ''s'' gλ''s'', λ's' = 0, correlation function of the electron of the adatom and s (1) is the spin quantum number. The energy attenuation at RA, []ωεÐ λ ()ω gλ , λ ()ω the electron energy level of the adatom can be written s s 's' in the form ∆ ≈ |V |2ρ, where ρ(W) is the density of {σRA, ω RA, ω } δ δ states of the electrons of the substrate (W is the conduc- Ð ∑ λs, λ''s''()gλ''s'', λ's'() = λλ' ss'. tion band width of the electrons of the film) and V is the λ''s'' potential of hybridization of the seed electron energy states of the adatom and the substrate. Here, λ stands for the quantum numbers characterizing ε ω the motion of electrons of an adatom, λ, s( ) is the At present, different versions of the AndersonÐ < Newns method with the use of model Hamiltonians energy of the adatom, gs is the correlation function of enjoy wide application. In particular, the charge an electron of the adatom, gR, A are the retarded and exchange in the course of interaction of atomic parti- advanced Green’s functions of the electron of the ada- < cles with the surface of a crystal was investigated in tom, σR, A are the corresponding mass operators, and σ the framework of a nonstationary model developed in is the generalized collision integral. The mass operator [7Ð11]. The distribution of the electron density in a , , σR, A σRA σRA σ monoatomic adsorbed layer was examined in [12]. consists of two parts: = 0 + a , where 0 is σ The derivation of quantum kinetic equations for a the mass operator of an isolated atom and a is the con- “crystal + adatom” system on the microscopic level tribution to the mass operator of the electron of the ada-

SPECIFIC FEATURES OF CHEMISORPTION 2197 tom due to the interaction with a thin film. This contri- Then, we obtain the expression for the spectral func- bution has the form tion: σ< ()RA, ()ω Γ ()ω a,,λs λ's' A ()ω = z ------iks -, (6) iks iks()ω 2 2 Γ2 ()ω < Ð Eiks + ziks iks ()ω ()RA, ()ω ()ω (2) = ∑ V λsi, ks'' Giks''; i'k's''' V i'k's''', λ's' , iks'' where i'k's''' ∂ Ð1 ()ε ()ω Σ ()ω ziks = 1 Ð ------iks + Re iks ω where G is the Green’s function of an electron of the ∂ω = Eiks film and i, k is the set of quantum numbers describing the motion of the electron in the film. is the renormalization constant. From expressions (1)Ð The equations of motion for the Green’s function of (6), we obtain the correlation function of the electron of an electron of the film are similar to Eqs. (1) and (2) the adatom in the following form: with appropriate substitution for the set of quantum numbers describing the motion of the electron. < ()ω 2 ()ω ()ω gs = V f as , (7) The system of equations (1) and (2), together with the equations of motion for the Green’s functions of the where the spectral function of the electron of the ada- electrons of the film, describes the general case with tom has the form allowance made for all possible interactions of adatoms ()ω with one another and with the substrate. Let us consider as the case when the interaction between adatoms is absent. Moreover, we assume that an adatom has a sin- ∑ A ()ω gle energy level (in what follows, we will drop the iks λ ik (8) index ). The correlation function of an electron of the = ------2. adatom, according to relationships (1), takes the fol-  []ωε()ω Λ ()ω 2 4 ()ω lowing form (we assume that σ is taken into account in Ð as Ð as + V ∑ Aiks 0  the determination of the energy of the electron of the ik atom in the absence of the interaction with the film): Here, < ()ω R()σω < ()ω A()ω gs = gs as, gs . (3) Λ ()ω as Under the assumption that the hybridization potential is < 2 ωεÐRe()ω Ð Σ ()ω constant, it can easily be shown that σ (ω) satisfies the = V ∑------iks iks -. ()ωε()ω Σ ()ω 2 Γ2 ()ω following expression: ik ÐReiks Ð iks + iks

< σ< ()ω 2 ()ω When analyzing the expressions for the electron energy as, = V ∑Giks . (4) of the adatom, the self-consistent energy is introduced ik ε ω according to the definition Eas = ( as( ) + Λ (ω))|ω , which, in turn, can be parametrized in For the correlation function of the electron of the film, as = Eas we obtain ε 〈 〉 the usual manner: Eas = a + U nÐs . < G ()ω = f ()ω A ()ω , (5) Further consideration depends on the concrete iks iks expression for the electronic spectrum of the thin film. where f(ω) is the FermiÐDirac distribution function This problem is considered in [1Ð3] in the absence of an and A is the spectral function, external magnetic field. Here, we analyze the case iks when an external magnetic field is present. Γ ()ω A ()ω = ------iks -. 3. Let us consider the case when the external mag- iks ()ωε()ω Σ ()ω 2 Γ2 ()ω ÐReiks Ð iks + iks netic field is directed parallel to the film surface. The OZ axis is perpendicular to the film surface. The mag- Here, Γ is the attenuation of single-particle states and netic field is assumed to be such that the magnetic ReΣ is the renormalization of single-particle states due Ð1/2 length lB = (h/eB) (h is the Planck constant, e is the to the interaction (the Coulomb and crystal potentials). elementary charge, and B is the magnetic field induc- The self-consistent energy of the electrons of the sub- tion) is of the same order of magnitude as the film thick- strate can be determined according to the expression ness L. Under the assumption that the film potential can ε ()ω Σ ()ω ω2 2 Eiks = iks = Eiks + Re iks = Eiks . be approximated by the expression V(z) = mz0 /2, we

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2198 MEŒLANOV et al.

〈n〉 〈n〉 0.200 0.2 0.174 0.134 0.100 0.1 0.080

0 20 60 120 0 1.1 1.9 3.5 L(B)/L0 B(L)/B0

Fig. 1. Dependence of 〈n〉 on the dimensionless thickness Fig. 2. Dependence of 〈n〉 on the dimensionless magnetic × × L(B)/L0 of the film in a constant magnetic field. B = 4 field B(L)/B0 at a constant thickness L of the film. B0 = 4 3 Ð7 3 Ð7 10 T, L0 = 10 cm. 10 T, L = 25L0, L0 = 10 cm. can exactly solve the Schrödinger equation and obtain For numerical calculations, it is necessary to estab- the following expression for the electron energy: lish a relationship between the chemical potential, the film thickness, and the magnetic field. Within the above 2 ω2 2 Γ px 0 py 1 approximation ( s 0), the number of electrons in E ,,,σ = ------++------ω˜ n + --- , px py n 2m ω˜ 2 2m 2 the film can be determined from the expression mL L ω˜ ω˜ where ω˜ = (ω2 + ω2 )1/2 and n = 0, 1, 2, … (ω = eB/m N = ------x -y ------()µn + 1 Ð,----()n + 1 (10) c 0 c π 2 ω F 2 F is the cyclotron frequency). h 0 Consequently, the spectral function takes the form where µ is the chemical potential. The condition of filling of the nth discrete state has the form µ(n) = ω˜ (n + Aσ()ω 1/2). From expression (10), we determine the chemical potential µ and, then, obtain the following condition: 1 n ωω˜ F Ð n + --- 2 zΠmL L ω˜ 2 π π ω x y  ω˜ 2 2 h N 0 = ------π ------ω- ∑ arctan ------Γ - + --- , = ------()-. (11) h 0 zΠ s 2 mLx Ly nn+ 1 n = 1  According to formula (11), we can determine the film ≡ zΠ ziks. thickness Ln at which the nth energy level is filled in a Γ 〈 〉 constant magnetic field B: L (B) = L (1 + B2/)B2 n(n + Finally, in the approximation s 0, ns can be rep- n 0 0 Ð1/3 resented by the following expression: 1), where L0 = (N/V0) , V0 is the volume of the system, and B = 2πh/eL 2 = 4.12 × 103 T. It is assumed that 3 0 0 n ω˜ n + --- Ð ε Ð Un〈〉 F 2 a Ðs ω = 2πh/mL2 . In the case when the film thickness L is 〈〉 za 0 0 ns = ---- ∑ arctan------π nρ|| constant, the magnetic field Bn corresponding to filling n = 1 of the nth energy level is determined by the relationship (9) B2()L /B2 + 1 = (L/L )/n(n + 1). As the film thickness 1 n 0 0 ω˜ n + --- Ð ε Ð Un〈〉 increases and becomes equal to L the number of dis- 2 a Ðs n Ð arctan------, crete states below the Fermi level also increases and nρ|| reaches n. The reverse situation is observed with an increase in the magnetic field: when the magnetic field reaches Bn, the number of filled discrete energy states where decreases by unity.

2 ω˜ mLx Ly Figure 1 presents the results of the numerical calcu- ρ|| = z zΠ V ------, 〈 〉 a ω ប2 lation of ns as a function of the film thickness in a con- 0 〈 〉 stant magnetic field. It is evident that the value of ns Ð1 []ε ()ω Λ()ω za = 1 Ð as + as ω = E remains constant until the next discrete energy level is as filled. This is due to the approximation used for the film 〈 〉 is the renormalization constant of the adatom, and nF is potential. Upon filling of the next energy level, ns the number of discrete energy states below the Fermi jumpwise increases, thus exhibiting an oscillating level. dependence on the film thickness; consequently, there

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 SPECIFIC FEATURES OF CHEMISORPTION 2199 appears an oscillating dependence of the chemisorption 5. The experimental observation of the aforemen- energy of an adatom on the film thickness. Figure 2 tioned effects under changes in the external magnetic 〈 〉 depicts the dependence of ns on the magnetic field at field and film thickness will make it possible to deter- a specified thickness of the film. It can be seen that, as mine the specific features of the energy characteristics 〈 〉 the magnetic field increases, the value of ns continu- of an adatom and the electronic subsystem of the film. ously decreases and jumps when the number of filled The decrease in the energy of interaction between the discrete energy levels decreases. Thus, an increase in adatom and the substrate in an external magnetic field the quantizing magnetic field is accompanied by a is of interest from the viewpoint of controlled changes decrease in the chemisorption energy. in the surface properties of thin films. The condition of 4. Let us now consider the case when the magnetic observation of oscillations in a magnetic field has the field is directed transversely to the film surface. The form ω˜ ӷ T, h/τ (T is the absolute temperature and τ is energy spectrum is totally quantized and has the form the relaxation time). Note that this condition is less ω ӷ τ 1 stringent than that for bulk samples: c T, h/ . E = ω n + --- + ε . Here, ε is the energy correspond- ni c 2 i i ing to the motion of an electron across the film. The REFERENCES spectral function of the adatom electron has the form 1. R. P. Meœlanov, Fiz. Tverd. Tela (Leningrad) 31 (7), 270 (1989) [Sov. Phys. Solid State 31, 1254 (1989)]. a ()ω = V 2ρ s ⊥ 2. R. P. Meœlanov, Fiz. Tverd. Tela (Leningrad) 32 (9), 2839 1 (1990) [Sov. Phys. Solid State 32, 1649 (1990)]. δωÐ ω n + --- Ð ε c2 i 3. R. P. Meœlanov, Poverkhnost, No. 3, 52 (1999). × ------. ∑ 2 4. P. W. Anderson, Phys. Rev. 124 (1), 419 (1961). ni, Γ , 5. D. M. Newns, Phys. Rev. 178 (3), 1123 (1969). ()ω Ð E 2 + V4 ρ2 ∑------ni - a ⊥ 2 6. T. L. Einstein, J. A. Hertz, and J. R. Schrieffer, in Theory 1 2 n'i' ωω ε Γ of Chemisorption, Ed. by J. R. Smith (Springer-Verlag, Ð cn + --- Ð i + n', i' 2 Berlin, 1980; Mir, Moscow, 1983). ω 7. Y. Muda and D. M. Newns, Phys. Rev. B 37 (12), 7048 ρ m c Lx Ly (1988). Here, ⊥ = zazΠ ------. 2បπ 8. D. C. Langreth and P. Nordlander, Phys. Rev. B 43 (4), 〈 〉 2541 (1991). In this case, ns takes the following form: 9. H. Shao, D. C. Langreth, and P. Nordlander, Phys. Rev. nF  B 49 (19), 13929 (1994). 〈〉 2 ρ ω 1 ns = V za ⊥∑ cn + --- 10. M. Yu. Gusev, D. V. Klushin, S. V. Shavrov, and  2 ni, I. F. Urazgil’din, Zh. Éksp. Teor. Fiz. 109 (2), 562 (1996) (12) [JETP 82, 299 (1996)]. Ð1 2 2  11. S. Yu. Davydov, Fiz. Tverd. Tela (St. Petersburg) 42 (7), ε ε 〈〉 2ρ2 V + i Ð a Ð UnÐσ + V ⊥------ . 1331 (2000) [Phys. Solid State 42, 1370 (2000)]. Γ2  ni, 12. S. Yu. Davydov, Fiz. Tverd. Tela (St. Petersburg) 41 (9), 1543 (1999) [Phys. Solid State 41, 1413 (1999)]. When deriving expression (12), we disregarded the ≈ Γ ω 2 Ӷ 13. R. P. Meœlanov, Poverkhnost, No. 12, 28 (1990). quantities of the order of magnitude ( n, i/ c) 1. The calculations demonstrate that, qualitatively, the 14. R. P. Meœlanov, Poverkhnost, No. 6, 37 (1994). 〈 〉 dependences of ns on the film thickness and the magnetic field, in this case, remain unchanged. Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2200Ð2203. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2101Ð2105. Original Russian Text Copyright © 2002 by Makarov, Sherman.

FULLERENES AND ATOMIC CLUSTERS

Low-Frequency Dispersion of the Negative Dielectric Permittivity in C70 Films V. V. Makarov and A. B. Sherman Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received November 2, 2001

Abstract—Measurements performed at frequencies of 0.1–10 kHz on ﬁlms of C70 fullerite revealed a negative dielectric permittivity ε' < Ð1000. The largest drop in ε' occurred in the temperature interval 170Ð270 K and was accompanied by frequency dispersion. The negative sign of the dielectric permittivity and its low-frequency dispersion are accounted for by the Lorentz correction to the local ﬁeld acting on the conduction electrons. This correction originates from the polarization of the electrons bound to impurity centers. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION achieved by vacuum annealing. Therefore, the only way in which one can study a fullerite with a low oxy- C70 fullerite is a semiconductor with a band gap of gen concentration lies in making measurements imme- 1.7Ð1.8 eV [1, 2]. Thermopower measurements made ° diately after the preparation of a sample, without taking after heat treatment at 200 C in vacuum showed C70 it out of the vacuum chamber (i.e., in situ). Indeed, the films to have n-type conduction [1]. The nature of the measurements of the conductivity of C60 carried out in donor centers remains unclear. The electrical conduc- σ σ this way showed E to be less by about one half and to tivity and its activation energy E depend strongly on be higher by several orders of magnitude in comparison the oxygen content in the C70 samples. Exposure to air with the results obtained after exposure to air [5]. We leads to an increase in E from 0.52 to 0.71 eV and a have not succeeded in finding any mention on in situ σ Ð9 ε σ decrease in at room temperature from 10 to measurements of ' in fullerites or of in C70. 10−12 Ω−1 cmÐ1 [2]. This suggests that oxygen creates The purpose of this study was to investigate the compensation centers in C70 which capture electrons from the donor centers and, thus, lower the Fermi level. dielectric permittivity and electrical conductivity of C70 ε fullerite films that were not subjected to ambient air. The low-frequency dielectric permittivity ' of C70 The low-frequency (0.1Ð10 kHz) measurements per- measured at low temperatures has a constant positive formed in this investigation revealed frequency-depen- value of about 4; above 300 K, this permittivity dent negative dielectric permittivity (Fig. 1a). increases to 20, which is accompanied by frequency dispersion [3]. This growth is believed [3] to be caused by the intrinsic dipole relaxation of C70. Moreover, a 2. EXPERIMENT weak anomaly of ε' was observed near 300 K, which has been associated [3] with a structural phase transi- The films used in our experiment were prepared in tion. The effect of oxygen on the dielectric permittivity vacuum at 10Ð5 Torr. The measurements were con- of C70 has not been studied. That this effect may be ducted in situ in the vacuum chamber immediately after quite strong is shown by measurements performed on deposition of C70 on a mica substrate. The 99%-pure samples of C60 fullerite, which is close in electrical C powder was produced by the company MER. The ε 70 properties to C70. For instance, ' of C60 measured at characteristic sublimation temperature was 750 K. 100 Hz can vary by up to a factor of five depending on Prior to this deposition, planar interdigitated electrodes the exposure time in air at room temperature [4]. Here, were deposited on the substrate. The substrate temper- we also note that all of the above data were obtained on ature during the film deposition was 470 K. X-ray samples that were subjected to ambient air after their structural characterization performed at 300 K showed preparation. At the same time, annealing in vacuum was that the films were single phase and had an hcp struc- shown not to restore completely the properties of sam- ture and cell parameters a = 10.593 Å and c = 17.262 Å. ples that underwent oxidation [5]. This is particularly The ac electric field strength between the electrodes valid in the case of C70, where this is believed [2] to be was 20 V/cm. The measurements were carried out in the reason why no one has succeeded in reaching values the temperature region 100Ð500 K with a rate of tem- of E less than 0.5 eV, whereas in C60, E = 0.2 eV was perature variation of 2 K/min.

LOW-FREQUENCY DISPERSION 2201

The dielectric permittivity ε' and the electrical con- σ 0.1 kHz ductivity of the film were calculated using the tech- 0 nique proposed in [6]. The film capacitance was found 1 kHz 10 kHz by subtracting the capacitance of the substrate (mea- σ F( c) sured before the film deposition) from the total capaci- –400 F(σ + 7σ ) tance measured after the deposition. c LT ' As already mentioned, the most interesting finding ε made in the measurements was the discovery of a high –800 negative dielectric permittivity and of its frequency dispersion in the low-frequency range (Fig. 1a). At low temperatures, ε' is positive, but as the temperature is –1200 increased, a fast drop in ε' is observed, accompanied by (a) a reversal of its sign and frequency dispersion. As the temperature is increased further, ε' saturates and the 100 200 300 400 dispersion disappears. In this stage, ε' becomes less T, K than Ð103. 10–1 The temperature and frequency dependences of σ of 0.1 kHz –2 1 kHz the C70 film (Fig. 1b) exhibit a typically semiconductor 10 10 kHz σ σ character. At low temperatures, is only weakly tem- F( c) ω0.9 σ σ perature-dependent and grows proportionally to 10–3 F( c + 7 LT) ω σ × Ð10ω0.9 ( is the circular frequency): LT = 9 10 . This σ –1 behavior of is characteristic of impurity conduction 10–4

[7]. In the high-temperature domain, σ is frequency- , Sm independent and, hence, coincides with the dc con- σ –5 σ σ σ 2 τ 10 ductivity c: HT = c = e0 n e/m*, where e0 is the electronic charge, τ is the electron mean free time, and e 10–6 m* is the effective electron mass. The conduction electron concentration n in the free-electron model for (b) parabolic bands is given by the relation n = 10–7 2 4 6 8 10 (1/4)π−1.5(2ប−2m*kT)3/2exp(ÐE/kT), where E is the acti- 1000/T, K–1 vation energy, which, as derived from the slope of the logσ(1000/T) plot, is 0.33 eV. The room-temperature Fig. 1. (a) Dielectric permittivity ε' and (b) electrical con- σ conductivity is 1.5 × 10Ð2 S/m. Thus, E measured in situ ductivity of a C70 film measured at the three frequencies specified in the graphs. The lines are plots of the calcula- in films is indeed substantially smaller than E measured tions made for different dependences of the Lorentz factor [2] after exposure to air and the conductivity is several F on conductivity. orders of magnitude higher. The total electrical conduc- σ σ σ tivity is the sum = LT + HT. Now, one can find the temperature dependence of 0 the mean free time. This quantity was found to scale τ × Ð13 Ð3/2 with temperature as e = 2.88 10 T . Such a dependence is typical of electron scattering from optical phonons. –500 The oxidation of films substantially affects the mag- ' nitude and sign of their dielectric permittivity and the ε –5 character of its temperature dependence. Figure 2 dis- 10 Torr plays temperature dependences of the dielectric permit- –3 –1000 10 Torr tivity of a film measured after its exposure to air at dif- –2 ferent pressures for 12 h. As the air pressure in the 10 Torr chamber increases, ε' of the film decreases in absolute 10–1 Torr ε value. After air is admitted to atmospheric pressure, ' 760 Torr becomes positive, as was observed earlier with the C70 –1500 fullerite acted upon by air [3]. 100 200 300 400 T, K As the film undergoes oxidation, its high-temperature conductivity decreases and the activation energy ε Fig. 2. Dielectric permittivity ' of a C70 film measured at increases. As a result, the boundary of high-tempera- 1 kHz after its successive exposure to air at different pres- ture conductivity shifts toward higher temperatures. sures.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2202 MAKAROV, SHERMAN

3. DISCUSSION approximately the same way and be frequency-independent. Because, as follows from experiment, this is Let us consider the conditions conducive to the ω ≠ onset of a negative dielectric permittivity. The relative not the case, we assume that F 0. In view of the fact complex dielectric permittivity ε* is given by the that the effect of the Lorentz correction on ε' becomes ω2 ω2 ω ω DrudeÐLorentz relation significant for F > , we replace eff with F; in this ε σ Ne2/()ε m case, we find the following expression relating ' to : ε ε ------0 * = ∞ + 2 2 2 , (1) σ σ ω ω ω ω τ F c HT 0 Ð F Ð + i / ε' = Ð------. (4) ε2 ω2 where ε∞ is the relative dielectric permittivity for ω 0 ∞, N is the concentration of charge carriers contributing As pointed out earlier, at high temperatures, we have ε ε to * Ð ∞, e is the charge of a carrier, m is the carrier σ = σ . Therefore, Eq. (4) takes the form ε HT c mass, 0 is the absolute dielectric permittivity of vacuum, ω is the circular resonance frequency character- σ2 0 ε F c ()ε 2 τ 'HT ==Ð------Ð c'' F. (5) izing the elastic restoring force, and is the relaxation ε2 ω2 0 (decay) time. The term ω2 = Fe2N/(ε m) is the Lorentz F 0 σ σ correction characterizing the local field enhancement As follows from Eq. (2b), for the equality HT = c to ω2 Ӷ ω τ caused by charge polarization; F is the Lorentz factor hold, the condition F / should be satisfied; this depending on the symmetry of the local environment inequality can be recast in the form and the extent of carrier wave function localization. For ε Ӷ localized charges in the case of local spherical symme- c'' 1/F. (6) try, we have F = 1/3, and for free electrons, we have ω This condition imposes certain constraints on the F = 0, i.e., F = 0 [8]. Equation (1) permits one to find the real part of choice of the F factor. dielectric permittivity ε' and the electrical conductivity In the region of ε' saturation (the high-temperature σ ε ε ω ε ε = '' 0 , where '' is the imaginary part of *: domain), according to Eq. (5), the following relation should be valid: 2 ω2 e N eff 2 ε' = ε∞ + ------, (2a) ()ε'' ε mω4 ω2 τ2 FF= 0 1/ c , (7) 0 eff + / where F = const and we have ε' = ÐF . A comparison 2 2 0 0 e N ω × 3 σ = ------, (2b) with experiment shows that F0 = 1.2 10 . mτ ω4 ω2 τ2 eff + / ε As the temperature decreases, c'' drops; therefore, ω2 ω2 ω2 ω2 ε'' ε'' where eff = 0 Ð F Ð . The necessary condition the product F c ~ 1/c increases and condition (6) for the onset of negative ε' is given by the inequality ε ӷ fails, to be replaced by the inverse inequality c'' 1/F ω2 ω2 ω2 ε ε > 0 Ð F . (Fig. 3). The expression for ' assumes the form ' = − ε 2 ε Let us see how this condition can be satisfied in our ()c'' /F0; i.e., ' depends on frequency, and its absolute experiment. value drops with decreasing temperature. The calcu- As follows from a comparison of the ε'(T) and σ(T) lated values of ε' in the region of dispersion almost plots (Fig. 1), within the temperature region where one coincide with the experimental figures (Fig. 1a). Simul- σ σ σ observes the largest drop and dispersion of ε', σ is taneously, the relation HT = c transforms to HT = determined primarily by the conduction electrons. This σ3 /(ε ωF )2, and as the temperature decreases, the cal- suggests that the behavior of ε' is likewise related to the c 0 0 inertia of free electrons. In this case, the quantities e, N, culated logσ (1000/T) plots deviate from a straight line m, and τ in Eqs. (2) should be replaced by the corre- toward smaller values, i.e., have a negative curvature. sponding quantities for the conduction electrons, i.e., Negative curvature is indeed observed in the experi- τ e0, n, m*, and e. At high temperatures, where one can mental plots. It is particularly pronounced below 250 K neglect ε∞, Eqs. (2a) and (2b) yield in the graph constructed for 10 kHz (Fig. 1b). The lesser curvature of the experimental graphs in comparison ε'ε ω 2 with that of the plots of Eq. (7) suggests that in actual ------0- = τ ------eff . (3) σ e 2 ε'' 2 HT ω fact F grows more slowly than proportional to (1/c ) . In the case of free electrons, as already mentioned, ω That the growth of F with decreasing temperature 0 must be limited is obvious, because the Lorentz factor ω ω2 ω2 and F are zero and eff = Ð . Then, as follows from can in no case be larger than unity. The constraint on the ε σ Eq. (3), Ð ' and HT should depend on temperature in values of F will appear in a natural way with decreasing

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 LOW-FREQUENCY DISPERSION 2203

permittivity is observed, the Lorentz factor is negligi- 1011 ble, 10Ð2Ð10Ð11 (Fig. 3).

–1 Oxidation of a sample reduces the absolute value of 9 F 0.1 kHz ε 10 ' and, hence, the coefficient F0. Simultaneously, the 1 kHz conductivity activation energy increases and the con- 7 10 kHz ductivity itself decreases. This means that oxygen cre- –1 10 ε'' ates in C70 compensating impurity centers capturing '', F

ε electrons from the donors, a circumstance which lowers 105 the Fermi level and increases the electron activation energy. Thus, the decrease in F0 observed during C70 103 oxidation correlates with the decrease in the electron concentration at the donor centers. One may, therefore, expect that it is the polarization of these electrons that 101 is responsible for the weak polarization of the conduc- 2 3 4 5 6 7 8 9 10 tion electrons and for the Lorentz correction. 1000/T, K–1 As C70 is oxidized still further, the positive lattice Fig. 3. Temperature behavior of ε'' and of the Lorentz factor F. permittivity begins to dominate and the total dielectric permittivity becomes positive. This provides an answer to the question of why negative ε' was not observed in [3], where well-oxidized C70 fullerite samples were ε ε measured. In the present study, the dielectric permittiv- temperature if we replace c'' in Eq. (7) with '', i.e., if σ σ ity of fullerites was measured for the first time on non- we replace the high-temperature conductivity HT = c σ σ σ oxidized samples (in situ). by the total conductivity = c + LT. In this case, as one approaches the region of impurity conduction, the growth of F will slow, subsequently stopping alto- ACKNOWLEDGMENTS gether. σ σ The authors are indebted to N.F. Kartenko for the x- The replacement of c by does indeed reduce the ray characterization of the films, to Yu.A. Stotskiœ for negative curvature somewhat, but this decrease is obvi- upgrading the experimental equipment, and to ously too small to reconcile the results with experiment. V.V. Lemanov for useful discussions. σ To enhance the effect of the term LT, we multiply it by The support of the Russian State Program a coefficient k > 1: “Fullerenes and Atomic Clusters” (project no. 3-2-98) FF= ()ε ω 2/()σ + kσ 2. (8) and program “Low-Dimensional Quantum Structures” 0 0 c LT of the Presidium of the Russian Academy of Sciences is The best fit of the experimental curves with the calcu- gratefully acknowledged. lations is obtained for k = 7 (Fig. 1b). At the same time, the agreement of the calculated with experimental curves for ε' is also improved (Fig. 1a). All this means REFERENCES that in reality the effect of impurity conductivity on the 1. D. Han, H. Habuchi, and S. Nitta, Phys. Rev. B 57 (7), Lorentz factor in the transition region is considerably 3773 (1998). stronger than in the case when only the contribution of 2. H. Habuchi, S. Nitta, D. Han, and S. Nanomura, J. Appl. the impurity conductivity to the total conductivity is Phys. 87 (12), 8580 (2000). taken into account. 3. P. Mondal, P. Lunkenheimer, and A. Loidl, Z. Phys. B 99, 527 (1996). 4. CONCLUSION 4. B. Pevzner, A. F. Hebard, and M. S. Dresselhaus, Phys. Rev. B 55 (24), 16439 (1997). Thus, the temperature and frequency dependences 5. T. Asakava, M. Sasaki, T. Shiraishi, and H. Koinuma, of the negative dielectric permittivity of C70 and of the Jpn. J. Appl. Phys. 34 (4), 1958 (1995). conductivity are fitted very well by the DrudeÐLorentz 6. O. G. Vendik, S. P. Zubko, and M. A. Nikolskiœ, Zh. model, which assumes the conduction electrons to be Tekh. Fiz. 69 (4), 1 (1999) [Tech. Phys. 44, 349 (1999)]. charge carriers. These electrons cannot, however, be 7. M. Pollak and T. H. Geballe, Phys. Rev. 122 (6), 1742 considered as completely free and, in order to correctly (1961). describe the results of an experiment, one should intro- 8. J. C. Slater, Quantum Theory of Molecules and Solids, duce the Lorentz correction. At low temperatures, the Vol. 3: Insulators, Semiconductors, and Metals (McGraw- value of the Lorentz factor is determined by the impu- Hill, New York, 1967; Mir, Moscow, 1969). rity conductivity, which drops with increasing temperature because of the increasing conductivity in the conduction band. In the temperature region where negative Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

Physics of the Solid State, Vol. 44, No. 11, 2002, pp. 2204Ð2209. Translated from Fizika Tverdogo Tela, Vol. 44, No. 11, 2002, pp. 2106Ð2111. Original Russian Text Copyright © 2002 by Razbirin, Starukhin, Chugreev, Zgoda, Smirnov, Grushko, Kolesnik, Coheur, Liévin, Colin.

FULLERENES AND ATOMIC CLUSTERS

Absorption Line Spectrum of the C60Cl24 Halofullerene B. S. Razbirin*, A. N. Starukhin*, A. V. Chugreev*, A. S. Zgoda*, V. P. Smirnov**, Yu. S. Grushko***, S. G. Kolesnik***, P.-F. Coheur****, J. Liévin****, and R. Colin**** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] ** St. Petersburg State Institute of Fine Mechanics and Optics, ul. Sablinskaya 14, St. Petersburg, 197101 Russia *** Konstantinov Institute of Nuclear Physics, Russian Academy of Sciences, Gatchina, Leningrad oblast, 188350 Russia **** Université Libre de Bruxelles, Brussels, 1050 Belgium Received December 29, 2001

Abstract—Optical spectra of the C60Cl24 halofullerene in the crystalline state, as well as of C60Cl24 matrix- isolated molecules, were studied. In both cases, a rich line structure was revealed in absorption spectra in the energy region 1.5Ð3.0 eV. An energy diagram of the electronic levels of the molecule which are responsible for the observed optical transitions is proposed. The parameters of the geometrical structure of the C60Cl24 molecule were calculated under the assumption of its having Th symmetry. These data were used in a theoretical study of the embedment of the C60Cl24 molecule in a toluene crystal matrix, which leads to the formation of a ﬁne spectral structure (an analog of the Shpol’skiœ effect) observed experimentally in this study. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. PREPARATION OF C60Cl24

Halogenated fullerenes belong to the so-called syn- C60Cl24 was synthesized by chlorinating crystalline thones, typical intermediate links in the preparation of C60 in a chlorine environment under heating. The C60 a number of fullerene derivatives that have application fullerene was isolated from fullerene-containing soot potential. Therefore, investigation of the physical prop- prepared from spectroscopically pure graphite rods by erties and, in particular, the electronic structure of the standard electric-arc technique of Krätschmer [4]. halofullerenes by optical methods is currently a prob- The isolation and purification were achieved by selec- lem of primary interest. Combining fluorine, chlorine, tive recrystallization, followed by flash chromatogra- or bromine with C60 produces stable halofullerene mol- phy on activated carbon and vacuum annealing. The ecules with different stoichiometries [1Ð3] with the for- purity of the C thus obtained, as derived from high- mation of strong covalent bonds, whereas the iodof- 60 performance liquid chromatography (HPLC) data ullerenes exhibit weak van der Waals bonds of molecular iodine with the carbon cage. obtained on a C18-silica gel stationary-phase column, was 99.9%+. The chlorine was produced by oxidizing Under normal conditions, the halofullerenes are HCl in a concentrated solution of chromic acid under molecular crystals. At the same time, they can be dis- heating, dried successively over CaCl2 and P2O5, and solved in a number of organic solvents with their purified of oxygen by passing over porous charcoal at ° ° molecular structure being conserved, which permits a 750 C. The C60 was heated in the dark at 310 C in a comparative study of their optical spectra both in the chlorine flow of 15 ml/min, with the sample weight crystalline state (in fullerites) and in the molecular controlled periodically. It was found that the weight of state, in the form of a frozen gas of matrix-isolated mol- the reaction product increased, to approach asymptoti- ecules. This communication presents the results of a cally the value corresponding to the C60Cl24 formula. study of optical electronic spectra made on the least We succeeded in reaching the maximum chlorine con- studied fullerene, C Cl , in both the crystalline and 60 24 tent in the molecule, which corresponded to an atomic molecular states. It is first established that, in contrast carbon/chlorine ratio of 60 : 23.97 (Fig. 1). These data to other halofullerenes, C60Cl24 has a distinct line struc- suggest that the original product obtained under the ture in the near-IR and visible regions of the spectrum. given conditions of preparation had no molecules con- We also briefly describe the method used to synthesize taining more than 24 chlorine atoms. C60Cl24 and the results obtained in a theoretical investigation of the geometrical structure of this molecule The product was a fine-grained, yellowish-brown (assuming it to have Th symmetry). powder well soluble in organic solvents.

ABSORPTION LINE SPECTRUM OF THE C60Cl24 HALOFULLERENE 2205

24 C60Cl24 n

Cl 18 60 c a 12 b

ratio in C d 60 6 e Cl/C

0 10 20 30 40 Duration of chlorination, hours

Fig. 1. Mass of the reaction product plotted as a function of the reaction time. Fig. 2. Geometric structure of the C60Cl24 molecule.

3. THE GEOMETRIC STRUCTURE 4. OPTICAL SPECTRA OF C60Cl24 OF THE C Cl MOLECULE 60 24 Spectroscopic studies were carried out on a DFS-12 The lines seen in vibrational spectra of the C60Cl24 double-monochromator-based setup with a reverse dis- halofullerene are either Raman-or IR-active [5], which persion of 0.5 nm/mm. Spectra were measured with an means that the molecule has inversion symmetry. Thus, FÉU-79 PM tube in the photon counting mode. A there are grounds to believe that the symmetry of the C60Cl24 polycrystalline sample was a layer of powdery C60Cl24 molecule is Th, similar to that of C60Br24. In this material 100Ð200 µm thick sandwiched between two structure, all the chlorine atoms are equivalent. glass plates. The samples of matrix-isolated molecules were prepared by freezing the fullerene solution in an We calculated the geometric structure of the C60Cl24 organic solvent placed in a glass ampoule. molecule using universal force field (UFF) parameteriza- tion [6]. Then, the C60Cl24 geometry was optimized at the Figure 3 displays a transmission spectrum of poly- HartreeÐFock semiempirical AM1 (Austin Model 1) crystalline C60Cl24 measured at T = 80 K. The spectrum quantum-chemical level with the use of the Gaussian 98 is seen to consist of a large number of absorption lines code [7]. of different widths, the widths increasing toward shorter wavelengths. The observed spectrum can be This molecule has C–C bonds of five types, 72 of which are single and the remaining 18 of which are double (Fig. 2). Table 1 lists the results of the calcula- Table 1. Geometrical structure and charge distribution in tions for C60Cl24 and compares them with the figures halofullerene molecules, obtained by quantum chemical calculations obtained for C60Br24 by us by using the same method and with experimental data from [1]. The single CÐC C60Cl24 C60Br24 bonds in the C60Cl24 molecule are longer than those in C60Br24, and the double bonds are somewhat shorter. C–C bond length, Å The calculated length of the C–Cl bond (1.7507 Å) is a 1.4876 1.4828 noticeably less than that of the corresponding CÐBr b 1.5272 1.5235 bond in C Br . An AM1 analysis of the populations of 60 24 c 1.5231 1.5187 the molecular orbitals shows the electron density in the d 1.3537 1.3559 C60Cl24 molecule to be shifted toward the chlorine atoms. Each chlorine atom donates to the fullerene cage e 1.3351 1.3367 an average charge of +0.0055 |e|, and this charge is dis- CÐX bond length, Å 2 tributed primarily over the carbon atoms with sp - 1.7507 1.9425 hybridized orbitals. In contrast to C Br , where all 60 24 Charge distribution carbon atoms have an excess positive charge as a result of the strongly shifted electron density, the carbon Average charge per +0.005488|e| +0.09325|e| atoms with sp3- and sp2-hybridized orbitals in C Cl halogen atom 60 24 | | | | are short of positive charge or have an excess of it, Total charge trans- +0.1317 e +2.238 e respectively. ferred to carbon cage

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002

2206 RAZBIRIN et al. divided, in a ﬁrst approximation, into three groups of lines. We denote them by A, B, and C. The long-wavelength group (A) is characterized by the smallest line- ≈ B transitions width (the FWHM is 10 meV) and is conﬁned to the energy interval 1.5Ð1.8 eV. The second group (B) is observed in the energy interval 2.3Ð2.5 eV. The line halfwidth is ≈20 meV. The third group (C) consists of A transitions C transitions two bands of halfwidth ≈50 meV. Within each group, ×5 one can isolate lines due to purely electronic transitions, as well as lines associated with their vibrational

Absorption, arb. units replicas. Table 2 lists the energies of all the line series observed (A, B, C) and their tentative assignment. The table speciﬁes the energies of the purely electronic tran- 1.6 2.0 2.4 2.8 sitions (E0Ð0) and of the corresponding vibrational Energy, eV quanta. We note that the line peaking at 1.817 eV is broader and apparently associated with more than one Fig. 3. Absorption spectrum of C60Cl24 crystalline powder. T = 80 K. vibrational mode with energy in the interval 50Ð 80 meV.

The C60Cl24 luminescence is extremely weak, which A implies that a considerable role is played by nonradia- E0–0 tive processes. At T = 80 K, we succeeded in observing 3 only three weak emission lines in the interval 1.55Ð 1.66 eV, which approximately coincide energywise 2 with the corresponding absorption lines (Fig. 4). ω h 4 hω hω 6 Our study of C60Cl24 in frozen organic solvents per- hω 3 hω hω 2 5 formed at T = 80 K showed that the spectra of matrix- 1 isolated molecules virtually do not differ from those of the C60Cl24 crystalline powder (Fig. 4). The only differ-

1 Absorption, arb. units Luminescence, arb. units ences that are observed are a slight line narrowing and some intensity redistribution between them. All this indicates a clearly pronounced molecular character of 1.6 1.7 1.8 1.9 the C60Cl24 crystal. Figures 4 and 5 illustrate in more Energy, eV detail the spectra of groups A and B. We note that the Fig. 4. Spectra of (1, 2) absorption and (3) luminescence of intensity of the long-wavelength line group A is approximately one-ﬁfth that of the short-wavelength lines B (2) C60Cl24 crystalline powder and of (1, 3) matrix-isolated C60Cl24 molecules in a matrix of 1,2,4-chlorobenzene. The and C. This suggests that type-A transitions occur with dips in the luminescence spectrum observed in the interval a lower probability. 1.69Ð1.76 eV are caused by reabsorption of the radiation. Theoretical calculations of electronic energy levels T = 80 K. of the C60Cl24 molecule are not found in the literature. Therefore, we propose a variant of the electronic level diagram of the molecule in the region of the HOMOÐ ω LUMO transitions which is based only on an analysis h 5 1 of our experimental data (Fig. 6). In accordance with B ω E0–0 hω h 8 hω this diagram, the line group A corresponds to transitions 3 9 C c ω E0-0 ω h 2 h 1 between levels 2 and 3; line group B, to the 1 3 ω transitions; and line group C, to the 1 4 transitions. h 1 The 2 4 transitions are manifest extremely weakly in the spectrum, only in the form of barely seen features in the absorption spectrum in the interval 1.8Ð1.9 eV. 2 The large linewidth observed in the 1 4 transitions Absorption, arb. units is possibly associated with short electron lifetimes on level 4 because of a fast excitation conversion to level 3 accompanied by the emission of a vibrational quantum 2.3 2.4 2.5 2.6 2.7 of the molecule with បω ≈ 0.33 eV (264 cmÐ1). Approx- Energy, eV imately the same vibrational quanta (បω ≈ 31 and Fig. 5. Fragment of the absorption spectrum of (1) crystal- 37 meV) were observed by us in the vibronic absorp- បω ≈ line powder and (2) matrix-isolated molecules of C60Cl24 in tion spectrum, as well as in the Raman spectrum ( bromobenzene. T = 80 K. 34 meV). The characteristic conversion time should be

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ABSORPTION LINE SPECTRUM OF THE C60Cl24 HALOFULLERENE 2207

Table 2. Energy positions of the lines observed in the absorption spectra of C60Cl24 at T = 80 K Transitions A Transitions B បω បω No. E, eV assignment i, meV No. E, eV assignment i, meV

1 1.522 E0Ð0 1 2.313 E0Ð0

2 1.538 2 2.340 E0Ð0*

3 1.549 3 2.357 E0Ð0 បω 4 1.566 4 2.377 E0Ð0* + 1 37 បω 5 1.579 5 2.392 E0Ð0* + 2 52 បω 6 1.614 6 2.409 E0Ð0* + 3 69 បω 7 1.632 7 2.422 E0Ð0* + 4 82 បω 8 1.650 E0Ð0 8 2.444 E0Ð0* + 5 104 បω 9 1.669 9 2.456 E0Ð0* + 6 116 បω 10 1.693 E0Ð0 10 2.471 E0Ð0* + 7 131 បω 11 1.734 11 2.485 E0Ð0* + 8 145 បω 12 1.753E0Ð0* 12 2.514 E0Ð0* + 9 174

13 1.761 E0Ð0 បω 14 1.784 E0Ð0* + 1 31 Transitions C បω 15 1.817 E0Ð0* + 2 64 * បω 16 1.846 E0Ð0 + 3 93 1 2.67 E0Ð0* បω បω 17 1.861 E0Ð0* + 4 108 2 2.77E0Ð0* + 1 100 បω 18 1.873 E0Ð0* + 5 120 បω 19 1.939 E0Ð0* + 6 186 20 1.979

on the order of 0.1 ps, which is in reasonable agreement In the C60Cl24Ðtoluene crystal matrix system, in with current ideas on the relaxation processes occurring samples cooled to a temperature of 2 K, we succeeded in molecules and crystals. We believe each of the levels in observing the formation of a fine structure in the of this diagram to have a complex structure, which may lines A0Ð0* (E = 1.753 eV) and B0Ð0* (E = 2.340 eV), as account for the presence of several E0Ð0 lines in the type-A and B transitions at T = 80 K. well as a vibrational satellite structure in the B0Ð0* line The width of the lines of groups A (10 meV) and B (Fig. 7). The linewidth of the fine structure of line A is (20 meV) is due to inhomogeneous broadening, as well 1 meV, and that of line B is 2 meV. This structure disappears as the temperature is raised to ≈30 K. There is as to interaction with acoustic vibrations of the C60Cl24 lattice or of the matrix. In certain cases, the broadening no fine structure in the C60Cl24Ðglass matrix system. We associated with the latter effect can be eliminated by believe that this structure of the C60Cl24 spectrum orig- properly choosing the corresponding matrix producing inates from the Shpol’skiœ effect. an optical counterpart of the Mössbauer effect [8, 9]. In It appears logical to assume that the fine structure in this case, spectra of a number of fullerene molecules the A* and B* lines is associated with the molecule (C70, C60, and some derivatives) reveal a fine line struc- 0Ð0 0Ð0 ture corresponding to pure electronic transitions [10]. occupying several inequivalent sites in the matrix. In This is similar to the Shpol’skiœ effect, which was connection with this, we carried out a theoretical study observed in aromatic molecules embedded in frozen of the structure of the C60Cl24 centers in a toluene solutions in organic solvents (n-paraffines) [11]. matrix.

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 2208 RAZBIRIN et al.

4 the cavity (for the C60Cl24 molecule in toluene) cannot be less than ~14.6 Å, because otherwise the repulsive 3 LUMO (positive) part of the energy with which fullerene atoms interact with atoms of the crystal matrix would become too high for the incorporation of the fullerene molecule AB C into the cavity of the crystal matrix to be energetically

1.52 eV favorable. However, the smaller the number of toluene molecules that have to be removed to create the cavity, HOMO the more energetically favorable its formation. There- 2 fore, in order to determine the position, dimensions, and shape of the cavity, we scanned the irreducible part of the primitive cell of the toluene crystal by a sphere of 1 radius 7.3 Å and found, for each position of the sphere’s center, the number of crystal molecules that have at Fig. 6. Electronic-level diagram of the C60Cl24 molecule least one atom located within the sphere (it is the accounting for the experimental spectra. removal of such atoms that produces a cavity for the C60Cl24 molecule); thereafter, we chose the variants of the cavity that require removal of the smallest number of toluene molecules from the crystal. Finally, we established that the most energetically preferable are cavities centered at a common point with the coordinates r = 0.636a + 0.227b + 0.477c and at equivalent A0–0 B0–0 c points; these cavities are obtained by removing 17 toluene molecules (the primitive cell of the crystal con- 2 tains eight molecules). In such a cavity, we found several deep energy minima (corresponding to different 1 ×5 positions of the C60Cl24 molecule in the matrix); it is with these minima that one should apparently identify experimental points in the optical spectra. Absorption, arb. units 4 The existence of a comparatively small number of 3 possible energetically favorable positions of the mole- 1.74 1.76 2.32 2.36 2.40 cule incorporated into the toluene crystal matrix corre- Energy, eV lates with the small number of lines due to purely electronic transitions that are observed in the spectrum of Fig. 7. (1, 2) Fine structure of absorption lines A and B of this system. Taking into account the relaxation of the the C60Cl24 molecule in the crystal matrix of toluene. (3, 4) crystal matrix molecules around the embedded C60Cl24 The spectrum of the molecule in the glassy toluene matrix molecule should permit a more detailed comparison of does not have a ﬁne structure. T = 80 K. the theoretical and experimental results.

The geometry of the C60Cl24 molecule was chosen in ACKNOWLEDGMENTS accordance with the Th symmetry and the parameters determined in this work and in [12]. The nonvalent The authors are indebted to J. Cornil for performing some calculations. interactions between the C60Cl24 molecule and the molecules of crystalline toluene were simulated by the This paper was partially supported by the Russian interatomic potentials taken from [13]. The interatomic Foundation for Basic Research, project no. 00-03- interaction in [13] had the form of the LennardÐJones 33016; the Ministry of Education of the Russian Feder- potential with parameters chosen [13] such as to obtain ation (RF), grant no. E00-5.0-338 (V.P.S.); the Ministry a good ﬁt between the calculated and experimental data of Science and Technology of the RF the (program for several model substances. The toluene crystal “Fullerenes and Atomic Clusters”); and INTAS, grant matrix was simulated by a 60-molecule fragment of it. nos. 97-11894 and YSF 00-59 (A.V.Ch.). The matrix represents a high-temperature modiﬁcation of the toluene crystal lattice, which has, according to ()5 REFERENCES [14], monoclinic symmetry P21/C C2h . 1. F. N. Tebbe, R. L. Harlow, D. B. Chase, et al., Science The position, dimensions, and shape of the crystal- 256, 822 (1992). matrix cavity for a C60Cl24 molecule were determined 2. P. R. Birkett, H. W. Kroto, R. Taylor, et al., Chem. Phys. by the following technique. The linear dimensions of Lett. 205, 399 (1993).

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002 ABSORPTION LINE SPECTRUM OF THE C60Cl24 HALOFULLERENE 2209 3. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 165, 10. B. S. Razbirin, A. N. Starukhin, A. V. Chugreev, et al., 977 (1995) [Phys. Usp. 38, 935 (1995)]. Pis’ma Zh. Éksp. Teor. Fiz. 60, 435 (1994) [JETP Lett. 4. W. Kraetschmer, K. Fostiropoulos, and D. R. Huffman, 60, 451 (1994)]; B. S. Razbirin, A. N. Starukhin, Chem. Phys. Lett. 170, 167 (1990). A. V. Chugreev, et al., Fiz. Tverd. Tela (St. Petersburg) 38, 943 (1996) [Phys. Solid State 38, 522 (1996)]. 5. M. F. Limonov, Yu. E. Kitaev, A. V. Chugreev, et al., Phys. Rev. B 57 (13), 7586 (1998). 11. É. V. Shpol’skiœ, Usp. Fiz. Nauk 77, 321 (1962) [Sov. Phys. Usp. 5, 522 (1962)]. 6. A. K. Rappé, C. J. Casewit, K. S. Colwell, et al., J. Am. 12. O. E. Kvyatkovski , M. G. Shelyapina, B. F. Shchegolev, Chem. Soc. 114, 10024 (1992). œ et al., Fiz. Tverd. Tela (St. Petersburg) 44, 557 (2002) 7. M. J. Frisch, G. W. Trucks, H. B. Schlegel, et al., Gaus- [Phys. Solid State 44, 585 (2002)]. sian 98 (Revision A.1) (Gaussuan Inc., Pittsburgh, 13. T. V. Timofeeva, N. Yu. Chernikova, and P. M. Zorkiœ, 1998). Usp. Khim. 49, 966 (1980). 8. E. D. Trifonov, Dokl. Akad. Nauk SSSR 147, 826 (1962) 14. M. Anderson, L. Bosto, J. Bruneaux-Poulle, and [Sov. Phys. Dokl. 7, 1105 (1963)]. R. Fourme, J. Chim. Phys. Phys. Chim. Biol. 74, 68 9. E. F. Gross, S. A. Permogorov, and B. S. Razbirin, Dokl. (1977). Akad. Nauk SSSR 147, 338 (1962) [Sov. Phys. Dokl. 7, 1012 (1963)]; Dokl. Akad. Nauk SSSR 154, 1306 (1964) [Sov. Phys. Dokl. 9, 164 (1964)]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 11 2002