The Fine Structure of FCC Nanocrystals in Al- and Ni-Based Alloys G

METALS AND SUPERCONDUCTORS

The Fine Structure of FCC Nanocrystals in Al- and Ni-Based Alloys G. E. Abrosimova and A. S. Aronin Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected] Received July 30, 2001

Abstract—The structural perfection of nanocrystals in alloys of different chemical composition is studied by x-ray diffraction and high-resolution electron microscopy. In all the alloys studied, crystallization of the amorphous phase produces a nanocrystalline structure. The nanocrystal size depends on the chemical composition of the alloy and varies in aluminum-based alloys from 5 nm in Al89Ni5Y6 to 12 nm in Al82Ni11Ce3Si4. Nanoc- rystals in nickel-based alloys vary in size from 15 to 25 nm. Al nanocrystals are predominantly defect-free, with microtwins observed only in some nanocrystals. The halfwidth of the diffraction lines is proportional to sec θ, which implies the small grain size provides the major contribution to the broadening. Nanocrystals in nickel alloys contain numerous twins, stacking faults, and dislocations. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. EXPERIMENTAL AlÐNiÐRe (Re = Y, Ce, Yb) and NiÐMoÐB amor- A nanocrystalline structure that forms during crys- phous alloys were prepared in the form of ribbons by tallization of an amorphous phase attracts considerable melt quenching on a rapidly rotating wheel. The cool- interest both from the viewpoint of the possibility of ing rate was ~106 K/s, and the ribbons thus obtained developing novel materials with promising physical were 3 mm wide and ~30 µm thick. X-ray measure- chemical properties and as an unusual material in ments were performed on a Siemens D-500 diffracto- which the structural constituents do not exceed a few meter with CuKα radiation. The microstructure was tens of nanometers in size [1, 2]. In most cases, the studied with a high-resolution JEM-4000EX electron nanocrystalline structure is two-phase and consists of microscope at an accelerating voltage 400 kV. A direct nanoparticles of a crystalline phase embedded in an image of the nanocrystal lattice was obtained by amorphous host [3]. In some cases, however, the nanoc- recording a series of images made with different defocusing, followed by computer processing. In this paper, rystalline structure may also be made up of nanoparti- we present photomicrographs obtained at an optimum cles of different crystalline phases [4]. The degree of defocusing δ = Ð46 nm corresponding to the Scherzer structural perfection of nanocrystals is one of the most δ ≅ 1/2 λ1/2 important characteristics of nanocrystalline materials. focus ( Ð1.2Cs , where Cs is the spherical-aber- λ It is known, for instance, that light nanocrystalline ration constant equal to 1 mm and is the electron alloys have a high strength [5, 6]. Aluminum-based wavelength). The foils for the electron microscopy studies were prepared by ion milling. The alloy compo- alloys containing 6Ð15% of a transition metal (Fe or Ni) sition was determined by x-ray spectral analysis. and a few percent of a rare-earth metal may feature a yield strength as high as 1.6 GPa [7] while having, at the same time, a good plasticity. To properly understand 3. RESULTS AND DISCUSSION the processes involved in the plastic deformation of The as-prepared samples were amorphous. The cor- such materials, one has to know the speciﬁc features of responding x-ray and electron diffraction patterns their structure (the fraction of nanocrystals, their exhibit only broad maxima, with no peaks due to crys- mutual arrangement) and the possibility of deforma- talline phases present. The high-resolution images of the structure of the as-prepared alloy feature only the tion, i.e., the degree of perfection of the nanocrystals. It mazy contrast typical of an amorphous structure. Fol- is important to compare this characteristic of perfection lowing controlled crystallization, a nanocrystalline in alloys of different mechanical properties and estab- structure formed in all the alloys. lish the relation connecting the structure with the prop- When heated, all the samples crystallized by the pri- erties of the material. This paper reports on a study of mary mechanism. In aluminum-based alloys, fcc Al the perfection of nanocrystals in aluminum- and nickel- crystals precipitate. At the end of the ﬁrst crystalliza- based alloys. tion stage, the structure of the AlÐNiÐRe alloys consists

1004 ABROSIMOVA, ARONIN

where L is the grain size, λ is the wavelength of the radiation used, θ is the reflection angle, and ∆(2θ) is the (111) halfwidth of the corresponding reflection. The halfwidth of the diffraction lines is usually denoted by B; this notation will be employed in the graphs below. As the halfwidth of diffraction reflections in nanocrystal- (200) Intensity line materials is large, one can neglect the instrumental (220) (311) broadening. (222) (331) (420) (422) θ θ (400) The broadening proportional to 1/cos (i.e., to sec ) can also be caused by stacking faults in the material. 40 60 80 100 120 140 For instance, a stacking fault in an fcc structure changes 2θ, deg the ...ABCABC... sequence of the {111} planes to ...ABCACABC... . The broadening caused by stacking Fig. 1. Diffraction pattern of the Al88Ni6Y6 nanocrystalline faults is similar to that originating from small particle alloy. size, but its magnitude depends on the defect concentration and interference indices; i.e., stacking faults can produce crystallographic anisotropy in the calculated B grain size, even if the true particle dimensions are inde- 2.6 pendent of crystallographic direction. 2 If crystals contain randomly distributed disloca- 2.2 tions, the displacement of atoms from lattice sites is determined by the superposition of displacements due 1 to each dislocation and, therefore, the action of dislo- 1.8 cation fields can be treated phenomenologically as a local change in the interplanar distance [9]. The dif- 3 ∆ θ 1.4 fraction line broadening 2 (2 ) (or B) can be written in this case as ∆()θ ()θ∆ 1.0 2 2 = 4 dm/d0 tan , ∆ where dm is the average maximum change in the dis- 0.6 tance between the hkl planes and d0 is the interplanar 0.4 distance in a perfect crystal. Thus, the broadening orig- 1.0 1.2 1.6 2.0 2.4 2.8 inating from a random dislocation distribution is pro- secθ portional to tanθ. By analyzing the angular dependence of the broad- Fig. 2. Halfwidth of the reflections of (1) the Al Ni Y , (2) ening, one can reveal the factor responsible for the 88 6 6 major contribution to it; indeed, if the broadening is Al88Ni10Y2, and (3) Al82Ni11Ce3Si4 nanocrystalline alloys θ plotted vs. secθ. proportional to sec , it is caused by the small crystallite size, but if it is proportional to tanθ, the broadening originates from dislocations and their clusters, which of fcc Al nanocrystals dispersed randomly throughout generate long-range distortion fields. the amorphous host, whose size depends on the chemi- Part of the reflections presented in Fig. 1 is due to cal alloy composition and varies from 5 nm in the superposition of reflections from the crystalline Al89Ni5Y6 to 12 nm in Al82Ni11Ce3Si4. phase and from nanocrystals; therefore, one should first Figure 1 presents a typical x-ray diffraction pattern separate them. For this purpose, special computer codes of an aluminum-based nanocrystalline alloy. The pat- were used which permit background correction, tern is seen to contain both broad lines due to fcc Al smoothening, and deconvolution of overlapping max- nanocrystals and a diffuse halo originating from the ima, as well as determination of the halfwidth of the amorphous phase. maxima. The shape of a diffuse maximum was fitted by 2 The width of the diffraction line may depend on a a Gaussian hexp(Ðax ), the fitting of the calculations to number of factors. Among them are the small size of the experimental data being preformed in both parameters crystals, the presence of various defects, and inhomo- h and a. geneities in the chemical composition. Figure 2 plots the dependence of the halfwidth B of θ The broadening due to the small grain size is diffraction reflections on sec for a sample of the described by the SelyakovÐScherrer expression [8] Al88Ni6Y6 alloy (curve 1) whose diffraction pattern is shown in Fig. 1. Similar curves for other aluminum- L = λ()1/cosθ /∆()2θ , based alloys are also presented in Fig. 2.

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THE FINE STRUCTURE OF FCC NANOCRYSTALS 1005

All three dependences can be fitted with good accu- B Σ 2 racy by straight lines, with the variance (y Ð yi) ( yi are 2.2 the experimental values, and y are obtained in calculations) not exceeding 0.005; the variance is 0.0008 for 1.8 Al88Ni6Y6, 0.0035 for Al88Ni10Y2, and 0.001 for Al82Ni11Ce3Si4. At the same time, the similar dependence of the halfwidth of diffraction peaks on tanθ is 1.4 obviously not linear (Fig. 3). An attempt to fit it with a linear function yields a substantially larger variance 1.0 (0.045). Thus, analysis of the angular dependences of dif- 0.6 fraction reflections allows one to conclude that the 0.2 0.6 1.0 1.4 1.8 2.2 2.6 broadening of diffraction lines in aluminum-based tanθ nanocrystalline alloys is dominated by the small grain size. This conclusion also correlates with the data obtained by high-resolution electron microscopy. The Fig. 3. Halfwidth of the reflections of the Al86Ni11Yb3 corresponding studies show the nanocrystals in alumi- nanocrystalline alloy plotted vs. tanθ. num-based alloys to be predominantly defect-free. Microtwins are observed only in rare cases. A typical image of nanocrystals in such a structure is displayed in Fig. 4. The nanocrystalline structure was also studied in nickel-based alloys. The studies were performed on three alloys of the following compositions: (Ni70Mo30)90B10, (Ni70Mo30)95B5, and (Ni65Mo35)90B10. The as-prepared alloys were amorphous; indeed, diffraction studies did not reveal any indication of crystalline phases and only diffuse maxima typical of an amorphous structure were observed. Heating amorphous alloys gives rise to their crystallization. The crystallization proceeds by the primary mechanism. By this mechanism, the amorphous matrix produces crystals (in our case, of a nickel-based solid solution) whose composition differs from that of the original alloy. The 2 nm composition of the matrix changes after the precipitation of the primary crystals. The lattice parameters of the fcc solid-solution grains of the alloys under study were changed depending on the time the alloys were Fig. 4. Microstructure of the Al88Ni6Y6 alloy. maintained isothermally at 600°C. After a 6 h annealing at 600°C, the average size of the nanocrystals was 13 nm for (Ni70Mo30)90B10 and (Ni65Mo35)90B10 and 17 nm for (Ni70Mo30)95B5. A typical x-ray diffraction pattern for a nickel-based nanocrystalline alloy is displayed in Fig. 5. It closely resembles the pattern shown in Fig. 1, which gives one grounds to assume that the (111) angular dependences of the diffraction peak halfwidth would also be the same. Figure 6 presents the dependence of the diffraction maximum halfwidth on secθ for the (Ni65Mo35)90B10 alloy obtained for two anneal Intensity (220) (311) times, 5 and 144 h. The average nanocrystal size after (200) (222) annealing for 144 h is 17 nm for (Ni Mo ) B , 19 nm (331)

70 30 90 10 (400) for (Ni70Mo30)95B5, and 27 nm for (Ni65Mo35)90B10. A 40 60 80 100 120 140 high-resolution image of such a microstructure is pre- θ sented in Fig. 7. The particle on the right reveals numer- 2 , deg ous stacking faults. It appears natural that as the thermal treatment duration increases, the size of the nanocrystals should also Fig. 5. Diffraction pattern of the (Ni65Mo35)90B10 alloy slightly increase and the halfwidth of the diffraction after annealing at 600°C for 5 h.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1006 ABROSIMOVA, ARONIN B 3.5 A question naturally arises as to the reason for such 1 differences in the nanocrystal structure. We consider 2 two main factors which can account for these differ- 2.5 ences. (1) Size factor. As already mentioned, aluminum 1.5 nanocrystals do not exceed 10 nm in size, whereas the size of the Ni(Mo) particles is about 20 nm. When dis- 0.5 locations form (by the FrankÐRead or another mecha- 0.8 1.2 1.6 2.0 2.4 2.8 nism), the dislocation loop size is about 102b (b is the B secθ 3.5 Burgers vector) [10]. The magnitude of the Burgers vector is b = a/2[110] = 0.349 nm for Al and 0.2501 nm 3 for Ni. We thus see that while in Ni(Mo) nanocrystals a 4 2.5 loop can still exist (because the nanocrystals are 70Ð 110b in size), the presence of loops ≈30b in size in aluminum nanocrystals is not likely. 1.5 (2) The formation of stacking faults depends on their energy. The stacking fault energy for aluminum is 0.5 135 erg cmÐ2; for nickel, 240 erg cmÐ2 [11]. One should, 0.2 0.6 1.0 1.4 1.8 2.2 2.6 θ however, bear in mind that nanocrystals in AlÐNiÐRe tan alloys are actually Al precipitates [12], but in NiÐMoÐ B alloys, they contain about 17 at. % Mo [3]. As follows Fig. 6. Angular dependences of the halfwidth of diffraction ° from [13], when fcc materials are alloyed, the stacking maxima of the (Ni65Mo35)90B10 alloy annealed at 600 C fault energy decreases with increasing electron concen- for (1, 3) 5 and (2, 4) 144 h. tration. For instance, in copper and silver alloys, the stacking fault energy decreases by about ten times, with an increase in the electron concentration of 20% [13, 14]. Adding 17 at. % Mo to nickel alloys also increases the electron concentration by about 20%. Therefore, if we assume that the dependence of stacking fault energy in nickel alloys on electron concentration is similar to that for other fcc alloys, we can expect a substantial decrease (by a few times) in this energy in the above alloys. In these conditions, the energy of the Ni(Mo) stacking faults will be lower than that of the Al stacking faults and, hence, the probability of stacking fault formation in these alloys will be higher. Both these factors can account for the differences in the structure of nanocrystals observed experimentally in our work, namely, the perfect structure of the Al nanocrystals and defects in the Ni(Mo) nanocrystals. Thus, our study shows that nanocrystals in alloys of 2 nm the AlÐNiÐRe systems (Re = Y, Ce, Yb) have a substantially better structural perfection than those in the nickel-based alloy.

Fig. 7. High-resolution image of the microstructure of the (Ni65Mo35)90B10 alloy. ACKNOWLEDGMENTS The study was supported by the Russian Foundation for Basic Research (project nos. 99-02-17459 and maxima should decrease. It is seen, however, that none 99-02-17477). of the relations presented is linear; in other words, both the small size of the nanocrystals and the dislocations contribute to the line broadening. The high-resolution REFERENCES electron microscope image also exhibits numerous 1. A. Inoue, Mater. Sci. Eng. A 179/180, 57 (1994). defects. Hence, both x-ray diffraction and electron 2. K. Nakazato, Y. Kawamura, A. P. Tsai, and A. Inoue, microscopy indicate the presence of numerous twins, Appl. Phys. Lett. 63, 2644 (1993). stacking faults, and dislocations in nanocrystals in the 3. G. Abrosimova, A. Aronin, Yu. V. Kir’janov, et al., NiÐMoÐB alloys. J. Mater. Sci. 34 (7), 1611 (1999).

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4. V. Cremaschi, B. Arcondo, H. Sirkin, et al., J. Mater. and Electron Microscopy (Metallurgiya, Moscow, Res. 15 (9), 1936 (2000). 1982). 5. Y. He, J. F. Poon, and G. Y. Shiﬂet, Science 241, 1640 10. J. P. Hirth and J. Lothe, Theory of Dislocations (1988). (McGraw-Hill, New York, 1967; Atomizdat, Moscow, 1972). 6. A. Inoue, T. Ochiai, Y. Horio, and T. Masumoto, Mater. 11. P. S. Dabson and L. M. Clarebrough, Philos. Mag. 9, 865 Sci. Eng. A 179/180, 649 (1994). (1964). 7. Y. H. Kim, A. Inoue, and T. Masumoto, Mater. Trans., 12. G. E. Abrosimova, A. S. Aronin, Yu. V. Kir’janov, et al., JIM 32, 331 (1991). Nanostruct. Mater. 12, 617 (1999). 13. P. C. J. Gallagher, Trans. AIME 1, 2429 (1970). 8. A. A. Rusakov, X-ray Diffraction of Metals (Atomizdat, Moscow, 1977). 14. A. Howie and P. R. Swann, Philos. Mag. 6, 1215 (1961). 9. Ya. S. Umanskiœ, Yu. A. Skakov, A. N. Ivanov, and L. N. Rastorguev, Crystallography, x-ray Diffraction, Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

METALS AND SUPERCONDUCTORS

Hyperfine Interactions of Tin Atoms in Samarium I. N. Rozantsev and V. P. Gor’kov Skobeltsyn Research Institute of Nuclear Physics, Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: [email protected] Received August 6, 2001

Abstract—The hyperfine interactions of 119Sn impurity atoms in samarium at temperatures from 5 to 70 K are investigated by Mössbauer spectroscopy. The distributions P of magnetic hyperfine fields Bhf for tin atoms at sites of the hexagonal [Ph(Bhf)] and cubic [Pc(Bhf)] samarium sublattices are determined from the experimental absorption spectra. Ion ordering in pairs of magnetic centers located in layers of the cubic sublattice is observed by Mössbauer spectroscopy for the first time. Each magnetic center involves ordered ions at the nearest neighbor sites of the tin atom replacing the samarium ion at the hexagonal lattice site. The quadrupole cou- 2 ± pling constant e qhQ = 0.59 0.12 mm/s is determined for tin atoms at the hexagonal sublattice sites of samarium. The quadrupole interaction of tin atoms in heavy rare-earth metals (from Tb to Er) with a hexagonal close- packed structure is discussed. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION TN1, the magnetic moments of samarium ions in the Hyperfine interactions of nonmagnetic impurity hexagonal sublattice are collinearly ordered with atoms in rare-earth metals have long since attracted the respect to the hexagonal axis c. These moments of particular attention of physicists. In heavy rare-earth samarium ions are ferromagnetically ordered in pairs of metals (from Gd to Er), the hyperfine interactions have adjacent h layers and antiferromagnetically alternate in been investigated by Mössbauer spectroscopy for tin pairs of h layers separated by the c layer. At tempera- atoms [1Ð8] and by perturbed-angular-correlation spec- tures below TN2, samarium ions in the c layers are troscopy for cadmium atoms [9, 10]. However, similar ordered to form an A1-type antiferromagnetic structure studies of hyperfine interactions in light rare-earth met- composed of pairs of adjacent ferromagnetic rows als have been hampered by the fact that these metals aligned parallel to the a2 axis. Pairs of these rows with exhibit a complex local magnetic structure. Some oppositely aligned moments of the samarium ions alter- insight into these difficulties can be gained, for exam- nate along the a1 axis (+ + – – …). The magnetic ple, from the results of Mössbauer measurements for moments of samarium ions at the c and h sites are col- tin in neodymium [11]. Samarium has a relatively sim- linear to the c axis. The magnetic structure of the h lay- ple magnetic structure [12]. The first investigations into ers is retained at temperatures below 14 K. hyperfine interactions of tin and cadmium atoms in In rare-earth metals and their compounds, the local- samarium revealed a few magnetic hyperfine fields ity of the magnetic hyperfine field Bhf at nonmagnetic Bhf (Sn) at Sn nuclei in the hexagonal sublattice at low atoms and the additivity of the contributions from indi- temperatures [13] and only one field Bhf at Cd nuclei in vidual rare-earth ions (at the nearest neighbor sites of the same sublattice over the entire range of magnetic these nonmagnetic atoms) to the field Bhf [16, 17] make ordering [14]. With the aim of resolving this contradic- it possible to predict changes in the field Bhf at a tin tion, we undertook the present investigation. Earlier, atom with variations in the local magnetic structure in Reœman and Rozantsev [15] determined the magnetic the neighborhood of this atom. The contributions of the hyperfine field Bhf (Sn) in samarium and used the result ferromagnetically ordered ions of a particular rare- obtained to analyze the dependence of the field B (Sn) hf earth element to the hyperfine field Bhf have the same in rare-earth metals on the projections of the spin and sign, and their total contribution is proportional to the the orbital angular momentum of rare-earth ions and number of these ions, because the sign of the contribu- the radial dependence of the partial contributions of tion of a rare-earth ion correlates with the direction of these ions to the field Bhf (Sn). its spin [15, 17]. The contributions of the antiferromag- Samarium has a rhomobohedral crystal lattice that netically ordered ions to the field Bhf are of opposite consists of nine layers with cubic (c) and hexagonal (h) sign. In this case, the total contribution is equal to the symmetries of the environment (chhchh…). For brevity, difference between the contributions of the ions with hereafter, the sites located in the c and h layers will be oppositely aligned moments and depends on the effec- referred to as the c and h sites, respectively. Samarium tive number of the ions inducing this field. The contri- is an antiferromagnet with two Néel temperatures: bution of an individual rare-earth ion to the field Bhf TN1 = 106 K and TN2 = 13.8 K. At temperatures below consists of the spin and orbital contributions, which are

HYPERFINE INTERACTIONS OF TIN ATOMS IN SAMARIUM 1009 proportional to the projections of the spin and the with one ordered ion at the h site free of a tin atom. This orbital momentum of this ion [15, 17]. The contribu- interaction, together with the interaction of ions in the tions of samarium ions located at sites with different c layer, should bring about their ordering at tempera- symmetries of the environment to the hyperfine field tures above TN2. Bhf are proportional to the projections of the sums of The aim of the present work was to determine the the spins and the orbital momenta of the ions at the rel- magnetic-hyperfine-field distributions for tin atoms at evant sites. Each of the sites in the samarium lattice has sites of the cubic [Pc(Bhf )] and hexagonal [Ph(Bhf )] 12 nearest neighbor sites, of which six sites are located samarium sublattices and the hyperfine parameters of in the same layer as the site under consideration, three tin atoms in samarium. sites occupy one of the two adjacent layers, and three sites occupy the other adjacent layer. Let us analyze how the ordering of samarium ions 2. EXPERIMENTAL TECHNIQUE affects the magnetic hyperfine field Bhf at tin atoms AND RESULTS whose environment retains the local magnetic structure The samples were prepared by melting samarium of pure samarium. In what follows, the magnetic hyper- (purity, 99.7%) with a calculated content of 1.5 at. % fine field B for tin atoms at the h sites will be desig- hf 119Sn in an induction furnace in an argon atmosphere. nated as Bhf (9). In the temperature range from TN1 to 119 TN2, the field Bhf (9) for tin atoms at the h sites is deter- The absorption Mössbauer spectra of Sn in mined by samarium ions with parallel magnetic samarium were measured in the temperature range 5Ð moments at the nine nearest neighbor h sites of each tin 70 K on a spectrometer operating in a constant-acceler- atom, because the samarium ions of two adjacent h lay- ation mode. The spectrometer was equipped with a ers are ordered ferromagnetically. The Mössbauer spec- CaSnO3 resonance detector. The samples under investi- trum should exhibit a magnetic sextet attributed to these gation were cooled in a helium-flow cryostat. A radio- tin atoms and a paramagnetic line of tin atoms at the c active source in the form of CaSnO3 was used at room sites. This line transforms into a sextet as the samarium temperature. ions in the c layers are ordered. Note that the field Bhf Figure 1 shows typical experimental Mössbauer at tin atoms in the aforementioned layers is effectively spectra. All the spectra were processed with the aim to determined by two samarium ions with parallel mag- determine the distributions P(Bhf ) in the form of histo- netic moments at the nearest neighbor c sites which are grams. The distribution function P(Bhf ) was determined occupied by four samarium ions with the same direc- by minimizing the χ2 functional according to the tion of the magnetic moments and two samarium ions FUMILI program. The calculated Mössbauer spectrum with opposite moments. The contributions of the was represented as a convolution of the distribution samarium ions of two adjacent h layers (on opposite P(B ) = P (B ) + P (B ) and an elementary magnetic sides of the c layer) to the field B at these tin atoms hf c hf h hf hf sextet with a Lorentzian linewidth of 1 mm/s. Apart cancel each other due to antiferromagnetic ordering of from the components of the distribution, we varied the the ions. ∆ ∆ quadrupole shifts c and h and the isomer shift. In the case when the samarium ions are ordered at Selected histograms are displayed in Fig. 2. In these the c sites, the magnetic hyperfine field Bhf (9) should histograms, one interval corresponds to two scale divi- split into four components. Each of these components sions in the spectrum. The distributions Pc and Ph are is induced either by nine ferromagnetically ordered shifted with respect to each other, because their edges ions at the nearest neighbor h sites of the tin atom and overlap at 5 K. For each of the histograms Pc and Ph, the three ions with parallel moments or, effectively, by one relative intensity is taken equal to unity. The widths of ion at the nearest neighbor c site. In the h layers, tin discrete components (including those falling in two atoms occupy equally probable positions in which the adjacent intervals), as a rule, do not exceed the width of net moments of the ions at the nearest neighbor c and h one interval of the histogram. The distribution P con- sites are parallel and antiparallel to each other, respec- c tains up to nine components P , which are numbered in tively, whereas the moments of three ions at the nearest i neighbor c sites are parallel to each other or the moment order of increasing magnetic hyperfine field Bhf (i). The of one of these three ions is antiparallel to the moments component P0 corresponds to hyperfine fields Bhf < of the two other ions. The Mössbauer spectra should 0.35 T. exhibit four sextets with identical intensities rather than The most intense component of the distribution Ph one sextet. at 70 K (Fig. 2) is assigned to the expected field Bhf (9), If the samarium ion at the h site is replaced by a tin which is induced by nine ferromagnetically ordered atom, the interactions between each of the three ions at ions at the nearest neighbor h sites of the tin atom. This the nearest neighbor c sites of this atom and the antifer- component is designated as P . The components P< are romagnetically ordered ions of the two adjacent h lay- 9 i ers do not cancel each other. The resultant interaction numbered in order of increasing hyperfine fields < > with ions of these layers is equivalent to the interaction Bhf()i < Bhf (9), and the components Pi are numbered

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1010 ROZANTSEV, GOR’KOV

T =5 K 3

15 K

20 K

35 K

50 K

–20 –10 0 10 20 V, mm/s

Fig. 1. The absorption Mössbauer spectra of 119Sn impurity atoms in samarium at temperatures from 50 to 5 K. For the Mössbauer spectrum measured at 5 K, solid lines represent (1) the spectrum corresponding to the distribution Pc(Bhf) for Sn atoms at sites of the cubic samarium sublattice, (2) the spectrum corresponding to the distribution Ph(Bhf) for Sn atoms at sites of the hexagonal samarium sublattice, and (3) the overall spectrum.

> ≥ in order of decreasing hyperfine field Bhf()i Bhf (9). correspond to both the quadrupole coupling constant 2 ± The components of the distribution Ph at temperatures for tin atoms at the h sites of samarium (e qhQ = 0.59 from 15 to 50 K are divided into two types reasoning 0.12 mm/s, where q is the electric-field gradient at the from the constancy of the sum of the intensities of all 119Sn nucleus and Q is the quadrupole moment of the > ± 119Sn nucleus) and the upper limit of its magnitude for the components Pi (0.35 0.01), including the P9 tin atoms at the c sites (|e2qQ| < 0.3 mm/s) in the case component. At 50 K, the separation of the Pi components from the P component is reflected in its shape. when the hyperfine field Bhf (Sn) is aligned parallel to 9 the hexagonal axis c. For both sublattices of samarium, the 119Sn isomer shifts coincide and are equal to 1.95 ± 0.05 mm/s at The distribution P even at 70 K includes at least ∆ h 5 K. The quadrupole shifts are as follows: c < four components. The number of components increases ∆ ± 0.08 mm/s and h = 0.15 0.03 mm/s. These quantities to 12 with a decrease in the temperature to 35 K. A fur-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 HYPERFINE INTERACTIONS OF TIN ATOMS IN SAMARIUM 1011

Pc Ph < < < < < < < > > > > > > P0 P1 P2 P3 P4 P5 P6 P7 P8 P1 P2 P3 P4 P5 P6 P7 P6 P5 P4 P3 P2 P1 0.2 5 K 0.1

0 < < < < < < < > > > > > > P0 P1 P2 P3 P4 P5 P6 P7 P1 P2 P3 P4 P5 P6 P7 P6 P5 P4 P3 P2 P1 0.2 15 K 0.1

0 < < < < < < > > > > > > P0 P1 P2 P3 P4 P5 P6 P7 P1 P2 P3 P4 P5 P6 P6 P5 P4 P3 P2 P1 0.2 20 K 0.1

0 < < < < < < > > > > > P0 P1 P2 P3 P4 P5 P1 P2 P3 P4 P5 P6 P9 P5 P4 P3 P2 P1

, rel. units 0.2 i P 35 K 0.1

0 < < < < < > > > P0 P1 P2 P3 P4 P5 P6 P1 P2 P3 P4 P5 P9 P3 P2 P1 0.2 50 K 0.1

0 < < > > P0 P1 P2 P3 P4 P5 P1 P2 P9 P2 P1 0.4 0.3 70 K 0.2 0.1

0 5 10 10 15 20 Bhf, T

Fig. 2. Histograms of the magnetic-hyperﬁne-ﬁeld distributions Pc(Bhf) and Ph(Bhf) for Sn atoms at sites of the cubic and hexagonal samarium sublattices at temperatures from 70 to 5 K. ther decrease in the temperature is accompanied by assumption underlies the subsequent discussion. How- redistribution of the intensities among the components. ever, the above replacement contributes to ion ordering at the c sites. At temperatures above TN2, this ordering ≠ 3. DISCUSSION manifests itself in the form of components at Bhf (i) 0 The replacement of samarium by tin (up to 1.5 at. %) in the Pc histograms and as additional components leaves the TN1 temperature unchanged and, most likely, (apart from the expected component P9) in the Ph histo- does not affect the ordering of ions in the h layers. This grams (Fig. 2).

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1012 ROZANTSEV, GOR’KOV

Parameters for pairs of Sn impurity atoms in the same and For these distances, we used the same designation (for adjacent h layers pairs in the adjacent h layers, ri is the projection of the iR, an ir, an distance Ri between the sites occupied by the atoms of i i i i a particular pair onto the c layer). 11 611/ 3 3 The probability that two atoms located at sites of the same h layer or at sites of the adjacent h layers (on 232/ 3 opposite sides of the c layer) are separated by a distance Ri is proportional to the number ni of sites located at this 26363 7/3 distance from an arbitrary site in the same or adjacent h layer. This probability increases with an increase in the 4613/3 concentration of atoms in the h layer and decreases with an increase in the distance Ri at a fixed number ni. The 32 654/ 3 3 shortest distances Ri and ri and the corresponding numbers n are listed in the table. 6619/3 i Reasoning from the concentration of tin atoms 412737 5/ 3 (1.5 at. %) introduced into the sample and their bino- mial distribution in the h layers, we calculated the prob- 862/ 7/3 abilities of finding these atoms at a distance Ri from each other, i.e., in the pairs characterized by the inter- 53 6 atomic distances Ri (in the same h layers) and ri (in the Note: Ri stands for the distances between the sites occupied by Sn adjacent h layers). We took into account the atomic atoms, ri is the distance between the centers of the nearest neigh- pairs in which the distances between the centers of the bor c sites of the Sn atoms (for pairs of Sn atoms in the same h nearest neighbor c sites of tin atoms were less than the layer, ri = Ri), ni is the number of sites located in a layer at the distance from each of these centers to a similar center distance R from any site in the same or adjacent h layer, and i is i of any other tin atom in the same c layer. The probabil- the number of the atomic pair in order of increasing distances Ri and ri (Ri and ri are given in terms of the lattice parameter a). ity of finding tin atoms in the pairs characterized by the distance R1 is equal to 0.080. The probabilities of finding tin atoms in the pairs characterized by the distances An increase in the number of components of the Ph Ri satisfy the following ratio: distribution with a decrease in the temperature to 35 K W(R ) : W(R ) : W(R ) : W(R ) : W(R ) and the constancy of the intensities of the extreme com- 1 2 3 4 5 ponents in the temperature range from 50 to 35 K = 1 : 0.86 : 0.76 : 1.15 : 0.50. (Fig. 2) suggest that the change in both the number and The probability of finding tin atoms in the pairs charac- the hyperfine field B of the components in this distri- hf terized by the distance r3 is equal to 0.033. The ratio of bution accounts for the ordering of samarium ions in the probabilities of finding tin atoms in the pairs char- the vicinity of the pairs of tin atoms whose nearest acterized by the distances r has the form neighbor c sites are located in the same c layer. The i interaction of samarium ions at the c sites near the tin W(r1) : W(r2) : W(r3) : W(r4) : W(r5) : W(r6) : W(r7) : W(r8) atoms of these pairs is responsible for ion ordering at = 0.61 : 0.53 : 1 : 0.70 : 0.33 : 0.60 : 0.22 : 0.38. higher temperatures compared to those of ion ordering in the vicinity of widely spaced atoms. A comparison of the probabilities of finding tin atoms in pairs and the intensities of the components of Since the c layer contains the nearest neighbor sites the Ph distribution at 35 K (and 50 K) enables us to of tin atoms of the two h layers adjacent to this c layer, assign the components P< ÐP< to the tin atoms in the it is necessary to consider the pairs of tin atoms located 1 5 > > > not only in the same layer but also in the neighboring h pairs with R1ÐR5 and the components P1 , P2 , and P3 layers on opposite sides of the c layer adjacent to these to the tin atoms in the pairs with r3, r4, and r6, respec- h layers. Let us assume that Ri is the distance between tively. We failed to reveal discrete components that the sites occupied by tin atoms of a pair. The pairs of the could be attributed to the tin atoms in the pairs with r , nearest neighbor tin atoms in the same and adjacent h 1 r2, and r5 at 50 and 35 K, even though the probabilities layers are characterized by the distances between the of finding tin atoms in these pairs correspond to com- centers of the nearest neighbor c sites of these atoms, ponent intensities that are sufficiently high for observa- i.e., by the distances Ri and ri, respectively. These cen- tion. At 35 K, the uncertainties in the component inten- ters coincide with the projections of the h sites (occu- < < pied by the tin atoms) onto the c layer. Hence, the dis- sities are equal to 20% for the P1 and P4 components; tance between the centers of the nearest neighbor c sites 30% for the P< and P> components; and 50, 60, and of the tin atoms located in the same layer is equal to the 2 1 < > > distance between the h sites occupied by these atoms. 75% for the P3 , P2 , and P3 components, respectively.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 HYPERFINE INTERACTIONS OF TIN ATOMS IN SAMARIUM 1013

The ordering of samarium ions in the c layer near a imum distances between the nearest neighbor c sites of pair of tin atoms depends not only on the distance the tin atoms in the pairs with R1 suggest that samarium between these atoms but also on the mutual arrange- ions at these sites can be ordered at a higher tempera- ment of their nearest neighbor c sites (located at verti- ture compared to that of ion ordering near the atoms in ces of equilateral triangles). In particular, the nearest other pairs. neighbor c sites of the tin atoms in the pairs with r1, r2, < r5, and r7 form figures with two mutually perpendicular For the Ph distribution at 70 K (Fig. 2), only the P1 axes of symmetry in their plane. One axis passes component is characterized by the relatively low inten- through the centers of the nearest neighbor c sites of the sity expected for tin atoms in the pairs with R1. Only for atoms linked by the ri segment in the pair. The other this component does the intensity remain constant in axis passes through the midpoint of the ri segment. The the temperature range from 70 to 5 K and is the hyper- latter axis is the boundary between the c sites that are fine field B< (1) minimum among the fields B of the located at a shorter distance from one or the other atom hf hf in the pair. The magnetic moments of samarium ions at components in the distribution Ph. It is this hyperfine the sites of this boundary can be ordered only along the field Bhf at the tin atoms in the pairs with r1 that can be normal to the c axis owing to the mutual compensation less than the fields Bhf at other atoms at the h sites, i.e., for their interaction with ions lying in the c layer on can correspond to the field B< (1). To accomplish this, opposite sides of the boundary due to antiferromagnetic hf ordering of the magnetic moments of the ions in the h the maximum contribution of three ions at the nearest layers adjacent to the atoms of the pair. Since the mag- neighbor c sites of the tin atom and the contribution of netic moments of samarium ions at the other c sites, as only eight ions at its nine nearest neighbor h sites (with in pure samarium, are collinearly ordered with respect the other atom of the pair) to the hyperfine field Bhf to the c axis, it is clear that the occurrence of this should have opposite signs; i.e., the magnetic moments boundary brings about a weakening of the interaction of three ions at the nearest neighbor c sites of the tin between the ions located on opposite sides of the atom should be antiparallel to the moments of the ions boundary and, hence, affects the ion ordering. at the nearest neighbor h sites of the atom. The minimum field Bhf automatically provides a constant inten- Among the tin atoms in the adjacent h layers, only sity of the component corresponding to this field with the atoms in the pairs with r1 and r2 have common near- variations in the temperature. Therefore, the constant est neighbor c sites. For an ion at this site, the interac- intensity of the P< component with a minimum field tions with antiferromagnetically ordered ions of these 1 < layers cancel each other. Each tin atom in the pair with Bhf (1) can be considered a consequence of antiparallel r1 (or r2) has two (or one) common nearest neighbor c ordering of the magnetic moments of samarium ions at sites. The line passing through these sites forms a the nearest neighbor sites of the tin atoms in the pairs boundary between the atoms of each pair such that the with R . magnetic moments of the ions at the sites of this bound- 1 ary can be ordered only along the normal to the c axis. The hyperfine fields Bhf of the components in the Pc Consequently, at three nearest neighbor c sites of each and Ph distributions at 5 K allow us to judge the contri- tin atom in the pairs with r1 (or r2), the magnetic butions of ions at the nearest neighbor sites of the rele- moments of only one (or two) samarium ion can be col- vant tin atoms. It can be assumed that the hyperfine linearly ordered with respect to the c axis. The temper- fields Bhf (7) and Bhf (8) are induced by five and six ions ature of ion ordering and the hyperfine field Bhf at the with parallel magnetic moments at the nearest neighbor tin atoms should substantially depend on the number of c sites of the tin atoms in the c layer. The hyperfine these ions, because their contribution to the field domi- fields B> (1) and B< (2) represent the sum and the dif- nates over the contribution from the ions with magnetic hf hf moments perpendicular to the c axis. The separation of ference of the contributions from nine ions at the near- the components associated with tin atoms in the pairs est neighbor h sites and three ions with parallel moments at the nearest neighbor c sites of the same tin with r1 and r2 can be complicated, because the hyperfine fields B at these atoms, in at least a certain tem- atoms. These inferences follow from the fact that the hf > ± < ± perature range, can be close to the fields Bhf of the other hyperfine fields Bhf (1) = 20.8 0.2 T, Bhf (2) = 9.1 ± components of the distribution Ph. 0.2 T, and Bhf (8) = 12.0 0.2 T satisfy the relationship > < Among the tin atoms in the same h layer, only the Bhf (8) = Bhf (1) Ð Bhf (2) and Bhf (7)/Bhf (8) = 5/6. Note atoms forming the pairs with R1 have a common nearest that B (9) = [B> (1) + B< (2)]/2 = 15.0 ± 0.2 T can be neighbor c site. For a samarium ion at this site, the hf hf hf interaction with antiferromagnetically ordered ions of regarded as an estimate of the magnetic hyperfine field two adjacent h layers is equivalent to the interaction induced by nine samarium ions at the nearest neighbor with two ions at its nearest neighbor h sites in the layer h sites of the tin atoms at 5 K. At this temperature, the < > without atoms of the pair. This interaction and the min- hyperfine fields Bhf (6) and Bhf (5) correspond to the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1014 ROZANTSEV, GOR’KOV difference and the sum of the contributions from nine give a more accurate account of the ordering of samar- samarium ions at the nearest neighbor h sites and one ium ions at the nearest neighbor c sites of the tin atoms. ion at three nearest neighbor c sites of the same tin atoms. The above interpretation of the distribution Ph(Bhf ) allows for the difference in the number of hyperfine The total contributions of samarium ions at the near- fields Bhf observed by the different methods at Sn and est neighbor c and h sites of the tin atoms to the hyper- Cd atoms in samarium. This difference is explained by < > fine fields Bhf (i) and Bhf (i) have opposite and identical different concentrations of nonmagnetic probing atoms signs, respectively. The correspondence between the used in the measurements. In the case when the hyper- < < fine interactions are investigated by perturbed-angular- extreme components of the Ph distribution (P1 , P5 and correlation spectroscopy, the concentration of these > > atoms is at least one order of magnitude (or, more fre- P1 , P3 ) and the atoms of pairs in the same and adjacent h layers indicates that there is a correlation between the quently, several orders of magnitude) less than that location of the pair of atoms in the same layer (or in two used in Mössbauer measurements. Consequently, the adjacent layers) and the relative signs of the total con- number of pairs with short interatomic distances is also tributions from samarium ions at the nearest neighbor c smaller in the former case. Forker and Fraise [14] did and h sites of the tin atoms to the hyperfine fields B (i). not reveal cadmium atoms in pairs with short distances hf between nonmagnetic impurity atoms. These authors For at least the shortest interatomic distances in the determined the temperature dependence of the hyper- pairs, the total contributions of samarium ions at the fine field B for cadmium atoms located at large dis- nearest neighbor c and h sites of the tin atoms in the hf pairs in the same and adjacent h layers to the fields tances from the other nonmagnetic impurity atoms. Bhf (i) have opposite and identical signs, respectively. All heavy rare-earth metals (from gadolinium to For collinear moments of all the samarium ions at the erbium) have a hexagonal close-packed structure. The nearest neighbor sites of the tin atoms, this implies that lattice parameter a decreases from 0.3636 nm for gad- the net moments of the samarium ions at the nearest olinium to 0.3560 nm for erbium. The lattice parameter neighbor c and h sites of the tin atoms are antiparallel ratio c/a varies from 1.590 to 1.570, respectively. and parallel, respectively. The distribution Ph at temper- For samarium, these parameters are as follows: a = atures from 15 to 50 K is asymmetric with respect to the 0.3663 nm and 2c/9a = 1.595 [18]. Since the changes in < the lattice parameters and their ratios are insignificant, sums of the intensities of all the components Pi and it can be expected that, in heavy rare-earth metals and P> (0.35 ± 0.01). This suggests that, at temperatures the hexagonal samarium sublattice, the electric-field i gradients q at nonmagnetic impurity atoms will be above 35 K, the aforementioned assignment of the P h h close to each other. This assumption is confirmed by the components to the pairs of the tin atoms in the same and experimental data for cadmium atoms in these rare- adjacent h layers can hold for larger values of Ri and ri, earth metals, according to which the electric-field gra- because the number of tin atoms in the pairs located in dient q changes only by a factor of 1.4 [10, 14]. the same h layers is twice as large as that in the adjacent h h layers. The shifts observed in the sextet components of the Mössbauer spectra for tin in terbium [4], dysprosium Thus, ion ordering in pairs of magnetic centers [3, 4], holmium [5, 6], and erbium [6Ð8] indicate that located in the c layers of samarium was observed for the Sn nuclei are involved not only in the magnetodipole first time. Each magnetic center involves ions at three interaction but also in the electric quadrupole interac- nearest neighbor c sites of the tin atom located at the h tion. The signs of the quadrupole shifts in these spectra site. These ions are ordered at temperatures above TN2 are in agreement with those expected from both the due to the disturbance of their interaction with antifer- determined sign of the electric-field gradient qh at tin romagnetically ordered (below TN1) ions of two adja- atoms in samarium and the available data on magnetic cent h layers on opposite sides of the c layer with a pair structures of rare-earth metals at low temperatures [18]. of magnetic centers. For tin atoms involved in pairs of The quadrupole shifts of the sextet components, which the same h layers, the hyperfine field Bhf is determined were roughly estimated from the known Mössbauer by the difference between the total contributions of spectra, proved to be of the same order of magnitude as samarium ions at the nearest neighbor c and h sites of those expected in the case of identical electric-field gra- the tin atoms (for at least the shortest interatomic dis- dients qh at tin atoms in the rare-earth metals under con- tances). sideration and samarium. Precision measurements of the Mössbauer spectra The Mössbauer spectra of tin in dysprosium [3, 4], of tin in samarium in the temperature range of magnetic holmium [5], and erbium [6Ð8] have been interpreted in ordering make it possible to determine the temperature terms of two fields whose strengths are related through ≅ dependences of the hyperfine fields Bhf at the tin atoms the expression Bhf (x) 0.75Bhf . It should be noted that, located at different distances from each other in the in either case, a lower field corresponds to a weaker same and adjacent h layers. These dependences can sextet with an intensity of several times less than the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 HYPERFINE INTERACTIONS OF TIN ATOMS IN SAMARIUM 1015 intensity of the principal sextet. These lower fields 5. P. V. Bogdanov, S. K. Godovikov, M. G. Kozin, and result from misinterpretation. When processing Möss- V. S. Shpinel, Hyperfine Interact. 5 (2), 333 (1978). bauer spectra whose components could not be unre- 6. S. K. Godovikov, Fiz. Tverd. Tela (Leningrad) 27 (5), solved even at low temperatures, the quadrupole shifts 1291 (1985) [Sov. Phys. Solid State 27, 782 (1985)]. of the sextet lines and their broadening due to the 7. S. K. Godovikov, N. I. Moreva, A. A. Sadovskiœ, and widths of the distributions P(Bhf ) were disregarded. A. I. Firsov, Izv. Akad. Nauk SSSR, Ser. Fiz. 54 (9), These spectra were fitted using two sextets in which all 1674 (1990). the lines had the same width. As a result, the sextet with 8. S. A. Nikitin, S. K. Godovikov, V. Yu. Bodryakov, and quadrupole shifts of the components was decomposed I. A. Avenarius, Izv. Akad. Nauk, Ser. Fiz. 58 (4), 15 into two sextets without quadrupole shifts but with dif- (1994). ferent isomer shifts. The location and intensity of the 9. M. Forker and A. Hammesfahr, Z. Phys. 263 (1), 33 principal sextet were determined from the extreme lines (1973). of the measured sextet. The parameters of the weak sex- 10. W. Witthuhn and W. Engel, in Hyperfine Interactions of tet were derived using the second and fifth lines remain- Radioactive Nuclei, Ed. by J. Christiansen (Springer- ing after the separation of the intense sextet. Certainly, Verlag, Berlin, 1983), Topics in Current Physics, Vol. 31, this procedure led to the aforementioned ratio between p. 205. the two fields. The isomer shifts of the two resultant 11. S. I. Reœman and N. I. Rokhlov, Fiz. Tverd. Tela (Lenin- sextets for tin in holmium and erbium at low tempera- grad) 27 (5), 1587 (1985) [Sov. Phys. Solid State 27, 960 tures differed by 0.5Ð1 mm/s [5Ð8]. (1985)]. 12. W. C. Koehler and R. M. Moon, Phys. Rev. Lett. 29 (21), ACKNOWLEDGMENTS 1468 (1972). 13. S. I. Reœman, N. I. Rokhlov, I. N. Rozantsev, and We are grateful to N.N. Delyagin for supplying the F. D. Khamdamov, Izv. Akad. Nauk SSSR, Ser. Fiz. 50 program used in processing the Mössbauer spectra and (12), 2392 (1986). helpful discussions of the results and S.I. Reœman for his participation in the initial stage of this work. 14. M. Forker and L. Fraise, Hyperfine Interact. 34 (2), 329 (1987). 15. S. I. Reœman and I. N. Rozantsev, Zh. Éksp. Teor. Fiz. REFERENCES 102 (2), 704 (1992) [Sov. Phys. JETP 75, 378 (1992)]. 1. V. Gotthardt, H. S. Moller, and R. L. Mössbauer, Phys. 16. K. Eckrich, E. Dorman, A. Oppelt, and K. H. J. Buschow, Lett. A 28 (7), 480 (1969). Z. Phys. B 23 (1), 157 (1976). 2. A. S. Kuchma, V. P. Parfenova, and V. S. Shpinel, Pis’ma 17. N. I. Delyagin, V. I. Krylov, N. I. Moreva, et al., Zh. Zh. Éksp. Teor. Fiz. 13 (2), 192 (1971) [JETP Lett. 13, Éksp. Teor. Fiz. 88 (1), 300 (1985) [Sov. Phys. JETP 61, 135 (1971)]. 176 (1985)]. 3. S. K. Godovikov, M. G. Kozin, V. V. Turovtsev, and 18. S. A. Nikitin, Magnetic Properties of Rare-Earth Metals V. S. Shpinel, Phys. Status Solidi B 78 (1), 103 (1976). and Alloys (Mosk. Gos. Univ., Moscow, 1989), p. 21. 4. P. V. Bogdanov, S. K. Godovikov, M. G. Kozin, et al., Zh. Éksp. Teor. Fiz. 72 (6), 2120 (1977) [Sov. Phys. JETP 45, 1113 (1977)]. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1016Ð1021. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 973Ð978. Original Russian Text Copyright © 2002 by Golubkov, Parfen’eva, Smirnov, Misiorek, Mucha, Jezowski, Ritter, Assmus.

METALS AND SUPERCONDUCTORS

Behavior of the Lorenz Number in the Light Heavy-Fermion System YbInCu4 A. V. Golubkov*, L. S. Parfen’eva*, I. A. Smirnov*, H. Misiorek**, J. Mucha**, A. Jezowski**, F. Ritter***, and W. Assmus*** *Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] **Institute of Low-Temperature and Structural Research, Polish Academy of Sciences, Wroclaw, 50-950 Poland ***Goethe University, Frankfurt-am-Main, 60054 Germany Received August 21, 2001

Abstract—The electrical resistivity and thermal conductivity of two polycrystalline YbInCu4 samples prepared by different techniques at the Ioffe Physicotechnical Institute, RAS (St. Petersburg, Russia), and the Goethe University (Frankfurt-am-Main, Germany) are studied within the temperature range 4.2Ð300 K. At Tv ~ 75Ð 78 K, these samples exhibited an isostructural phase transition from a state with an integer valence (at T > Tv) to a state with an intermediate valence (at T < Tv) of the Yb ions. It is shown that at T < Tv ; i.e., in the temperature range where YbInCu4 is assumed to be a light heavy-fermion compound, the Lorenz number behaves as it should in a classical heavy-fermion system. At T > Tv , where YbInCu4 is a semimetal, the Lorenz number has a value characteristic of standard metals. © 2002 MAIK “Nauka/Interperiodica”.

κ In the last decade, the physical properties of We measured the total thermal conductivity tot and YbInCu4 have received increasing interest from the electrical resistivity ρ of two polycrystalline sam- researchers in the leading laboratories of the USA, Ger- ples of YbInCu4 within the temperature range 4.2Ð many, and Japan.1 300 K on a setup similar to the one employed in [9].

At Tv ~ 40Ð80 K and atmospheric pressure, YbInCu4 The samples were prepared at the Ioffe Physicotech- undergoes an isostructural phase transition from a nical Institute (St. Petersburg, Russia) and at the Goethe CurieÐWeiss paramagnet with localized magnetic University (Frankfurt-am-Main, Germany) by different techniques [10Ð14]. YbInCu was rf melted in sealed moments (at T > Tv) to a Pauli paramagnet with a non- 4 magnetic Fermi-liquid state and intermediate valence tantalum crucibles. However, the starting materials used in the synthesis in the two laboratories differed in of the Yb ions (at T < Tv). purity. In what follows, we shall label the sample pre- At the phase transition, the Yb valence changes from pared in St. Petersburg by 1P and the one made in 2 3 (T > Tv) to 2.9 (T < Tv). Frankfurt-an-Main by 2F. The high- and low-temperature phases are a semi- The samples were subjected to x-ray diffraction metal and metal, with the Yb 4f electrons weakly and characterization on a DRON-2 setup (CuKα radiation) strongly hybridized with the conduction electrons, and found to be single-phase with an AuBe5-type structure and the lattice constant a equal to 7.133(4) Å (sam- respectively. At T < Tv, YbInCu4 possesses a high density of states at the Fermi level, a feature characteristic ple 1P) and 7.139(5) Å (sample 2F). of heavy-fermion systems and compounds with inter- The YbÐInÐCu system is homogeneous within a mediate rare-earth ion valence. The parameter γ (the fairly broad range of compositions. It can be presented coefﬁcient of the linear-in-temperature term in the elec- as YbIn1 Ð xCu4 + x [11, 12, 15]. The composition of a tronic speciﬁc heat) for the low-temperature phase is sample depends substantially on the method by which ~50 mJ/mol K2 [4Ð6], which indicates a large carrier it was prepared. Within the homogeneity region, the effective mass. The YbInCu4 system is classed among phase transition temperature T varies from 40 to 70Ð light heavy-fermion systems [2, 7]. 80 K. Tv = 40 K corresponds to the YbInCu4 stoichiometry. The YbIn Cu composition with the highest YbInCu crystallizes in an AuBe cubic lattice 0.8 4.2 4 5 melting temperature has T ~ 70Ð80 K. For the 2 v (C15b structure, space group F43m(T a ) [8]). YbIn1 − xCu4 + x samples prepared by the techniques 1 2 κ ρ References to numerous publications dealing with YbInCu4 can The experimental data on tot and of sample 2F were used ear- be found in [1Ð3]. lier in [3].

BEHAVIOR OF THE LORENZ NUMBER 1017 employed in St. Petersburg and Frankfurt-am-Main, the melt starts to solidify with the formation of crystals corresponding to the transition at Tv ~ 70 K [11, 12]. According to [16],3 our samples with a = 7.133Ð 10 7.139 Å were close in composition to YbIn0.83Cu4.17. The thermal conductivity of YbInCu4 was studied in [3, 17]; however, no detailed analysis of the data was made. 8 1 The main purpose of this work was to study the behavior of the Lorenz number (L) of the light heavy- 2 fermion system YbInCu4 for T < Tv. We were interested in whether this behavior would exhibit features charac- 6 teristic of classical heavy-fermion systems [18, 19]. , W/m K tot

Earlier, we found L to behave in a manner typical of κ such systems in the light heavy-fermion system 4 YbIn0.7Ag0.3Cu4 [20]. Figures 1 and 2 display our experimental results on κ ρ tot(T) and (T) obtained on YbIn0.83Cu4.17 samples 1P Tv ~ 75–78 K and 2F at temperatures extending from 300 to 4 K. 2 Measurements performed in the reverse run (4 to 300 K) revealed a highly hysteretic behavior of ρ(T) κ [23] and tot(T) at T > Tv, which is due to defects associated with the lattice stresses forming as the tempera- 4 ture passes through Tv. 0 100 200 300 Despite the difference in the starting material purity T, K and in the techniques employed in the preparation of Fig. 1. Temperature dependence of κ for samples (1) 1P κ ρ tot samples 1P and 2F, their tot(T) and (T) dependences and (2) 2F. were found to be sufﬁciently similar. The average value of Tv estimated from these dependences is ~75Ð78 K for both samples. We made an attempt to estimate the 200 composition of samples 1P and 2F using the data on (a) Tv(x) obtained in [22] for the YbIn1 Ð xCu4 + x system (Fig. 2b). It was again found that samples 1P and 2F are 3 close in composition to YbIn0.83Cu4.17. To simplify 160 exposition of the experimental material, we shall assume that, on the average, the samples studied by us 2 have the same composition, although, as seen from Figs. 1 and 2 (as well as from Figs. 3, 4), these samples 1 κ κ 120 differ slightly in the magnitude of tot, ph (the lattice component of thermal conductivity), and ρ. This differ- cm ence can be apparently assigned to the fact that the sam- µΩ , (b) ples do differ in composition (though very slightly) and ρ 80 80 2 probably contain different amounts of residual impurities. 1 , K According to Hall effect measurements [2, 24], both v

T 40 YbIn Cu phases (at T > Tv and T < Tv) have a fairly high carrier 1 – x 4 + x κ concentration, such that tot should have both lattice 40 0 κ 5 and electronic ( e) components: 0 0.1 0.2 x κ κ κ tot = e + ph. (1)

3 He et al. [16] presented the dependence of a on x measured for 0 100 200 300 YbIn1 Ð xCu4 + x. T, K 4 We will devote a separate paper to the study and discussion of the κ behavior of tot(T) in the direct and reverse measurement runs. ρ 5 Fig. 2. (a) Temperature dependence of for YbIn0.83Cu4.17 Because YbInCu4 is a semimetal at T > Tv, one could also expect κ κ samples (1) 1P and (2) 2F and (3) for LuInCu4, a compound the bipolar thermal conductivity bip to contribute to tot at certain band parameters of this material [21, 25], but, as follows with no phase transition [21]. (b) Dependence of Tv on x in κ κ from [3], the component bip in tot of YbInCu4 at T < 300 K is the YbIn1 Ð xCu4 + x system [22]. The dashed lines are drawn negligible. to determine the values of x for samples (1) 1P and (2) 2F.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1018 GOLUBKOV et al. According to the classical theory of the thermal con- κ (a) (b) ductivity of solids, e should obey the WiedemannÐ 8 6 Franz law 1 κ ρ 4 e = LT/ . (2) , W/m K

ph 2 At T տ Θ/3 (Θ is the Debye temperature) and very low 6 κ 2 T temperatures in pure metals, as well as at low and high Tv v temperatures in dirty metals, we have L = L0 [26], 0 100 200 300 where L0 is the Sommerfeld value of the Lorenz num- T, K 1 , W/m K × Ð8 Ω 2 4 ber (L0 = 2.45 10 W /K ). The YbIn0.83Cu4.17 sam- ph

κ Tv ples belong neither to very pure metals nor to semimet- 2 als, and, thus, one may accept, as the first approximation, L = L0 throughout the temperature range (4Ð 2 300 K) studied by us. κ0 Figure 3a plots the κ (T) relation calculated from ph ph Eqs. (1) and (2) under the assumption of L = L0. As is Tv evident from this figure, as well as from Fig. 4, within κ 0 100 200 300 the temperature interval ~120Ð300 K, ph follows a κ 0.28 κ 0.34 T, K power law: ph ~ T for sample 1P and ph ~ T for Fig. 3. (a) Temperature dependence of κ for κ ph sample 2F. As the temperature is lowered, ph passes YbIn0.83Cu4.17 samples (1) 1P and (2) 2F (see Fig. 4 for the through a minimum, then reaches a maximum, and κ0 κ explanation of ph ). (b) Temperature dependence of ph of finally falls off to zero. (1) sample 1P and (2) the YbIn0.7Ag0.3Cu4 sample [20]. What could be the explanation for this behavior of κ ph? First, we consider the data obtained for T > Tv. At 5.0 these temperatures, our compound is not a heavy-fermion system but rather a semimetal; therefore, using 0.28 κ T 1 the value L = L0 in calculations of e is fully valid. How- κ ever, the reasons for the growth of ph with increasing temperature remain unclear. This behavior of κ (T) is 2 ph characteristic of amorphous or heavily defected materials. In our case, defects can form in YbIn0.83Cu4.17 T 0.34 through copper substituting for indium [11] (we κ stressed this point when analyzing data on ph(T) for YbIn0.7Ag0.3Cu4 [20]). Such defects should affect the 2.0 κ behavior of ph(T) noticeably for both T > Tv and T < T . κ0 v ph κ A growth of ph with temperature was also observed to occur in a number of other compounds which can be

, W/m K κ0 classed among heavy-fermion systems (light ph ph κ YbIn0.7Ag0.3Cu4 [20], moderate YbAgCu4 [27] and UInCu5 [28], and classical CeAl3 [29]) (Fig. 5). It 1.0 appears highly unlikely that in all the above com- κ pounds, the increase in ph(T) is associated only with a high defect concentration.6 It is possible that this behavior of κ (T) is characteristic of a certain class of T < Tv T > Tv ph heavy-fermion systems. Tv κ Now, we consider the behavior of ph(T) for T < Tv. It is important to understand the reason for such a 0.5 6 κ n In [28, 29], the growth of ph as ~T is explained in terms of the theoretical model developed in [30]. We believe, however, that 5 10 20 50 100 200 300 this interpretation is unsatisfactory, because, as shown in [30], κ κ 0.3 T, K ph ~ T (in our case, ph ~ T ) and the relation proposed in [30] is valid only at very low temperatures (T < Θ/20), whereas in our κ Fig. 4. Temperature dependence of ph for YbIn0.83Cu4.17 experiment, the effect is observed at substantially higher temper- samples (1) 1P and (2) 2F. atures (up to 300 K).

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 BEHAVIOR OF THE LORENZ NUMBER 1019 κ strong increase in ph, more speciﬁcally, whether it κ stems from a real growth of ph inherent in this compound or originates from our having wrongly taken into 20 κ account the quantity L when calculating e from 6 Eq. (2). One cannot exclude the possibility that L, 4 15 rather than being equal to L0, varies in a complicated way with temperature. The soundness of the latter conjecture is argued for by the following observations. 5 10 κ 5 (1) To explain the sharp growth of ph at T > Tv, one has to assume that a mechanism causing strong phonon 5 TK scattering becomes operative at this temperature. We 1 have not found thus far a reasonable explanation for the 4 0 nature of such a mechanism. At T < Tv, the valence of T the Yb ion changes from 3 to 2.9, the Yb ionic radius N Tv 2 increases, and the lattice becomes looser and, hence, , W/m K , W/m K ph 3 ph

κ κ κ more defected, which should entail a decrease in ph(T) rather than its growth. 3 (2) According to the elementary theory of thermal Tv conductivity, we have 2 κ ∼ Cvl, (3) Tv ph 6 where C, v, and l are the speciﬁc heat, sound velocity, 1 and phonon mean free path, respectively. In YbInCu4, C [5] and v [10] vary abruptly within a narrow temperature interval around Tv. The quantities C and v decrease smoothly at T < Tv and increase as smoothly at T > Tv; i.e., if one excludes the narrow interval near the phase 0 50 100 150 200 250 300 transition (around Tv), one observes a smooth variation T, K of C(T) and v(T) throughout the temperature range 7 studied by us. This is naturally only an indirect indica- Fig. 5. Temperature dependence of κ for YbIn Cu κ ph 0.83 4.17 tion of the absence of a bell-shaped behavior of ph(T) samples (1) 1P and (2) 2F, (3) YbIn0.7Ag0.3Cu4 [20], (4) plotted in Fig. 3 for samples 1P and 2F in the T < Tv YbAgCu4 [27], (5) UInCu5 [28], and (6) CeAl3 [29]. For all range. κ0 curves, the dashed lines plot ph , which was derived by (3) YbIn0.7Ag0.3Cu4 undergoes a gradual phase tran- extrapolating the high-temperature data by the power laws sition (without an abrupt change in ρ, a, and other found for each compound. TK and TN are, respectively, the Kondo and Néel temperatures. parameters at Tv) which is similar in nature to the one observed in YbInCu4. Assuming L = L0 throughout the κ temperature range studied, we isolated ph(T) from κ As already mentioned more than once, tot(T) using Eqs. (1) and (2) and found that, at T < Tv, κ YbIn0.83Cu4.17 transfers to the state corresponding to a ph(T) of this compound follows a bell-shaped pattern κ light heavy-fermion system at T < Tv. The behavior of similar to the one obtained for ph(T) of YbIn0.83Cu4.17 κ the Lorenz number in a classical heavy-fermion system (Fig. 3). In [20], we explained this behavior of ph(T) as being due to our having incorrectly taken the Lorenz differs substantially both in magnitude and in the tem- κ perature dependence from that in both pure and dirty number into account in e; it was found that the Lorenz number varies in a complex way with temperature and metals. According to [18, 19], Lx/L0 for this system is substantially larger than L0. It is possible that we also increases from unity (at T ~ 0), then passes through a have a similar situation in YbIn0.83Cu4.17. maximum and falls off down to 0.648, and then grows κ again to reach unity at T ~ T (T is the Kondo temper- Thus, the above arguments suggest that e, rather K K κ than ph, is most likely responsible in our case for the ature). strong increase in thermal conductivity at T < Tv. To ﬁnd Lx/L0(T) for T < Tv, we shall assume that Now, we consider the behavior of L in YbIn Cu κ n 0.83 4.17 ph ~ T (where n, as pointed out earlier, is 0.28 and at T < Tv. 0.34 for samples 1P and 2F, respectively) throughout 7 the temperature range covered by us (4Ð300 K). To do Unfortunately, we have not been able to directly estimate the κ character of l variation within the temperature range covered. this, we extrapolate ph(T) using this relation from the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1020 GOLUBKOV et al.

Lx/L0(T) determination (Fig. 5) proposed in this work and, as it seems, have obtained more accurate information on the behavior of the Lorenz number in this com- 7 pound. Note, however, the speciﬁc features in the behavior of Lx/L0(T) in YbAgCu4 which still remain unclear. We come up against a paradox. Indeed, in light heavy-fermion systems, such as YbInCu4 and γ Ӎ 2 YbIn0.7Ag0.3Cu4, whose parameter 50 mJ/mol K , 5 Lx/L0(T) behaves as it should in classical heavy-fer- 3 mion systems (with γ ≥ 400 mJ/mol K2). At the same

0 γ

L time, in moderately heavy-fermion systems with ~ / 2 x 2 L 200Ð250 mJ/mol K , to which YbAgCu4 belongs, the Lx/L0(T) ratio (determined in the temperature range 4Ð 3 300 K) exhibits only a slight deviation from unity (in 1 the region T < TK).

ACKNOWLEDGMENTS 4 The authors express gratitude to N.F. Kartenko and 1 N.V. Sharenkova for x-ray structural measurements. This work was conducted within bilateral agree- 0 ments between the Russian Academy of Sciences, 0 20 40 60 80 100 Deutsche Forschungsgesellschaft, and the Polish Acad- T, K emy of Sciences and was supported by the Russian Foundation for Basic Research (project no. 99-02- Fig. 6. Temperature dependences of Lx/L0 for the 18078) and the Polish Committee for Scientiﬁc YbIn0.83Cu4.17 samples (1) 1P and (2) 2F, (3) Research (grant no. 2PO3B 129-19). YbIn0.7Ag0.3Cu4 [20], and (4) YbAgCu4 [27].

REFERENCES κ0 region T > Tv to T < Tv (Fig. 4; ph in Figs. 3 and 4 and 1. J. L. Sarrao, C. D. Immer, Z. Fisk, et al., Phys. Rev. B 59 dashed lines in Fig. 5)8 and apply the relation (10), 6855 (1999). 2. A. V. Goltsev and G. Bruls, Phys. Rev. B 63 (15), 155109 κ κ κ0 e = tot Ð ph (4) (2001). 3. I. A. Smirnov, L. S. Parfen’eva, A. Jezowski, et al., Fiz. for the range 4Ð50 K to determine Lx/L0(T) (we Tverd. Tela (St. Petersburg) 41 (9), 1548 (1999) [Phys. excluded the temperature interval around Tv) from con- Solid State 41, 1418 (1999)]. sideration. The results of this calculation are plotted in 4. A. L. Cornelius, J. M. Lawrence, J. L. Sarrao, et al., Fig. 6. Phys. Rev. B 56 (13), 7993 (1997). We readily see from Fig. 6 that the behavior of 5. J. L. Sarrao, A. P. Ramírez, T. W. Darling, et al., Phys. Lx/L0(T) in samples 1P and 2F ﬁts the above theoretical Rev. B 58 (1), 409 (1998). pattern of the behavior of the Lorenz number in a 6. N. Pillmayer, E. Bauer, and K. Yoshimura, J. Magn. heavy-fermion system [18]. Magn. Mater. 104Ð107, 639 (1992). We may thus conclude that the Lorenz number 7. I. Felner, I. Nowik, D. Vakin, et al., Phys. Rev. B 35 (13), behaves similarly in a classical and a light heavy-fer- 6956 (1987). mion system. 8. R. Kojima, Y. Nakai, T. Susuki, et al., J. Phys. Soc. Jpn. 59 (3), 792 (1990). For comparison, Fig. 6 presents data for Lx/L0(T) of YbIn0.7Ag0.3Cu4 [20] and our reﬁned data for YbAgCu4 9. A. Jezowski, J. Mucha, and G. Pompe, J. Phys. D 20, 1500 (1987). from [27]. In [27], we determined the ratio Lx/L0(T) using a slightly different technique. By contrast, here 10. B. Kindler, D. Finsterbusch, R. Graf, et al., Phys. Rev. B 50 (2), 704 (1994). we have applied to YbAgCu the method of κ0 (T) and 4 ph 11. A. Löffert, M. L. Aigner, F. Ritter, and W. Assmus, Cryst. Res. Technol. 34 (2), 267 (1999). 8 κ0 That ph of the YbIn0.83Cu4.17 samples can behave in this way at 12. A. Löffert, S. Hautsch, F. Ritter, and W. Assmus, Physica low temperatures follows indirectly from the data published in B (Amsterdam) 259Ð261, 134 (1999). κ [28, 29], which testify that ph(T) for YbIn0.83Cu4.17 behaves 13. E. Feschbach, A. Löffert, F. Ritter, and W. Assmus, exactly as it does in CeAl3 and UInCu5 (Fig. 5). Cryst. Res. Technol. 33, 267 (1998).

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14. A. V. Golubkov, T. B. Zhukova, and V. M. Sergeeva, Izv. 22. K. Yoshimura, N. Tsujii, K. Sorada, et al., Physica B Akad. Nauk SSSR, Neorg. Mater. 2, 77 (1966). (Amsterdam) 281/282, 141 (2000). 15. A. V. Golubkov, L. S. Parfen’eva, I. A. Smirnov, et al., 23. J. L. Sarrao, C. D. Immer, C. L. Benton, et al., Phys. Rev. Fiz. Tverd. Tela (St. Petersburg) 44 (6), 973 (2002) B 54 (17), 12207 (1996). [Phys. Solid State 44 (2002)]. 24. Y. Itoh, H. Kadomatsu, J. Sakurai, and H. Fujiwara, Phys. Status Solidi A 118, 513 (1990). 16. J. He, N. Tsujii, K. Yoshimura, et al., J. Phys. Soc. Jpn. 66 (8), 2481 (1997). 25. A. V. Golubkov, L. S. Parfen’eva, I. A. Smirnov, et al., Fiz. Tverd. Tela (St. Petersburg) 42 (11), 1938 (2000) 17. E. Bauer, E. Gratz, G. Hutﬂesz, et al., Physica B [Phys. Solid State 42, 1990 (2000)]. (Amsterdam) 186/188, 494 (1993). 26. I. A. Smirnov and V. I. Tamarchenko, Electron Heat 18. V. I. Belitsky and A. V. Goltsev, Physica B (Amsterdam) Conductivity in Metals and Semiconductors (Nauka, 172, 459 (1991). Leningrad, 1977). 19. I. A. Smirnov and V. S. Oskotskii, in Handbook on the 27. A. V. Golubkov, L. S. Parfen’eva, I. A. Smirnov, et al., Physics and Chemistry of Rare Earth, Ed. by Fiz. Tverd. Tela (St. Petersburg) 43 (2), 210 (2001) K. A. Gschneidner, Jr. and L. Eyring (Elsevier, Amster- [Phys. Solid State 43, 218 (2001)]. dam, 1993), Vol. 16, p. 107. 28. D. Kaczorowski, R. Troc, A. Czopnik, et al., Phys. Rev. B 63, 144401 (2001). 20. A. V. Golubkov, L. S. Parfen’eva, I. A. Smirnov, et al., 29. H. R. Ott, O. Marti, and F. Hulliger, Solid State Com- Fiz. Tverd. Tela (St. Petersburg) 43 (10), 1739 (2001) mun. 49 (12), 1129 (1984). [Phys. Solid State 43, 1811 (2001)]. 30. I. E. Zimmerman, J. Phys. Chem. Solids 11, 299 (1959). 21. A. V. Golubkov, L. S. Parfen’eva, I. A. Smirnov, et al., Fiz. Tverd. Tela (St. Petersburg) 42 (8), 1357 (2000) [Phys. Solid State 42, 1394 (2000)]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1022Ð1027. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 979Ð984. Original Russian Text Copyright © 2002 by Avdyukhina, Katsnel’son, A. Olemskoœ, D. Olemskoœ, Revkevich. METALS AND SUPERCONDUCTORS

Evolution of the PdÐTaÐH Alloy Structure in the Edwards Thermodynamic Representation V. M. Avdyukhina, A. A. Katsnel’son, A. I. Olemskoœ, D. A. Olemskoœ, and G. P. Revkevich Moscow State University, Vorob’evy gory, Moscow, 119899 Russia e-mail: albert@solst.phys.msu.su Received August 24, 2001

Abstract—It is shown that the time evolution of x-ray diffraction patterns of a deformed PdÐTa alloy after its saturation with hydrogen can be determined by the multiwell energy proﬁle of the states of the system. Within the Lorenz synergetic approach, a phenomenological model is proposed in which the evolution of the alloy structure is represented as a random walk of the nonergodic system from one internal-energy minimum to another. In this case, the order parameter is the fraction of states with minimum energy occupied by the system, the conjugate ﬁeld is associated with the Edwards entropy, and the control parameter is the internal energy. The evolution of the PdÐTaÐH alloy structure is interpreted as that of a complex nonergodic system in terms of thermodynamics. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION internal stresses are produced by hydrogen atoms and the state of the system can become nonequilibrium as in Considerable recent attention has been focused on the case under an external load. Investigations of complex nonergodic systems, such as spin and struc- annealed PdÐWÐH [6] and deformed PdÐSmÐH, PdÐ tural glasses, disordered heteropolymers, granular ErÐH, PdÐMoÐH, and PdÐTaÐH alloys [7Ð13] revealed media, and transport flows (see review [1]). The main that these alloys can undergo nonmonotonic nonregular feature of such systems is that their phase space is sep- transformations. The main features of the behavior of arated into isolated domains, each of which corre- such systems have been explained in terms of syner- sponds to a metastable thermodynamic state, and the getic models [6, 10]. However, these models cannot number of these domains N0 is much larger than the adequately describe the evolution of the PdÐTaÐH alloy total number of (quasi)particles N and exponentially structure (see also [12]). We deal with this problem in increases as the latter tends to infinity: N0 = exp(sN), the present paper. where s is the so-called Edwards entropy per particle (complexity) [2, 3]. In contrast to the Boltzmann measure, which characterizes the disorder in a given statis- 2. EXPERIMENTAL DATA tical ensemble, the Edwards entropy characterizes the disorder in the distribution of states of a nonergodic The experimental technique is described in detail in system over internal-energy minima, each of which [9Ð12]; here, we only point out that a deformed PdÐTa (7 at. % Ta) alloy was first saturated with hydrogen corresponds to a statistical ensemble. In complex sys- 2 tems, statistical ensembles play the role of particles and electrolytically (current density 40 mA/cm ) for 15 min. After relaxation for 176 h, the sample was sat- the distribution over these ensembles is characterized 2 by the effective temperature T and entropy s (intro- urated once again for the same time at 80 mA/cm . An duced by Edwards [3]). For example, at T = 0, the com- x-ray diffraction study was carried out using a comput- plex system corresponds to a granular medium charac- erized diffractometer (CuKα1 doublet component); the terized by a flat distribution over all energy minima. (220) and (311) diffraction maxima were analyzed and their deconvolution was performed using the Origin The Edwards systems have an exponentially large computer code and assuming that the components are ӷ number of energy minima N0 1, in contrast to ergodic Lorentzian-shaped [14, 15]. systems, for which N0 = 1. It is of interest to consider an The x-ray diffraction study revealed that, after the intermediate case where the number of energy minima deformed PdÐTa alloy was saturated with hydrogen, a N0 is not exponentially large but exceeds unity. regular shift in the position of diffraction maxima In this paper, it will be shown that this is the case occurred, which indicated that the crystal lattice with deformed PdÐTa alloys saturated electrolytically expanded and then anisotropically contracted. During with hydrogen. When heavily deformed, a solid can be the relaxation, stochastic changes were observed in the so far from its equilibrium state that self-organization positions, widths, and symmetry of diffraction maxima; effects become significant and dissipative structures their shape also varied in a random fashion, with sev- arise [4, 5]. In a metal saturated with hydrogen, strong eral peaks appearing and disappearing. Figure 1 shows

EVOLUTION OF THE PdÐTaÐH ALLOY STRUCTURE 1023

I 28 h (f)

1 h (a) 30.5 h (g)

3.5 h (b) 33 h (h)

6 h (c) 73.5 h (i) , rel. units I

11 h (d) 97 h (j)

25.5 h (e) 148 h (k)

67 68 69 67 68 69 2 ϑ, deg 2 ϑ, deg

Fig. 1. Time variations in position and shape of the (220) diffraction line after the second saturation. I is the initial state. Vertical lines indicate the position corresponding to the initial state (after the first saturation and subsequent relaxation for 173 h). diffraction curves and their deconvolutions for the highly nonequilibrium [5]. According to the theory of (220) maximum at different instants of time after the the glasslike state [1], the phase space of a nonequilib- second sample saturation. It is clearly seen that there is rium system is characterized by a complex energy pro- a complex structure of diffraction peaks which vary file in this case. This energy profile is dictated by the nonmonotonically in time. The parameters of the dif- initial defect structure and the atom distribution in the fraction curves are listed in the table. alloy, which are a result of a prior mechanical treatment, saturation with hydrogen, and subsequent relax- Since a homogeneous phase is represented by a bell- ation. The difference in time variation between the shaped diffraction peak, one can conclude that this type regions producing the (220) and (311) coherent-scatter- of diffraction pattern variation is a consequence of ing maxima and differing in orientation relative to the interconversions of several phases associated with sample surface is indicative of energy transfer from redistribution of Pd, Ta, and H atoms among different some degrees of freedom to others, which can give rise regions of the system. A characteristic feature of these to diffusion-flow turbulence [12]. structural changes is that they can be repeated but are not periodic. On the other hand, different widths of the deconvoluted components and their nonmonotonic 3. THEORETICAL MODEL variations with time imply that these processes are The nonmonotonic structural transformations accompanied by a nonmonotonic evolution of the described above can be explained if one assumes that defect structure and of the elastic fields produced by the saturation of a deformed PdÐTa alloy with hydrogen this structure. The presence of long-range fields, as well produces a set of metastable states corresponding to as the marked difference in the binding energy of different phases and defect structures. In this case, the hydrogen to the perfect and defect lattices of palladium migration of the system from one internal-energy min- [16], causes the state of the system under study to be imum to another, corresponding to the states mentioned

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1024 AVDYUKHINA et al.

Time dependence of the number of peaks of the (220) diffrac- direct and indirect. The former action is determined by tion maximum the conjugate field, while the latter depends on the control parameter. For example, in a synergetic model for Elapsed time Number Relative number Panel from saturation, h of peaks N of peaks n of Fig. 1 alloy ordering, the order parameter is the usual long- order parameter, the conjugate field corresponds to the 1.0 4 0.8 a difference in the chemical potential of the alloy compo- 3.5 2 0.4 b nents, and the control parameter is the difference in the 6.0 4 0.8 c unit-cell site occupation number of atoms of different 11.0 2 0.4 d species [19]. The usual dissipative regime of a phase transition is realized when the relaxation times of the 25.5 5 1 e order parameter and of the conjugate field are much 28.0 1 0.2 f longer than that of the control parameter. In the opposite 30.5 4 0.8 g extreme, the transition to the strange-attractor regime, 33.0 2 0.4 h rather than to an ordered state, occurs in the system 73.5 4 0.8 i when the steady-state value of the control parameter 97.0 2 0.4 j becomes larger than a certain critical value [20]. 148 2 0.4 k We will explain the nonmonotonic variation in the phase composition of a PdÐTaÐH alloy under the assumption that its saturation with hydrogen leads to above, will lead to the stochastic structural transforma- the formation of a complex internal-energy profile of tions observed experimentally. the states of the system with a great number of minima The essential point is that the structural evolution is separated by energy barriers (multivalley structure). nonmonotonic but is not periodic. The variations in This allows us to explain the critical slowing-down of structure bear a resemblance to the behavior of a the structural transformation in systems of the PdÐH strange attractor, which can be described in terms of the type [21]. This slowing-down is due to the hierarchic Lorenz model [17]. Using this model, we have structure of the energy profile: the deep minima are ser- explained [6] the nonmonotonic behavior of a two- rated, having profiles with a set of shallower minima, phase PdÐErÐH alloy. The parameters of the system which, in turn, have profiles with still shallower min- were taken to be the volume fraction of the Er-rich ima, and so on. In its time evolution, the system has to phase, concentration of Er atoms in this phase, and con- occupy the shallowest minima first, then deeper ones, centration of erbium-atom traps in the matrix. then still deeper ones, and so on, down to the deepest minimum, which determines the behavior of the system Such parametrization cannot be realized in a mulas a whole. tiphase PdÐTaÐH alloy, where the situation is much more complex. At first glance, it would be reasonable In our case, it will suffice to consider only one class not to describe the behavior of many phases but, of minima which differ in depth only slightly. The min- instead, to introduce parameters averaged over these ima are numbered by index α, running the values 1, 2, phases or to consider only the dominant phase sepa- …, N0, and the evolution of the system is described by rately. In this case, however, one does not take into the time-dependent probability distribution pα over account an essential feature of the evolution of a mul- these minima. The dependence of pα on time t can be tiphase PdÐTaÐH alloy, namely, the fact that not only found by solving the FokkerÐPlanck equation, which is the relationship among the volumes of the coexisting a very complicated problem [22]. However, as a prelim- phases and their composition but also their number vary inary, it will suffice to find integral quantities which in the alloy with time. Therefore, selection of the proper characterize the distribution pα and the thermodynamic parameters is of importance in order for a synergetic behavior of the system as a whole. Such quantities are model to describe the stochastic behavior of the PdÐTaÐ (a) the number N of internal-energy minima occupied H system. by the system at a given instant of time (this number According to the RuelleÐTakens theorem [18], such characterizes the halfwidth of the spread of probabili- a system exhibits a nontrivial behavior when the num- ties pα); (b) the entropy, characterizing the spread of the ber of the degrees of freedom characterized by the system over the minima, parameters of the system is not less than three. The behavior of a system in phase transitions is controlled Sp= Ð∑ α ln pα; (1) by a hydrodynamic mode, with its amplitude being an α order parameter whose magnitude is determined by the α thermostat. The characteristic feature of a self-organiz- and (c) the specific internal energy eα at a minimum , ing (synergetic) system is that both the influence of the from which the total specific energy can be found as thermostat on the selected subsystem and the backward action of the subsystem on the thermostat are of funda- ep= ∑ αeα. (2) mental importance. The backward action can be both α

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

EVOLUTION OF THE PdÐTaÐH ALLOY STRUCTURE 1025

The order parameter is defined as the ratio n = N/N0 is due to coupling between the relative number of min- of the number N of internal-energy minima occupied by ima n and internal energy e and causes the entropy s to the system to the total number of minima N0. Then, the increase, whereas the negative feedback in Eq. (5) is specific Edwards entropy s = S/N0, which characterizes due to correlation between the number of minima n and the disorder in the distribution over the minima α, is the entropy s and tends to decrease the internal energy e. conjugate field and the internal-energy density e is a In general, the set of differential equations (3)Ð(5) control parameter. cannot be solved analytically. These equations can be In order to describe the evolution of the system phe- conveniently expressed in terms of dimensionless vari- nomenologically, one should relate the rates dn/dt, ables n, s, e, and t, which are normalized to the respec- ds/dt, and de/dt of changes in the basic parameters to tive quantities the parameters n, s, and e themselves. The Lorenz model has the advantage that it corresponds to the sim- ()τ 1/2()τ 1/2 nm nm = sgs ege , smτ------, plest Hamiltonian in the corresponding microscopic ngn (6) representation [5]. The linear Lorenz equation has the ()τ Ð1()τ Ð1 τ form em = ngn sgs , n. dn n As a result, Eqs. (3)Ð(5) take the simple form ------Ð= ---- + g s, (3) dt τ n dn n ------Ð= ns+ , (7) where the first term on the right-hand side describes the dt Debye relaxation of n(t) to the value n = 0 with charac- ds teristic time τ and the second term (with positive coef- τ----- = Ðsne+ , (8) n dt ficient gn) is an increase in the number of internal- energy minima occupied by the system, which is de θ----- = ()EeÐ Ð sn, (9) accompanied by an increase in the entropy s that char- dt acterizes the spread over these minima. where we have also introduced the dimensionless The equations for the rates ds/dt and de/dt, in con- parameters trast to Eq. (3), contain nonlinear terms which describe the backward action of the selected subsystem on the τ τ e τ ===----s , θ ----e , E -----0 . (10) thermostat mentioned above (i.e., the feedback). The τ τ e change in the entropy is described by the equation n n m In the adiabatic regime τ, θ Ӷ 1, one can employ the ds s slaving principle, which asserts that the conjugate field -----= Ð ---τ- + gsne, (4) dt s s(t) and control parameter e(t) follow the order param- where the first term on the right-hand side describes the eter n(t) in their variations [18]. In this case, the left- Debye relaxation to the value s = 0 characterized by the hand sides of Eqs. (8) and (9) can be taken equal to τ zero, which gives relaxation time s. In accordance with the second law of thermodynamics, the nonlinear term with positive coef- En E s ==------, e ------. (11) ficient gs is the increase in entropy s due to the system 1 + n2 1 + n2 spreading over less deep minima of the internal energy. Finally, the equation for the rate of change in the These equations imply that, in accordance with the sec- internal energy is ond law of thermodynamics, the Edwards entropy monotonically increases and the internal energy de eeÐ 0 decreases with increasing number of phases. Eliminat------Ð= ------Ð g sn. (5) dt τ e ing the variable n between Eqs. (11), we obtain a simple e relation between the entropy and internal energy, Here, we have taken into account that the Debye relax- τ () ation with characteristic time e leads not to a zero value seEe= Ð . (12) of the internal energy e but rather to a finite value e0 Using the relation defining the temperature determined by the position of the system in the phase ∂ diagram and by the prior treatment of the alloy. The ≡ e T -----∂ , (13) nonlinear term with positive coefficient ge describes s negative feedback, which means that a redistribution we arrive at the expression over the internal-energy minima must lead to a decreased total energy e because the system goes over E Ð1 E T = Ð 1 Ð ------Ð. 1 (14) to deeper minima. It should be noted that the nonlinear 2e e terms are of fundamental importance; the behavior of the self-organizing system is a compromise between It follows from Eq. (14) that in the internal-energy their competing effects: the positive feedback in Eq. (4) range E/2 < e < E, the Edwards temperature T is nega-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1026 AVDYUKHINA et al. tive and the system becomes self-organizing. Indeed, system during its prior treatment exceeds the critical according to the definition of Eq. (13), if T < 0, any value em given by the next to last expression in Eq. (6). increase in the internal energy δe > 0 leads to a decrease From this expression, it follows that the systems char- in entropy δs < 0, i.e., to ordering. acterized by large positive linear (g ) and nonlinear (g ) n τ s By substituting the first of Eqs. (11) into Eq. (7), the coupling constants and by long relaxation times n and τ Lorenz equations are reduced to the LandauÐKhalatni- s are prone to stochastic behavior. kov equation The analysis performed above revealed that a stochastic structural transformation is a compromise ∂n ∂W between the competing effects of the order parameter -----∂ = Ð------∂ -, (15) t n on the conjugate field and on the control parameter of where the role of the free energy is played by the syn- the self-organizing system. In the case of the mul- ergetic potential tiphase PdÐTaÐH alloy considered above, the last two parameters are the specific entropy s and internal 1 2 E 2 W = ---n Ð --- ln()1 + n . (16) energy e and the competition between them is due to 2 2 the thermodynamic identity If the internal energy E acquired by the system during feTs= Ð , (19) its prior treatment is small, then the function W(n) has a minimum at n = 0 and increases monotonically; that where T is the effective Edwards temperature. This is, a multiphase state is synergetically unfavorable. As identity is derived from the definition of the Edwards the energy increases and exceeds the critical value E = temperature given by Eq. (13) and from the definition 1, the minimum of the synergetic potential W(n) is of its conjugate (entropy) shifted to the point ∂ f s = Ð------. (20) ∂T n0 = E Ð.1 (17) In this case, a dissipative relaxation process in which Recently, one of the present authors (A.I.O.) showed the number of phases in the system varies and becomes [23] that these definitions [and, therefore, Eq. (19)] fol- ≠ low from a simple field model in which the behavior of equal to n0 0 is favorable; the Edwards temperature given by Eq. (14) has a steady-state value a self-organizing system is described by three two- component fields: the order parameter, conjugate field, E Ð 1 and control parameter. The first components of these T = Ð------, (18) 0 1 Ð E/2 fields are reduced to the quantities considered above, namely, to the relative number of occupied minima n, which is negative for supercritical energy values E > 1 the entropy s, and the internal energy e of the noner- and decreases monotonically with increasing internal godic system. The second components are generalized energy acquired by the system during its prior treat- fluxes conjugate to the quantities indicated above: the ment. probability density flux q characterizing the redistribu- In the above-described adiabatic regime τ, θ Ӷ 1, tion over the internal-energy minima; the thermody- the dissipative system monotonically goes over to a namic force Ð∇f, which is the negative of the free- nonergodic steady state. In this paper, however, our energy gradient; and the negative of the temperature interest is in describing the nonmonotonic behavior; gradient Ð∇T. Changes in these components become therefore, we should assume that the characteristic- essential when the system is in a nonsteady state (such time ratios τ and θ are not small. In this case, the Lorenz as the state of a multiphase PdÐTaÐH alloy exhibiting equations cannot be solved analytically. However, an nonmonotonic behavior). There is reason to believe that analysis performed in [20] revealed that the regime of a the synergetic model proposed in this paper is a variant strange attractor of a self-organizing system (character- of the thermodynamic theory of highly nonequilibrium istic of the experimental situation in question) is real- systems, which is far from being complete [24, 25]. ized in the case of θ > τ > 1. Therefore, the internal- As shown above, the kinematic condition for the energy relaxation time must be longer than the relax- system to exhibit nonmonotonic behavior is that the ation times of the entropy and of the number of minima relaxation times of the internal energy and temperature occupied by the system. This condition is likely to be gradient exceed those of the entropy and free-energy satisfied in the experiment discussed above [12]. gradient, as well as the relaxation times for the number In addition to the inequality for the relaxation times of minima occupied by the system and for the conjugate indicated above, the condition E > 1 must be satisfied probability flux. This condition explains the anoma- for a strange attractor to occur [20]. Therefore, stochas- lously long periods of time over which the system tic changes in structure caused by redistribution of the exhibits stochastic behavior. The dynamic condition is states of the system over the energy minima corre- that an increase in entropy produce a noticeable sponding to different phases will occur only under the increase in the growth rate of the number of minima condition that the internal energy e0 acquired by the occupied by the system and, on the other hand, that the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EVOLUTION OF THE PdÐTaÐH ALLOY STRUCTURE 1027 decrease in the internal energy due to the redistribution 5. A. I. Olemskoi, Theory of Structure Transformations in of the system over these minima give rise to a signifi- Non-Equilibrium Condensed Matter (Nova Science, cant increase in the entropy itself. This condition can be New York, 1999). satisfied in the case of large values of the parameters gn 6. A. A. Katsnel’son, A. I. Olemskoœ, I. V. Sukhorukova, and gs, that is, in the case where the barriers separating and G. P. Revkevich, Usp. Fiz. Nauk 165 (3), 331 (1995) the energy minima are low and the internal-energy pro- [Phys. Usp. 38, 317 (1995)]; Vestn. Mosk. Univ., Ser. 3: file is not “hard.” Finally, the thermodynamic condition Fiz., Astron. 35 (3), 94 (1994). is that the system be far from the equilibrium state, 7. V. M. Avdyukhina, A. A. Katsnel’son, and G. P. Rev- which can be realized by subjecting the system to prior kevich, Poverkhnost, No. 2, 30 (1999). treatment, such as quenching, irradiation, or heavy 8. V. M. Avdyukhina, A. A. Katsnel’son, and G. P. Rev- deformation. In the case of the PdÐTaÐH alloy under kevich, Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 38 (3), study, the treatment employed was prior deformation 44 (1997). and saturation with hydrogen, which produced elastic 9. V. M. Avdyukhina, A. A. Katsnel’son, and G. P. Rev- stresses as high as roughly 10% of the characteristic kevich, Kristallografiya 44 (1), 49 (1999) [Crystallogr. elastic modulus. Therefore, one can assume that the Rep. 44, 44 (1999)]; Vestn. Mosk. Univ., Ser. 3: Fiz., Astron. 40 (5), 44 (1999). internal energy em acquired by the system during this treatment is also high. 10. V. M. Avdyukhina, A. A. Katsnel’son, A. I. Olemskoœ,

In closing, we note that the critical slowing-down of et al., Fiz. Met. Metalloved. 88 (6), 63 (1999). the structural transformation observed in the PdÐH sys- 11. A. V. Knyaginichev, Khan Kha Sok, V. M. Avdyukhina, tem [21] may also be expected to occur in a PdÐTaÐH et al., Fiz. Tverd. Tela (St. Petersburg) 43 (2), 200 (2001) alloy. Our preliminary data obtained after repeated [Phys. Solid State 43, 207 (2001)]. hydrogenation lend support to this conjecture. This 12. V. M. Avdyukhina, A. A. Katsnel’son, G. P. Revkevich, means that when the number of repeated hydrogena- et al., Al’tern. Énerg. Ékol., No. 1, 11 (2000). tions is large, the multivalley structure of the energy 13. V. M. Avdyukhina, A. A. Katsnel’son, N. A. Prokof’ev, proﬁle (with minima of equal depth) is transformed into and G. P. Revkevich, Vestn. Mosk. Univ., Ser. 3: Fiz., a multilevel hierarchical structure, which leads to the Astron. 39 (2), 94 (1998). slowing-down of the alloy evolution reported in [21]. 14. D. M. Vasil’ev, Diffraction Methods for Study of Struc- Another remark should be made with reference to tures (Metallurgiya, Moscow, 1957). the deformed multiphase PdÐMoÐH alloy, which also 15. M. A. Krivoglaz, Diffraction of X-rays and Neutrons in undergoes nonmonotonic structural transformations Nonideal Crystals (Naukova Dumka, Kiev, 1983). similar to those considered above [26]. Here, as in the 16. S. M. Myers, M. I. Baskers, and H. K. Birbaum, Rev. PdÐTaÐH alloy, a regular shift in the position of diffrac- Mod. Phys. 64 (2), 559 (1992). tion maxima occurred immediately after saturation 17. E. Lorenz, J. Atmos. Sci. 20, 130 (1963). with hydrogen, which is indicative of expansion and 18. H. Haken, Synergetics: An Introduction (Springer, Ber- subsequent anisotropic contraction of the crystal lat- lin, 1977; Mir, Moscow, 1980). tice. Thereafter, stochastic changes in the number, posi- 19. A. A. Katsnel’son and A. I. Olemskoœ, The Microscopic tion, width, and intensity of the components of diffrac- Theory of Inhomogeneous Structures (Mosk. Gos. Univ., tion maxima were observed. Therefore, it can be con- Moscow, 1987). cluded that the nonregular behavior of a highly 20. A. I. Olemskoœ and A. V. Khomenko, Zh. Éksp. Teor. Fiz. nonequilibrium multiphase system is inherent in struc- 110, 2144 (1996) [JETP 83, 1180 (1996)]. tural transformations of alloys saturated with hydrogen. 21. A. A. Katsnel’son, M. A. Knyazeva, and A. I. Olemskoœ, Fiz. Tverd. Tela (St. Petersburg) 41 (9), 1621 (1999) [Phys. Solid State 41, 1486 (1999)]; Fiz. Met. Metall- ACKNOWLEDGMENTS oved. 89 (6), 5 (2000). This study was supported by the Russian Founda- 22. H. Risken, The Fokker-Planck Equation (Springer, Ber- tion for Basic Research, project no. 99-02-16135. lin, 1989). 23. A. I. Olemskoi, cond-mat/0106270. REFERENCES 24. J. Marro and R. Dickmann, Nonequilibrium Phase Tran- sitions in Lattice Models (Cambridge Univ. Press, Cam- 1. J. P. Bouchaud, L. F. Gugliandolo, and J. Kurchan, in bridge, 1999). Spin Glasses and Random Fields, Ed. by A. P. Young 25. D. Helbing, cond-mat/0012229. (World Scientiﬁc, Singapore, 1998). 2. S. F. Edwards and A. Metha, J. Phys. (Paris) 50, 2489 26. V. M. Avdyukhina, A. A. Anishchenko, A. A. Kats- (1989). nel’son, and G. P. Revkevich, in Abstracts of RSNÉ-2001 Conference, Moscow, 2001, p. 133; Perspekt. Mater. 6, 3. S. F. Edwards, in Granular Matter: An Interdisciplinary 52 (2001). Approach, Ed. by A. Metha (Springer, New York, 1994). 4. A. I. Olemskoœ and I. A. Sklyar, Usp. Fiz. Nauk 162 (6), 29 (1992) [Sov. Phys. Usp. 35, 455 (1992)]. Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1028Ð1030. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 985Ð987. Original Russian Text Copyright © 2002 by Shaposhnikov, Gritsenko, Zhidomirov, Roger.

SEMICONDUCTORS AND DIELECTRICS

Hole Trapping on the Twofold-Coordinated Silicon Atom in SiO2 A. V. Shaposhnikov*, V. A. Gritsenko*, G. M. Zhidomirov**, and M. Roger*** * Institute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 13, Novosibirsk, 630090 Russia ** Boreskov Institute of Catalysis, Siberian Division, Russian Academy of Sciences, pr. Akademika Lavrent’eva 5, Novosibirsk, 630090 Russia *** DRECAM, SPEC, Orme des Merisiers CEA Saclay, Gif sur Yvette Cedex, 91191 France Received August 1, 2001

Abstract—The ability of a neutral diamagnetic twofold-coordinated silicon atom with two paired electrons (=Si: silylene center) in SiO2 to capture charge carriers is investigated by the ab initio density-functional method. It is found that this defect is a hole trap in SiO2. Hole trapping brings about the formation of paramagnetic twofold-coordinated silicon atoms with an unpaired electron =Siá. According to this prediction, the silylene center and the siliconÐsilicon bond can be responsible for the accumulation of the positive charge in metalÐoxideÐsemiconductor structures under ionizing radiation. © 2002 MAIK “Nauka/Interperiodica”.

The understanding of the trap nature in amorphous center. However, some authors also attributed the B2 SiO2 is a key factor in the development of reliable radi- absorption band to the SiÐSi bond [7, 11]. ation-resistant metalÐoxideÐsemiconductor devices. Considerable theoretical and experimental efforts have A blue band at 2.7 eV was observed for a SiO2 ther- + + + been made to investigate the atomic structure of intrin- mal layer on Si [12], B -, P -, and Ar -implanted SiO2 sic defects in SiO2 [1Ð3]. Four of the most important thermal layers [13, 14], chemically deposited SiO2 neutral intrinsic defects in SiO2 have been studied to [15], and an oxide prepared by oxygen implantation date. These are the ≡SiÐSi≡ bond or the oxygen into Si [16]. The 2.7-eV blue band was also observed in ≡ vacancy [2Ð4], the nonbridging oxygen SiOá [1, 4Ð6], the luminescence spectrum of the SiOxNy silicon oxyni- the superoxide radical ≡SiOOá [6], and the twofold- tride [17]. The origin of this band in all the above cases coordinated silicon atom with two unpaired electrons can be associated with the silylene center. (silylene center) =Si: [7, 8]. We carried out ab initio calculations in the frame- Here, symbols (Ð) and (á) denote the chemical bond work of the density-functional theory (DFT) according to the Amsterdam Density Functional (ADF) program and an unpaired electron, respectively. At present, the optical (absorption and luminescence) and magnetic [18]. All the calculations were performed in the cluster (electron paramagnetic resonance) properties of these approximation. The structure of SiO2 was simulated defects have been investigated in sufﬁcient detail. How- using fragments of crystalline α-quartz. Dangling ever, the ability to localize (or to capture) electrons and bonds at the cluster boundary were saturated with holes has been analyzed only for the SiÐSi bond. It was hydrogen. In order to simulate the bulk of SiO2 and the shown that the SiÐSi bond can capture a hole through =Si: defect, we used the Si5O16H12 and Si3O8H6 clus- the reaction ≡SiÐSi≡ + hole ≡Si+áSi≡ with the for- ters, respectively. mation of a positively charged paramagnetic E' center [3, 9, 10]. In the present work, we investigate the ability The KohnÐSham molecular orbitals were constructed using Slater atomic orbitals. The basis set of the twofold-coordinated silicon atom in SiO2 to capture electrons and holes. involved the double-zeta basis set with 3d polarization functions for all the silicon atoms, which corresponds Skuja [7] experimentally observed the absorption to the basis set III in the ADF terminology [18]. The band at an energy of 5.0 eV (B2 absorption band) for location of all the Si and O atoms was optimized using SiO2. Excitation into this band leads to the excitation of a gradient-corrected DFT potential that included the luminescence with energies of 2.7 eV (blue band) and form for the exchange term [19] and the form for elec- 4.4 eV (ultraviolet band). According to [7, 8], the B2 tron correlation [20]. The positions of hydrogen atoms absorption band of SiO2 can be assigned to the silylene were ﬁxed. The energy gain due to charge carrier trap-

HOLE TRAPPING ON THE TWOFOLD-COORDINATED SILICON ATOM 1029 ping on a defect was determined from the following E, eV formulas: –30 –20 –10 0 ∆ e ()0 Ð ()Ð 0 E = Ebulk + Edef Ð Ebulk + Edef , (1) Si L2, 3

∆ h ()0 + ()+ 0 E = Ebulk + Edef Ð Ebulk + Edef . (2)

0 Ð + 80 100 Here, Ebulk , Ebulk , and Ebulk are the energies of neutral, negatively charged, and positively charged clusters Si 0 Ð + K which simulate the bulk and Edef , Edef , and Edef are the energies of neutral, negatively charged, and positively charged clusters which simulate the defect. In the case 1820 1840 of negative ∆Eh (∆Ee), the capture of a hole (electron) X-ray emission spectra is energetically favorable. O K With the aim of verifying the reliability of the cluster model used for the simulation of the electronic structure of the SiO2 bulk, theoretical x-ray emission 510 530 spectra, namely, the Si K, Si L2, 3, and O K spectra, were calculated and compared with the experimental data. The calculated discrete spectrum was broadened using Experimental (thick line) and theoretically calculated (thin line) x-ray emission Si K, Si L , and O K spectra of SiO . a Lorentzian curve (0.5 eV in width) in order to obtain 2, 3 2 a continuous spectrum. As follows from the results of calculations, better energy of electron polarization of the lattice can be esti- agreement with the experiment can be achieved with mated from the relationship the inclusion of Si 3d polarization functions in the basis []ε2 ()π []Ð1 εÐ1 set. The calculations also demonstrated that the contri- E p = Ð q /8 R 0 Ð ∞ , (3) bution of cross transitions must be taken into account in ε ε the case of the Si L2, 3 spectrum, even though these tran- where 0 is the permittivity of free space and ∞ is the sitions can be ignored for the Si K and O K emission permittivity of the medium. From relationships (1) and spectra. The ﬁgure shows the results of the calculations (2), we obtain the following expression for the correc- in comparison with experimental data for the Si K, Si tion to the energy gain ∆Eh: L2, 3, and O K spectra of thermally grown SiO2 on sili- ()2 π []εÐ1 Ð1 []Ð1 εÐ1 con [21]. By and large, the calculated and experimental E p = Ð q /8 Rdef Ð Rbulk 0 Ð ∞ . (4) spectra are in good agreement. Here, 2R ~ 5.5 Å and 2R ~ 8 Å correspond to the In order to answer the question as to whether the def bulk sizes of the defect and bulk clusters used in our calcu- silylene center in SiO2 can capture an electron and (or) ε ε lations. For SiO2, we accept ∞ ~ 2.25 0 and obtain a hole, we calculated the binding energy for negatively ∆ and positively charged clusters. The results obtained Ep = Ð0.45 eV. This estimate shows that the correction indicate that electron trapping on the =Si: defect is to the energy gain for lattice polarization is relatively energetically unfavorable. For hole trapping, the energy small compared with the energy gain due to local elec- gain is equal to Ð3.2 eV. Moreover, the calculations pre- tronic and atomic relaxation. dict that the silylene center can capture a hole in SiO2 It is generally believed that the accumulation of the through the reaction =Si: + h =Siá. According to positive charge in metalÐoxideÐsemiconductor devices our prediction, the capture of a hole by a neutral dia- is associated with hole trapping on a neutral SiÐSi bond magnetic silylene center leads to the formation of the with the formation of the positively charged E' center positively charged paramagnetic atom of twofold-coor- [3, 9, 10]. However, the experiments have revealed a dinated silicon with an unpaired electron =Siá. continuous distribution of hole traps in SiO2 in the energy range from 1.1 to 2.2 eV with two well-resolved However, our model ignores the long-range Cou- peaks at E1 ~ 1.2 eV and E1 ~ 1.9 eV [22]. This two- lomb polarization induced in the lattice by a positively peak structure can be due to the presence of traps of two charged defect. The correction to the energy gain ∆Eh types. Our calculated values (~1 and ~3.2 eV) were for lattice polarization can be estimated in the frame- obtained for the trapping energies of the SiÐSi bond and work of the classical Born model in the following way. the silylene center, respectively. Taking into account the Under the assumption that the charge of a trapped car- error in our theoretical calculations, we assume that rier is distributed within a sphere of radius R, the total these values correspond qualitatively to these two

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1030 SHAPOSHNIKOV et al. experimental peaks. The veriﬁcation of this hypothesis 8. G. Pacchioni and R. Ferrario, Phys. Rev. B 58 (10), 6090 calls for further experimental investigation. (1998). 9. J. K. Rudra and W. B. Fowler, Phys. Rev. B 35 (15), 8223 We assume that the silylene center in SiO2 and the SiÐSi bond can be responsible for the accumulation of (1987). the positive charge in metalÐoxideÐsemiconductor 10. T. E. Tsai, D. L. Griscom, and E. J. Frieble, Phys. Rev. structures under ionizing radiation and injection of hot Lett. 61 (4), 444 (1988). holes. According to the data obtained, hole trapping on 11. R. Tohmon, Y. Shimogaichi, H. Mizuno, et al., Phys. the silylene center leads to the formation of the para- Rev. Lett. 62 (12), 1388 (1989). 12. W. C. Choi, M. S. Lee, E. K. Kim, et al., Appl. Phys. magnetic =Siá center. This effect can be revealed in Lett. 69 (22), 3402 (1996). experiments on electron paramagnetic resonance. 13. H. Nishikawa, E. Watanabe, D. Ito, et al., J. Appl. Phys. 78 (2), 842 (1995). ACKNOWLEDGMENTS 14. B. Carrido, J. Samitier, S. Bota, et al., J. Appl. Phys. 81 (1), 126 (1997). This work was supported by the International Asso- 15. T. Kanashima, R. Nagayoshi, M. Okuyama, and ciation of Assistance for the promotion of cooperation Y. Hamakawa, J. Appl. Phys. 74 (9), 5742 (1993). with scientists from the New Independent States of the 16. M. Martini, F. Meinardi, E. Rosetta, et al., IEEE Trans. former Soviet Union, project INTAS no. 97-0347. Nucl. Sci. 45 (7), 1396 (1998). 17. V. A. Gritsenko, Yu. G. Shavalgin, P. A. Pundur, et al., REFERENCES Philos. Mag. B 80 (10), 1857 (2000). 1. T. E. Tsai and D. L. Griscom, Phys. Rev. Lett. 67 (8), 18. C. Fonseca Guerra, J. G. Snijders, G. Te Velde, and 2517 (1991). E. J. Baerends, Theor. Chem. Acc. 99 (2), 391 (1998). 2. V. A. Gritsenko, J. B. Xu, R. W. M. Kwok, et al., Phys. 19. A. D. Becke, Phys. Rev. A 38 (6), 3098 (1988). Rev. Lett. 81 (5), 1054 (1998). 20. C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37 (2), 785 3. G. Pacchioni and G. Ierano, Phys. Rev. Lett. 79 (4), 753 (1988). (1997). 21. V. A. Gritsenko, R. M. Ivanov, and Yu. N. Morokov, 4. G. Pacchioni and G. Ierano, Phys. Rev. B 57 (2), 818 Zh. Éksp. Teor. Fiz. 108 (12), 2216 (1995) [JETP 81, (1998). 1208 (1995)]. 5. L. N. Skuja, J. Non-Cryst. Solids 179 (3), 51 (1994). 22. S. L. Miller, D. M. Fleetwood, and P. J. McWhorter, Phys. Rev. Lett. 69 (5), 820 (1992). 6. E. J. Friebele, D. L. Griscom, M. Stapelbroek, and R. A. Weeks, Phys. Rev. Lett. 42 (20), 1346 (1979). 7. L. N. Skuja, J. Non-Cryst. Solids 239 (6), 16 (1998). Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1031Ð1034. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 988Ð991. Original Russian Text Copyright © 2002 by Golubkov, Parfen’eva, Smirnov, Misiorek, Mucha, Jezowski.

SEMICONDUCTORS AND DIELECTRICS

Lattice Thermal Conductivity of Compounds with Inhomogeneous Intermediate Rare-Earth Ion Valence A. V. Golubkov*, L. S. Parfen’eva*, I. A. Smirnov*, H. Misiorek**, J. Mucha**, and A. Jezowski** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] ** Institute of Low-Temperature and Structural Research, Polish Academy of Sciences, Wroclaw, 50-950 Poland Received August 21, 2001

Abstract—The thermal conductivity κ (within the range 4Ð300 K) and electrical conductivity σ (from 80 to 300 K) of polycrystalline Sm3S4 with the lattice parameter a = 8.505 Å (with a slight off-stoichiometry toward Sm2S3) are measured. For T > 95 K, charge transfer is shown to occur, as in stoichiometric Sm3S4 samples, by the hopping mechanism (σ ~ exp(Ð∆E/kT) with ∆E ~ 0.13 eV). At low temperatures [up to the maximum in the κ κ 2.6 κ Ð1.2 lattice thermal conductivity ph(T)], ph ~ T ; in the range 20Ð50 K, ph ~ T ; and for T > 95 K, where the κ Ð0.3 hopping charge-transfer mechanism sets in, ph ~ T and a noticeable residual thermal resistivity is observed. It is concluded that in compounds with inhomogeneous intermediate rare-earthion valence, to which Sm3S4 belongs, electron hopping from Sm2+ (ion with a larger radius) to Sm3+ (ion with a smaller radius) and back generates local stresses in the crystal lattice which bring about a change in the thermal conductivity scaling of κ Ð1.2 Ð0.3 ph from T to T and the formation of an appreciable residual thermal resistivity. © 2002 MAIK “Nauka/Interperiodica”.

Researchers in many laboratories of the world have tron hopping between them. Such systems bear the still not lost interest in the behavior of thermal conduc- name of materials with static MV. tivity in systems with mixed (intermediate) valence (MV) [1Ð3]. The MV phenomenon is accounted for by Materials with homogeneous MV are exemplified the rare-earth (RE) ions in these compounds. by SmB6, SmS (under hydrostatic pressure), and Sm1 − xLnxS (Ln = Gd, Y, etc.); those with inhomoge- All intermediate-valence materials can be divided neous MV, by Sm X (X = S, Se, Te), etc., and static MV into three groups according to the crystallographic 3 4 positions the RE ions occupy, as well as to the nature of compounds, by Eu3O4, etc. their electronic relations [2]. Let us see how the homogeneous, inhomogeneous, The first two of these groups (1a and 1b) combine and static MVs become manifest in the total thermal κ κ κ compounds whose RE ions sit at crystallographically conductivity tot(T) = ph + e of a compound (where κ κ equivalent sites in the lattice. In materials of group 1a, ph and e are the lattice and electronic components of the 4f electrons transfer between RE-ion configurations the thermal conductivity, respectively). with different average numbers of f electrons at the center, i.e., with different valence [transition between The literature does not mention anything unusual in n n Ð 1 κ the 4f and 4f + (sd)]. The frequency of these tran- the behavior of ph in materials with homogeneous MV; 15 κ sitions does not depend on temperature and is ~10 Hz. they exhibit only some features in e(a nontypical tem- These materials are called compounds with homoge- perature dependence and magnitude of the Lorenz neous MV. number) [1]. Systems with static MV behave as usual classic crystalline materials and do not reveal any fea- Materials of group 1b exhibit electron hopping κ κ between cations in different valence states with a tem- tures in e and ph associated with the MV of their RE perature-dependent frequency from zero to ~1011 Hz (at ions. 300 K). Such systems are referred to as materials with There are practically no publications on the thermal inhomogeneous MV. conductivity of materials with inhomogeneous MV. Finally, the third group comprises compounds in The purpose of this work was to study whether or not which cations in different valence states occupy electron hopping between ions with different valence κ inequivalent crystallographic lattice sites, with no elec- affects ph.

1032 GOLUBKOV et al. Because of σ being small, the magnitude of κ of (b) (a) e 8.55 Sm3S4 is negligible; thus, what we measure in an exper- 10 κ κ 8.53 Sm2S3 – Sm3S4 iment is tot = ph. 1 8.51 Sm3S4 + SmS The polycrystalline Sm3S4 sample was prepared (as

8 Å , Two-phase in [4]) by rf melting in a tantalum crucible which was a 8.49 region placed in a sealed molybdenum container. X-ray dif- 6 8.47

, W/m K fraction analysis showed the sample to be single-phase

tot 8.45 with a well-deﬁned crystal structure and a lattice con- κ 4 0.97 0.99 1.01 x stant a = 8.505 Å. According to [5], this value of a cor- Sm2S3 Sm3S4 responds not to the stoichiometric formula Sm3S4 2 SmxS1.5 SmxS1.333 (SmS1.333) but rather to a composition shifted slightly toward Sm S ; as a result, our sample turned out to have 0 2 3 0 50 100 150 200 250 300 the composition Sm0.995S1.333 (Fig. 1a). T, K κ κ The measurements of tot = ph (within the range 4Ð 300 K) and of σ (from 80 to 300 K) were carried out on

Fig. 1. (a) Dependence of the lattice constant of Sm3S4 a setup similar to that used in [6]. (SmxS1.333) on x [5]. (1) The a = 8.505 Å value corresponds Figures 1b and 2 present experimental temperature κ to the sample studied. (b) Temperature dependence of tot = dependences of κ and σ for Sm S . The data for κ tot 0.995 1.333 ph of Sm0.995S1.333. κ tot obtained in the range 80Ð3000 K agree fairly well κ with our earlier measurements of in Sm3S4 [7]. The σ electrical conductivity of Sm0.995S1.333 grows exponentially in the range from 110 to 300 K: σ ~ exp(−∆E/kT), with ∆E ~ 0.13 eV (Fig. 2). The temper- 100 1 ature interval within which σ shows activated behavior 2 and the magnitude of the activation energy coincide fairly well with those obtained on stoichiometric sam- σ ples of Sm3S4 [2, 3] (see the (T) curve in Fig. 2 relat-

–1 ing to one of the Sm3S4 samples studied in [3]). This shows that charge transfer in Sm S , as in stoichi- cm 10–2 0.995 1.333 –1 3 ometric Sm3S4, occurs by the hopping mechanism. Ω , κ σ Figure 3 presents a logÐlog plot of ph(T), and Fig. 4a shows the temperature dependences of the ther- κ mal resistivity Wph = 1/ ph for Sm0.995S1.333. T0 10–4 Consider again Fig. 1b. It would seem that the dependence of thermal conductivity on temperature κ should not have any anomalies, because ph(T) is a κ smooth curve. It was found, however, that the ph(T) 2 4 6 8 10 12 14 relation plotted in the coordinates of Figs. 3 and 4a does 103/T, K–1 exhibit a number of interesting features. As is evident from Fig. 3, in the low-temperature σ 3 Fig. 2. log vs. 10 /T plot for the Sm0.995S1.333 sample domain [up to the maximum in κ (T)], κ ~ T 2.6, as is studied. (1, 2) Direct (300Ð80 K) and reverse temperature ph ph runs, respectively, and (3) data for Sm S (a = 8.5396 Å) the case with most of the crystalline materials, and for 3 4 κ Ð1.2 from [3]. temperatures from 20 to 50 K, ph ~ T , which is a behavior likewise characteristic of sufﬁciently perfect κ Ð1 solids, for which theory suggests ph ~ T . As the tem- For the subject of our study, we chose Sm3S4, a com- κ Ð0.3 perature is raised still further (T > T0), ph ~ T , which pound that crystallizes in the Th3P4 cubic structure. The is a feature characteristic of heavily defected materials. 2+ 3+ ≥ divalent (Sm ) and trivalent (Sm ) samarium ions are It thus appears that for T T0, a new, fairly efﬁcient statistically distributed over the equivalent lattice sites phonon scattering mechanism becomes operative. This in a 1 : 2 ratio (Sm2+Sm3+S2Ð ). Sm S is an n-type semi- conclusion is also borne out by the data in Fig. 4. It is 2 4 3 4 well known that the residual thermal resistivity of per- conductor with a fairly high carrier concentration and 0 low electrical conductivity, which grows exponentially fect solids Wph = 0 and, conversely, defected materials with temperature (with an activation energy ~0.13 eV) exhibit a fairly high residual thermal resistivity due to the hopping mechanism of charge transfer. (Fig. 4b).

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 LATTICE THERMAL CONDUCTIVITY OF COMPOUNDS 1033

10 (a) 1.0

Ӎ T0 95 K

0.8

T–1.2 0.6

2.6 , m K/W T ph W

0.4

, W/m K (b) 1 ph Wph Θ κ 2 2 0.2 0 Wph T0 0 T 0 T–0.3 0 50 100 150 200 250 300 T, K

1 Fig. 4. (a) Temperature dependence of the lattice thermal κ resistivity (Wph = 1/ ph) of the Sm0.995S1.333 sample studied, and (b) schematic of the temperature dependence of thermal resistivity of (1) a defect-free sample and (2) a sam- 0.7 0 ple with defects. Wph is the residual thermal resistivity for a defected sample, and Θ is the Debye temperature. 5 10 20 50 100 200 300 T, K

κ Fig. 3. Temperature dependence of ph for the The ionic radius of Sm2+ is considerably larger than Sm0.995S1.333 sample studied. that of Sm3+. Electron hopping from Sm2+ to Sm3+ changes their ionic radii, and this results in a local lattice rearrangement generating stresses (compressive The above is illustrated by the experimental Wph(T) and tensile) around these ions. The lattice breathes, as relation for an Sm0.995S1.333 sample. The behavior of it were, and cannot apparently relax completely in one Wph in the range 22Ð55 K remains, however, puzzling. electron hopping cycle. The local stresses thus created It is unclear why W tends to zero, i.e., W0 = 0 (as possibly act as additional phonon scatterers; this ph ph accounts for the formation of the noticeable thermal though the crystal becomes perfect at low tempera- resistivity. tures), although one could expect a ﬁnite residual ther- One may thus conclude that electron hopping from mal resistivity due to the presence of Sm vacancies in Sm2+ to Sm3+ in compounds with inhomogeneous MV, the sample, because Sm0.995S1.333 is not a stoichiometric to which Sm3S4 belongs, generates local stresses in the composition but is shifted toward Sm2S3. lattice, resulting in a change of the temperature dependence from κ ~ TÐ1.2 to κ ~ TÐ0.3 and in an apprecia- At T = T0 (Fig. 4b), as already mentioned, some new ph ph mechanism of phonon scattering starts to operate. What ble thermal resistivity. could its nature be? ACKNOWLEDGMENTS The results displayed in Fig. 2 for Sm0.995S1.333 and an analysis of published data permit one to conclude The authors express gratitude to N.F. Kartenko and that the new phonon scattering mechanism becomes N.V. Sharenkova for x-ray structural measurements. active after the charge transfer by hopping has become This work was conducted within a bilateral agree- a dominant process. ment between the Russian and Polish Academies of

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1034 GOLUBKOV et al.

Sciences and was supported by the Russian Foundation 3. L. N. Vasil’ev, V. V. Kaminskiœ, and M. V. Romanova, for Basic Research (project no. 99-02-18078) and the Fiz. Tverd. Tela (St. Petersburg) 38 (7), 2034 (1996) Polish Committee for Scientiﬁc Research (grant [Phys. Solid State 38, 1122 (1996)]. no. 2PO3B 129-19). 4. A. V. Golubkov, T. B. Zhukova, and V. M. Sergeeva, Izv. Akad. Nauk SSSR, Neorg. Mater. 2, 77 (1966). 5. E. Kaldis, J. Less-Common Met. 76, 163 (1980). REFERENCES 6. A. Jezowski, J. Mucha, and G. Pompe, J. Phys. D 20, 1500 (1987). 1. I. A. Smirnov and V. S. Oskotski, in Handbook on Physics and Chemistry of Rare Earth, Ed. by K. A. Gsh- 7. I. A. Smirnov, L. S. Parfen’eva, V. Ya. Khusnutdinova, neidner, Jr. and L. Eyring (Elsevier, Amsterdam, 1993), and V. M. Sergeeva, Fiz. Tverd. Tela (Leningrad) 14 (9), Vol. 16, p. 107. 2783 (1972) [Sov. Phys. Solid State 14, 2412 (1972)]. 2. B. Battlog, E. Kaldis, A. Schlegel, et al., Solid State Commun. 19, 673 (1976). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

SEMICONDUCTORS AND DIELECTRICS

Effect of the Isotope Composition on the Energy Bands in a Semiconductor: Universal Relation for Monatomic Crystals A. P. Zhernov Kurchatov Institute Russian Research Center, pl. Kurchatova 1, Moscow, 123182 Russia Received April 13, 2001; in ﬁnal form, September 27, 2001

Abstract—The effect of the isotopic composition of a semiconductor compound on its energy band structure is considered. The role of the changes in the lattice unit-cell volume and of the renormalization of the electronÐ phonon interaction that are caused by isotopic-composition variations is discussed. A universal relation describing the dependence of the energy bands on isotopic composition and temperature is derived for monatomic crystals in the virtual-crystal approximation. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 〈 〉 k ents replaced by their mean values M = Mc = Considerable recent attention has been given to the ∑ck Mk , where k specifies atoms in the unit cell and ck study of the properties of chemically pure and structur- i i ally perfect semiconductor single crystals of different i isotopic content. There has been a great deal of work is the concentration of the ith isotope of the given ele- devoted to the study of classical, commercially impor- ment. It is also assumed that the relative mean square 〈 2〉 〈 〉2 〈 〉2 tant monatomic semiconductors such as diamond, sili- fluctuation in the atomic mass G2 = ( M Ð M )/ M con, and germanium. That work was favored by the is small. In the virtual-crystal model, the isotopic effect availability of fabricated, virtually defect-free bulk iso- can take place directly or through anharmonic phononÐ tope-rich single crystals of 12C, 13C, 28Si, and 70Ge and phonon and electronÐphonon interactions. 76Ge, as well as of crystals with an isotopic composition different from the natural abundance ratio (see, e.g., [1Ð In this paper, we consider the effect of the isotopic 5]). We note that diamond crystals of different isotope composition on the structure and position of energy composition have been grown in a laboratory of the bands in a semiconductor within the virtual-crystal General Electric Company (USA) and that germanium model in the quasi-harmonic approximation. crystals are the result of joint efforts of research groups at the Molecular-Physics Institute of the Russian It is well known that the temperature dependence of Research Center Kurchatov Institute (Russia) and at the the energy bands and of the optical characteristics of a Lawrence National Laboratory (Berkeley, USA). Sili- crystal is due to two basic effects. First, the En, f bands con crystals of high isotopic content are the result of depend on the unit-cell volume of the crystal, which collaborations of Russian, German, and Japanese varies because of thermal expansion of the lattice, and, researchers. second, the energy En, f varies with increasing tempera- The effects of a variation in the isotopic composition ture T because of electronÐphonon interaction (EPI). of a compound can be linear in the isotopic-mass differ- Due to EPI, the contribution from the elastic channel is ence (first-order effects) or proportional to the mean renormalized, because the true amplitude of electronÐ square fluctuation in the atomic mass (second-order ion interaction contains the dynamic DebyeÐWaller effects). The first-order effects can reveal themselves in (DW) factor in addition to the static component. The static and thermodynamic properties. Strong first- and contributions from inelastic intraband and interband second-order effects should be observed in the behavior EPI processes are also renormalized [9, 10]. of the kinetic parameters and optical spectra (see, e.g., [6Ð8]). Therefore, when the isotopic composition is varied As a first approximation, one can describe harmonic and the phonon spectrum is distorted, the energy bands phonon modes in terms of the virtual-crystal model. In should also be markedly affected. The effect of the iso- this case, a real lattice with randomly distributed isotopic composition on the energy bands (isotope shift in topes is replaced by a lattice without isotopic disorder, energy) is due to a change in the unit-cell volume and but with the atomic masses of the compound constitu- to the EPI renormalization (the corresponding contribu-

1036 ZHERNOV tions will be designated by indices DΩ and EP, respec- 2. THE DEPENDENCE OF THE ENERGY BANDS tively). At some temperature T, we can write ON THE ISOTOPE COMPOSITION AND TEMPERATURE ∂E ∂E ∂E ∂E 2.1. Volume Changes ∂------==∂------∂------+ ∂------. M M M tot M DΩ M EP Let us find the renormalization of the energy band E due to the change in the lattice unit-cell volume associ- It should be noted that in recent years, the effect of ated with phonon spectrum distortions that are pro- the isotopic composition on the electronic band struc- duced by variations in the isotope composition. For this ture has been studied experimentally in considerable purpose, we calculate the derivative of the energy with detail by Cardona and coworkers (the relevant refer- respect to the atomic mass of a compound constituent ences are cited in Section 4). As for theoretical papers for a fixed value of temperature. We have (see also [14]) on the subject under discussion, the present author ∂E ∂E ∂lnΩ ------= ÐB ------, (1) knows of only three such publications. In [11], calcula- ∂ k ∂P ∂ k tions were carried out for carbon and germanium using Mc DΩ V Mc P the NLPP method for electrons and the bond charge where Ω is the unit-cell volume of the crystal, B = − ∂ ∂ ∂ ∂ model for phonons; the changes in the renormalized V/( P/ V)T is the bulk modulus, and ( E/ P)V charac- interband transition energies produced by a change in terizes the pressure dependence of the energy band. We the isotopic composition and associated with the term note that the bulk modulus B and the coefficient ∂ ∂ (∂E/∂P) usually depend on T only slightly. ( E/ M)EP were found. In [12, 13], the contributions V ∂ ∂ ∂ ∂ from both terms ( E/ M)DΩ and ( E/ M)EP were calcu- In the quasi-harmonic approximation, lated for Ge, GaAs, and ZnSe using the NLPP and 1 Ω()T = Ω+ --- ∑γ ()l ⑀()l , (2) LCAO methods; the phonon modes were described in 0 B terms of the bond charge model, as well as in the shell l Ω model and the rigid-ion model. where 0 is the unit-cell volume of the “frozen” (static) lattice; the index l = {q, j} specifies vibrational modes, In this paper, we show that there are universal rela- with q and j being the quasi-momentum and polariza- tions that relate interband transition parameters for tion index of a phonon mode, respectively; γ(l) is the monatomic crystals of different isotope composition at partial Grüneisen constant for the lth vibrational mode any temperature. These relations allow one to find the (these constants account for the fact that the depen- ω parameters of isotope-enriched crystals when data for dences of mode frequencies c(l) on the unit-cell vol- crystals of natural isotope composition is given. ume are different); and ⑀(l) is the contribution from this mode to the thermal energy. We have In Section 2, we summarize the published results of ∂ω ()∂Ω studies of the influence of the temperature and isotopic γ () c l / ⑀() បω () () 1 l = Ð------ω ()Ω , l = c l nl +,--- (3) composition on the energy band structure. The effects c l / 2 of both volume changes and EPI are considered. A uni- where n(l) is the BoseÐEinstein factor. We note that the versal relation describing the dependence of the energy ω frequencies c(l) are functions of the volume, temper- bands of monatomic crystals on isotopic composition ature, and atomic masses. and temperature is derived in Section 3. This relation k The derivative (∂E/∂M ) Ω given by Eq. (1) is posi- has a simple form, because in such crystals, the polar- D tive definite as a rule, because the unit-cell volume Ω ization vectors of phonon modes are independent of the usually decreases with increasing isotope mass. atomic mass and the mass dependence of the phonon k Ð1/2 ∂ ∂ frequencies is ω(l) ~ M . In the case of polyatomic Given the quantity ( E/ Mc )DΩ, one can directly crystals, the situation is much more complicated, calculate the difference in the energy band of crystals because the isotope shifts in frequency are proportional k k ∆ k with atomic masses Mc and Mc + M from the for- to the square of the magnitude of the corresponding mula polarization vector (see Appendix), which, in turn, ()∆∂ ˜ DΩ ()∆k DΩ()∆k ∆ k DΩ()k depends on the atomic masses. This case will be ana- E M = E Mc + M Ð E Mc lyzed separately. In Section 4, we discuss the data ∂E k (4) (obtained by linear and nonlinear spectroscopy meth- = ------∆M . ∂ k ods) on the effect of the composition on the isotopic Mc DΩ energy shifts of direct and indirect interband electron From Eqs. (1) and (4), it follows that the energy in the transitions and on the position of critical points in the band increases with increasing isotope mass because optical spectra. the crystal volume decreases.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EFFECT OF THE ISOTOPE COMPOSITION ON THE ENERGY BANDS 1037

It should be noted that for monatomic semiconduc- Let us consider how the electronic structure is tors C, Si, and Ge, the quantities involved in Eq. (4) affected by the linear and quadratic terms (in dynamic have been investigated in detail. The values of coeffi- atomic displacements) H and H , respectively, that are ∂ ∂ 1 2 cients such as ( E/ P)T can be found in [10]. As for the present in the electronÐion interaction Hamiltonian. By ∂ Ω ∂ k quantity ( ln / M )V, its behavior in a wide range of definition, we have parameter values was investigated theoretically within ∂ microscopical models, e.g., in [15Ð18]. V k α H ==H + H ∑------u , int 1 2 ∂ ()α0 m k From the above discussion, it follows that the energy m, k Rm (6) renormalization due to the change in the unit-cell vol- ∂2 ume is given by 1 V k α β … + --- ∑ ------()α ()βum, kum', k' + , 2 ∂R 0 ∂R 0 ∂ Ω() Ω m, k; m' m m' DΩ Ef, n T Ð 0 ∆E , ∝ ÐB ------f n ∂P Ω 0 α β V 0 where Vk = Vk(r Ð Rm ) and and are Cartesian coor- (5) ∂ dinates. Ef, n γ ()⑀() = Ð------∂ - ∑ l l . In the adiabatic approximation [to the order of P V 1/2 l ~(me/M) ], one can consider the atomic displacements to be classical quantities when calculating the renor- ∆ DΩ This formula for Ef, n is used in Section 3 when deriv- malized spectrum. In this case, using the time-indepen- ing a universal relation for the electronic energy bands. dent perturbation theory, the energy renormalization of a particle in the state |f, n〉 can be written as ∆ EP(){} 〈〉,, 2.2. The DebyeÐWaller Factor and Inelastic Ef, n um, k = f nH1 + H2 f n ElectronÐPhonon Scattering 〈〉,,2 (7) f ' n' H1 f n In this subsection, we summarize the published + ∑ ε------() ε() η. n f Ð n' f ' + i results of investigations of the influence of the temper- f,,n ≠ f' n' ature and isotope composition on EPI and on the energy 〈 〉 band renormalization caused by this interaction. In the Next, we perform thermal ensemble averaging … of virtual-crystal approximation, this energy renormaliza- Eq. (7) over small dynamic thermal atomic displace- tion was calculated in [9]. In fact, Allen and Heine [9] ments, which allows one to directly find the tempera- laid the foundation for the theory of the temperature ture dependence of the energy bands, i.e., to go over dependence of the electronic band structure. They cal- from Ef, n({um, k}) to Ef, n(T). In the harmonic approximation to atomic oscillations [9], we obtain culated the energy bands Ef, n(T) in the adiabatic approximation using second-order perturbation theory ε ∆ EP Ef, n()T = n()f + Ef, n(),T in the atomic displacements. (8) EP() ∆DW() ∆SE () We assume the crystalline potential V(r, u) to be the Ef, n T = Ef, n T + Ef, n T , sum of the potentials V of individual ions. The position k where of an ion in the lattice is determined by the equilibrium () 0 2 position vector of the unit cell R and, within this cell, ∂ α β m ∆ EP() 1 ,,V k 〈〉 () Ef, n T = ---∑ f n ------f n um, kum, k ˜ ()0 2 ∂ ()α0 ∂ ()β0 by index k. We write V k = V k rRÐ m Ð umk , where m Rm Rm vectors umk are dynamic atomic displacements. The ˜ ∂V ∂V potentials V k can be expanded in a power series in dis- f,,n ------k f' n' f',,n' ------k'- f n ∂ ()α0 ∂ ()β0 placements umk. Then, we find the states of electrons Rm Rm' moving in the potential V(r, u = 0) of the frozen lattice. + ∑ ∑ ------ε () ε() η -(9) n f Ð n' f' + i The corresponding one-electron energy eigenvalues m, k; m', k' f,,n ≠ f' n' ε (f) and eigenfunctions |f, n〉 of the KohnÐSham type n × 〈〉α β are specified by the quasi-momentum f and band um, kum', k' . index n. The spin index will be dropped for the sake of It will be recalled that the dynamic atomic displace- simplicity. ment operator is defined as It is generally agreed that such one-electron states describe long-lived excitations. Although the concept ប u = ------of quasiparticles is substantiated only when their m, k ∑ k 2NM ω ()l energy is close to the Fermi energy, experiment sug- qi c c (10) gests that the limits of applicability of this concept are ()0 ()0 × []c()iqRm c*()ÐiqRm wider. e kle bl + e kle bl* ,

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1038 ZHERNOV ω | where c(l) and e(k l) are the frequency and polariza- 1 c c × ------eα()k Ð q j eβ()k' q j . tion vector of phonon mode l = {q j}, respectively, and k k' 2Mc Mc bl and bl* are the annihilation and creation operators of a quasiparticle, respectively. Here, the quantity Q is There are two terms on the right-hand side of (),, , Eq. (9). The first of them describes the renormalization Qfnn'; qj k through the DebyeÐWaller factor, i.e., due to elastic 1/2 ប (15) scattering in which an electron in the state |f, n〉 creates 〈〉,,∇ = ω------(), f ' n' V k fn, and destroys a phonon of wave vector q and polariza- c q j tion j. We denote the corresponding correction to the electronic energy spectrum of the crystal with frozen where f' = f + q + G, with G being a reciprocal lattice ∆ DW atomic displacements as Ef, n . The second term vector. describes the second-order contribution to electronÐ The difference in the energy bands of crystals with phonon interaction from inelastic processes (including k k ∆ k interband and intraband transitions). The correspond- atomic masses Mc and Mc + M that is associated ∆ SE with EPI can be written as ing correction Ef, n to the electronic energy spectrum EP is a complex quantity; its real part gives the change in ∆E˜ ()∆∆Mk = EEP()∆Mk + ∆Mk Ð EEP()Mk the effective electron mass, and the imaginary part c c determines the electron lifetime. EP (16) ∂E k ------∆ Thus, Eq. (9) can be represented in the form = k M . ∂M c TV, EP() ∆DW ∆ SE Γ Ef, n T = Ef, n + Ef, n + i f, n. (11) We note that the derivative (∂EEP/∂Mk ) is positive. Pick et al. [19] have derived the so-called acoustic c T, V sum rule for metals and insulators in the long-wave- The energy renormalization due to EPI decreases with length limit; this sum rule is a consequence of the elec- increasing atomic mass. In fact, crystal vibrations trical neutrality condition of the system. Based on this become frozen out as the atomic mass increases. The sum rule, one can redefine the first term in Eq. (9), interband transition energies are increased in this case. which represents the renormalization through the DW Thus, we have investigated the renormalization of factor [9Ð11]. Using Eq. (10) for the dynamic atomic the electronic energy spectrum due to EPI at a fixed displacements, one can thus obtain the following crystal volume. Based on Eqs. (8), (9), and (12)Ð(14), expression instead of Eq. (9) [12, 20]: one can analyze the influence of the isotope composition on the energy bands through EPI over a wide tem- ∂ ∂ ∆ EP() Ef, n Ef, n perature range. Ef, n T = ∑------∂ (), + ------∂ (), n q j DW n q j SE q, j (12) 3. A UNIVERSAL RELATION DESCRIBING × (), 1 n q j + --- , THE EFFECT OF THE ISOTOPE COMPOSITION 2 ON THE ENERGY BANDS: MONATOMIC CRYSTALS ∂ EP Ef, n ------We consider the isotope-composition dependence of ∂n()q, j DW the energy bands Ef, n of a crystal composed of atoms of ,, , , (),, , the same element. It will be shown that in the virtual- 1 Qα*()f nn'0jkQβ f n' n; 0 jk' crystal approximation, this dependence is described by = Ð------∑ ------ε () ε() η (13) 2N n f Ð n' f + i a simple relation. n'; kk, ' The basic equation determining the eigenfrequen- × 1 c ()c ()1 c ()c ()cies and polarization vectors of the vibrational modes ------k eα k Ð q j eβ k q j + ------k-'eα k'Ðq j eβ k' q j , Mc Mc l = {q j} of a monatomic crystal of an arbitrary isotopic composition (specified by index c) has the form ∂E , ------f n ω2() c () Φc ()c () ∂ (), c l eα kl = ∑ αα' kk' q eα' k' l . (17) n q j SE k', α' ,, , , (),, , 1 Qα()f nn' q jkQβ f nn'; q jk' Φc | = ---- ∑ ------ε () ε() (14) Here, αα' (kk' q) is the dynamic matrix of the crystal N n f Ð n' fq+ n'; kk, ' and α and α' specify the Cartesian coordinate axes. The

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EFFECT OF THE ISOTOPE COMPOSITION ON THE ENERGY BANDS 1039 matrix Φ has the form Next, we consider the contribution associated with EPI and described by Eqs. (12)Ð(14). In this case, the quan- Φc ()1 Φ˜ () tity dependent on the mean atomic mass Mc is αα' kk' q = ------αα' kk' q Mc បω () ()2 (), 1 c l 1 (18) Xc lT = ------ω ()- n------+ --- 1 ()0 ()0 Mc c l kBT 2 = ------∑ϕαα ()mk, m'k' exp()iqR()Ð R . (23) NM ' m m' បω () c c l 1 mm' ∼ω ()l n ------+.--- c k T 2 ϕ B Here, αα'(mk, m'k') is the matrix of second-order force constants and N is the number of unit cells. We note that Thus, for an arbitrary phonon spectrum, the energy band renormalizations associated with the unit-cell vol- Φ˜ the matrix is independent of the mean atomic mass Mc. ume and EPI have the same dependence on the mean atomic mass. Using Eq. (20), we find In the case of a monatomic crystal, as shown in the () () appendix, the frequencies of vibrational modes are sub- X i ()lT, = M /M X i ()lT, ' , c c0 c c0 ject to the relation (24) T' = TM/M . c c0 ω2() ∆ 2 ∆ 〈〉2 d ln c l M ,,M u Therefore, the changes in the energy bands caused by ------Ð1= + O------2 , (19) dMln c Mc Mc a isotope composition variations are described by the universal relation from which it follows that () ()c ∆E c ()T = M /M ∆E 0 ()T' . (25) fn c0 c fn Ð1/2 ω ()l = wl()M . (20) Γ c c The quantity f, n (inversely proportional to the electron lifetime) involved in Eq. (11) is also described by By definition, w(l) is independent of Mc. a universal relation similar to Eq. (25). Substituting Eqs. (20) and (18) into Eq. (17) yields In the practically important case of very low temperatures, as follows from Eqs. (22) and (23), the atomic- mass dependences of the energy and decay constant of 2() c () Φ˜ c ()c () w l eα kl = ∑ αα' kk' q eα' k' l . (21) an electron in the state (f, n) are given by k', α' ()c C1 ()c C2 E ()εT = 0 ==()f + ------, Γ , ------. (26) It is immediately obvious from Eq. (21) that, in contrast fn n f n Mc Mc c to the frequencies, the polarization vectors eα (k|l) are ε We recall that n(f) is the energy of an electron in the independent of the atomic mass for a specified isotope frozen lattice. This energy and the constants C1 and C2 composition of the crystal. are independent of the atomic mass. We will designate the parameters of the crystal of a It should be noted that in the classical limit of high temperatures, the electronic energy spectrum does not certain isotope composition by index c and in the case 0 depend on the isotope composition, as follows immedi- of an arbitrary isotope composition, by index c, as ately from Eqs. (22)Ð(25). before. Let us consider the changes in the energy bands Universal relations are useful in analyzing the caused by isotope composition variations and associ- experimental dependences of crystal parameters on the ated with the altered volume and EPI. isotope composition. Given data for crystals of the natural isotope composition c = c , one can predict the First, we consider the energy band renormalization 0 nat values of the parameters for isotope-enriched crystals associated with the change in the unit-cell volume and γ and compare them with the respective experimental described by Eq. (5). Differentiating (l) with respect values. γ to Mc and using Eq. (19), one can verify that (l) is inde- The universal relations derived above are simple in pendent of Mc. Therefore, in the sum over l in Eq. (5), structure and take place in monatomic crystals, because only the following quantity depends on the mean the polarization vectors are independent of the atomic ω Ð1/2 atomic mass of the crystal at T < TD [through the fre- mass and the frequencies vary as (l) ~ M . In poly- ω atomic crystals, the situation is more complicated, quency c(l)]: because the isotopic frequency shifts are proportional បω () to the square of the magnitudes of the corresponding ()1 ()ω, () c l 1 Xc lT = c l n------+.--- (22) polarization vectors (see Appendix) and the polariza- kBT 2 tion vectors themselves also depend on the atomic

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1040 ZHERNOV mass. The case of polyatomic crystals calls for special 4. DISCUSSION investigation. OF THE EXPERIMENTAL DATA In some papers (see, e.g., [10, 21, 22]), the temper- To date, precise measurements have been made ature and atomic-mass dependences of the energy which provide reliable data on the energies of the inter- bands Ei were described introducing a mean phonon band transitions and critical points of optical spectra of ϑ frequency c and a mean BoseÐEinstein factor nc and monatomic semiconductors. the following formula was used: The effect of the isotope composition on the optical 1/2 0 M properties of semiconducting germanium was first E = E Ð B ------nat- ()2n + 1 , (27) i i iM c investigated in [23]. Only two sets of samples were c investigated: isotope-enriched, 75.7Ge crystals (84% ϑ 0 76Ge, 15% 74Ge, and less than 0.2% of each of the other where nc = 1/[exp( c/T) Ð 1], Ei is the nonrenormalized ϑ isotopes) and a crystal of natural isotope composition. energy gap, and Bi is a constant. By definition, c = The near-gap photoluminescence spectra were mea- ϑ 1/2 nat(Mnat/Mc) . The electron damping constant due to sured in their short-wavelength region, and the trans- EPI was determined in much the same way. At T > TD, mission spectra were recorded in the region of direct we have exciton transitions at T = 1.7 K. Excitonic absorption spectra were also measured in the region of direct near- 0 T E ()TT> = E Ð 2B ------, (28) gap optical transitions. From these experimental data, i D i iT ∆ D the isotopic shifts of the band gap Eg were found at the Γ and L points of the Brillouin zone. where E is independent of the atomic mass. There- i The luminescence spectra of diamond crystals in the fore, the constant Bi can be determined from the depen- frequency region of indirect interband transitions of the dence of Ei on T at high temperatures. We note that Eg type were measured in [24] (see also [25]). In [24], Eq. (27) is an analog of the empirical Varshni formula. measurements at liquid-nitrogen temperatures were also made only on two sets of samples, namely, on In connection with Eq. (27), we will make a few 13 remarks with reference to the results of the studies pre- those enriched with C and those of the natural isotope composition (98.9% 12C, 1.1% 13C). The shifts in posi- sented in [12, 13], where the energy gap Eg in germanium was calculated as a function of isotope composition of the peaks corresponding to free excitons of all tion using the NLPP and LCAO methods. (The calcula- three types (A, B, C) were determined in the lumines- tions made in [13] using the LCAO method seem to be cence spectra. A shift was also observed in the position ∂ ∂ of the peaks that correspond to excitons localized on more accurate. In particular, the derivative ( Eg/ T)tot, which determines the temperature dependence of the neutral boron impurities. energy gap, was calculated to an appropriate degree of The role of the change in volume and of EPI was accuracy.) The isotopic energy gap shifts due to the considered qualitatively in [23, 24], and it was found change in the DW factor (elastic processes) and to EPI that the isotopic energy shifts are basically due to EPI. (inelastic processes) were calculated, and the role of Further, in [14], the energies of direct transitions optical and acoustic phonon modes was analyzed. It Γ+ ΓÐ Γ+ + was shown that the isotopic energy shifts due to elastic E0(8 Ð 7 ) and of indirect transitions Eg(8 Ð L6 ) processes and to EPI are of the same order of magni- and their isotope-composition dependences were deter- tude. The influence of the acoustic and optical phonon mined for Ge at liquid-helium temperatures using the modes on EPI was also found to be almost the same; modulation spectroscopy method. The measurements acoustic phonons affect the energy spectrum through were made on four isotope-enriched samples of germa- elastic scattering; optical phonons, through inelastic nium (70Ge, 72.9Ge, 73.9Ge, 75.6Ge) and on a sample of the processes. We note that, according to [18], the value of natural isotope composition. The energy E0 was deter- the derivative (∂lnΩ/∂Mk ) and, therefore, the energy mined from measured spectra of photomodulated c P reflectance. The energy of indirect transitions E was shift due to the change in volume are determined in g found from data on photoluminescence spectra and large measure by optical phonon modes. measurements of the electric-field-modulated transmis- A comparison was also made between the isotopic sion coefficient. According to the results presented in energy shifts due to the change in volume and to EPI; [14], the isotope-composition dependences of the elec- they are of the same order of magnitude for E0 and Eg tronic-spectrum parameters corresponding to the tran- bands. In the case of critical points of the E1 type, the sition energies E0 and Eg can be closely approximated EPI mechanism is dominant. by the relation E = E∞ + B/M with B < 0. Thus, it can be concluded that Eq. (27) is a very rough and, in general, inadequate approximation of the In [26], the behavior of the critical points of the E1 dependence of energy bands on isotope composition type was investigated for samples of 70Ge, 75.6Ge, and of and temperature. the natural isotope composition. The dielectric function

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EFFECT OF THE ISOTOPE COMPOSITION ON THE ENERGY BANDS 1041 ε 2 was measured using an ellipsometric technique. We crystal potential only slightly. It is agreed that the val- note that, according to energy band calculations for ger- ues of ∆ in a free atom and in an atom in a crystal differ manium, the critical E1 points (of the type of a 2D min- by roughly ten percent. The character of changes in the imum and a saddle point) on the curve of the imaginary coupling constant ∆ in an atom due to the isotopic part of the dielectric constant are located in the range energy shift is well known [27]. However, according to ε 1.8Ð22.6 meV. The structure and magnitude of 2 in this the experimental data presented in [26], the isotope Λ effect for ∆ in a crystal is much more profound than in energy range are mainly determined by the doublet 3Ð Λ a free atom. Furthermore, the coupling constant ∆ has 1 transitions. The energy ranges of other transitions overlap, and their contributions cannot be separated. been found to be strongly temperature-dependent in GaSb and α-Sn crystals (see references in [26]). There- As in the case considered above, the isotope-compo- fore, the question concerning the role of spinÐorbit cou- sition dependence of E1 can be described by the relation pling calls for special investigation. ∞ E1 = E1 + B/M with B < 0. A similar relation also It should be noted that in monatomic semiconductor takes place for the line widths Γ(M). crystals, the structure and magnitude of the density of In [14, 26], the contributions due to the change in states of the upper valence bands and lower conduction volume and EPI to the empirical parameter B were eval- bands are mainly determined by s and p states slightly uated. The former contribution was determined from hybridized with d states (in germanium, the typical dif- the experimental data on (1/V)(dV/dM) and hydrostatic ference in energy between the p and d states of the ∆ ∝ valence bands is roughly 20 eV). Therefore, the d orbit- deformation potential Vg using the relation E ∆ ∆ als are certain to be of minor importance in optical tran- Vg(1/V)( V/ M). The contribution from EPI was estimated as the difference between the experimental value sitions in these crystals. However, in some compounds, of B and the contribution due to the volume change. It such as CuCl(Br) and CdS, the difference in energy between the atomic p and d states is of the order of 1 eV was found that the isotopic shifts in the energies E0 and E of optical transitions associated with the volume [28, 29]. In this case, the hybridization between the p g and d orbitals is of considerable importance and the change and EPI are of the same order of magnitude, upper valence bands are significantly affected by the d while the isotopic shifts of the critical points E1 are states. In such compounds, the influence of a change in mainly due to EPI. the isotope composition can be described as follows Thus, the theory is in reasonable agreement with the [28]. In the matrix element Vpd responsible for the experimental data for diamond and germanium and, hybridization between the p and d states, the cation therefore, there are situations in which the universal (copper) pseudopotential is dominant, while the contri- relations are valid. The influence of spin–orbit coupling bution from the anion pseudopotential is negligible and the possible specific role of the d band are dis- [28]. An increase in the cation atomic mass causes the cussed briefly in the following section. DW factor exp(ÐW) to decrease because exp(ÐW) ∝ [1 Ð (1/2)〈u2〉G2]. At the same time, the magnitudes of 5. CONCLUSION the effective cation pseudopotential and of the matrix element Vpd are increased. In this case, the interband In this paper, we investigated the effect of the iso- transition energy E decreases; i.e., we have ∂E/∂M < tope composition of a crystal on the energy bands E 0 c f, n 0. As a consequence, the contribution to the energy within the virtual-crystal model in the quasi-harmonic band renormalization from EPI (which is of opposite approximation. The role of the change in the unit-cell sign, ∂E/∂M > 0) is partly or even completely canceled. volume of the lattice and of the EPI renormalization c produced by a change in the isotope composition (i.e., A change in the anion atomic mass affects the matrix the role of the elastic and inelastic channels, respec- element Vpd only slightly; the isotope shift in energy is mainly determined, as before, by EPI and, hence, tively) was discussed. For monatomic crystals, a uni- ∂ ∂ E/ Mc > 0. The question concerning the role of versal relation describing the dependence of Ef, n on the isotope composition and temperature was derived. A hybridization between p and d states also calls for comparison was made with the experimental data on detailed investigation in this case. the interband transition energies and the critical points The dependences of the electronic energy spectrum in optical spectra. on temperature and isotope composition are of the same The actual electronic energy spectra are signifi- origin. Knowledge of the behavior of the isotope shifts cantly affected by spinÐorbit coupling (except in the in energy is of importance, because it provides addi- case of crystals such as diamond composed of atoms of tional insight into the origin of the electronic structure. light elements). SpinÐorbit coupling leads to doublet splitting of p and d states at certain points of the Bril- ACKNOWLEDGMENTS louin zone. Since this coupling takes place in the region of the atom core, the spinÐorbit coupling constant ∆ The author is grateful to L.A. Maksimov for his should be determined, in principle, by core electron interest in this work and helpful suggestions, to states, which, generally speaking, are affected by the Yu.M. Kagan for his encouragement, and to A.V. Inyu-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1042 ZHERNOV shkin and D.A. Zhernov for their assistance throughout known. Let us find the isotopic shift in frequency as one this work. goes over to the crystal with isotopic content c1. It is well known that, in order to calculate an eigen- APPENDIX value to a certain order in a perturbation, one needs the eigenfunctions to be calculated through the unity lower We consider the isotope shift in frequency of a order. The isotopic shift in an eigenvalue (square of a vibrational mode in a crystal whose unit cell contains frequency) calculated to the first order is equal to the atoms of different elements. The elements are assigned corresponding diagonal matrix element of the perturba- variable mean atomic masses, and the atomic masses of tion energy between the unperturbed eigenfunctions. the isotopes of the same element are assumed to differ Therefore, we can write little in value. We consider the effects linear in the isotope mass difference. In this approximation, the point symmetry group of the crystal lattice remains ∆ω2()l ------unchanged and so does its vibration spectrum; there- k1 ∆M fore, the degeneracy is not lifted. In actuality, in a crys- c (A5) tal lattice with isotopic disorder, there are fields of static c ∆Φαα ()kk' q atomic displacements due to zero-point oscillations *()' c () = ∑∑ eα kl------eα' k' l . which vary with atomic mass. In the presence of these k1 , α , α ∆M c fields, the local symmetry is reduced. However, in con- k k' ' ventional (not quantum) crystals, these displacements From Eq. (A3), it follows that are proportional to an additional small parameter, 〈u2〉/a2, and can be neglected. δ δ ∆Φαα ()kk' q ' 1Φc ()kk1 k1k' We consider polyatomic crystals composed of iso- ------= Ð--- αα' kk' q ------. (A6) k1 2 k k' topes of different elements and assume that the isotopic ∆M c Mc Mc content of one element (specified by index k1) is varied. Let two crystals differ in the isotopic content of ele- We substitute Eq. (A6) into Eq. (A5) and take into ment k1. We will specify these crystals by indices c and account that c1. The mean atomic mass of element k1 is equal to 2 c c c ω ()l eα()kl = ∑ Φαα ()kk' q eα ()k' l . (A7) k1 k1 k1 c ' ' Mc = ∑ci Mi (A1) k', α' i In addition, since the matrix Φ(q) is Hermitian, the k1 (ci is the concentration of the ith isotope of the given polarization vectors satisfy the orthonormality and element) and is assumed to differ little from the mean completeness conditions atomic mass of this element in the crystal with isotopic c*()c ()δ content c1; that is, ∑eα k q j eα k q j' = jj', k, α k1 k1 k1 ∆ k1 k1 Ӷ (A8) Mc Ð/Mc1 Mc = M /Mc 1. (A2) c*()c ()δ δ The dynamic matrix of a polyatomic crystal ∑eα k q j eα' k' q j = kk' αα'. Φ | αα'(kk' q) can be written as j Φ ()1 1 We can also replace ∆ by the differential sign, αα' kk' q = ------N k k' because the mean atomic mass varies continuously. The Mc Mc (A3) final result is ×ϕ(), ()()()0 ()0 ω2() ∑ αα' mk m'k' exp iqRm Ð Rm' , d ln c l c 2 ------= Ð∑ eα()kl . (A9) mm' k dMln c α ϕ where αα'(mk, m'k') is the matrix of the second-order force constants. The dynamic matrix is a Hermitian As seen from Eq. (A9), the shift in the frequency of 3s × 3s matrix (s is the number of atoms in the unit cell) a vibrational mode caused by a variation of the mean and, therefore, satisfies the condition atomic mass of one of the components of the polyatomic crystal is proportional to the square of the mag- Φ ()Φ() αα' kk' q = α*'α k'k q . (A4) nitude of the corresponding polarization vector. ω The eigenfrequencies c(l) and orthonormal polar- Relations of such type were first derived in [30], ization vectors ec(k|l) of the dynamic matrix Φ of where the effect of the isotope composition on the prop- Eq. (A3) for the virtual crystal are assumed to be erties of fullerenes was studied.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EFFECT OF THE ISOTOPE COMPOSITION ON THE ENERGY BANDS 1043

In the case of a monatomic crystal, the unit cell con- 14. C. Parks, A. K. Ramdas, S. Rodríguez, et al., Phys. Rev. sists of atoms of the same element and Eq. (A9) takes B 49 (20), 14244 (1994). the form 15. P. Pavone and S. Baroni, Solid State Commun. 90 (5), 295 (1994). 2 d lnω ()l ∆M 2 ∆M 〈〉u2 ------c -Ð1= + O ------, ------. (A10) 16. G.-M. Rignanese, J.-P. Michenaud, and X. Gonze, Phys.  2 Rev. B 53 (8), 4488 (1996). dMln c Mc Mc a 17. A. P. Zhernov, Zh. Éksp. Teor. Fiz. 114 (2), 654 (1998) [JETP 87, 357 (1998)]. REFERENCES 18. A. P. Zhernov, Fiz. Nizk. Temp. 26 (12), 1226 (2000) [Low Temp. Phys. 26, 908 (2000)]. 1. H. Holloway, K. C. Hass, M. A. Tamor, et al., Phys. Rev. B 44 (13), 7123 (1991). 19. R. M. Pick, M. H. Cohen, and R. M. Martin, Phys. Rev. B 1 (2), 910 (1970). 2. W. S. Capinski, H. J. Maris, E. Bauser, et al., Appl. Phys. Lett. 71 (15), 2109 (1997). 20. P. Lautenschlager, P. B. Allen, and M. Cardona, Phys. Rev. B 31 (4), 2163 (1985). 3. T. Ruf, R. W. Henn, M. Asen-Palmer, et al., Solid State Commun. 115 (5), 243 (2000). 21. L.F. Lastras-Martínez, T. Ruf, M. Konuma, et al., Phys. Rev. B 61 (19), 12946 (2000). 4. H. D. Fuchs, C. H. Grein, R. I. Devlen, et al., Phys. Rev. B 44 (16), 8633 (1991). 22. S. D. Yoo, D. E. Aspnes, L. F. Lastras-Martínez, et al., Phys. Status Solidi B 220 (1), 117 (2000). 5. V. I. Ozhogin, A. V. Inyushkin, A. N. Taldenkov, et al., Pis’ma Zh. Éksp. Teor. Fiz. 63 (6), 463 (1996) [JETP 23. V. F. Agekyan, V. M. Asnin, A. M. Kryukov, et al., Fiz. Lett. 63, 490 (1996)]. Tverd. Tela (Leningrad) 31 (12), 101 (1989) [Sov. Phys. Solid State 31, 2-82 (1989)]. 6. H. Bettger, Principles of the Theory of Lattice Dynamics (Akademie, Berlin, 1983). 24. A. T. Collins, S. C. Lawson, G. Davies, and H. Kanda, Phys. Rev. Lett. 65 (7), 891 (1990). 7. G. Leibfried, in Handbuch der Physik, Ed. by S. Flugge 25. T. Ruf, M. Cardona, H. Sternshlute, et al., Solid State (Springer, Berlin, 1955; Fizmatgiz, Moscow, 1963), Vol. Commun. 105 (5), 311 (1998). 7, Part 1. 26. D. Rönnow, L. F. Lastras-Martínez, and M. Cardona, 8. V. Cardona, Physica B (Amsterdam) 263Ð264, 376 Eur. Phys. J. B 5 (1), 29 (1998). (1999). 27. I. I. Sobelman, Atomic Spectra and Radiative Transi- 9. P. B. Allen and V. Heine, J. Phys. C 9, 2305 (1976). tions (Fizmatgiz, Moscow, 1963; Springer, Berlin, 10. V. V. Sobolev and V. V. Nemoshkalenko, The Methods of 1979). Computational Physics in the Theory of Solid State. 28. A. Göbel, T. Ruf, M. Cardona, et al., Phys. Rev. B 57 Electronic Structure of Semiconductors (Naukova (24), 15183 (1998). Dumka, Kiev, 1988). 29. J. M. Zhang, T. Ruf, R. Lauck, and M. Cardona, Phys. 11. S. Zollner, V. Cardona, and S. Gopalan, Phys. Rev. B 45 Rev. B 57 (16), 9716 (1998). (7), 3376 (1992). 30. J. Menéndez, J. B. Page, and S. Guha, Philos. Mag. B 70 12. N. Garro, A. Cantarero, M. Cardona, et al., Phys. Rev. B (3), 651 (1994). 54 (7), 4732 (1996). 13. D. Olguín, A. Cantarero, and M. Cardona, Phys. Status Solidi B 220 (1), 33 (2000). Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1044Ð1049. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1001Ð1005. Original Russian Text Copyright © 2002 by Mikushev, Ulyashev, Charnaya, Chandoul.

SEMICONDUCTORS AND DIELECTRICS

Temperature Dependence of the SpinÐLattice Relaxation Time for Quadrupole Nuclei under Conditions of NMR Line Saturation V. M. Mikushev*, A. M. Ulyashev*, E. V. Charnaya*, and A. Chandoul** * Research Institute of Physics, St. Petersburg State University, Ul’yanovskaya ul. 1, Petrodvorets, St. Petersburg, 198904 Russia ** Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunis, 1080 Tunisie e-mail: [email protected] Received September 13, 2001

Abstract—The contributions of different mechanisms of nuclear spinÐlattice relaxation are experimentally separated for 69Ga and 71Ga nuclei in GaAs crystals (nominally pure and doped with copper and chromium), 23Na nuclei in a nominally pure NaCl crystal, and 27Al nuclei in nominally pure and lightly chromium-doped Al2O3 crystals in the temperature range 80Ð300 K. The contribution of impurities to spinÐlattice relaxation is separated under the condition of additional stationary saturation of the nuclear magnetic resonance (NMR) line in magnetic and electric resonance ﬁelds. It is demonstrated that, upon suppression of the impurity mechanism of spinÐlattice relaxation, the temperature dependence of the spinÐlattice relaxation time T1 for GaAs and NaCl crystals is described within the model of two-phonon Raman processes in the Debye approximation, whereas the temperature dependence of T1 for corundum crystals deviates from the theoretical curve for relaxation due to the spinÐphonon interaction. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION spinÐlattice relaxation rate, their comparison with theoretical models is not entirely correct because of an SpinÐlattice relaxation of quadrupole nuclei (with uncertainty introduced by the impurity contribution I > 1/2) in solid dielectrics is governed by two mecha- imp nisms, namely, the lattice mechanism responsible for T 1 , which, even for nominally pure crystals, remains relaxation in perfect samples and the impurity mecha- signiﬁcant due to spin diffusion. Unlike the lattice com- nism associated with the presence of impurity para- imp magnetic ions, radiation-induced centers, and other ponent, the impurity component T 1 characterizes the structural point defects in real crystals [1]. The overall degree of perfection of a particular sample rather than process of spinÐlattice relaxation is characterized by the structure of the material. A theoretical evaluation of the time the impurity contribution is substantially complicated by the presence of undetectable paramagnetic centers, Σ ()()lat Ð1 ()imp Ð1 Ð1 imp T 1 = T 1 + T 1 , (1) the dependence of T 1 on the external quantizing ﬁeld

lat imp [2], the complex dependence of the spin relaxation rate where T 1 and T 1 are the lattice and impurity com- of paramagnetic centers on the temperature and con- ponents, respectively. The former mechanism of spinÐ centration, etc. lattice relaxation in perfect crystals predominantly occurs through the modulation of the internuclear sep- Direct experimental measurements of the lattice lat aration in the crystal lattice by thermal vibrations and, component T 1 became possible with the advent of the consequently, through changes in the electric-field gra- new technique proposed in our earlier works [4, 5]. This dient at a nucleus. Since the density of phonon states at technique consists in suppressing the impurity contri- frequencies close to the Larmor frequency is relatively bution to nuclear spinÐlattice relaxation for samples low, Raman processes involving all phonons of the with a low relative concentration of paramagnetic cen- spectrum more efficiently manifest themselves in spinÐ ters (<10Ð5) through additional stationary saturation of lattice relaxation. The ratio between the efficiencies of the nuclear magnetic resonance (NMR) line. The ther- different Raman processes, which, in the general case, modynamic idea underlying the proposed method is as should lead to different temperature dependences of the follows. The impurity mechanism of relaxation pro- lat lattice component T 1 , has been discussed, for exam- ceeding through spin diffusion is efficient provided the α ple, in [2, 3]. Despite a large amount of available exper- reciprocal of the local spin temperature loc in the vicin- imental data on the temperature dependence of the ity of defects is closer to the reciprocal of the lattice

TEMPERATURE DEPENDENCE OF THE SPINÐLATTICE RELAXATION TIME 1045 α temperature l as compared to the reciprocal of the Saturating-pulse mean spin temperature 〈α〉 in other regions of the sam- sequence 90° Σ ple [1]. If the spinÐlattice relaxation time T 1 is measured in the course of the nuclear magnetization recovery after complete saturation of the NMR line by a Steady-state sequence of rf pulses [6], the impurity relaxation makes magnetic A(∆t) a contribution to spinÐlattice relaxation under the fol- or electric field lowing condition: Time ∆t α > 〈〉α loc . (2) Strong additional stationary acoustical, magnetic, or Fig. 1. A scheme of measuring the recovery time τ of nuclear magnetization to a steady-state value. A(∆t) is the electric saturation of the NMR line can result in local ∆ overheating of the nuclear spin system in the vicinity of free-induction signal measured in the time interval t after α 〈α〉 rf-pulse saturation of the nuclear spin system. defects to loc = 0, whereas becomes equal to a cer- 〈α〉 tain steady-state value st > 0 [7, 8]. The effect of strong local heating was experimentally established in Then, the spinÐlattice relaxation time T1 was calculated our earlier works [9, 10]. The phenomenological theory from the relationship [4, 14] of this effect was developed in [11]. In the case when τ st the regions near defects are overheated, inequality (2) T 1 = /Z , (3) becomes invalid and the impurity contribution to spinÐ lattice relaxation is suppressed. Under these conditions, where Zst is the stationary-saturation factor defined as st 〈α〉 α st the recovery of nuclear magnetization in the sample is Z = st/ l [14]. The factor Z was measured as the governed by the lattice mechanism and the experimen- ratio of the free-nuclear-induction signal after a 90° Σ pulse in an extra saturating field to the signal in the tally measured spinÐlattice relaxation time T corre- 1 absence of an extra field. The thermal stabilization was sponds to spinÐlattice relaxation in a perfect sample. accurate to better than 0.2 K. The degree of local heating of the nuclear spin system depends on the nature of the paramagnetic centers in High-resistance crystals of gallium arsenide the sample. However, as was shown in [4, 8, 12, 13], an (including a nominally pure GaAs sample and copper- increase in the saturation in a stationary resonance field and chromium-doped GaAs samples with an impurity 〈α〉 ion concentration of 1018 cmÐ3) were grown by the Czo- (i.e., a decrease in st) can bring about suppression of imp chralski method from the melt. The electrical resistivity 8 the impurity contribution T 1 of all the paramagnetic of GaAs samples was approximately equal to 10 Ω cm. centers involved in impurity relaxation. Thus, the pro- Crystals of gallium arsenide have a cubic symmetry posed technique made it possible for the first time to without an inversion center. As a consequence, the investigate separately the temperature dependence of NMR spectra of 69Ga and 71Ga isotopes with spin I = the lattice contribution to spinÐlattice relaxation in real 3/2 exhibit a Zeeman structure; however, the transitions crystals. between spin levels can be excited in an external elec- In the present work, we measured temperature tric field at twice the Larmor frequency [9]. All three lat samples were cut out normally to the [001] crystallo- dependences of the spinÐlattice relaxation time T 1 in graphic axis in the form of 1-mm-thick plates. Elec- the temperature range 78Ð300 K for 71Ga and 69Ga trodes applied to the samples induced a stationary ac nuclei in nominally pure and doped GaAs crystals, 23Na electric field with an amplitude up to 103 V cmÐ1 in the nuclei in a nominally pure NaCl crystal, and 27Al nuclei sample bulk. The [001] cubic axis was oriented along in nominally pure and chromium-doped Al2O3 crystals the magnetic field of the spectrometer. under the conditions of stationary saturation of the A nominally pure crystal of sodium chloride was NMR lines. grown by zone melting from the melt with the use of high-purity materials. The sample was prepared in the form of a parallelepiped with edges aligned parallel to 2. SAMPLE PREPARATION the crystallographic cubic axes. In measurements, the AND EXPERIMENTAL TECHNIQUE cubic axis of the crystal was directed along the dc field The experiments were performed on an ISP-1 of the spectrometer. Stationary saturation of the 23Na pulsed NMR spectrometer operating at a frequency of NMR lines (I = 3/2) was achieved in an extra magnetic 5.5 MHz. Under the conditions of stationary saturation field at the Larmor frequency with the use of an addi- of the NMR line in extra magnetic or electric resonance tional coil wound on the sample. The field amplitude fields, the recovery time τ of nuclear magnetization was was controlled in the range from 0 to 0.1 Oe. measured immediately after saturation of the spin sys- A nominally pure single crystal of aluminum oxide tem by a sequence of rf nonselective pulses (Fig. 1). was grown in a molybdenum crucible according to the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1046 MIKUSHEV et al. special procedure devised at the State Optical Institute measured by the conventional technique are equal to (St. Petersburg, Russia). A chromium-doped Al2O3 27.0 ± 0.4 s for the copper-doped crystal, 27.8 ± 0.4 s sample was grown by the Verneuil method. The chro- for the chromium-doped crystal, and 31.4 ± 0.5 s for the mium concentration in the melt was approximately nominally pure sample. equal to 5 × 1016 cmÐ3. The samples were prepared in the form of cylinders (~2 cm3 in volume) whose axes It can be seen from Fig. 2 that, as the strength of the were oriented perpendicular to the C3 crystallographic saturating electric field increases, the spin–lattice relax- 27 ation times for doped crystals increase to the relaxation axis. In a hexagonal Al2O3 crystal, the Al levels (I = 5/2) are shifted as a result of quadrupole interaction. It time of 71Ga nuclei in the nominally pure crystal. As a should be noted that the relaxation of the difference in result, under the conditions of strong saturation of the the populations of a particular pair of spin levels is gen- NMR signals (Zst < 0.4), the spinÐlattice relaxation erally a superposition of several exponential processes times for the nominally pure crystal and both doped and cannot be described by a sole spinÐlattice relax- samples become equal to each other. Therefore, this time ation time [7, 15], which makes the quantitative interpretation of the experimental results considerably more can be treated as a relaxation time associated only with the lattice mechanism of spinÐlattice relaxation, i.e., as difficult. For this reason, the measurements of Al2O3 lat crystals were performed at a magic angle between the the lattice component T 1 . By using relationship (1), it C3 crystallographic axis of the sample and the direction is possible to estimate the impurity contributions for of the dc magnetic field such that the spin levels were imp ± equidistant and the recovery of nuclear magnetization chromium-doped (T 1 = 240 40 s) and copper-doped occurred in an exponential manner [2]. As for NaCl imp ± crystals, the contributions of the mechanisms of 27Al (T 1 = 190 40 s) crystals at the same concentration spinÐlattice relaxation were separated under the condi- of impurity ions in the samples. Within the limits of tions of magnetic saturation of the NMR line. experimental error, the T1 time for the nominally pure sample remains constant with a change in the Zst factor. 3. RESULTS AND DISCUSSION This implies that undetectable paramagnetic centers contained in the pure sample do not contribute signifi- Figure 2 depicts the experimental curves of the Σ lat spinÐlattice relaxation time for 71Ga nuclei in nomi- cantly to the relaxation rate; hence, T 1 = T 1 . Similar nally pure and doped GaAs crystals at different degrees results are obtained for 69Ga isotopes at T = 78 K. Rea- of stationary electric saturation of the NMR signal at a soning from a comparison of the spinÐlattice relaxation temperature of 78 K. In the absence of an extra saturat- Σ lat Σ ± st time T 1 and the lattice component T 1 = 12.4 0.5 s, ing field (Z = 1), the spinÐlattice relaxation times T 1 imp 69 the impurity relaxation times T 1 for Ga nuclei in Σ chromium-doped (T = 11.5 ± 0.2 s) and copper-doped 35 1 Σ ± ± (T 1 = 11.4 0.2 s) samples are estimated at 160 50 and 140 ± 50 s, respectively. As the temperature of the crystal increases, the lat- lat

, s 30 tice relaxation time T for both Ga isotopes decreases

1 1 T imp more rapidly than the impurity relaxation time T 1 . At ≥ Σ 1 temperatures T 100 K, the total relaxation time T 1 2 lat 25 3 becomes equal to the lattice relaxation time T 1 ; as a result, the spinÐlattice relaxation times for all three 0 0.2 0.4 0.6 0.8 1.0 samples are identical and do not change at any value of st Z Zst. The experimental temperature dependences of the Σ 71 Fig. 2. Dependences of the spinÐlattice relaxation time T1 spinÐlattice relaxation time T 1 for Ga nuclei in nom- for 71Ga nuclei on the electric-stationary-saturation factor inally pure and doped crystals are displayed in Fig. 3. st Z for (1) nominally pure, (2) chromium-doped, and (3) Similar dependences for 69Ga nuclei in nominally pure copper-doped gallium arsenide samples at a temperature of 78 K. The straight line corresponds to the lattice component and chromium-doped samples are plotted in Fig. 4. The lat lat T1 . The error bars are shown selectively. theoretical curves T 1 (T) depicted by solid lines in

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TEMPERATURE DEPENDENCE OF THE SPINÐLATTICE RELAXATION TIME 1047

Figs. 3 and 4 are calculated according to the standard 40 relationship [2] 1 Ω 32 Ð2 2 lat Ð1 6 បω បω ()T = C ω exp ------exp ------Ð 1 dω, (4) 3 1 ∫ kT kT 0 24 , s 1 where k is the Boltzmann constant, Ω is the Debye fre- T quency, and C is the temperature-independent numeri- 16 cal coefficient. Relationship (4) is derived for relaxation within the model of two-phonon Raman pro- 8 cesses. In this case, the thermal-phonon spectrum is described in the framework of the Debye approximation. Since the numerical coefficient C has defied exact 0 50 100 200 300 calculation, the C value used in our case is chosen from T, K the condition of equality between the theoretical and lat experimental times T 1 at a temperature of 78 K. The Fig. 3. Temperature dependences of the spinÐlattice relax- 71 Debye frequency for gallium arsenide is calculated ation time T1 for Ga nuclei in (1) nominally pure, (2) from the Debye temperature Θ = 345 K taken from chromium-doped, and (3) copper-doped gallium arsenide [16]. As can be seen from Figs. 3 and 4, the temperature crystals. The solid line represents the theoretical tempera- lat dependences of the lattice component of the spinÐlat- ture dependence of the lattice component T . tice relaxation time for Ga nuclei are adequately 1 described within the chosen model.

As was shown above, the nominally pure GaAs 25 crystal exhibits an ideal behavior from the standpoint of nuclear spinÐlattice relaxation. In the nominally pure 1 NaCl sample, unlike the GaAs crystal, the impurity 2 mechanism of spinÐlattice relaxation of 23Na nuclei is 20 comparable in efﬁciency to the lattice mechanism due to the presence of undetectable paramagnetic centers at a relative concentration of the order of 10Ð5. This 15 , s

explains the wide scatter of the available data on the 1 spinÐlattice relaxation rate of Na nuclei [17]. Although T the contributions of different relaxation mechanisms 10 for different paramagnetic centers at certain temperatures have been experimentally separated under the conditions of acoustical [8] and magnetic [12] satura- 5 tion of the 23Na NMR line, the temperature dependence lat of the lattice component T 1 has not been conclusively 0 established. In particular, Bakhramov et al. [18] 50 100 150 200 250 300 Σ T, K obtained the experimental dependences T 1 (T) for nominally pure and copper-doped NaCl crystals and ana- Fig. 4. Temperature dependences of the spinÐlattice relax- 69 lyzed the impurity mechanism within the microscopic ation time T1 for Ga nuclei in (1) nominally pure and (2) approach. The results obtained in [18] are in close chromium-doped gallium arsenide crystals. The solid line agreement with the dependence represented by rela- represents the theoretical temperature dependence of the lat tionship (4), even though these authors accepted a sublattice component T . stantial impurity contribution to the spinÐlattice relax- 1 Σ ation time T 1 . On the other hand, considerable devia- lat 23 tions from the dependence described by relationship (4) nent T 1 for Na nuclei in a nominally pure NaCl sam- were revealed experimentally in a number of works and ple, which were measured under the conditions of mag- assigned to the contribution of optical branches of the netic saturation of the NMR signal. The theoretical phonon spectrum [19], covalence, [20], antiscreening curve was calculated from relationship (4) for the effects [21], etc. Raman process of quadrupole spinÐlattice relaxation at Θ lat Figure 5 shows the temperature dependences of the the Debye temperature = 275 K [16] and T 1 = 540 s Σ spinÐlattice relaxation time T 1 and the lattice compo- at a sample temperature of 78 K. The calculated time

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1048 MIKUSHEV et al.

300 500 1 2 1 2 400 3

300 , s 1 T 200 200 100

0 , s

100 200 300 1 T, K T

Fig. 5. Temperature dependences of (1) the spinÐlattice Σ relaxation time T1 measured by a conventional technique 100 lat 23 and (2) the lattice component T1 for Na nuclei in nominally pure NaCl crystals. The solid line represents the theoretical temperature dependence of the lattice component lat T1 .

lat T 1 coincides with the lattice relaxation times obtained under the conditions of suppression of the impurity 0 100 200 300 relaxation mechanism in different NaCl crystals [8], T, K including those exposed to gamma-ray irradiation [12]. Fig. 6. Temperature dependences of the spinÐlattice relax- As is clearly seen, the experimental dependence is con- Σ sistent with the chosen model of spinÐlattice relaxation. ation time T1 measured by a conventional technique for It should be noted that, within the limits of experimen- 27Al nuclei in (1) nominally pure and (2) chromium-doped tal error and with allowance made for the sample orien- Al2O3 crystals. (3) Temperature dependence of the lattice lat lat 27 tation in the spectrometer, the dependence T 1 (T) is component T1 for Al nuclei in a chromium-doped characteristic of the material and can be used to esti- Al2O3 crystal (measurements under the conditions of mag- imp netic suppression of the impurity contribution). The solid mate the impurity component T 1 for other samples line represents the theoretical temperature dependence of Σ lat from the spinÐlattice relaxation time T 1 determined by the lattice component T 1 for Raman processes within the the conventional technique. Debye approximation of the phonon spectrum. The dashed lat line corresponds to the dependence T ∝ T Ð0.6. The actual phonon spectrum in noncubic corundum 1 crystals corresponds to the Debye approximation to a smaller extent [16]. Although the temperature depen- Σ ± 27 dence of the heat capacity of Al2O3, as a whole, is well measured time T 1 = 240 5 s for Al nuclei in this Σ described in the framework of this approximation, it is crystal at T = 78 K exceeds any of the T times avail- of interest to clarify whether the Debye approximation 1 can be applied to the description of the nuclear spinÐlat- able in the literature for Al2O3. Under the conditions of tice relaxation rate. Note that doped Al2O3 crystals have magnetic suppression, we separated the contributions been extensively studied as an example of materials of the mechanisms of 27Al spinÐlattice relaxation in this with impurity relaxation proceeding through spin diffu- crystal and demonstrated that the role played by the sion. Since the contribution of the lattice mechanism is impurity mechanism is insigniﬁcant at temperatures relatively small, conventional techniques have failed to lat lat higher than 78 K and that the lattice component T 1 is estimate T 1 even for specially synthesized pure Al2O3 Σ samples. In our recent work [13], we studied a leu- equal to the spinÐlattice relaxation time T 1 at tempera- cosapphire crystal grown using a special procedure tures above the liquid-nitrogen temperature. In the devised at the State Optical Institute and found that the present work, we measured the temperature depen-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TEMPERATURE DEPENDENCE OF THE SPINÐLATTICE RELAXATION TIME 1049

Σ 4. P. Yu. Eﬁtsenko, V. M. Mikushev, and E. V. Charnaya, dence of the spinÐlattice relaxation time T 1 for the Pis’ma Zh. Éksp. Teor. Fiz. 54 (10), 583 (1991) [JETP same corundum crystal with the use of the conventional Lett. 54, 587 (1991)]. technique without additional stationary saturation of 5. I. Mavlonazarov, V. M. Mikushev, and E. V. Charnaya, the spin system. The experimental results obtained are Pis’ma Zh. Éksp. Teor. Fiz. 56 (1), 15 (1992) [JETP Lett. represented in Fig. 6. The temperature dependence of 56, 13 (1992)]. Σ the spinÐlattice relaxation time T for 27Al nuclei in a 6. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of 1 NMR in One and Two Dimensions (Clarendon, Oxford, lightly doped sample indicates that the impurity com- 1987; Mir, Moscow, 1990). imp ponent T 1 makes a considerable contribution over the 7. E. V. Charnaya, V. M. Mikushev, and E. S. Shabanova, entire temperature range under investigation. The tem- J. Phys.: Condens. Matter 6, 7581 (1994). lat 8. E. V. Charnaya, I. Mavlonazarov, and V. M. Mikushev, perature dependence of the lattice component T 1 for J. Magn. Reson., Ser. A 112, 96 (1995). the doped Al2O3 sample, which was obtained under the 9. A. A. Kuleshov, V. M. Mikushev, A. L. Stolypko, et al., conditions of magnetic suppression of the impurity Fiz. Tverd. Tela (Leningrad) 28 (11), 3262 (1986) [Sov. relaxation, virtually coincides, to within the experimen- Phys. Solid State 28, 1837 (1986)]. Σ 10. A. A. Kuleshov, A. L. Stolypko, E. V. Charnaya, and tal error, with the dependence T 1 (T) for the leucosap- V. A. Shutilov, Dokl. Akad. Nauk SSSR 293 (6), 1361 lat (1987) [Sov. Phys. Dokl. 32, 308 (1987)]. phire sample. The time T = 260 ± 20 s for Al O at 1 2 3 11. G. L. Antokol’skiœ and E. V. Charnaya, Fiz. Tverd. Tela 78 K agrees well with the results obtained in [7, 13]. (Leningrad) 17 (6), 1552 (1975) [Sov. Phys. Solid State However, the temperature dependence of the spinÐlat- 17, 1019 (1975)]. tice relaxation time is very weak and can be represented 12. E. V. Charnaya, V. M. Mikushev, A. M. Ulyashev, and lat ∝ Ð0.6 D. A. Yas’kov, Physica B (Amsterdam) 292, 109 (2000). as T 1 T (dashed line in Fig. 6). At the same time, 13. A. Chandoul, E. V. Charnaya, A. A. Kuleshov, et al., lat according to [2], the lattice relaxation time T 1 should J. Magn. Reson. 135, 113 (1998). change inversely with temperature even in the case of 14. E. R. Andrew, Nuclear Magnetic Resonance (Cambridge direct one-phonon transitions whose contribution to Univ. Press, London, 1955; Inostrannaya Literatura, relaxation is insigniﬁcant [2]. For comparison, the Moscow, 1957). dependence calculated from relationship (4) at the 15. W. W. Simmons, W. J. O’Sullivan, and W. A. Robinson, Debye characteristic temperature Θ = 1042 K [16] is Phys. Rev. 127 (4), 1168 (1962). shown by the solid line in Fig. 6. Thus, our results dem- 16. Handbook of Physical Quantities, Ed. by I. S. Grigoriev onstrated that the model of Raman processes in the and E. Z. Meilikhov (Énergoatomizdat, Moscow, 1991; Debye approximation for the phonon spectrum is inap- CRC Press, Boca Raton, 1997). plicable to the description of spinÐlattice relaxation of 17. V. M. Mikushev and E. V. Charnaya, Nuclear Magnetic Resonance in Solids (Sankt.-Peterb. Gos. Univ., Al nuclei in corundum. St. Petersburg, 1995). 18. A. Bakhramov, A. L. Stolypko, E. V. Charnaya, and V. A. Shutilov, Fiz. Tverd. Tela (Leningrad) 28 (3), 844 REFERENCES (1986) [Sov. Phys. Solid State 28, 470 (1986)]. 19. B. I. Kochelaev, Zh. Éksp. Teor. Fiz. 37 (1), 242 (1959) 1. G. R. Khutsishvili, Usp. Fiz. Nauk 87 (2), 211 (1965) [Sov. Phys. JETP 10, 171 (1960)]. [Sov. Phys. Usp. 8, 743 (1966)]. 20. E. G. Wikner and E. L. Hahn, Bull. Am. Phys. Soc. 3 (5), 2. A. Abragam, The Principles of Nuclear Magnetism 325 (1958). (Clarendon, Oxford, 1961; Inostrannaya Literatura, 21. J. van Kranandonk, Physica (Amsterdam) 29 (10), 781 Moscow, 1963). (1954). 3. A. R. Kessel’, Nuclear Acoustic Resonance (Nauka, Moscow, 1969). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1050Ð1054. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1006Ð1010. Original Russian Text Copyright © 2002 by Mart’yanov, Shandarov, Litvinov.

SEMICONDUCTORS AND DIELECTRICS

Interaction of Light Waves on a Reflecting Holographic Grating in Cubic Photorefractive Crystals A. G. Mart’yanov, S. M. Shandarov, and R. V. Litvinov Tomsk State University of Control Systems and Radio Electronics, Tomsk, 634050 Russia e-mail: [email protected] Received July 2, 2001

Abstract—The interaction of two antiparallel light waves on a reﬂecting grating is analyzed theoretically in arbitrarily oriented optically active cubic crystals of point group 23. The effect of nonunidirectional energy transfer on the interaction efﬁciency is investigated in the undepleted-pump-power approximation in (100)-, () (111)-, and 112 -cut Bi12TiO20 crystals. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION of two circularly polarized eigenwaves (with allowance for their attenuation): The interaction of light waves on reflecting photorefractive gratings is of interest for the development of E ()x = [C ()x e exp()Ðik n x holographic-interferometry devices [1] and narrow- P P1 1 0 1 band optical filters [2]. In sillenites, reflecting gratings α (1) + C ()x e exp()]Ðik n x exp Ð--- x , have been investigated only for (111) cuts [1] and (100) P2 2 0 2 2 cuts [3] of Bi12TiO20 and Bi12SiO20 crystals. In this () [ () () paper, the interaction of two light beams on reflecting ES x = CS1 x e1*exp ik0n1 x gratings is investigated in arbitrarily oriented optically α (2) active cubic crystals of point group 23. + C ()x e*exp()]ik n x exp --- x , S2 2 0 2 2 2. RESULTS AND DISCUSSION ± where e1, 2 = (y0 iz0)/2 are the polarization vectors ± ρ We consider two antiparallel (signal and pump) of the eigenwaves; n1, 2 = n0 /k0 are the refractive π λ waves of intensities I and I , respectively, propagating indices for these waves; k0 = 2 / is the wavenumber in S P α along the x axis in a photorefractive crystal of point vacuum; n0 and are the refractive index and the group 23 (Fig. 1). The crystal end faces (x = 0, x = Ðd) absorption coefficient of the crystal, respectively; and ρ are assumed to be antireflection-coated; therefore, there is its rotatory power. The interference pattern formed in are no reflected waves in the crystal. The signal and the crystal is characterized by the grating vector K = pump light fields in the crystal are represented as sums 2k0n0x0 and by the contrast

C C* + C C* mx()= 2------S1 P2 S2 P1 - (3) ()α2 2 ()()α2 2 () CP1 + CP2 exp Ð x + CS1 + CS2 exp x

and gives rise to a charge redistribution over defect cen- where ED = (kBT/e)K is the diffusion field and Eq = Ӷ ε | | π Λ ters. In the case of a low contrast m 1, the space eNA/ K is the trap saturation field, with K = K = 2 / charge field (which forms through diffusion) contains and Λ being the spatial period of the photorefractive only the first spatial harmonic with vector K. Within the grating. For an arbitrary crystal orientation relative to single-level band model of the photorefractive crystal, the grating vector K (Fig. 1), the change in the optical the first-harmonic amplitude is given by [4] properties of the medium is due to both the linear electrooptic effect and an additional photoelastic contribution associated with elastic strains of the crystal that are ED produced through the piezoelectric effect [5]. Using E1 ==ÐimESC Ðim------, (4) 1 + ED/Eq well-known formulas from [6, 7] and Eq. (4), the total

1063-7834/02/4406-1050 $22.00 © 2002 MAIK “Nauka/Interperiodica” INTERACTION OF LIGHT WAVES ON A REFLECTING HOLOGRAPHIC GRATING 1051 amplitude of the change in the permittivity tensor in the z eS ∆θ e electrooptic crystal can be found to be S S0

4 S 4 ∆ε ==Ðn r ∆b E imn E ∆b , IP mn 0 41 mn 1 0 SC mn –d 1 E (5) ∆b = δ m +,------()P m γ e m m mn mnp p S mnkl l ki pir p r 0 r41 eS0 y S E θ where rmnp and pmnkl are the electrooptic tensor of the 0 clamped crystal and the photoelastic tensor measured x γ in a static electric field, respectively; ki is the tensor IS Γ E E inverse of ik = (Cijkl mjml); Cijkl and epir are the elastic eP0 moduli tensor and the piezoelectric constants, respec- θ δ 0 tively; mnp is the unit antisymmetric tensor of third rank; and mp are the direction cosines of the grating Fig. 1. Schematic diagram of two-wave interaction. vector K, which in our case is taken to be parallel to the unit vector x0 of the crystallographic coordinate frame. Within the slowly varying amplitude approximation, where the wave equation for gyrotropic media can be reduced to the following equations for coupled waves describ- γxC* C exp()i2ρx Ð 1 P1 P2 ing two-wave interaction on the reflecting grating: Gx( ) = Ð----- gE + 2Im gI ------ρ - , (8) 2 IP0 2 x γ dCS1 []α()ρ () ------= Ð---mg*I CP1 exp Ði2 x + gECP2 exp Ð x , x dx 4 γ Φ()x = Ð------∫ exp[]G()ξ dC γ 2IP0 (9) S2 []α()ρ () 0 ------= Ð---mgECP1 + gICP2 exp i2 x exp Ð x , dx 4 × []2 ()ρξ 2 ()ρξ ξ g*I CP1 exp Ði2 Ð gICP2 exp i2 d , γ dCP1 []α()ρ () and I = |C |2 + |C |2 is the pumping wave intensity at ------= Ð---m* gICS1 exp i2 x + gECS2 exp x , P0 P1 P2 dx 4 x = 0. The signal-wave gain due to the reflecting grating and the polarization of this wave are determined by the dC γ ------P-2 = Ð---m*[]αg C + g*C exp()Ð2i ρx exp()x , contractions gE and gI, as seen from Eqs. (2), (3), and dx 4 E S1 I S2 (7)Ð(9). Using Eq. (5), it can be easily verified that (6) when the grating vector points along a 〈110〉 direction γ 3 S we have gE = gI = 0 and there is no interaction between where = k0 n0r41 ESC is the coupling constant and gI = the waves. ∆ ∆ ∆ (e1* be2) and gE = (e1* be1) = (e2* be2) are the tensor In (111)-cut crystals with K || [111], we have contractions describing the contributions from intra- ∆ ∆ mode (without a change in the refractive-index eigen- gI ==0, gE b11 Ð b12 values) and intermode processes, respectively, to the E 1 2e ()p ++p p Ð 2 p (10) interaction between waves propagating in opposite = Ð------1 Ð.------14 11 12 13 44 - directions. S ()E 3 r41 C11 ++2C12 4C44 In the undepleted pumping wave approximation, the With the values of the electrooptic constant for a eigenwave amplitudes CP1 and CP2 are independent of S the coordinate x and a solution for the components of clamped crystal r41 , piezoelectric and photoelastic the signal light wave field can be found to be E constants e14 and pmn , respectively, and elastic moduli () () E CS1 x = CS1 0 Cmn of bismuth titanate, which can be found in [8], we () obtain gE = Ð0.266. In this case, efficient energy transfer m 0 {}[]() Φ() + ------CP2 expGx Ð 1 + C*P1 x , occurs between the left-circularly polarized signal 2 (7) ≠ wave (CS1(0) 0, CS2(0) = 0) and the right-circularly () () ≠ CS2 x = CS2 0 polarized pumping wave (CP2 0, CP1 = 0): γα m()0 c gE Ð + ------{}C []expGx()Ð 1 Ð C* Φ()x , E ()x = e*C ()0 exp Ð------x exp()ik n x . (11) 2 P1 P2 S 1 S1 2 0 1

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1052 MART’YANOV et al.

2.5 P 1 I ()Ðd 1 Γ = --- ln ------S - ; (12) 2.0 d 0() IS Ðd –1 1.5 2 in the case at hand, it is found to be

1.0 c 2π 3 S gEED Γ[]==g γ ------n r ------. (13) Gain, cm 111 E λ 0 41 3 1 + ED/Eq 0.5 The same signal-wave gain characterizes the interaction between the signal and pump waves linearly polar- 0 5 10 15 20 ized in the same direction d, mm l Γ[]= g γ ; (14) Fig. 2. Dependence of the signal-wave gain due to two- 111 E wave interaction in a Bi12TiO20 crystal on its thickness for the polarization structure of the signal wave also the grating vector K pointing along the [100] axis and for Ð1 remains unchanged. the coupling constant γ equal to (1) 2, (2) 4, and (3) 6 cm . || || In (100)-cut crystals, for y0 [010] and z0 [001], we have Gain, cm–1 7 3 g ===0, g Ði∆b Ði. (15) (a) E I 23 3' 5 2 It should be noted that when the pumping wave is circularly polarized (CP1 = 0 or CP2 = 0), there is no signal- 3 1 wave gain in (100)-cut crystals since G(x) = 0. If the 1 signal and pump waves are of the same circular polarization (say, right-handed, i.e., CP1 = 0, CS1 = 0), they –1 1' do not interact, because there is no interference –3 between them, m(0) = 0. In the case where the pumping 2' wave is left-circularly polarized and the signal wave at –5 the entrance of the crystal is of the right-hand polarization (CS2(0) = 0), the amplitudes of the eigenwaves in –7 the crystal can be found from Eqs. (7) to be 7 3 C ()x = C ()0 , 3' (b) S1 S1 5 2 γ (16) () () []()ρ CS2 x = CS1 0 ------expi2 x Ð 1 . 3 1 4ρ 1 1' Thus, the left-circularly polarized signal wave, in com- 30 60 90 120150 180 bination with the pumping wave, produces a photore- –1 fractive grating in this case but is not involved directly –3 in the energy transfer. The energy of the pumping wave 2' is transferred, through its diffraction by grating, to the –5 right-circularly polarized component of the signal θ , deg wave. The corresponding gain due to two-wave interac- –7 0 tion Fig. 3. Signal-wave gain due to two-wave interaction calcu- γ 2 Γc 1  2 ()ρ lated without (dashed curves) and with allowance for non- []100 = --- ln 1 + ------sin d (17) unidirectional energy transfer (solid curves) as a function of d 2ρ θ the angle 0 for Bi12TiO20 crystals of thickness d equal to (a) 2 and (b) 6 mm with the grating vector K pointing along is independent of the sign of the electrooptical constant the [100] axis. The coupling constant γ is equal to (1, 1') 2, and characterizes the nonunidirectional energy transfer (2, 2') 4, and (3, 3') 6 cmÐ1. effect, which was considered earlier only in the transmission geometry of wave interaction [9]. For crystals Γc The polarization of the signal wave remains unchanged. with low rotatory power, the gain []100 can reach high The signal-wave gain due to the two-wave interaction is Γc values. Figure 2 shows the dependence of []100 on the P ρ ° deﬁned in terms of the signal-wave intensities (IS (Ðd)) Bi12TiO20 sample thickness ( = 6.5 /mm for the wave- 0 length λ = 633 nm) for different values of the coupling and (IS (Ðd)) in the presence and absence of the pump- constant γ. We note that the maximum efﬁciency of the ing wave, respectively, as nonunidirectional energy transfer is reached at certain

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 INTERACTION OF LIGHT WAVES ON A REFLECTING HOLOGRAPHIC GRATING 1053

20 (a) 4 3 3 (a) 10 2 3 2 0 3' 30 60 90 120 150 180 2 –10 –1 1 4 ' –20 1 1 2 90 120 –30 1' Gain, cm 0 30 60 150 180 –40 –1 –50 θ , deg

, deg 0

S 40 –2 3' (b) 30 60 90 120 150 180 ∆θ 20 0 2' 0 (b) 30 60 90 120 150 180 –20 2 –20 1' 4' –40 4 , deg

S –40

–60 ∆θ 1 –80 –60 –100 θ 0, deg 3 θ –80 0, deg

Fig. 4. Polarization rotation angle of the signal wave as a Fig. 5. (a) Signal-wave gain due to two-wave interaction θ function of the angle 0 in Bi12TiO20 crystals of thickness d calculated without (dashed curves) and with allowance for equal to (a) 2 and (b) 6 mm with grating vector K pointing nonunidirectional energy transfer (solid curves) and (b) along the [100] axis. The coupling constant γ is equal to polarization rotation angle of the signal wave as functions θ (4) 0, (1, 1') 2, (2, 2') 4, and (3, 3') 6 cmÐ1. of the angle 0 in a Bi12TiO20 crystal of thickness d = 6 mm with grating vector K pointing along the [112 ] axis. The coupling constant γ is equal to (4) 0, (1, 1') 2, (2, 2') 4, and optimal values of the interaction length, which decrease (3, 3') 6 cmÐ1. with increasing coupling constant. In the case of linearly polarized waves interacting in (100)-cut crystals, the signal-wave amplification is due pointing along the same direction at the x = 0 crystal to both unidirectional and nonunidirectional energy face, eS0 = eP0], the polarization of the signal wave transfer. The polarization dependences of the gain remains linear everywhere over the crystal. However, as l the signal wave propagates through the crystal, its Γ[] due to two-wave interaction as calculated from 100 polarization vector e rotates, which is due to both the Eqs. (7)Ð(9) are shown in Fig. 3. The signal and pump S waves at x = 0 are assumed to have identical polariza- optical activity of the crystal and the interaction with θ the pumping wave. The dependence of the angle ∆θ tion vectors eS0 = eP0, making an angle 0 with the y0 S axis. It should be noted that in a thin sample (d = 2 mm, through which the vector eS rotates as the signal wave Fig. 3a), the positions of the extrema in the polarization propagates from the x = 0 to the x = Ðd crystal face at an θ dependences calculated without (dashed curves) and angle 0 (defining the direction of this vector at x = 0) with allowance for nonunidirectional energy transfer is shown in Fig. 4. Dashed lines in this figure corre- (solid curves) are almost the same and correspond to spond to the absence of wave interaction (γ = 0) and, || 〈 〉 the polarization vector orientations eP 011 in the therefore, represent the angle of the polarization vector middle of the crystal. In a thicker crystal (d = 6 mm, || rotation due to optical activity of the crystal. In the Fig. 3b), the polarization vector orientation eP [011] in θ ° ρ presence of the pumping wave, the polarization rotation the middle of the crystal (for 0 = 45 + d/2) differs angle of the signal wave increases with the coupling from its optimal orientation, the difference increasing constant and with the crystal thickness. When γ = 6 cmÐ1 with the coupling constant γ. We note that the nonuni- and d = 6 mm (Fig. 4b), the maximum rotation angle is directional energy transfer (always proceeding from a ° strong wave to a weak one) broadens the range of θ roughly 70 . We note that the maximum effect of the 0 reflecting grating on the signal-wave polarization takes Γl θ values in which the gain []100 ( 0) is positive. place in the case where the signal-wave gain is minimal In the case under discussion [a (100)-cut crystal and in magnitude. This agrees with the results presented in linearly polarized waves with their polarization vectors [10] for the case of wave interaction on a transmission

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1054 MART’YANOV et al. grating characterized by vector K || []110 in optically REFERENCES active (110)-cut crystals. 1. N. V. Kukhtatev, B. S. Chen, P. Venkateswarlu, et al., In general, both intramode and intermode interac- Opt. Commun. 104, 23 (1993). tions between waves take place. In the case of the 2. I. F. Kanaev, V. K. Malinovskiœ, and N. V. Surovtsev, Fiz. reflecting-grating vector orientation K || []112 in bis- Tverd. Tela (St. Petersburg) 42 (11), 2079 (2000) [Phys. Solid State 42, 2142 (2000)]. muth titanate, we have gI = 0.467 and gE = 0.216 for || [] || [] 3. S. Mallick, M. Miteva, and L. Nikolova, J. Opt. Soc. Am. y0 110 and z0 111 . Figure 5 shows numerically B 14 (5), 1179 (1997). Γl calculated polarization dependences of the gain []112 4. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, et al., Fer- due to two-wave interaction and of the polarization roelectrics 22, 949 (1979). ∆θ rotation angle S for a linearly polarized signal and 5. A. A. Izvanov, A. E. Mandel’, N. D. Khat’kov, and pump waves with their polarization vectors pointing S. M. Shandarov, Avtometriya, No. 2, 79 (1986). along the same direction at x = 0 (eS0 = eP0). We note 6. S. I. Stepanov, S. M. Shandarov, and N. D. Khat’kov, that in this case, the positions of the extrema of the Fiz. Tverd. Tela (Leningrad) 29 (10), 3054 (1987) [Sov. polarization rotation angle of the signal wave (Fig. 5b) Phys. Solid State 29, 1754 (1987)]. coincide with those of the rate of change in the gain Γl θ 7. B. I. Sturman, E. V. Podivilov, K. H. Ringhofer, et al., ( 0) (Fig. 5a). Phys. Rev. E 60, 3332 (1999). 8. O. V. Kobozev, S. M. Shandarov, R. V. Litvinov, et al., 3. CONCLUSION Neorg. Mater. 34 (12), 1486 (1998). Thus, within the undepleted-pumping-wave approx- 9. V. Yu. Krasnoperov, R. V. Litvinov, and S. M. Shandarov, imation, we have derived relations that describe the Fiz. Tverd. Tela (St. Petersburg) 41 (4), 632 (1999) interaction of two light waves on a reflecting photore- [Phys. Solid State 41, 568 (1999)]. fractive grating in cubic gyrotropic crystals. It was 10. R. V. Litvinov, S. M. Shandarov, and S. G. Chistyakov, shown that the nonunidirectional energy transfer can Fiz. Tverd. Tela (St. Petersburg) 42 (8), 1397 (2000) significantly affect the wave interaction efficiency. The [Phys. Solid State 42, 1435 (2000)]. changes in the polarization of the signal wave in the presence of the pumping wave depend on the differently cut crystals and their thicknesses. Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1055Ð1060. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1011Ð1016. Original Russian Text Copyright © 2002 by Vishnevskaya, Frolova, Yulmetyev.

SEMICONDUCTORS AND DIELECTRICS

Manifestation of Liquid-State and Solid-State Properties in Electron Relaxation of Paramagnetic Ions in Solutions: Non-Markovian Processes G. P. Vishnevskaya*, E. N. Frolova*, and R. M. Yulmetyev** * Zavoœskiœ Physicotechnical Institute, Russian Academy of Sciences, Sibirskiœ trakt 10/7, Kazan 29, 420029 Tatarstan, Russia ** Kazan State Pedagogical University, Kazan, Tatarstan, Russia e-mail: [email protected] Received July 3, 2001

δ 3+ 2+ 3+ Abstract—The linewidth H and the spinÐspin relaxation time T2 for Gd , Mn , and Cr ions in aqueous, waterÐglycerol, and waterÐpoly(ethylene glycol) solutions at paramagnetic ion concentrations providing the dipoleÐdipole mechanism of spin relaxation are measured using two independent methods, namely, electron paramagnetic resonance (EPR) and nonresonance paramagnetic absorption in parallel fields. Analysis of the experimental results indicates a gradual crossover from pure liquid-state (diffusion) to quasi-solid-state (rigid lattice) spin relaxation. It is demonstrated that the limiting cases are adequately described by standard, universally accepted formulas for dipoleÐdipole interactions in the liquid-state (the correlation time of translational τ Ӷ τ ӷ motion satisfies the condition c T2) and solid-state ( c T2) approximations. A complete theoretical treatment of the experimental dependences (including the observed gradual crossover of spin relaxation) is performed in the framework of the non-Markovian theory of spin relaxation in disordered media, which is proposed by one of the authors. Within this approach, the collective memory effects for spin and molecular (lattice) variables are taken into account using the first-order and second-order memory functions for spin–spin and spinÐlattice interactions. A correlation between the spin magnitude and the temperatureÐviscosity conditions corresponding to the crossover to non-Markovian relaxation is revealed, and the situations in which structural transformations occurring in the solutions favor the crossover to solid-state spin relaxation are analyzed. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION implies a gradual crossover from the liquid-state to the solid-state mechanism of spin relaxation. It is well known that electron relaxation of paramagnetic ions in a condensed medium substantially depends on the spinÐspin interactions and the proper- 2. EXPERIMENTAL TECHNIQUE ties of the medium. Intermolecular (dipoleÐdipole and exchange) interactions are governed by the distance The solutions used in the experiments were pre- between paramagnetic centers and the velocity of trans- pared from chromium, manganese, and gadolinium lational motion of particles in the solution. Radiospec- nitrate salts (analytical grade). The concentrations of troscopic investigations of the intermolecular interac- Cr3+ and Gd3+ ions in the solutions prepared varied in tions can provide valuable information on the spatial the range from 0.01 M to concentrations close to the distribution of paramagnetic particles in the liquid-state solubility limits of these salts in aqueous solutions (2.7 lattice and, consequently, on the structure of the studied and 3.6 M, respectively). It should be noted that inter- solutions and changes in the molecular mobility of the molecular interactions occurring in these solutions are paramagnetic complexes in these solutions at different governed only by dipoleÐdipole interactions. However, viscosities and concentrations. the solutions with a high concentration of Mn2+ are characterized by exchange interactions in addition to The main objective of this work was to analyze the dipoleÐdipole interactions. For this reason, the concen- conditions corresponding to a change in the ratio tration of Mn2+ ions was chosen so as to exclude the between the spinÐspin relaxation time T2 and the corre- occurrence of exchange interactions (0.5 M). The τ lation time of molecular motion c in the following sit- required changes in the viscosity of the solution were ӷ τ τ Ӷ τ uations: T2 c, T2 ~ c, and T2 c. According to the accomplished by varying either the salt concentration concept of non-Markovian random processes, this or the content of glycerol and poly(ethylene glycol) in change is associated with a gradual crossover of a spin the solvents. The viscosity was measured using an Ost- system from Markovian to quasi-Markovian and then wald viscometer or a viscometer designed by Gon- to non-Markovian spin relaxation. Physically, this charov [1].

1056 VISHNEVSKAYA et al.

The time and rate of the spinÐspin relaxation were and the relaxation time T2 on the viscosity are shown in measured using two independent methods at room tem- Figs. 2Ð6. As can be seen from Fig. 1, the concentra- δ perature: (i) nonresonance paramagnetic absorption in tion-independent contribution H0 is equal to 204 G for parallel fields [2, 3] and (ii) electron paramagnetic res- Gd(NO3)3 aqueous solutions and 150 G for Cr(NO3)3 onance (EPR) in the X band. Note that, even at an insig- aqueous solutions. In the case when the measurements nificant water content, the absorption signals observed were performed at a constant concentration of para- in waterÐglycerol and waterÐpoly(ethylene glycol) magnetic ions and the required viscosities were solutions are attributed to aqua ions [3, 4]. achieved by varying the content of glycerol or poly(eth- In order to determine the linewidth assigned to the ylene glycol) in the solutions (Figs. 3, 5), the concen- δ δ dipoleÐdipole interaction Hdd, the concentration-inde- tration-independent contribution H0 was determined δ pendent contribution H0, which is associated with from the EPR spectra of diluted solutions (~0.01 M) intramolecular mechanisms of spin relaxation and, pos- with a similar composition of the solvent and virtually sibly, with an incompletely averaged fine structure, was the same viscosity. subtracted from the measured total linewidth δH. The It is seen from Figs. 2Ð6 that the experimental δ linewidth Hdd was calculated from the following for- dependences of T , δH , and δH /C on the viscosity δ δ δ 2dd dd dd mulas: Hdd = H Ð H0 for a Lorentzian line shape and for solutions of ions with different spins exhibit a simi- δH = (δH2 Ð δH2 )1/2 for a Gaussian line shape. lar behavior. As the viscosity increases in the low-vis- dd 0 cosity solutions, the linewidth δH increases, whereas According to our estimates, the resonance line has a dd Lorentzian shape for aqueous and waterÐglycerol solu- the spinÐspin relaxation time T2 correspondingly tions in which the concentration of Gd3+ or Cr3+ ions decreases and gradually reaches a steady-state value does not exceed 2.5Ð2.7 M. At higher concentrations of independent of further change in the viscosity. Except 3+ in the range of the observed gradual crossover, the Gd ions, the resonance line has a Gaussian shape. In experimental dependences can be adequately described the case of nonresonance paramagnetic absorption by standard, universally accepted formulas [5Ð7]: measurements, there is no need to determine the concentration-independent contribution, because nonreso- 4 4 1 23g β η nance absorption due to spinÐlattice (T ) and spinÐspin ----- = 1.78× 10 ------CS() S + 1 (1) 1 T 2 T (T2) relaxations is measured separately; furthermore, 2 h k the dipoleÐdipole interactions make a dominant contribution to the spinÐspin relaxation time T and do not in the liquid-state approximation (the correlation time 2 of translational Brownian motion satisfies the condition affect the spinÐlattice relaxation time T1 [3]. τ c < T2) and

2 2 1 20g β 3. RESULTS AND DISCUSSION ----- = 26.29× 10 ------CSS()+ 1 (2) T ប The experimental dependences of the total EPR 2 linewidth on the concentration of paramagnetic centers in the solid-state approximation when the dipoleÐ are depicted in Fig. 1. The dependences of the linewidth dipole interactions are not averaged through molecular motion (the correlation time of translational Brownian τ η motion satisfies the condition c > T2). Here, is the 700 macroscopic viscosity of the solution and C is the con- 1 centration of paramagnetic ions in terms of M. The cor- τ λ2 λ 600 relation time is defined as c = /(12D), where is the root-mean-square length of hopping in an elementary 500 diffusion event, D = kT/(6πηa) is the diffusion coefficient described by the Stokes relationship, and a is the , G

H radius of the complex. The correlation time can be cal- δ 400 2 culated from the formula 300 π 3η τ 2 a c = ------. (3) 200 kT τ 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 This formula gives the most reliable estimate of c in C, å the framework of the diffusion and hopping models of particle motion in the course of spin-exchange interaction due to collisions [7]. Fig. 1. Dependences of the total EPR linewidth on the concentration of (1) Gd(NO3)3 and (2) Cr(NO3)3 in aqueous At high concentrations of paramagnetic ions, the solutions at 298 K. relaxation rate in the solid-state approximation for a

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MANIFESTATION OF LIQUID-STATE AND SOLID-STATE PROPERTIES 1057

200 180

160 180 140 160 , G , G/M 120 C dd / H dd δ H δ 100 140 80 120 60 1.0 1.5 2.0 2.5 3.0 3.5 4.0 η 0 5 10 15 20 25 , cP η, cP δ δ Fig. 2. Dependence of Hdd/C on the viscosity of Gd(NO3)3 Fig. 3. Dependence of Hdd on the viscosity of Gd(NO3)3 in aqueous solutions at 298 K. Points are the experimental in 1.2 M waterÐglycerol solutions at 298 K. The viscosities data. The solid line represents the theoretical results η correspond to the specified concentrations of glycerol in obtained in the liquid-state approximation. the solutions.

10 20 18 8 16 14 s 6 s 12 –10 –10

, 10 10 2 , 10 2 T 4 T 8 6 2 3 3 1 4 2 2 2 1 0 2 4 6 8 10 12 η, cP 0 20 40 60 80 100 η, cP Fig. 4. Dependences of the spinÐspin relaxation time on the viscosity of Gd(NO3)3 in aqueous solutions at 298 K. The Fig. 5. Dependences of the spinÐspin relaxation time on the η viscosities correspond to the speciﬁed concentrations of viscosity of Mn(NO3)2 in 0.5 M waterÐpoly(ethylene gly- Gd(NO3)3 in the solutions. Points are the experimental data. col) solutions at 298 K. The viscosities η correspond to the Solid lines represent the theoretical results obtained in the speciﬁed concentrations of poly(ethylene glycol) in the framework of (1) liquid-state and (2) solid-state approxima- solutions. Points are the experimental data. Designations of tions and (3) our theory. the curves are the same as in Fig. 4.

Gaussian line shape can be expressed through the sec- If required, the linewidths and the spinÐspin relax- ond moment; that is, ation times can be recalculated from the following rela- 1/2 tionships: 4β4 1 3g ()Ð6 -----= 2 ------2 SS+ 1 ∑r jk , (4) T 2 5 ប k δ 1 Ӷ H = ------αγ -, T 2 T 1, where r is the distance between the jth and kth para- T 2 jk (5) magnetic centers (r ~ CÐ1/3). For a simple cubic lattice δ 2 ≤ Ð6 Ð6 H = ------αγ -, T 2 T 1, with parameter d, we have ∑k r jk = 8.5d [6]. T 2

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1058 VISHNEVSKAYA et al.

90 structural transformations in solutions. For example, it is this situation that occurs in Cr(NO3)3 aqueous solu- 80 2 tions in which stable multilayer hydrate shells are 3 formed around Cr3+ ions [3, 8]. As a consequence, com- 70 plete hydration—all water molecules are in the bound state—is attained in Cr(NO3)3 aqueous solutions even 60 at concentrations in the range 2.2Ð2.3 M and viscosities η , G/M = 10Ð12 cP. For the concentrations and viscosities

C 50 / corresponding to complete hydration and above, the dd

H 1 structure of solutions should be treated as a disordered δ 40 pseudocrystalline structure. In these solutions, the 30 dipoleÐdipole interactions are well described within the solid-state approximation (Fig. 6). 20 At intermediate concentrations, many aqueous solutions of salts undergo phase separation into microregions with a structure of the free solvent and microre- 2 4 6 8 10 12 14 gions whose structure is similar to that of the crystal η, cP hydrate [9, 10]. Inside microregions of the latter type, ions are closely spaced and their translational motion is δ Fig. 6. Dependences of Hdd/C on the viscosity of hindered. In this case, the dipoleÐdipole interactions Cr(NO3)3 in aqueous solutions of at 298 K. The viscosities can be described within the static approximation. This η correspond to the specified concentrations of Cr(NO3)3 in model was applied earlier to the description of the spe- the solutions. Points are the experimental data. Designa- ciﬁc features of the dipole–dipole interaction in Mn2+ tions of the curves are the same as in Fig. 4. aqueous solutions at concentrations above 2 M [3]. A similar situation arises in aqueous solutions of gadolin- α ium nitrate (Fig. 5). The solid-state approximation for where = 3 for a Lorentzian line shape and 1.18 for dipoleÐdipole interactions is also valid at lower concen- a Gaussian line shape, γ is the gyromagnetic ratio, and trations of paramagnetic ions in waterÐglycerol and T1 is the spinÐlattice relaxation time. waterÐpoly(ethylene glycol) viscous solutions (Figs. 2, By using the measured macroscopic viscosities and 4). However, the experimentally observed gradual the radii a = 4.15 Å for Cr3+ aqua complexes, a = 3 Å crossover from liquid-state (diffusion) to solid-state behavior of the system has deﬁed description with the for Mn2+ aqua complexes, and a = 3.8 Å for Gd3+ τ use of relationships (1), (2), and (4). aqua complexes, we calculated the correlation times c according to formula (3) and compared the obtained In general, the experimental dependences (including values with the experimental relaxation times T (see the observed gradual crossover) can be interpreted 2 within the statistical non-Markovian spin relaxation table). It is seen from the table that, at the viscosities η ≥ 3+ η ≥ 2+ theory developed by one of the authors of the present 5 cP for Cr aqua complexes, 10 cP for Mn work. This theory is based on the ZwanzigÐMori η ≥ 3+ aqua complexes, and 3 cP for Gd aqua complexes, kinetic equations [11, 12] for normalized spin time cor- τ the ratio between the correlation time c and the spinÐ relation functions. It should be noted that statistical τ τ spin relaxation time T2 changes from c < T2 to c ~ T2 memory effects manifest themselves in many random τ and c > T2. These results are in agreement with the processes of different physical nature. For example, the experimental dependences depicted in Figs. 2Ð5. non-Markovian processes have been revealed in Therefore, we can make the inference that the dipoleÐ nuclear magnetic resonance, infrared spectroscopy, and dipole interactions in systems with a low viscosity are dielectric relaxation [13Ð15]. The non-Markovian spin averaged through diffusion motion. In the case when relaxation theory makes allowance for collective mem- the viscosity is higher than the aforementioned values, ory effects of the molecular (lattice) and spin variables τ the inequality c < T2 becomes invalid and the molecu- with the use of the memory function for the description lar motion does not lead to the averaging of the dipoleÐ of the averaged dynamics of spinÐspin and spinÐlattice dipole interactions, even though the lattice system interactions. By using the results obtained in [16, 17], remains in the liquid state. Note that the greater the spin the kinetic equations for spin time correlation functions and the larger the radius of the complex, the lower the can be written in the following form: viscosity at which the system exhibits solid-state dµα()t η ≥ 2+ αω µ () behavior (in particular, at 20 cP in solutions of Mn ------= Ði 0 α t ions with S = 5/2 and at η ≥ 4Ð6 cP in solutions of Gd3+ dt ions with S = 7/2). t (6) τ[]µ()τ ()τ ()τ It is quite possible that, in some cases, the crossover Ð ∫d ka t Ð + Nα t Ð α , to the solid-state behavior can be caused by different 0

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MANIFESTATION OF LIQUID-STATE AND SOLID-STATE PROPERTIES 1059

τ 3+ Correlation times c calculated according to formula (3) and experimental relaxation times T2 for aqua complexes of Cr (T2 2+ 3+ are determined from the EPR linewidths), Mn , and Gd (T2 are measured using nonresonance paramagnetic absorption) Cr3+ Mn2+ Gd3+ η, cP τ Ð10 Ð10 τ Ð10 Ð10 τ Ð10 Ð10 c, 10 s T2, 10 s c, 10 s T2, 10 s c, 10 s T2, 10 s 1.2 1.38 0.50 18.0 1.00 10.00 1.5 1.63 0.62 13.5 1.26 4.75 2.2 2.40 26.0 0.90 9.5 1.84 3.60 2.5 2.72 20.8 1.03 8.3 2.09 3.30 3.5 3.81 14.0 1.44 6.4 2.93 2.40 4.0 4.36 11.5 1.65 6.0 3.35 2.10 6.0 6.54 5.0 2.47 5.0 5.03 1.80 8.8 9.60 3.8 3.62 4.5 7.90 1.70 10.0 10.90 3.5 4.12 4.2 8.38 1.60 12.0 13.10 3.0 4.94 4.0 10.80 1.60 15.0 16.35 2.9 6.18 3.9 12.55 1.60 20.0 8.24 3.9 16.76 1.64 30.0 12.36 3.6 40.0 16.50 4.0 60.0 24.70 4.0

α ± where = 0 and 1 are the components of the total spin dhβ()t (the indices α = 0 and ±1 refer to the longitudinal and ------= Ðiωβhβ()t ω dt transverse components, respectively), 0 is the Larmor t (10) frequency, and kα(t) is the nonsymmetric part of the τγ[]2 () Ω2 πτ() ()τ spin memory function [Nα(t) 0]. For the transverse Ð ∫d βhβ t + β hβ t Ð . spin components and the transverse component of the 0 memory function, we have In this equation, we introduced two relaxation frequen- () σ2 () 2 k+ t = ∑ 1βhβ t , (7) cies, γβ and Ωβ. The former frequency squared γ β corresponds in dimension to the ratio of the components of where the fourth moment to the components of the second 2 moments of the EPR lines (γ β ≈ M /M ) for the quasi- 2 〈〉[]Sˆ Ð, Hˆ β* , []Hˆ β, Sˆ + 4 2 σ β = ------, Ω 1 2 2 solid-state lattice. The latter frequency β is related to ប 〈〉Sˆ +()0 τ π τ (8) the characteristic diffusion time c. The function ( ) is 〈〉[]Sˆ Ð, Hˆ β* exp{}iLˆ t []Hˆ β, Sˆ + the normalized time correlation function for the veloc- hβ()t = ------. 〈〉[]ˆ , ˆ * , []ˆ , ˆ ity (momentum) of particles in the liquid. Therefore, SÐ Hβ Hβ S+ owing to the hierarchical structure of the spin memory β functions, the spin relaxation is described by two con- Here, hβ(t) is the th component of the first-order nor- tributions, namely, the molecular diffusion (liquid- 2 malized memory function, σαβ stands for the compo- state) and static spin (solid-state) contributions. nents of the second moment in magnetic resonance, Lˆ The calculation of the relaxation rate gives the fol- is the total Liouvillian of the whole system (including lowing formula: spins and motion), and ωβ is the frequency related to the ˆ ˆ 1 σ2 τ Zeeman part L0 of the operator L through the expres------= ∑ 1β c*. (11) sion T 2 τ* Lˆ 0[]ωHˆ β, Sˆ + = Ð β[]Hˆ β, Sˆ + . (9) Here, c is the effective correlation time, which characterizes the coexistence of two spin memory channels After a number of simplifications, for the first-order associated with molecular diffusion motion and spinÐ memory function, we obtain the kinetic equation spin interactions in the quasi-solid-state lattice; that is,

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1060 VISHNEVSKAYA et al.

1 2 2 Tatar Foundation for Research and Development τ* = ------[]14+1γ βτ Ð . (12) c γ 2 τ c [project no. 14-78/2000 (F)]. 2 β c In the two limiting cases, we obtain REFERENCES τ τ τ γ  c at c 0or c β 0 1. V. A. Goncharov, RF Patent No. 1749776. τ* =  (13) c Ð1 2. B. A. Volkov, G. P. Vishnevskaya, V. A. Gorozhanin, and τγ = γ β at τ ∞ or τ γ β ∞. c c R. T. Ramazanov, Prib. Tekh. Éksp., No. 4, 167 (1973). τ 3. G. P. Vishnevskaya, Radiospectroscopy of Condensed Here, the effective correlation time c* corresponds to either the liquid-state relaxation or the solid-state relax- Matter (Nauka, Moscow, 1990), p. 13. ation behavior. For the limiting Markovian or non- 4. G. P. Vishnevskaya, F. M. Gumerov, E. N. Frolova, and Markovian behavior of the system, the proposed theory A. R. Fakhrutdinov, Structure and Dynamics of Molecu- is reduced to the standard relationships [1Ð3]. In actual lar Systems (Inst. Fiz. Khim. Ross. Akad. Nauk, Mos- fact, within the liquid-state (Markovian) approxima- cow, 2000), Vol. VII, p. 329. τ* 5. A. Abragam, The Principles of Nuclear Magnetism tion, the correlation time c is determined by the for- (Clarendon, Oxford, 1961; Inostrannaya Literatura, σ2 Ð1 τ Moscow, 1963). mulas ∑ 1β = M2 and T 2 = cM2 [3]. Within the solid-state (non-Markovian) approximation, for a 6. S. A. Altshuler and B. M. Kozyrev, Electron Paramag- netic Resonance in Compounds of Transition Elements τ Lorentzian line shape, we have γ = M2 /M4 and (Nauka, Moscow, 1972; Halsted, New York, 1975). T Ð1 = M M /.M 7. K. I. Zamaraev, Yu. N. Molin, and K. M. Salikhov, Spin 2 2 2 4 Exchange (Nauka, Novosibirsk, 1977). Figures 4Ð6 (curves 3) represent the results of theo- 8. R. Caminiti, G. Licheri, G. Piccaluga, and G. Pinna, retical calculations with the use of formulas (11) and J. Chem. Phys. 69 (1), 1 (1978). τ × Ð10 3+ τ × (12) at γ = 4.54 10 s for Gd solutions, γ = 2.2 9. O. Ya. Samoœlov, Structure of Aqueous Electrolytic Solu- Ð10 2+ Ð10 10 s for Mn solutions, and τγ = 17.8 × 10 s for tions and Ion Hydration (Akad. Nauk SSSR, Moscow, 3+ τ Cr solutions and the correlation times c calculated 1957). from formula (3). 10. T. Erdey-Grüz, Transport Phenomena in Aqueous Solu- tions (Akadémiai Kiadö, Budapest, 1974; Mir, Moscow, 1976). 4. CONCLUSION 11. R. Zwanzig, Phys. Rev. 124, 983 (1961). Thus, the proposed theory provides the most ade- 12. H. Mori, Prog. Theor. Phys. 34, 765 (1965). quate qualitative and quantitative description of the experimental results obtained for all the studied ions, 13. B. B. Laird, J. Budimir, and J. L. Skinner, J. Chem. Phys. including the range of the observed gradual crossover 94, 4391 (1991). from Markovian to quasi-Markovian and non-Mark- 14. A. G. Hernández, S. Velasco, and F. Mauricio, J. Chem. ovian behavior. It is in this range that both the liquid- Phys. 86, 4597 (1987); 86, 4607 (1987). state and solid-state properties manifest themselves in 15. R. G. Abbot and D. W. Oxtoby, J. Chem. Phys. 72, 3972 electron relaxation of paramagnetic ions in the studied (1980). solutions. 16. R. M. Yulmetyev, Theor. Math. Phys. 30, 264 (1977). 17. V. Yu. Shurygin and R. M. Yulmetyev, Phys. Lett. A 174, ACKNOWLEDGMENTS 433 (1990). This work was supported by the Russian Foundation for Basic Research (project no. 00-03-32915a) and the Translated by O. Borovik-Romanova

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Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1061Ð1066. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1017Ð1022. Original Russian Text Copyright © 2002 by Bogomolov, Parfen’eva, Sorokin, Smirnov, Misiorek, Jezowski, Hutchison.

SEMICONDUCTORS AND DIELECTRICS

Structural and Thermal Properties of the OpalÐEpoxy Resin Nanocomposite V. N. Bogomolov*, L. S. Parfen’eva*, L. M. Sorokin*, I. A. Smirnov*, H. Misiorek**, A. Jezowski**, and J. Hutchison*** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] ** W. Trzebiatowski Institute of Low-Temperature and Structural Research, Polish Academy of Sciences, Wroclaw, 50-950 Poland *** Department of Materials, University of Oxford, Oxford OX1 3PH, UK Received August 21, 2001

Abstract—Thermal conductivity of the opalÐepoxy resin nanocomposite is measured in the range 4.2Ð250 K, and the material is studied by electron microscopy at 300 K. An analysis of the electron microscope images permits a conclusion on the character of opal void filling by the epoxy resin. It is shown that the thermal conductivity of the nanocomposite within the range 40–160 K can be fitted fairly well by the corresponding standard expressions for composites. For T < 40 K and T > 160 K, the experimental values of the nanocomposite thermal conductivity deviate strongly from the calculated figures. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ter a = 300Ð400 nm. Depending on the actual method used to fill the opal voids by various fillers and on the Synthetic opals are extremely interesting subjects to quality of the opal single crystals, one may conceive at physicists. They possess a unique fractal crystalline least three versions of preparation of an opal-based structure [1Ð3]. Opals consist of closely packed amor- nanocomposite: (a) the opal voids are completely filled phous SiO2 spheres, most frequently, 200Ð250 nm in (to 100%); (b) the voids are filled in an island pattern diameter (first-order spheres). These spheres contain an (to less than 100%), with the regularly filled voids in array of closely packed amorphous spheres of a smaller the opal crystal separated by the filler-free opal host; diameter, 30Ð40 nm (second-order spheres), which are and (c) the filler can be randomly distributed over the again made up of closely packed particles of amor- opal voids, likewise filled to less than 100%. phous SiO , now ~10 nm in diameter (third-order 2 Therefore, to ensure correct interpretation of the spheres). experimental data on the thermal conductivity of opal- The array of closely packed spheres has octahedral based nanocomposites, one needs to know the real pat- and tetrahedral voids which are interconnected by tern of void filling. Electron microscopy is capable of “channels.” The first-order voids (and spheres of amor- providing an answer to this problem. phous SiO2) make up a regular cubic array with a giant The thermal conductivity of opal and opal-based period a = 300Ð400 nm. The diameters of the first-order nanocomposites with PbSe and NaCl crystalline fillers octahedral and tetrahedral voids and channels are 80, is dealt with in [3, 5Ð7], and electron microscopic stud- 40, and 30 nm, respectively. The total theoretical opal ies of nanocomposites of this type, with Te, GaAs, and porosity is 59% (the first-order voids add up to 26%). InSb crystalline fillers, are discussed in [8, 9]. However, in actual fact, the total porosity of the opal It was of interest to carry out simultaneous electron single crystals grown by us is ~46Ð50% [4, 5]. The microscopy and thermal-physical measurements on the lower real porosity compared to the theoretical value is same samples of the opal-based nanocomposite with primarily due to the sintering of the second- and third- voids filled by an amorphous substance, epoxy resin, order spheres, as a result of which they are no longer in whose thermal conductivity is substantially lower (par- point contact. However, the first-order voids remain, as ticularly, for T > 40 K) than that of amorphous quartz. a rule, close in volume to the theoretical figure (26%). There are methods (chemical techniques under pressure, melt injection, impregnation of the sample by a 2. PREPARATION OF SAMPLES filler material) by which one can fill the first-order AND EXPERIMENTAL TECHNIQUE voids by metals, semiconductors, and dielectrics to The opal used in the work was perfect in terms of the design an opal-based regular three-dimensional com- ordering of its structural elements, i.e., silicate spheres. posite with a cubic filler cluster array with the parame- Epoxy resin was introduced into the opal host voids in

1062 BOGOMOLOV et al. the following way. A parallelepiped-shaped opal sam- slightly in intensity, permits one to conclude that all the ple (this shape was chosen to facilitate determination of observed excess elements, with the exception of cop- the sample volume in subsequent density measure- per, originate from the epoxy resin introduced into the ments and thermal conductivity studies) was placed in opal pores. We readily see that most of the resin com- a quartz ampoule evacuated to 10Ð2 mm Hg. Next, ponents are light elements which are close in scattering epoxy resin of a liquid consistency was poured into the capacity and absorption to silicon. For this reason, elec- ampoule, which enveloped the sample over all of its tron microscopic images of this composite cannot surface. After this, air at atmospheric pressure was reveal the presence of epoxy resin in the opal pores admitted into the ampoule. The time during which the because of the zero diffraction contrast inherent to crys- sample was maintained in the epoxy resin determined talline objects only; indeed, the difference in absorption the extent to which the opal first-order voids became between the silicate spheres and epoxy clusters is filled. Next, the sample was removed from the epoxy small. The EDX spectra of the original opal contain resin which had not yet set, and after it became hard, a lines of oxygen and silicon only. layer of epoxy resin was cut off down to the opal sur- An attempt was made to reveal the presence of face. Measurements of the sample density showed that epoxy clusters by the Fresnel phase contrast technique, the free host volume was filled to about 60%. The accu- which has been recently employed to advantage in racy with which the percentage opal filling by the determining the oxygen depth profile in the oxide layer epoxy resin was inferred from the change in the sample in the initial stage of silicon oxidation [12]. density was not sufficiently high because the densities The Fresnel method is based on an analysis of both of the epoxy resin and of the opal were nearly equal. the shape and contrast of Fresnel fringes, which are We also measured the thermal conductivity of the seen at the interface lying perpendicular to the projec- epoxy resin used in the work. The corresponding sam- tion plane, from a series of electron microscope images ple was also parallelepiped-shaped. made with different defocusing on both sides of the The samples intended for electron microscopy stud- exact focus. These are analogs of the fringes formed in ies were prepared by the standard technique [10], optical systems in studies on end-on opaque objects which included grinding to a thickness of 150 µm, cut- under Fresnel diffraction conditions. ting out a square-shaped sample with an ~3-mm diago- In electron microscopy, any interface separating nal, dimpling it to a thickness of ~50Ð60 µm, and sub- materials in a sample which differ in scattering capacity sequent thinning by ion milling until a pinhole formed. acts as a source of Fresnel contrast. The fringes are EM 4000EXII and JEM 2010FX electron micro- most visible if the interface is planar and nearly parallel scopes were employed with an attachment providing to the electron beam; however, even if the surface is for elemental analysis of the clusters. inclined and wavy, such interfaces will produce a The temperature dependences of the thermal con- noticeable contrast. Because Fresnel fringes are formed ductivity of the opalÐepoxy resin and epoxy resin sam- in interference between electrons that have suffered ples were obtained within the range 4.2Ð250 K by the phase changes in elastic scattering from sample regions technique described in [11]. having different scattering potentials, it appears only natural to expect that the fringes will contain informa- Locating the epoxy clusters proved a hard task. The tion on the nature of the material on both sides of the structural elements of the opal host, namely, the silicate interface. Obtaining quantitative information with this spheres, are amorphous. Viewed from the structural technique would require, however, its refinement and standpoint, the epoxy clusters are likewise amorphous. taking account of the specific features of each object. In Therefore, revealing the presence of epoxy resin in opal this work, we made use of the qualitative aspects of the voids from microdiffraction patterns was found to be Fresnel method only. impossible, because both the resin and the silicate spheres produce a diffuse halo near the trace of the primary beam. Energy-dispersive x-ray (EDX) spectra 3. EXPERIMENTAL RESULTS contain, in addition to the lines of silicon and oxygen The presence of any other material besides silicate present in the silicate spheres, the lines of C, N, K, S, spheres could be revealed only in very thick samples Ca, and Cl, with the first three elements dominant. The with a large wedge near the edge. No other information spectrum of epoxy resin also exhibits Zn, Fe, and Cu in could be extracted from these images because of the trace amounts. It should be pointed out that all EDX weak and uniform contrast. Defocusing smeared out spectra of samples with fillers have Cu lines. Control the images even more. experiments (with the sample holder having a copper receptacle replaced by a beryllium base holder) estab- The images obtained on thin samples (1Ð2 silicate lished unambiguously that the Cu peak is produced by sphere layers) studied with off-focussing at a few hun- the holder, which had copper components in the imme- dred nanometers were the most informative. diate vicinity of the sample base. The good reproduc- Remarkably, the pores (both octahedral and tetrahe- ibility of EDX spectra obtained from different points of dral) between the silicate spheres, which are projected the same sample, whose elemental peaks varied only as squares and triangles with concave sides, seem, in

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STRUCTURAL AND THERMAL PROPERTIES 1063

0.2 µm

Fig. 1. Electron microscope image of the opal filled to 60% by epoxy resin. The sample thickness is of the order of the diameter of the silicate sphere. The central parts of the pores (A) do not contain the filler. Fresnel contrast is observed at the silicate-sphereÐvoid interface. most cases, to be filled only partially, with no filler in above regions cannot be identified with the sphere con- the middle of the pores. This is reliably established tact area becoming extended rather than pointed due to from the absence of Fresnel fringes at the filler–void the opal having undergone deformation during the sin- interface (Fig. 1). tering (which is required to impart strength to the mate- The contrast in the silicate spheres is very complex, rial). As the spheres have a nearly perfect surface, the and it differs substantially from that of these spheres in extended sphere contact region should be circular. pure opal. In most cases, three regions of irregular These regions are centered at approximately one half shape are found to be projected onto the image of a sil- the sphere radius. To understand their origin, we con- icate sphere. They are seen against the background of sider three spheres lying in the (111) plane and a fourth the lobes of the rosette formed by the superposition of placed on top (Fig. 3). The points of contact of the top the spheres. In the region of the lobes, the total sample sphere with the three bottom ones are denoted by A, B, thickness is larger than the thickness of a sphere and C. They are located on the triangle corners at the because of the sphere superposition; therefore, the con- middle of the sphere radius. If we bound some regions trast of a rosette lobe is darker as a result of larger a fraction of the sphere diameter in size by an irregular absorption than that of a region one sphere thick. The contour, the pattern thus obtained will resemble the irregularly shaped regions that attracted our attention electron microscope image of the composite near the seem to be still darker in contrast. This indicates the sphere contact points. Thus, one can conjecture that presence of an additional mass of material in these these regions formed in the following way. The epoxy regions, which increases absorption and, hence, pro- resin entering the opal leaks into the pores through the duces a darker contrast (Fig. 2). The irregular regions channels they are interconnected by, then flows over the are bounded by a very thin and dark, sharply contrasted sphere surface, and stops near the sphere contact points; line, which is apparently a Fresnel fringe. In about 50% in other words, the resin clusters form primarily in the of the observations, these regions are interconnected by vicinity of the sphere contacts. The central part of the channels that are 2Ð20 nm wide in the narrowest part. pores may remain unfilled if there is not enough resin. The narrowest channels are seen as one sharp and dark It should be pointed out that the voids in the opal can be Fresnel fringe. The edges of broad channels are approximated by octahedra and tetrahedra only in the bounded by Fresnel fringes, the space within the chan- first approximation. The facets of these voids are the nels having the same contrast as the main regions under sphere surfaces, which converge at very small angles at study. These regions are, on the average, 0.2Ð0.25 of the points of contact. These parts of the voids will the silicate sphere diameter, i.e., 40Ð50 nm in size. The exhibit capillary properties near the points of contact. It

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1064 BOGOMOLOV et al.

0.2 µm

Fig. 2. Electron microscope image of the opal filled to 60% by epoxy resin (the region of superposition of two silicate-sphere layers). A is the interface between the filler material and the region (void) not occupied by it within the tetrahedral (octahedral) structural void. The interface appears due to the Fresnel contrast under defocusing. is these points that the filler will be drawn to above all. pore filling by epoxy resin was revealed in a one- or At the same time, we may recall that this pattern of the two-layer opal prepared by ion milling. These regions with epoxy resin observed to exist in opal samples one or two silicate-sphere layers thick are located between the surfaces of the spheres, near the points they meet at; therefore, they are inaccessible for EDX analysis. As already mentioned, very thick samples exhibit a uniform fringe along all of the perforation perimeter. This is apparently the part of the compound by which the original opal was impregnated. In thicker regions, superposition of several silicate-sphere layers and the weak difference in electron absorption by these layers and by the epoxy resin precluded distinct dis- crimination of individual spheres and of the filler clus- B ters. An analysis of the electron microscope image contrast in thick and thin samples of opal with the filler suggests the conclusion that the absence of the filler in the middle of the voids in thin samples is apparently A C due both to the opal pores being not filled to 100% of their volume and to the action of the ion beam used in the milling. It is possible that epoxy resin is sputtered by the ion beam at a higher rate than the silicate spheres. Figure 4 shows the experimental data obtained for the opalÐepoxy resin nanocomposite, epoxy resin,1 and the data [5] for opal into which epoxy resin was intro- Fig. 3. Projection of closely packed silicate spheres in the duced. opal array onto the (111) plane. Diagram explaining the epoxy cluster formation. A, B, and C are the points of con- 1 The values obtained by us for κ of epoxy resin are in good agree- tact of the silicate spheres near which the epoxy clusters ment with the numerous figures available in the literature (see, form. e.g., [13]).

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

STRUCTURAL AND THERMAL PROPERTIES 1065

The opalÐepoxy resin composite and epoxy resin are insulators. Therefore, the thermal conductivity mea- 3 sured in an experiment is that of the crystal lattice. The 0.7 dashed lines in Fig. 4 relate to the thermal conductivity κ 4 eff of the opalÐepoxy resin composite calculated for 5 the case of the first-order opal voids filled by epoxy 6 κ resin to 60% (curve 5), 50% (6), and 90% (4). eff was calculated from the well-known simple expression 0.5 derived by Odelevskiœ [14] for a standard composite: 2 κ κ {}[]ν () eff/ mat = 1 Ð m/1/1Ð1Ð Ð m /3 , (1) , W/m K κ ν κ κ 0.3 where = fill/ mat, m is the volume occupied by the κ κ filler material, and mat and fill are the thermal conduc- 1 tivities of the host matrix and the filler (epoxy resin), respectively. As already mentioned, the volume of the first-order voids is 26%, and 60% of these pores is 0.1 occupied by the epoxy resin. Thus, in our case, m = κ 15.6%. mat was calculated from the equation [15] 0 50 100 150 200 250 κ = κ ()[]()1 Ð P 1 Ð P , (2) mat SiO2 amorph T, K where P is the porosity of the opal matrix with the Fig. 4. Temperature dependences of the thermal conductivity of (1) epoxy resin, (2) synthetic opal [5], (3) opalÐ60% epoxy resin. For the opalÐ60% epoxy resin composite, κ epoxy resin composite, and calculated eff for the cases of P = 0.324. When using Eq. (2), the data for κ () SiO2 amorph the opal first-order voids filled by epoxy resin to (4) 90, were taken from [16, 17].2 (5) 60, and (6) 50%. κ As seen from Fig. 4, the eff(T) relation calculated for the opal–60% epoxy resin composite fits the exper- at the Ioffe Institute [18]. These points will be dealt κ with in a separate publication. imental exp(T) curve within the range 40Ð160 K approximately well. For T < 40 K and T > 160 K, the experimental data deviate noticeably from the calcula- ACKNOWLEDGMENTS tions. The reasons for these deviations remain unclear, but we intend to look more closely into the matter. The The authors are indebted to L.I. Arutyunyan and κ (T) curve for the opalÐepoxy resin composite A.V. Prokof’ev for preparation of the opalÐepoxy resin exp nanocomposite. resembles the temperature dependence of thermal conductivity in amorphous materials. No features associ- This study was supported by the Russian Founda- ated with the onset of coherent effects, which could tion for Basic Research (project no. 00-02-16883), the have been expected because of the regular arrangement Polish Committee for Scientific Research (grant no. 2 of the amorphous filler in the opal pores, were P03B 127-19 KBN), and the Royal London Society. observed. REFERENCES We note in conclusion that the above pattern of the epoxy resin in the opal concentrating at the interfaces 1. V. N. Bogomolov and T. M. Pavlova, Fiz. Tekh. Polupro- between the amorphous SiO spheres will permit us to vodn. (St. Petersburg) 29 (5Ð6), 826 (1995) [Semicon- 2 ductors 29, 428 (1995)]. carry out a more comprehensive analysis of the behavior of κ(T) of our nanocomposite; in that work, we will 2. V. G. Balakirev, V. N. Bogomolov, V. V. Zhuravlev, et al., Kristallografiya 38 (3), 111 (1993) [Crystallogr. Rep. 38, not use the simple relations for the thermal conductivity 348 (1993)]. of composite materials [14, 15] that were employed 3. V. N. Bogomolov, L. S. Parfen’eva, A. V. Prokof’ev, here but more sophisticated concepts and methods of et al., Fiz. Tverd. Tela (St. Petersburg) 37 (11), 3411 calculation of the thermal conductivity of opals and (1995) [Phys. Solid State 37, 1874 (1995)]. opal-based nanocomposites which are being developed 4. V. V. Ratnikov, Fiz. Tverd. Tela (St. Petersburg) 39 (5), 2 κ 956 (1997) [Phys. Solid State 39, 856 (1997)]. The calculations of eff shown in Fig. 4 for the cases of 50 and 90% opal filling by the epoxy resin were performed as in the case 5. V. N. Bogomolov, D. A. Kurdyukov, L. S. Parfen’eva, of 60% filling, but with due account of the values of m and P cor- et al., Fiz. Tverd. Tela (St. Petersburg) 39 (2), 392 (1997) responding to these compositions. [Phys. Solid State 39, 341 (1997)].

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1066 BOGOMOLOV et al.

6. L. I. Arutyunyan, V. N. Bogomolov, N. F. Kartenko, 13. K. W. Garrett and H. M. Rosenberg, J. Phys. D 7, 1247 et al., Fiz. Tverd. Tela (St. Petersburg) 39 (3), 586 (1997) (1974). [Phys. Solid State 39, 510 (1997)]. 14. G. N. Dul’nev and Yu. P. Zarichnyak, Heat Conductivity 7. V. N. Bogomolov, N. F. Kartenko, D. A. Kurdyukov, of Blends and Composite Materials (Énergiya, Lenin- et al., Fiz. Tverd. Tela (St. Petersburg) 41 (2), 348 (1999) grad, 1974). [Phys. Solid State 41, 313 (1999)]. 15. E. Ya. Litovskiœ, Izv. Akad. Nauk SSSR, Neorg. Mater. 8. V. N. Bogomolov, D. A. Kurdyukov, L. M. Sorokin, 16 (3), 559 (1980). et al., Inst. Phys. Conf. Ser. 160, 95 (1997). 9. V. N. Bogomolov, J. L. Hutchison, S. M. Samoilovich, 16. R. C. Zeller and R. O. Pohl, Phys. Rev. B 4 (6), 2029 et al., Inst. Phys. Conf. Ser. 157, 35 (1997). (1971). 10. V. N. Bogomolov, L. M. Sorokin, V. A. Kurdyukov, 17. R. B. Stephens, Phys. Rev. B 8 (6), 2896 (1973).

et al., Fiz. Tverd. Tela (St. Petersburg) 39 (11), 2090 18. V. N. Bogomolov, L. S. Parfen’eva, I. A. Smirnov, et al., (1997) [Phys. Solid State 39, 1869 (1997)]. Fiz. Tverd. Tela (St. Petersburg) 44 (1), 175 (2002) 11. A. Jezowski, J. Mucha, and G. Pompe, J. Phys. D 20, [Phys. Solid State 44, 181 (2002)]. 1500 (1987). 12. F. M. Ross and W. M. Stobbs, Philos. Mag. A 63, 1 (1991). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1067Ð1070. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1023Ð1025. Original Russian Text Copyright © 2002 by Luguev, Lugueva, Ismailov.

SEMICONDUCTORS AND DIELECTRICS

The Debye Temperature and Grüneisen Parameter of the CaLa2S4ÐLa2S3 System S. M. Luguev, N. V. Lugueva, and Sh. M. Ismailov Institute of Physics, Dagestan Scientiﬁc Center, Russian Academy of Sciences, ul. 26 Bakinskikh Komissarov 94, Makhachkala, 367003 Dagestan, Russia e-mail: [email protected] Received August 30, 2001

Abstract—The thermal expansion coefficient of solid solutions in the CaLa2S4ÐLa2S3 system at a temperature of 300 K is investigated experimentally. The Debye temperature, the Grüneisen parameter, and the isothermal compressibility coefficient of solid solutions in the system under investigation are determined from the experimental thermal expansion coefficient. It is demonstrated that, upon substitution of calcium ions for cation vacancies in La2S3, the Debye temperature decreases, the isothermal compressibility coefficient increases, and the Grüneisen parameter remains constant for all compositions in the CaLa2S4ÐLa2S3 system. A correlation between the ionic radii of Ca2+ and La3+, the concentration of cation vacancies, and the rigidity of the lattice, on the one hand, and the Debye temperature, the Grüneisen parameter, and the isothermal compressibility coefficient, on the other, is revealed for the studied samples. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION lent alkaline-earth metal ions. In the case when the cation vacancies are filled with trivalent rare-earth metal Investigation into the thermal expansion coefficient 3+ α can provide valuable information on the Debye tem- ions, specifically with La ions, two valence electrons perature Θ and the Grüneisen parameter γ. As is known, are involved in chemical bonding and the third electron the Debye temperature is determined by the highest fre- passes into the conduction band. As a consequence, the quency of crystal lattice vibrations. Since the Debye La3S4 tetrasulfide in which all the cation vacancies are completely filled exhibits metallic conduction. If the temperature obeys the relationship Θ ~ β/M (where cation vacancies are filled with divalent alkaline-earth β is the quasi-elastic force coefficient and M is the metal ions, for example, with Ca2+ ions, the compound average atomic weight), the value of Θ can be used to contains no free electrons and remains dielectric. All judge the strength of chemical bonding in the crystal compositions in the CaLa2S4ÐLa2S3 system have a structure [1]. The Grüneisen parameter characterizes Th3P4-type structure [5]. Compounds with a Th3P4-type the anharmonicity of atomic vibrations in the crystal structure are characterized by ion-covalent interatomic lattice (in the harmonic approximation, γ = 0). From the bonds [3]. In these compounds, metalÐsulfur ion-cova- aforesaid, it is clear that an analysis of the Debye tem- lent bonds are combined with sulfurÐsulfur covalent perature Θ and the Grüneisen parameter γ enables one bonds. When the cation vacancies are filled with diva- to reveal the specific features of interatomic bonds and lent calcium ions, the newly formed calciumÐsulfur to gain a deeper insight into the dynamic properties of ion-covalent bonds accept electrons from sulfurÐsulfur crystals. covalent bonds. This leads to a weakening of chemical Crystalline materials based on solid solutions in the bonds in the compounds. CaLa2S4ÐLa2S3 system show considerable promise for In the present work, we measured the thermal use in infrared technology [2]; however, their properties expansion coefficient α of solid solutions in the are not clearly understood. To our knowledge, reliable CaLa2S4ÐLa2S3 system in order to obtain reliable data data on the Debye temperature and the Grüneisen on the Debye temperature Θ and the Grüneisen param- parameter for solid solutions in this system are not eter γ, to elucidate the nature of interatomic bonds, and available in the literature, except for the Debye temper- to reveal the dependence of these characteristics on the ature of the boundary composition La2S3 [3, 4]. composition of compounds in the CaLa2S4ÐLa2S3 system. Lanthanum trisulfide La2S3 crystallizes in a Th3P4- type structure with a large number of vacancies in the cation sublattice in such a way that each ninth cation 2. SAMPLES AND EXPERIMENTAL TECHNIQUE site is vacant. Electrically, the La2S3 trisulfide is a dielectric. The cation vacancies can be filled either with The samples used in the measurements were pre- trivalent and divalent rare-earth metal ions or with diva- pared according to the procedure described earlier in

1068 LUGUEV et al. [6]. X-ray diffraction analysis revealed that all the stud- the lattice parameter decreases and interatomic bonds ied samples have a Th3P4-type structure. The thermal become weaker. This explains the lack of any correla- expansion coefﬁcient α was measured using a quartz tion in the composition dependences of the thermal dilatometer with a capacitive sensor [7]. expansion coefﬁcient α and the lattice parameter a for compounds in the studied system.

The Debye temperature of CaLa2S4 was determined 3. RESULTS AND DISCUSSION from the experimental data on the thermal expansion Figure 1 presents the results of our measurements of coefficient α and the data available in the literature [9] the thermal expansion coefficient α for samples in the on the elastic constants for this compound. The thermal expansion coefficient and the Debye temperature are CaLa2S4ÐLa2S3 system and the data available in the lit- related by the following expression [10, 11]: erature on the thermal expansion coefficient for La2S3 [8] and CaLa S [2]. It is seen that our results are in 2 4 Θ C close agreement with the available data. This figure also = ------, (1) 1/3 α shows the dependence of the lattice parameter a on the V a A composition of solid solutions in the CaLa2S4ÐLa2S3 system [6]. As can be seen from Fig. 1, a decrease in the where C is the constant for isostructural compounds, A is the mean-square atomic weight, and V is the volume lattice parameter for the CaLa2S4ÐLa2S3 system is a accompanied by an increase in the thermal expansion per atom. The coefficient C was determined from both coefficient. In general, a decrease in the lattice parame- the empirical dependence of the thermal expansion ter a corresponds to an increase in the interatomic inter- coefficient on the melting temperature [12], which was action, which, in turn, results in a decrease in the ther- derived for rare-earth sesquichalcogenides with a mal expansion coefficient α. Upon addition of CaS to Th3P4-type structure, and the Lindemann formula [13]. 2+ We calculated the Debye temperature from formula (1) La2S3 in the system under investigation, Ca ions occupy cation vacancies and partially substitute for using experimental data on the thermal expansion coefficient for CaLa S and obtained Θ = 273 K. On the La3+ ions. Since the ionic radius of Ca2+ is considerably 2 4 other hand, the Debye temperature can be expressed smaller that the ionic radius of La3+, the interatomic dis- through the elastic constants according to the formula tance decreases. At the same time, the difference [14] between the electronegativities of calcium and sulfur is larger than that of lanthanum and sulfur. Consequently, Θ = BχÐ1/2MÐ1/3ρÐ1/6. (2) the calciumÐsulfur bond exhibits a more pronounced ionic character as compared to the lanthanumÐsulfur Here, B is the constant, χ is the isothermal compress- bond. Therefore, in the case when the calcium ions ibility coefficient, M is the average atomic weight, and occupy cation vacancies in the CaLa2S4ÐLa2S3 system, ρ is the density. In our case, the density used in the calculation was taken from [9]. The isothermal compressibility coefficient χ was calculated from the Poisson ratio ν and Young’s modulus E according to the formula [15] 31()Ð 2ν 8.72 χ = Ð------. (3) 14 1 E Å

8.70 , The Poisson ratio ν and Young’s modulus E for a

–1 CaLa2S4 were taken from [2]. By substituting these data K into formulas (2) and (3), we calculated the Debye tem- –6 8.68 Θ 12 perature of the CaLa2S4 compound: = 284 K. It is , 10 Θ α 2 seen that this result agrees satisfactorily with the temperature found from the experimental data on α. On 3 this basis, we used relationship (1) to determine the Debye temperature of other compounds in the 10 4 CaLa2S4ÐLa2S3 system for which data on the elastic constants are unavailable.

CaLa2S4 0.6 0.8 La2S3 The calculated dependence of the Debye temperature Θ on the number of cation vacancies per unit cell for the studied samples in the CaLa2S4ÐLa2S3 system is Fig. 1. (1) Composition dependence of the lattice parameter shown in Fig. 2. It can be seen from Fig. 2 that, upon [6] and composition dependences of the thermal expansion substitution of calcium ions for cation vacancies, the coefﬁcient for (2) solid solutions in the CaLa2S4ÐLa2S3 sys- Debye temperature decreases. A similar decrease in the tem (our data), (3) CaLa2S4 [2], and (4) La2S3 [8]. Debye temperature Θ is observed upon ﬁlling of cation

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 THE DEBYE TEMPERATURE AND GRÜNEISEN PARAMETER 1069 vacancies in the La3S4ÐLa2S3 [4] and La3Te4ÐLa2Te3 300 [16] systems, which are isostructural to the CaLa2S4Ð La2S3 system. At first glance, the fact that the Debye temperature Θ decreases with a decrease in the degree of structural imperfection seems to be unexpected. 1 However, as was noted above, this behavior of the 250 Debye temperature can be associated with a decrease in , K the number of covalent bonds upon filling of cation Θ vacancies. 2 Figure 2 also depicts the dependence of the Debye temperature Θ on the number of cation vacancies per unit cell for samples in the La3S4ÐLa2S3 system [4]. In 200 this case, the Debye temperature was determined from the heat capacity measurements [4]. It was found that, 0 0.5 1.0 1.5 V upon filling of cation vacancies with La3+ ions in the La3S4ÐLa2S3 system, the Debye temperature decreases drastically with a decrease in the number of vacancies Fig. 2. Dependences of the Debye temperature on the number of cation vacancies per unit cell for solid solutions in beginning from the composition LaS . According to 1.42 (1) CaLa2S4ÐLa2S3 (our data) and (2) La3S4ÐLa2S3 [4] sys- Ikeda et al. [4], the decrease observed in Θ upon chang- tems. ing over from La2S3 with a high content of defects to defect-free La3S4 is associated with a weakening of chemical bonds due to the screening of interatomic reason, the heat capacity Cv was calculated from the interactions by free electrons whose concentration Debye interpolation formula [17]: increases upon filling of cation vacancies with La3+ ions. In the range where the number of vacancies per Θ Θ Θ unit cell is larger than 0.8, the Debye temperatures for Cv = 3kD---- Ð ----D'---- , (5) both systems are close in magnitude and change in a T T T similar manner with a change in the number of vacancies. In this range of compositions, the concentration of where k is the Boltzmann constant and D is the Debye free electrons in the La2S4ÐLa2S3 system is still not very function. The satisfactory agreement between the high and the electron screening of chemical bonds is of results of our calculations of the heat capacity Cv little importance. The observed decrease in the Debye according to formula (5) and the experimental data on temperature in the aforementioned composition range the heat capacity of La2S3 [18] lends support to the is due primarily to an increase in the average atomic validity of the use of formula (5) in our case. The cal- weight upon filling of the cation vacancies with La3+ culation of the Grüneisen parameter from relationship (4) using the elastic constants taken from [2], experi- ions. In the CaLa2S4ÐLa2S3 system, no screening of α interatomic bonds by free electrons occurs and the aver- mental data on the thermal expansion coefficient , and the heat capacity Cv calculated from formula (5) dem- age atomic weight decreases upon filling of the cation γ vacancies. A decrease in the average atomic weight onstrated that, for CaLa2S4, the Grüneisen parameter should lead to an increase in the Debye temperature Θ. is equal to 1.60. For other compounds in the CaLa2S4Ð It seems likely that the weakening of chemical bonds La2S3 system whose elastic constants are unknown, the upon filling of the cation vacancies with Ca2+ ions isothermal compressibility coefficient χ was deter- would be more pronounced than that in the case of La3+ mined from the following relationship [10]: ions without regard for the screening of chemical bonds αV = Kχ, (6) by free electrons. a where K is the constant for the given class of materials. The Grüneisen parameter was determined from the The numerical value of K was obtained from the ther- relationship mal expansion coefficient α and the isothermal com- χ α pressibility coefficient for CaLa2S4. The Grüneisen γ 3 V a parameter γ for the other compounds studied in the = ------χ , (4) Cv CaLa2S4ÐLa2S3 system was determined using the isothermal compressibility coefficient χ calculated from where Cv is the atomic heat capacity at a constant vol- relationship (6). The calculated data on χ and γ for solid ume. As far as we know, experimental data on the heat solutions in the CaLa2S4ÐLa2S3 system are given in the capacity of the compounds under investigation are table. Our calculations proved that the Grüneisen unavailable, except for the composition La2S3. For this parameter for compounds in the CaLa2S4ÐLa2S3 system

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1070 LUGUEV et al.

Debye temperatures Θ, Grüneisen parameters γ, and isother- REFERENCES mal compressibility coefficients χ for solid solutions in the CaLa S ÐLa S system at 300 K 1. A. I. Anselm, Introduction to Semiconductor Theory 2 4 2 3 (Nauka, Moscow, 1978; Prentice-Hall, Englewood Composition Θ, K γχ, 10Ð11 PaÐ1 Cliffs, 1981). 2. J. A. Savage and K. L. Lewis, Proc. SPIE 683, 79 (1986). La2S3 297 1.60 1.17 3. Physical Properties of Rare-Earth Chalcogenides 0.9La2S3Ð0.1CaS 288 1.60 1.23 (Nauka, Leningrad, 1973). 4. K. Ikeda, K. A. Gschneidner, B. J. Beaudry, and U. Atz- 0.8La2S3Ð0.2CaS 281 1.59 1.28 mony, Phys. Rev. B 25 (7), 4604 (1982). 0.7La S Ð0.3CaS 277 1.60 1.34 2 3 5. J. Flahaut, Progress in the Science and Technology of the 0.6La2S3Ð0.4CaS 275 1.59 1.42 Rare Earths (Pergamon, Oxford, 1968), Vol. 3, p. 209. CaLa2S4 273 1.60 1.50 6. O. V. Andreev, A. V. Kertman, and G. N. Dronova, in Physics and Chemistry of Rare-Earth Semiconductors (Nauka, Novosibirsk, 1990), p. 143. remains virtually constant. The constancy of γ can be 7. M.-R. M. Magomedov, I. K. Kamilov, Sh. M. Ismailov, explained by the fact that the decrease in the number of M. M. Rasulov, and M. M. Khamidov, in Thermoelec- vacancies in the crystal leads to a decrease in the aver- trics and Their Applications (St. Petersburg, 2000), age atomic weight (the factor responsible for the p. 241. decrease in the degree of anharmonicity of interatomic 8. V. I. Marchenko and G. V. Samsonov, Fiz. Met. Metall- interactions) and an increase in the degree of ionicity of oved. 15 (4), 631 (1963). the chemical bonds (the factor responsible for the 9. K. J. Saunders, T. V. Wong, T. M. Hartnett, et al., Proc. increase in the degree of anharmonicity). SPIE 683, 72 (1986). 10. S. I. Novikova, Thermal Expansion of Solids (Nauka, Moscow, 1974). 4. CONCLUSIONS 11. M. N. Abdusalyamova, Zh. Vses. Khim. O-va. 26 (6), Thus, we performed the experimental investigation 673 (1981). of the thermal expansion coefficient. The results 12. E. M. Dudnik, G. V. Lashkarev, Yu. B. Paderno, and obtained were used to determine the Debye characteris- V. A. Obolonchik, Izv. Akad. Nauk SSSR, Neorg. Mater. tic temperature, the Grüneisen parameter, and the iso- 2 (6), 980 (1966). thermal compressibility coefficient for compounds in 13. J. M. Ziman, Electrons and Phonons (Clarendon, Oxford, 1960; Inostrannaya Literatura, Moscow, 1962). the CaLa2S4ÐLa2S3 system. The decrease observed in the Debye temperature for the studied system indicates 14. D J. R. Drabble and H. J. Goldsmid, Thermal Conduc- tion in Semiconductors (Pergamon, Oxford, 1961; Inos- that, upon transition from La2S3 to CaLa2S4, the substi- trannaya Literatura, Moscow, 1963). 2+ tution of Ca ions for cation vacancies brings about an 15. L. D. Landau and E. M. Lifshitz, Course of Theoretical increase in the ionicity and a weakening of interatomic Physics: Theory of Elasticity (Nauka, Moscow, 1965; bonds. The isothermal compressibility coefficient Pergamon, New York, 1986), Vol. 7. increases as the cation vacancies are filled with Ca2+ 16. V. V. Tikhonov, V. N. Bystrova, R. G. Mitarov, and ions, and the Grüneisen parameter remains virtually I. A. Smirnov, Fiz. Tverd. Tela (Leningrad) 17 (4), 1225 insensitive to changes in the composition. (1975) [Sov. Phys. Solid State 17, 795 (1975)]. 17. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics: Statistical Physics (Nauka, Moscow, 1964; Per- ACKNOWLEDGMENTS gamon, Oxford, 1980), Vol. 5. We would like to thank G.N. Dronova for supplying 18. I. E. Paukov, V. V. Nogteva, and E. M. Yarembash, Zh. the samples used in our investigations. Fiz. Khim. 43 (9), 2351 (1969). This work was supported by the Russian Foundation for Basic Research, project no. 01-02-16283. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1071Ð1076. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1026Ð1031. Original Russian Text Copyright © 2002 by Bazhenov, Gorbunov, Aldushin, Masalov, Emel’chenko.

SEMICONDUCTORS AND DIELECTRICS

Optical Properties of Thin Films of Closely Packed SiO2 Spheres A. V. Bazhenov, A. V. Gorbunov, K. A. Aldushin, V. M. Masalov, and G. A. Emel’chenko Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected] Received September 4, 2001

Abstract—Ordered, closely packed, defect-free one-, two-, and three-layer thick ﬁlms of SiO2 spheres of diameter D varying from 0.6 to 1.4 µm were obtained. Their optical transmittance and reﬂectance spectra were measured in the range 0.3Ð2.5 eV. The one-layer structures reveal a transmittance minimum whose spectral position is described by the Bragg law with the plane separation equal to the sphere radius D/2. As the number of the layers increases, a spectral feature appears which signals the formation of a photonic gap in the 〈111〉 direction of the fcc crystal lattice and is determined by the distance between the {111} planes equal to 0.816D. The spectra of two- and three-layer structures measured with a diverging light beam contain additional lines originating from the formation of photonic gaps by the {211} and {221} planes. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION area. The maximum dimensions of ordered regions µ Materials with a photonic gap, called photonic crys- (domains) in a film reach 600 m. At the substrate tals, have been recently attracting increasing interest, periphery, one observes the formation of multilayer which stems from their application potential in opto- structures (two or more particle layers). electronics [1]. The numerous studies of natural and Here, we report an a study of the optical transmit- synthetic opals, which represent three-dimensional tance and reflectance spectra in the spectral range 0.3– arrays of closely packed amorphous SiO2 spheres vary- 2.5 eV from one-layer defect-free films as a function of ing in diameter from 0.2 to 0.5 µm, have shown that in the size of SiO2 spheres, which was varied from 0.5 to the visible region, they exhibit properties characteristic 1.4 µm. We also studied the variation of optical spectra of photonic crystals [2Ð6]. An intense search is pres- observed to occur as one crosses over from a one- to ently under way for photonic structures with gaps near two- and three-layer defect-free structure of SiO2 1.5 µm [7, 8]. spheres 1.05 µm in diameter. To develop such opal-based photonic crystals, SiO µ 2 spheres about 1 m in diameter are required. The pro- 2. EXPERIMENTAL TECHNIQUES cesses of sedimentation and ordering of large SiO2 particles are poorly studied. Moreover, some authors main- Closeness in size of the nano- and microspheres of tain that particles 0.4 µm in size and larger do not order silicon oxide is a crucial condition of preparation of in the course of sedimentation [9, 10]. Structural stud- ordered opal structures. In this work, we prepared uni- ies of bulk opal samples showed them to be polycrys- form silicate spheres through the hydrolysis of tetra- talline materials with single-crystal domains typically ethyl orthosilicate (TEOS) in a solution of ethyl alcohol tens of microns in size [3]. The largest ordered regions in the presence of ammonium hydroxide, a process in the best bulk samples do not exceed 200 µm in size known as Stober’s method [13]. [3]. The mechanism by which structural defects form in Using this method, we prepared spheres over a bulk opal samples remains unclear. broad range of diameters (0.1 to 1.6 µm), with a devia- Preparation of thin, large-area, close-packed films tion of no more than 5% from the average size. Spheres with diameters above 0.5 µm were obtained in two of SiO2 spheres is, in our opinion, a more controllable, bottom technological process than sedimentation of stages. In the first stage, spheres ranging in diameter bulk samples and would broaden their application from 0.1 to 0.5 µm were obtained, which were subse- potential in the near future. Considerable progress has quently used as seeds to produce particles up to 1.6 µm been demonstrated in recent years in this area [11]. in size. Moreover, investigation of thin films of SiO2 spheres Uniform films of SiO2 spheres were prepared by a could make it possible to follow the formation of struc- combination of natural sedimentation with capillary tural defects in the transition to bulk opal samples and extraction. The experiments were carried out at 20 ± shed light on the mechanism of their formation. 2°C. Glass plates measuring 70 × 25 × 2 mm served as In a recent publication [12], we reported on the the substrates. preparation of uniform monolayers of ordered SiO2 The structure of the films thus prepared was studied spheres on glass substrates a few square centimeters in with a scanning electron microscope (SEM); in the case

1072 BAZHENOV et al. method. It makes use of an optical microscope, and the spectra were obtained from a region 80 µm in size. Within the spectral range from 4 to 1.1 µm (0.3 to 1.1 eV), the spectra were measured with an IR microscope of a Fourier spectrometer. In the region from 1.1 to 0.5 µm (1.1 to 2.5 eV), another spectral setup provided with an optical microscope was employed. In this case, in contrast to the ﬁrst method, optical transmittance and reﬂectance spectra were measured in a con- verging light beam. The light cone angle was ±16°.

3. EXPERIMENTAL RESULTS AND DISCUSSION Figure 1 presents an SEM image of a one-layer film µ of SiO2 spheres 1.05 m in diameter. Numerous examinations of one-layer films have shown that they always feature the closest packed structure (triangular arrangement of spheres). In an fcc lattice, these layers correspond to the {111} face. The domains in a film typically vary in size from 10 to 50 µm. The largest domains may be as large as a few hundred microns. Fig. 1. A region of a one-layer film of close-packed SiO2 spheres measuring 1.05 µm. The image was obtained with a The microstructure of a two-layer film is the closest scanning electron microscope. hexagonal packing of spheres, with the packing in the first and second layers being correlated. However, along the whole boundary separating regions with one- and two-layer films, one observes sparsely packed square structures (Fig. 2). Such structures are also seen at transitions between two- and three-layer films, etc. In the case of thin films, a close-packed perfect crystalline structure is seen to form. We obtained perfect monolayers about 0.5 × 0.5 mm in size. The dimensions of single-domain regions in three-layer structures were of the order of 0.1 mm. Figure 3 shows optical transmittance spectra of monolayers of SiO2 spheres, with diameter D from 0.6 to 1.4 µm, measured by the microscopicÐspectral method. The decrease in sample transparency with increasing photon energy is associated with the Ray- leigh (elastic) light scattering from the spheres. In the limit of large wavelengths λ, the cross section of light scattering from a particle of size D is small and propor- λ 4 Fig. 2. Part of the boundary between a one- and a two-layer tional to (D/ ) . Scattering losses become noticeable µ film of SiO2 spheres measuring 1.05 m, with square struc- when the particle size is comparable to the wavelength. tures forming along the whole boundary. The image was Therefore, as the sphere diameter decreases, the trans- obtained with a scanning electron microscope. mission edge associated with scattering shifts naturally toward the violet region. of SiO spheres about 1 µm in diameter or greater, with All spectra in Fig. 3 have an absorption line at 2 ≅ Ð1 a conventional optical microscope as well. 0.42 eV ( 3400 cm ), which does not vary in spectral position with variation of the sphere diameter. This line Spectral measurements were performed at room is of an extrinsic nature and will be discussed below. temperature by two methods. In the first method, opti- Against the background of slowly varying transmit- cal transmittance spectra of monolayers of SiO2 tance, a broader line is observed whose spectral posi- spheres in a parallel light beam were recorded in the µ µ tion depends on the diameter of the SiO2 spheres. When spectral region from 0.6 m (2.07 eV) to 2.3 m measured in a parallel light beam, its halfwidth is (0.54 eV). The analyzed region of the film was about 50 nm. If, however, the spectra are obtained with a 1 mm in size. microscope, the halfwidth increases to 90 nm, which is Most of the measurements reported in this work accounted for by the position of this transmittance min- were carried out by the second, microscopicÐspectral imum being angle-dependent.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

OPTICAL PROPERTIES OF THIN FILMS OF CLOSELY PACKED SiO2 SPHERES 1073 λ Figure 4 plots the dependence of the wavelength c 43 2 1 corresponding to this minimum on the sphere diameter D. The experimental values fit, with high accuracy, to a λ straight line described by Bragg’s diffraction law N c = 2n dsinθB, where N is the diffraction order, n is the eff eff 1.5 effective refractive index of the SiO2 spheres/air composite, d is the interplanar distance, and θB is the Bragg angle (at normal incidence, θB = 90°). Thus, the line changing its spectral position with variation of the SiO2 sphere diameter results from interference of the light 1.0 wave striking the sample and the wave that has under- T gone Bragg scattering from a monolayer of SiO2 spheres. Assuming N = 1, as is usually observed in opal-type photonic crystals, we come to neff = 1.316 and d = D/2 = r, where r is the sphere radius. 0.5 This value of neff is noticeably less than the effective refractive index 1.348 of a bulk fcc crystal made up of SiO2 spheres, whose material has a refractive index ≅ nsph 1.45. This discrepancy can be readily assigned to 0 the smaller volume filling factor f of a monolayer com- 0.5 1.0 1.5 2.0 2.5 pared with that for a bulk crystal, in which the spheres Photon energy, eV of each subsequent layer fit into the depressions between the spheres of the preceding layer. Calculation π × Fig. 3. Optical transmittance spectra T of a monolayer of of f for a closely packed monolayer yields f = /(3 SiO spheres on a glass substrate. The spectra were trans- 1/2 π × 1/2 2 3 ), which differs from f = /(3 2 ) for a closely lated vertically for convenience and were obtained with packed three-dimensional structure. The value of neff sphere diameters of 1.40, 1.12, 1.05, 0.96, 0.86, 0.80, 0.70, for a monolayer can be calculated roughly by averaging and 0.60 µm, bottom to top, respectively. ε 2 2 2 the dielectric function = neff = nsph fn + a (1 Ð f). Here, na is the refractive index of the medium filling the space between the spheres (for air, na = 1). Then, we obtain neff = 1.29, which is not very far from the exper- 2.0 imental value of 1.316. That the relation displayed in Fig. 4 is determined by the interplanar distance d equal to the radius r rather than to the diameter of the spheres D might, at first 1.6 m

glance, appear strange. It would seem that it is the µ , thickness D of a film of SiO2 spheres and its effective c λ refractive index neff that should determine the path dif- 1.2 ference for the two interfering waves. Assuming that in actual fact d = D, the observed transmittance minima should be due to second-order diffraction (N = 2). In this case, one should expect the presence of deeper 0.8 transmittance minima originating from first-order diffraction and lying at wavelengths equal to twice the 0.4 0.8 1.2 1.6 λ µ experimental values for c in Fig. 4. Therefore, optical D, m transmittance spectra of the samples under study were λ also measured in the IR range up to 4 µm (the absorp- Fig. 4. Wavelength c of the minima in the optical transmit- tion edge of the glass substrate). In Fig. 3, the crosses tance spectra of a monolayer of SiO2 spheres plotted vs. identify the assumed positions of the transmittance their diameter D. The straight line is a linear fit to the exper- minima corresponding to the first diffraction order for imental data. an interplanar distance d = D. As is evident from the experimental spectra, if such minima do exist, their tion features seen in Figs. 3 and 4 are due to first-order depth is substantially less and width is much larger than diffraction at an interplanar distance d in Bragg’s rela- those of the minima associated with second-order dif- tion, which is equal to the sphere radius r. fraction. Further, the dependence of the spectral position of the weak transmittance minima on the sphere This result can be explained in terms of the theory of diameter does not obey Bragg’s law. Thus, the diffrac- diffraction of a plane electromagnetic wave from a

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1074 BAZHENOV et al.

0.6 tion with variation of the sphere diameter. As already (a) mentioned, attempts to interpret line B as being due to 0.4 ﬁrst-order diffraction with an interplanar distance equal to the sphere diameter fail, because its spectral position 0.2 A C does not vary regularly with the sphere diameter. More- B over, estimation of the effective refractive index for this 0 ≅ line yields a very low value of neff 1.20. It should be (b) pointed out that light reﬂectance spectra of a monolayer 0.6 0.8 of SiO2 spheres exhibit no maxima in the region of line 0.6 B and even of line C, which is certainly of diffraction 0.4

T 0.4 nature. This is apparently associated with extremely R weak light reﬂection from this structure.

–ln 0.2 E 0.2 A line with hν = 0.52Ð0.53 eV, to which a maximum 0 0 in the reflectance spectrum corresponds, appears in the (c) optical density spectrum of the closely packed perfect 0.8 0.6 structure of two SiO2 sphere layers (Fig. 5b) in the 0.6 region of the B line. The fact that the maxima in the 0.4 G optical density and reflectance spectra coincide argues 0.4 F for the diffraction nature of this line. Its spectral posi- 0.2 0.2 tion fits, with good accuracy, Bragg’s law with neff = 1.34, which is close to the effective refractive index of 0 0 0.2 0.4 0.6 0.8 1.0 a bulk fcc crystal constructed of SiO2 spheres (neff = Photon energy, eV 1.348). In this case, the interplanar distance d = (2/3)1/2D ≅ 0.8165D corresponds to the distance Fig. 5. Spectra of the optical density ÐlnT (solid lines) and between the adjacent {111} planes in the close-packed of reflectance R (dot-and-dash curves) for a structure of (a) fcc structure of SiO2 spheres with diameter D. one, (b) two, and (c) three perfect, closely packed layers of µ When the second layer is deposited, line C shifts SiO2 spheres 1.05 m in diameter. Dashed lines show a toward higher energies by about 0.03 eV (hν ≅ decomposition of the spectra into constituents in the region of impurity line A and line C. 0.94 eV). Thus, deposition of the second layer of spheres results in a decrease in the effective refractive index (by about 3%) for the diffraction feature deter- homogeneous sphere. The fundamentals of this theory mined by the substrate plane and the plane passing were published by G. Mie in 1908. The specific features through the centers of the first-layer spheres. This effect of diffraction from a homogeneous sphere are may be interpreted as a result of the first-layer spheres described, for instance, in [14]. It was established that being pushed apart by those of the second layer. the light scattered by a sphere represents a set of spher- A broad feature E appears in the optical density ical waves emanating from its center. Thus, in our case, spectrum of the two-layer structure and is accompanied the planes featured in Bragg’s law are the plane passing by the corresponding broad line in the reflectance spec- through the centers of the SiO2 spheres and the surface trum. of the glass substrate on which these spheres are located. As the third layer of SiO2 spheres is deposited (Fig. 5c), the intensity of the hν = 0.52-eV line To study the intensities of the lines observed in the increases in both the optical-density and reflectance transmittance spectra and their variation with increas- spectra. It is this behavior that should be expected in the ing number of SiO2 sphere layers, we measured transformation of a photonic gap with increasing the number mittance and reflectance spectra of perfect structures of SiO2 sphere layers. Line C slightly narrows com- consisting of one, two, and three layers of SiO2 spheres pared with the spectra of the mono- and bilayer struc- 1050 nm in diameter. Films with larger spheres are here tures, but the position of its maximum (hν = 0.94 eV) preferable, because, in contrast to spheres of a smaller remains the same as in the two-layer structure. Signifi- diameter, one can readily check here the number of lay- cantly, the intensity of this line in the optical-density ers and the degree of their perfection with an optical and reflectance spectra does not grow with increasing microscope. The results of these studies are presented the number of layers in the structure, in contrast to the in Fig. 5 in the form of optical density spectra ÐlnT, line with hν = 0.52 eV, which is produced by diffraction where T is the transmittance, as well as of reflectance from the {111} planes. This is an additional argument spectra R. for line C not being a result of second-order diffraction The monolayer optical density spectrum (Fig. 5a) in a structure with the interplanar distance correspond- exhibits a broad line B (hν = 0.49 eV) in addition to the ing to the {111} planes. This feature forms only within ν impurity line A (photon energy h = 0.42 eV) and a fea- the first layer of SiO2 spheres and is determined by the ture C (hν = 0.91 eV) which changes its spectral posi- interplanar distance being equal to the sphere radius r.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 OPTICAL PROPERTIES OF THIN FILMS OF CLOSELY PACKED SiO2 SPHERES 1075

After a third layer of SiO2 spheres has been depos- λ, µm ited, the E feature, which formed in the two-layer struc- 4 3 2 1.5 ture, splits into two lines, F (hν = 0.69 eV) and G (hν = 0.84 eV), in both the optical-density and reﬂectance 1.4 spectra. The most probable origin of these lines is diffraction caused by the formation of additional planes inclined to the {111} plane. 1.0 To check this assumption, we studied the changes in the transmittance spectra of the three-layer structure produced by variation of the cone angle γ of the light beam passing through the sample. The angle γ was var- T 0.6

ied by means of an iris diaphragm built into the measur- –ln ing system. As seen from Fig. 6, in reducing γ from ±16° to ±7°, the limit, determined in our case by the design of the microscope mirror objective, virtually 0.4 does not affect the 0.52 and 0.94-eV lines. At the same time, line G disappears and line F shifts toward shorter wavelengths, with its intensity also decreasing with G decreasing γ. This behavior of the F and G lines sug- 0.2 gests that they are related to light diffraction from other F planes of the structure which are inclined relative to 0 {111}. 0.3 0.5 0.7 0.9 1.0 A rough calculation shows that such planes can be Photon energy, eV {211} and {221} for lines F and G, respectively. Fig. 6. Optical-density spectra (ÐlnT) of a perfect three- Because the {211} and {221} planes are inclined to µ ° ° layer film of SiO2 spheres 1.05 m in diameter measured {111} by 19.5 and 15.8 , respectively, the light beams γ ± ° B with the light cone angle reduced from 16 (bottom) to striking these planes at a close to normal angle (θ ≅ ±7° (top). The spectra were translated vertically by 0.2 for 90°) should provide a substantial contribution to the the sake of convenience. transmittance spectrum for large γ. As γ decreases to ±7°, the light beam diffraction angle decreases to about θB ≅ 70°. In accordance with Bragg’s relation, this proportional to their diameter. On the other hand, the λ intensity of this line in the optical-density spectra is should entail a decrease in c, which is seen clearly to occur with line F. Moreover, it is well known that a proportional, for a given sphere diameter, to the number decrease in θB also reduces the intensity of the diffrac- of SiO2 sphere layers, with the line shape remaining the tion minimum in the transmittance spectrum of opal [2, same with variation of the number of layers (Fig. 5). 6]. This is why the intensities of the F and G lines fall Thus, this line can be used in estimating the number of off with decreasing γ. In the first diffraction order (N = layers of SiO2 spheres in structures with both large and 1), the maxima in the optical-density spectrum (Fig. 6) small spheres. that are determined by the {211} and {221} planes should be observed at hν = 0.75 and 0.93 eV, respectively. These values compare satisfactorily with the 4. CONCLUSION experimental values hν = 0.69 eV (line F) and hν = 0.84 eV (line G). The slight disagreement can be asso- In summary, we have prepared ordered, closely packed one-, two-, and three-layer structures of SiO ciated with the fact that the estimation was based on the µ 2 linear Bragg approximation. This discrepancy will be spheres 0.6 to 1.4 m in diameter on glass substrates smaller if the dynamic approximation is employed. and measured their optical transmittance and reflec- ν tance spectra in the range extending from 0.3 to 2.5 eV. In closing, we consider line A with h = 0.42 eV In one-layer structures, a reflectance minimum has ≅ Ð1 ( 3400 cm ), whose spectral position does not change been revealed whose position fits to Bragg’s law with with variation of the SiO2 sphere diameter (Fig. 3). the interplanar distance equal to the sphere radius. It Obviously enough, this line originates from an impurity was established that this diffraction feature forms in the containing OÐH groups. The feature with hν = 0.45 eV interference of light waves diffracted from SiO2 spheres seen on its high-energy wing is apparently associated with waves reflected from the substrate surface. As the with the manifestation of NÐH bonds. The presence of number of layers in a structure increases, a feature these impurities in the SiO2 spheres is a consequence of appears in its optical spectra, which indicates the for- the use of alcohol and NH3 in the preparation of the mation of a photonic gap in the 〈111〉 direction of the structures. It was found that while the halfwidth of this fcc crystal lattice and is determined by the separation line does not change under variation of the SiO2 spheres between the {111} planes. The spectra of two- and in size, its intensity in the optical-density spectrum is three-layer structures measured with a diverging light

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1076 BAZHENOV et al. beam exhibit additional lines caused by the formation 4. H. Migue, C. Lopez, F. Meseguer, et al., Appl. Phys. of photonic gaps due to the {211} and {221} planes. Lett. 71, 1148 (1997). These additional lines disappear when a close-to-paral- 5. H. Migue, F. Meseguer, C. Lopez, et al., Langmuir 13, lel light beam is used. 6009 (1997). An absorption line originating from impurities 6. V. N. Astratov, Ju. A. Vlasov, O. Z. Karimov, et al., Phys. which contain OÐH and NÐH bonds was observed in Lett. A 222, 349 (1996). the spectra of the structures studied. It was shown that 7. J. K. Hwang, H. Y. Ryu, D. S. Song, et al., Appl. Phys. this line can be used to estimate the number of layers in Lett. 76, 2982 (2000). ordered structures of SiO spheres. 8. A. Blanco, E. Chromski, S. Grabtchak, et al., Nature 2 405, 437 (2000). 9. S. Emmett, S. C. Lubetkin, and B. Vincent, Colloids ACKNOWLEDGMENTS Surf. 42, 139 (1989). This study was partially supported by the Russian 10. K. Davis, W. B. Russel, and W. J. Glantschnig, J. Chem. Foundation for Basic Research (project no. 01-02- Soc., Faraday Trans. 87, 411 (1991). 97024). 11. P. Jiang, J. F. Bertone, K. S. Hwang, and V. L. Colvin, Chem. Mater. 11, 2132 (1999). 12. V. M. Masalov, K. A. Aldushin, P. V. Dolganov, and REFERENCES G. A. Emelchenko, Phys. Low-Dimens. Struct. 5Ð6, 93 (2001). 1. T. E. Krauss and R. M. De La Rue, Prog. Quantum Elec- tron. 23, 51 (1999). 13. W. Stober, A. Fink, and E. Bohn, J. Colloid Interface Sci. 26, 62 (1968). 2. V. N. Bogomolov, D. A. Kurdyukov, A. V. Prokof’ev, and 14. M. Born and E. Wolf, Principles of Optics (Pergamon, S. M. Samoœlovich, Pis’ma Zh. Éksp. Teor. Fiz. 63, 496 Oxford, 1969; Nauka, Moscow 1970). (1996) [JETP Lett. 63, 520 (1996)]. 3. Ju. A. Vlasov, V. N. Astratov, A. V. Baryshev, et al., Phys. Rev. E 61, 5784 (2000). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1077Ð1084. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1032Ð1038. Original Russian Text Copyright © 2002 by Ivanov, Sukhovsky, Aleksandrova, Totz, Michel

SEMICONDUCTORS AND DIELECTRICS

Mechanism of Protonic Conductivity in an NH4HSeO4 Crystal Yu. N. Ivanov*, A. A. Sukhovsky*, I. P. Aleksandrova*, J. Totz**, and D. Michel** * Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia ** Leipzig University, D04103 Leipzig, Germany e-mail: [email protected] Received July 20, 2001; in ﬁnal form, October 25, 2001

Abstract—The chemical exchange of deuterons in a partly deuterated ammonium hydrogen selenate crystal is investigated by deuteron magnetic resonance (2H NMR) spectroscopy over a wide range of temperatures. The changes observed in the line shape of the NMR spectra at temperatures above 350 K are characteristic of chemical exchange processes. The exchange processes are thoroughly examined by two-dimensional 2H NMR spectroscopy. It is established that, over the entire temperature range, only deuterons of hydrogen bonds are involved in the exchange and the rates of exchange between deuterons of all types are nearly identical. No deuteron exchange between the ND4 groups and hydrogen bonds is found. A new model of proton transport in ammonium hydrogen selenate is proposed on the basis of the experimental data. This model makes it possible, within a uniﬁed context, to explain all the available experimental data, including macroscopic measurements of the electrical conductivity. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In the present work, we analyzed our results with the aim of elucidating the microscopic mechanism of pro- Considerable recent interest expressed by research- ton transport in AHS crystals. ers in crystals with a high ionic conductivity stems from both important practical applications of these compounds and the basic problems concerning electrical 2. SAMPLE PREPARATION conductivity in superionic crystals [1Ð4]. In this AND EXPERIMENTAL TECHNIQUE respect, crystals whose structure involves quasi-one- Partly deuterated (25%) AHS crystals were grown dimensional chains of hydrogen bonds are of particular from an aqueous solution containing an excess of importance. These crystals are good model objects for H2SeO4 and the appropriate amount of heavy water. use in verifying different assumptions on microscopic Protons involved both in ammonium groups and in mechanisms of ionic conductivity. Ammonium hydro- hydrogen bonds were partly replaced by deuterons. The gen selenate (AHS) NH4HSeO4 belongs to these crys- degree of deuteration was chosen reasoning from the tals. In the structure of AHS crystals, inﬁnite quasi-one- speciﬁc features of the phase diagram of the dimensional chains are formed by SeO4 tetrahedra NH4HSeO4 compound, which, at a growth temperature joined through protons of hydrogen bonds. The deuter- of 30°C and a degree of deuteration higher than 45%, ation of AHS makes it possible to apply the powerful crystallizes in another phase [8]. The nuclear magnetic method of nuclear magnetic resonance (NMR) at qua- resonance and dielectric measurements were per- drupole nuclei to perform research into proton (deu- formed with the same samples. The 2H NMR investiga- teron) transport. In addition to conventional Fourier- tions were carried out on a BRUKER MSL 300 NMR transform NMR spectroscopy, elementary processes of spectrometer operating at a Larmor frequency of deuteron chemical exchange have been investigated by 46.073 MHz. The width of a 90° pulse was equal to two-dimensional (2D) NMR spectroscopy. As a rule, approximately 4 µs. A spin echo sequence with a time the 2D NMR data are compared with the results of interval of 25 µs between pulses was used in order to dielectric measurements performed over a wide range exclude the effect of the dead time of the NMR spec- of frequencies (10Ð2Ð106 Hz). Dielectric measurements trometer receiver. Moreover, proton decoupling was at very low frequencies permit one to increase apprecia- applied to suppress the broadening of 2H NMR lines bly the accuracy in determination of the dc conductivity due to the dipoleÐdipole interaction with the remaining σ dc and to compare quantitatively the results of dielec- protons. The two-dimensional NMR measurements tric and NMR measurements for AHS crystals. The were performed using the following spin echo π π τ π τ π τ structure and properties of AHS single crystals have sequence: ( /2)xÐt1Ð( /2)ÐxÐ mÐ( /2)xÐ Ð( /2)yÐ Ðt2, been described thoroughly in our earlier works [4Ð7]. where t1 is the evolution time, t2 is the measurement

1078 IVANOV et al. τ τ time, is the time interval between pulses, and m is the hydrogen bonds occurs in the paraelectric phase. These mixing time. The dielectric susceptibility was mea- authors proposed a microscopic mechanism of proton sured on a Schlumberger Solartron 1255 HF Frequency transport through a correlated reorientation of SeO4 Response Analyzer in the frequency range from 10Ð2 to groups with a sequential exchange of protons between 6 10 Hz. Samples approximately 0.8 mm thick were cut SeO4 groups along an infinite chain of hydrogen bonds. from the AHS single crystal. For dielectric measure- In this case, the activation energy for reorientational ments, electrodes were prepared in the form of thin motion of SeO4 groups is a controlling factor of the pro- gold films applied to the sample surface under vacuum. ton hopping rate. It should be noted that the activation energy for reorientational motion of SeO4 groups is approximately equal to the activation energy for isotro- 3. RESULTS AND DISCUSSION pic diffusion of ammonium groups which contribute The most interesting features inherent in the AHS significantly to the electrical conductivity of the crystal crystal are as follows: the ferroelectric state associated [1, 2]. However, for the most part, all these assumptions with ordering of protons involved in hydrogen bonds, are based on analyzing the temperature dependence of the incommensurate phase [6, 7], and the protonic con- the second moment (linewidth) of the 1H NMR spectra. ductivity [1Ð3]. In the paraelectric phase, the AHS crys- It is known that, over the entire temperature range, the tal is characterized by a monoclinic unit cell with space 1H NMR spectrum consists of a single line whose sec- group B2 and the lattice parameters a = 19.745 Å, b = ond moment is predominantly determined by the 4.611 Å, c = 7.552 Å, and γ = 102.56° [9]. The crystal dipoleÐdipole interactions between protons of ammo- structure is built up of SeO2Ð tetrahedral ions joined by nium groups [1]. In our opinion, these investigations 4 cannot provide detailed information on the microscopic hydrogen bonds into infinite chains aligned along the mechanism of proton transport in the AHS crystal. ferroelectric axis b (Fig. 1). The SeO4 groups are linked by ammonium ions along the two other axes a and c. In this work, the microscopic characteristics of Hydrogen bonds between the different structural ammonium hydrogen selenate are determined from the 2 groups SeO4 considerably differ from each other. The H NMR spectra of a partly deuterated AHS crystal. Unlike protons, deuterium nuclei possess a quadrupole length of the hydrogen bonds between the Se(1)O4 groups (α bonds) is equal to 2.56 Å, and the length of moment. Nuclear magnetic resonance at quadrupole β nuclei provides valuable information on the magnitude the hydrogen bonds between the Se(2)O4 groups ( bonds) is 2.59 Å (Fig. 1). In the paraelectric phase, pro- and symmetry of crystal-electric-field gradients at the tons of the α bonds are disordered dynamically [6, 7]. studied nucleus. For a strong external magnetic field B0, Reasoning from the analysis of the 1H NMR spectra when the Zeeman interaction energy substantially 1 exceeds the energy of interaction between the nuclear and measurements of the H spinÐlattice relaxation quadrupole moment and the crystal field, the crystal time, Moskvich et al. [1] assumed that the isotropic dif- field brings about a perturbation of equidistant Zeeman fusive motion of ammonium groups and protons of levels and a splitting of the NMR line into 2I components (I is the nuclear spin), which are symmetrically located with respect to the Larmor precession fre- ν quency 0 in the magnetic field B0 [10]. Consequently, a the NMR spectrum of deuterons (ID = 1) consists of doublets whose number for a single-crystal sample in b O(5)' NH4(2) the general case is equal to the number of magnetically nonequivalent deuterium nuclei. According to Pound O(2) ν ν Se(1) [10], the quadrupole splitting ( 2 Ð 1) can be repre- β' sented by the relationship α O(1) O(3) a* 6eQ LAB O(6) ν Ð ν ==------V Φ , (1) NH (1) 2 1 4h zz zz O(1)' Se(2) 4 β'' O(5) O(4) where Q is the nuclear quadrupole moment, e is the elementary charge, h is the Planck constant, and Vzz is the zth component (the magnetic field B0 is aligned along the z axis) of the electric-field gradient at the nucleus. All the components Vij of the electric-field gradient tensors for each structurally nonequivalent position of deuterium in the crystal in the laboratory coordinate Fig. 1. Structure of NH4HSeO4 in the paraelectric phase. A system can be determined from the orientation depen- half of the unit cell is shown. Wavy lines indicate hydrogen dences of the quadrupole splitting within the frame- bonds between the oxygen atoms of the SeO4 groups. work of the well-known Volkoff method [11]. The elec-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MECHANISM OF PROTONIC CONDUCTIVITY 1079 tric-field gradient tensor (for simplicity, the values of Φ ij in frequency units will be used instead of Vij)—a (a) symmetric second-rank tensor with zero spur— accounts for the point symmetry at the position of the nucleus under investigation. Specifically, for the α- and 400 K β-type hydrogen bonds in the AHS crystal, the principal Φ axis 33 of the electric-field gradient tensor approximately coincides with the direction of the OÐHáááO 390 K Φ hydrogen bond and the principal axis 22 is perpendicular to the plane of the SeÐOáááO bond. On this basis, each electric-field gradient tensor can be assigned to a 370 K particular hydrogen bond in the crystal. Thus, the magnetic resonance at 2H nuclei in a partly deuterated AHS crystal appreciably extends the capabilities of the NMR 300–350 K technique and makes it possible to determine the indi- –3–2–10123 vidual dynamic characteristics of protons involved in f, kHz hydrogen bonds and ammonium groups. Figure 2 displays typical temperature dependences of the 2H NMR spectra for an AHS crystal in the range 300Ð400 K. These spectra were measured for a crystal (b) orientation at which the b axis is perpendicular to the external magnetic field B0 and the angle between the a* ° axis and the field B0 is equal to 15 . For this orientation, 400 K the 2H NMR spectra contain two groups of lines. The central doublets (Fig. 2a) are assigned to the deuterons 390 K of the ammonium groups, and three doublets with splittings larger than 20 kHz (Fig. 2b) are attributed to the 370 K deuterons of the hydrogen bonds. As is clearly seen from Fig. 2a, no significant changes occur in the central portion of the spectrum over the entire temperature 300–350 K range of the existence of the paraelectric phase. Two electric-field gradient tensors for the deuterons of the –30 –20 –10 0 10 20 30 ammonium groups (Table 1) were determined from the f, kHz angular dependences of the 2H NMR spectra at temperatures T = 300 and 390 K. One tensor is similar to an Fig. 2. Temperature dependences of the 2H NMR spectra of axially symmetrical tensor and corresponds to the deu- an AHS crystal in the paraelectric phase: (a) the central qua- terons of the ammonium groups in special positions. drupole doublets assigned to the deuterons of the ammo- The other tensor is assigned to the deuterons of the nium groups and (b) the side peaks attributed to the deuterons of the hydrogen bonds. The spectra are measured for a ammonium groups in general positions. The small qua- crystal orientation with the b axis perpendicular to the drupole coupling constant for these deuterons indicates external magnetic field B0. The angle between the a* axis ° a fast reorientation of the ammonium groups in the and the field B0 is equal to 15 . paraelectric phase. Therefore, in this case, the parameters of the electric-field gradient tensor account for an effective distortion of the ammonium group due to its environment. As follows from Table 1, an increase in aforementioned data and provides additional informa- the temperature leads to an insignificant change in the tion on the deuteron motion. The dominant contribution 2 parameters of both electric-field gradient tensors in the to the width of the H NMR lines assigned to the ammo- paraelectric phase of the AHS crystal. These findings nium groups is made by the intermolecular dipoleÐ unambiguously demonstrate the absence of chemical dipole interaction of the ammonium deuterons with exchange between two structurally nonequivalent each other and the dipoleÐdipole interaction of the ammonium groups over the entire range of the exist- ammonium deuterons with deuterons of the hydrogen ence of the paraelectric phase. Consequently, the bonds. The dipole broadening of the 2H NMR lines due hypothesis proposed in [1] regarding isotropic diffusion to the dipole interaction of the deuterons with the of ammonium groups at 390 K should be revised. Anal- remaining protons is suppressed by proton decoupling, ysis of the second moments of the 2H NMR lines broad- whereas the intramolecular dipole interactions of the ened at the expense of dipoleÐdipole interactions (here- deuterons involved in the ammonium groups are aver- after, these lines will be referred to as the dipole-broad- aged through fast reorientation of these groups. Taking ened lines) gives results that are in agreement with the into account the deuterons (with due regard for the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1080 IVANOV et al.

Table 1. Parameters of the electric-ﬁeld gradient tensors for two structurally nonequivalent ammonium groups in the AHS crystal

NH4(1) NH4(2) Tempera- principal values direction cosines (magnitudes) principal values direction cosines (magnitudes) ture, K of electric-ﬁeld with respect to crystallographic axes of electric-ﬁeld with respect to crystallographic axes gradient tensors gradient tensors Φ Φ ii, Hz a* bcii, Hz a* bc Φ Φ 300 11 = Ð1196 0.51 0.77 0.39 11 = Ð1350 0 0 1 Φ Φ 22 = Ð402 0.4 0.61 0.68 22 = Ð1329 0.73 0.68 0 Φ Φ 33 = 1598 0.76 0.19 0.62 33 = 2679 0.68 0.73 0 Φ Φ 390 11 = Ð920 0.26 0.93 0.26 11 = Ð850 0 0 1 Φ Φ 22 = Ð519 0.47 0.35 0.81 22 = Ð1306 0.72 0.69 0 Φ Φ 33 = 1439 0.84 0.09 0.53 33 = 2156 0.69 0.72 0

Table 2. Theoretical (calculated with inclusion of all the magnetic nuclei, except for protons, and a random distribution of deuterons) and experimental (at 300 K) second moments of the 2H NMR lines attributed to the ammonium groups in the AHS crystal

Group NH4(1) Group NH4(2) Orientation 2 2 M2, Hz M2, Hz of ﬁeld B0 Calculation Experiment Calculation Experiment ||a* 1.95 × 104 (2 ± 0.2) × 104 1.58 × 104 (1.7 ± 0.2) × 104 ||b 1.96 × 104 (2 ± 0.2) × 104 1.81 × 104 (2 ± 0.2) × 104 ||c 1.5 × 104 (1.7 ± 0.2) × 104 1.1 × 104 (1.2 ± 0.2) × 104

Table 3. Theoretical (calculated as described in Table 2, except for the contribution of the dipoleÐdipole interaction between the deuterons of ammonium groups and the deuterons of hydrogen bonds) and experimental (at 390 K) second moments of the 2H NMR lines attributed to the ammonium groups in the AHS crystal

Group NH4(1) Group NH4(2) Orientation 2 2 M2, Hz M2, Hz of ﬁeld B0 calculation experiment calculation experiment ||a* 6.89 × 103 (1.0 ± 0.2) × 104 6.72 × 103 (0.9 ± 0.2) × 104 ||b 1.5 × 104 (1.6 ± 0.2) × 104 1.48 × 104 (1.6 ± 0.2) × 104 ||c 7.61 × 103 (1.0 ± 0.2) × 104 7.63 × 103 (0.9 ± 0.2) × 104 degree of deuteration) and other magnetic nuclei, dipole interaction with the deuterons of the hydrogen except for protons, we calculated the second moments bonds. This suggests a fast diffusive motion of these of the 2H NMR lines. The lattice sums were calculated protons (deuterons). More detailed information can be in a sphere of radius 40 Å. The calculated second obtained from analyzing the relevant lines of the NMR moments are in agreement with the experimental data spectra. at 300 K (Table 2). An increase in the temperature leads It can be seen from Fig. 2b that the side components to a decrease in the experimental second moments of of the NMR spectra do not exhibit noticeable changes the 2H NMR lines associated with the ammonium in the temperature range from 300 to 350 K. The groups (Table 3). It is interesting to note that the exper- parameters of two electric-ﬁeld gradient tensors for two imental second moments at 390 K agree well with the structurally nonequivalent positions of the protons theoretical moments calculated without regard for the involved in the α- and β-type hydrogen bonds were cal-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MECHANISM OF PROTONIC CONDUCTIVITY 1081 culated from the orientation dependences of the quadrupole splittings at 300 K. The results of calculations coincide with the data obtained in our earlier work [12]. –24 At temperatures above 350 K, the side components of β'' the NMR spectrum become broadened (Fig. 2b). This indicates that chemical exchange occurs in a system of β' hydrogen bonds in the crystal. However, the contribu- –16 tion of this exchange is substantially less than the qua- α drupole splitting up to the temperature of phase transi-

–8 , kHz tion to the superionic phase (417 K). Hence, the mech- f anism of proton motion cannot be judged from these spectra. It is only possible to estimate the rate of chem- 0 ical exchange from the NMR linewidth in the framework of the well-known Anderson theory (see, for example, [13]). The calculated rates of chemical 8 exchange are as follows: 0.5 × 103 sÐ1 at 370 K, 2 × 24 16 8 0 103 sÐ1 at 390 K, and 4 × 103 sÐ1 at 400 K. In order to f, kHz obtain information on the microscopic mechanism of 2 the deuteron mobility, the exchange rate, and the acti- Fig. 3. Two-dimensional H NMR exchange spectrum of the AHS crystal at a temperature of 350 K and a mixing time vation energy of this process, we used two-dimensional of 3 ms (only the upper left quadrant is shown). The crystal 2H NMR spectroscopy. The mathematical formalism of orientation is the same as in Fig. 2. The exchange is charac- the chemical exchange processes and numerical calcu- terized by the off-diagonal peaks and occurs between deu- lations of the exchange rates from two-dimensional terons at the positions α, β', and β''. NMR spectra have been described in a number of well- known works [14, 15]. Here, we outline this method only brieﬂy. The exchange kinetics, as a rule, is characterized by the probability (rate) pij of transferring an atom from the position i to the position j in a unit time. 103 The chemical exchange can be represented by the basic –1 equation (see, for example, [13]) 102 n ∂n i 2D NMR data ------= ∑ pijn j (2) 101 ∂t 1D NMR data j Exchange rate, s or, in the matrix form, as nú = pn with the solution 1 2.6 2.8 3.0 3.2 3.4 ()⋅ n()t ==exp p t n0 At()n0. (3) 103/T, K–1

Here, the components n0i of the vector n0{n01, …, n0i} are equal to the number of deuterons at the ith position Fig. 4. Temperature dependence of the deuteron exchange at the instant of time t = 0 and the components ni of the rate for an AHS crystal according to 1D and 2D NMR data. Ð1 × 14 Ð1 vector n(t) = {n1, …, ni} are equal to the number of deu- Ea = 81.1 kJ mol and p0 = 1.9 10 s . τ terons at the same position at the instant t = m. The components Aij(t) of the exchange matrix A(t) in relationship (3) completely determine the kinetics of deu- were measured at the same orientation as for the one- teron (proton) exchange in the crystal and can be dimensional (1D) 2H NMR spectra displayed in Fig. 2. obtained from the intensities of the corresponding The typical off-diagonal peaks (see, for example, [14, peaks in the two-dimensional 2H NMR spectra [14, 15]. 15]) in Fig. 3 indicate deuteron exchange between In our recent work [4], the AHS crystal was thoroughly hydrogen bonds of two types (the α and β bonds) and investigated by two-dimensional 2H NMR spectros- between magnetically nonequivalent positions of the copy. Here, we present only the results that are essential adjacent bonds. Note that the rates of these processes to the understanding of the microscopic mechanism of are approximately equal to each other. The two-dimen- protonic conductivity. The two-dimensional 2H NMR sional NMR spectra unambiguously demonstrate that experiments with the AHS crystal were performed in no chemical exchange between the deuterons of the the temperature range 300Ð350 K. Figure 3 shows the ammonium groups and the deuterons of the hydrogen left upper quadrant of the typical total two-dimensional bonds occurs over the entire temperature range of the 2H NMR spectrum of the AHS crystal (at a temperature existence of the paraelectric phase. Figure 4 depicts the of 350 K and a mixing time of 3 ms). These spectra temperature dependence of the exchange rate according

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1082 IVANOV et al. one chain of hydrogen bonds to the adjacent chain. (a) Therefore, the conductivity anisotropy in the AHS crystal should be insignificant. From the results of dielectric measurements [5], we determined the conductivity parallel and perpendicular to the crystallographic axis b (the direction of hydrogen bond chains). Within the limits of experimental error, no conductivity anisotropy is revealed in the temperature range from 300 to 350 K. 418 K For proton exchange, the activation energies determined from our NMR data and the temperature dependence of the conductivity coincide to within good accu- 416 K racy [5]. At temperatures above 360 K, the conductivity for both directions deviates from the Arrhenius behav- 410 K ior toward larger values. It should be noted that these –2 –1 0 1 2 deviations vary from sample to sample for the same ori- f, kHz entation. This can be explained by the high hygroscop- icity of the AHS crystal. All the samples are characterized by a sharp increase in the conductivity upon transition to the superionic phase. (b) At 417 K, the AHS crystal undergoes a transition to the high-temperature superionic phase. Unfortunately, the crystal in this phase rapidly loses protons and is 418 K destroyed. For this reason, the superionic phase cannot be investigated thoroughly by nuclear magnetic resonance and diffraction methods. There exist only the 416 K assumptions that the superionic phase has the P2/n [16] 2 or P21/b [17] symmetry. We analyzed the H NMR spectra for several orientations of the AHS crystal in the 410 K temperature range 410Ð420 K. For each new orientation, we used a particular sample. Figure 5 displays the temperature dependences of the 2H NMR spectra for –30 –20 –10 0 10 20 30 the AHS crystal oriented in such a manner that the b f, kHz axis is perpendicular to the external magnetic field B0 and the angle between the a* axis and the field B0 is Fig. 5. Temperature dependences of the 2H NMR spectra of equal to 15°. The changes in the 2H NMR spectra are an AHS crystal in the vicinity of the phase transition to the superionic state: (a) the central quadrupole doublets observed at temperatures close to 417 K. A single dou- assigned to the deuterons of the ammonium groups and (b) blet with extremely narrow components appears the side peaks attributed to the deuterons of the hydrogen instead of the dipole-broadened NMR lines assigned to bonds. The crystal orientation is the same as in Fig. 2. the deuterons of the hydrogen bonds. As the temperature increases, the intensity of the narrow lines increases, whereas the intensity of the dipole-broadened lines decreases to zero (the phase coexistence typ- to the data of one-dimensional and two-dimensional ical of first-order phase transitions). The results NMR spectroscopy. The solid line in Fig. 4 represents obtained indicate that deuterons (protons) in the superi- the approximation of this dependence by the Arrhenius onic phase exhibit a high diffusive mobility. This leads equation with the activation energy Ea: to complete averaging of the dipoleÐdipole interactions () of the deuterons involved in the hydrogen bonds. The pT()= p0 exp Ea/RT . (4) central NMR lines attributed to the deuterons of the 2 ammonium groups also change at a temperature of It should be noted that the 2D H NMR data and the 417 K. The NMR quartet observed in the paraelectric estimates made from the 1D 2H NMR spectra at high phase for the given orientation transforms into a dou- temperatures are in good agreement and lead to the fol- blet with a splitting of approximately 200 Hz and a lin- lowing parameters of the exchange process: the activa- ewidth of 100 Hz for each component. Therefore, the tion energy is approximately equal to 80 kJ molÐ1 and transition to the superionic phase is accompanied by the × the preexponential factor is estimated as p0 = 1.9 change in the structural positions of the ammonium 1014 sÐ1. This suggests that the proton mobility in the groups (one position instead of two structurally non- paraelectric phase occurs through a sole mechanism, equivalent positions) and, possibly, their diffusive namely, through sequential hoppings of protons from motion. However, the rate of this diffusion is relatively

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MECHANISM OF PROTONIC CONDUCTIVITY 1083 low and cannot exceed the rate corresponding to the lin- tric-field gradient tensor. This suggests a high diffusive ewidth (~100 Hz). Consequently, the contribution of mobility of these protons. the ammonium groups to the conductivity of the AHS crystal in the superionic phase is negligible and the conductivity in this phase, as in the paraelectric phase, is 4. CONCLUSIONS completely governed by the mobility of protons Thus, proton exchange between hydrogen bond involved in the hydrogen bonds. chains extended along the b axis is the main mechanism Let us now consider structural characteristics that of proton transport in the paraelectric phase of the AHS are important for proton transport in the AHS crystal. In crystal. Unlike the model proposed in [1, 2], the mech- the paraelectric phase, the shortest distance between the anism considered above is not limited by the potential protons in equivalent positions is equal to the unit cell barrier to reorientation of the SeO4 groups and does not parameter b (4.61 Å). This distance exceeds the shortest require the simultaneous breaking of two hydrogen distance (3.91 Å) between the positions of protons in bonds. This mechanism makes it possible, within a uni- the adjacent chains of the hydrogen bonds (Fig. 1). In fied context, to explain all the available experimental data, including macroscopic measurements of electrical the adjacent tetrahedra SeO4, the shortest distance between the oxygen atoms, which are not involved in conductivity. The superionic conductivity in the high- the formation of hydrogen bonds, is approximately temperature phase of the AHS crystal is most likely equal to 3.3 Å. Moreover, protons can occupy posi- associated with orientation disordering of the SeO4 tions, for example, between the O(3) and O(4) atoms groups, which results in an increase in the number of separated by a distance of 3.18 Å. Owing to thermal sites providing proton diffusion. vibrations (librations) of the SeO4 groups, these dis- It can be assumed that proton chemical exchange tances can be even shorter. As a result, hydrogen bonds similar to that observed in the paraelectric phase of the with a short lifetime can be formed between the rele- AHS crystal occurs in other crystals with chains of vant O atoms. The aforementioned structural features, hydrogen bonds. In this respect, it would be very inter- the results of 2D 2H NMR investigations, and the esting to carry out similar investigations with crystals absence of conductivity anisotropy allow us to con- characterized by another configuration of the hydro- clude that the main mechanism of proton transport in gen-bond system, especially with a KHSeO4 crystal in the paraelectric phase of the AHS crystal is associated which layers of hydrogen-bond chains alternate with with proton hoppings between the adjacent chains of layers of closed dimers consisting of SeO4 groups [20]. hydrogen bonds. At present, we are involved in such investigations. A drastic increase in the conductivity of the AHS crystal at temperatures above 417 K can be caused by a ACKNOWLEDGMENTS jumpwise decrease in the height of potential barriers to proton diffusion and also by a disordering of the oxy- This work was supported by the Russian Foundation for Basic Research, project no. 00-15-96790. gen atoms of the SeO4 groups in the superionic phase, as has been observed in (NH4)3H(SeO4)2 crystals [18, 19]. This assumption is confirmed by the data obtained REFERENCES by Dvorak et al. [16], who explained the overall 1. Yu. N. Moskvich, A. A. Sukhovsky, and O. V. Rozanov, sequence of phase transitions in the AHS crystal and its Fiz. Tverd. Tela (Leningrad) 26, 38 (1984) [Sov. Phys. deuterated analog as the result of small distortions of Solid State 26, 21 (1984)]. the praphase with the space group of symmetry Immm 2. R. Blinc, J. Dolinsek, G. Lahajnar, et al., Phys. Status and one formula unit per unit cell. In this phase, the Se Solidi B 123, K83 (1984). and N atoms (the centers of the SeO4 and NH4 groups) 3. A. I. Baranov, R. M. Fedosyuk, N. M. Schagina, and occupy the special positions (0 0 0) and (1/2 1/2 0), L. A. Shuvalov, Ferroelectr. Lett. Sect. 2, 25 (1984). respectively. According to [16], the space groups Immm 4. Yu. N. Ivanov, J. Totz, D. Michel, et al., J. Phys.: Con- and P2/n describe only an averaged symmetry of the dens. Matter 11, 3151 (1999). AHS crystal. In particular, the positions of the tetrahe- 5. J. Totz, D. Michel, Yu. N. Ivanov, et al., Appl. Magn. dral groups in the praphase can have an inversion center Reson. 17, 243 (1999). only due to disordering of the oxygen atoms through 6. I. P. Aleksandrova, O. V. Rozanov, A. A. Sukhovskii, and reorientation of the SeO4 groups. In any case, the num- Yu. N. Moskvich, Phys. Lett. A 95, 339 (1983). ber of possible relative positions of the oxygen atoms in 7. I. P. Aleksandrova, Ph. Colomban, F. Denoyer, et al., the SeO4 groups and the number of possible positions Phys. Status Solidi A 114, 531 (1989). of the protons in the hydrogen bonds considerably 8. A. A. Sukhovsky, Yu. N. Moskvich, O. V. Rozanov, and increase compared to those in the paraelectric phase. As I. P. Aleksandrova, Ferroelectr. Lett. Sect. 3, 45 (1984). a consequence, the conductivity of the high-tempera- 9. K. S. Aleksandrov, A. I. Kruglik, S. V. Misyul’, and ture phase increases jumpwise. According to the 2H M. A. Simonov, Kristallografiya 25, 1142 (1980) [Sov. NMR data, all protons of the hydrogen bonds in the Phys. Crystallogr. 25, 654 (1980)]. superionic phase are described by a sole averaged elec- 10. R. V. Pound, Phys. Rev. 79 (4), 685 (1950).

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1084 IVANOV et al.

11. G. V. Volkoff, H. E. Petch, and D. W. Smellie, Phys. Rev. 17. A. Onodera, A. Rozycki, and F. Denoyer, Ferroelectr. 84, 602 (1951). Lett. Sect. 9, 77 (1988). 12. Yu. N. Moskvich, O. V. Rozanov, A. A. Sukhovsky, and 18. B. V. Merinov, M. Yu. Antipin, A. I. Baranov, et al., Kri- I. P. Aleksandrova, Ferroelectrics 63, 83 (1985). stallograﬁya 36, 872 (1991) [Sov. Phys. Crystallogr. 36, 13. A. Abragam, The Principles of Nuclear Magnetism 488 (1991)]. (Clarendon, Oxford, 1961; Inostrannaya Literatura, Moscow, 1963). 19. A. Piertaszko, B. Hilczer, and A. Pawlowski, Solid State 14. C. Schmidt, B. Blümich, and H. W. Spiess, J. Magn. Ionics 119, 281 (1999). Reson. 79, 269 (1988). 20. J. Baran and T. Lis, Acta Crystallogr., Sect. C C42, 270 15. S. Kaufmann, S. Weﬁng, D. Schaefer, and H. W. Spiess, (1986). J. Chem. Phys. 93, 197 (1990). 16. V. Dvorak, M. Quilichini, N. Le Calvé, et al., J. Phys. I 1, 1481 (1991). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1085Ð1092. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1039Ð1047. Original Russian Text Copyright © 2002 by Ogorodnikov, Yakovlev, Kruzhalov, Isaenko.

SEMICONDUCTORS AND DIELECTRICS

Transient Optical Absorption and Luminescence in Li2B4O7 Lithium Tetraborate I. N. Ogorodnikov*, V. Yu. Yakovlev**, A. V. Kruzhalov*, and L. I. Isaenko*** *Ural State Technical University, ul. Mira 19, Yekaterinburg, 620002 Russia e-mail: [email protected] **Tomsk Polytechnical University, Tomsk, 634021 Russia ***Technological Institute of Single Crystals, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630058 Russia Received June 1, 2001

Abstract—This paper reports on a study of the transient optical absorption exhibited by Li2B4O7 (LTB) in the visible and UV spectral regions. Using absorption optical spectroscopy with nanosecond time resolution, it is established that the transient optical absorption (TOA) in these crystals originates from optical transitions in hole centers and that the kinetics of the optical-density relaxation is controlled by interdefect tunneling recombination, which involves these hole centers and electronic Li0 centers representing neutral lithium atoms. At 290 K, the Li0 centers migrate in a thermally stimulated, one-dimensional manner, without carrier ejection into the conduction or valence band. The kinetics of the pulsed LTB cathodoluminescence is shown to be controlled by a relaxation process connected with tunneling electron transfer from a deep center to a small hole polaron migrating nearby, a process followed by the formation of a self-trapped exciton (STE) in an excited state. Radi- ative annihilation of the STE accounts for the characteristic σ-polarized LTB luminescence at 3.6 eV, whose kinetics is rate-limited by the tunneling electron transfer. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION high optical quality [2Ð5], including the methods of Czochralski [6] and Bridgman [7] and the low-temper- Progress in solid-state laser technology has resulted ature hydrothermal method [8]. At room temperature, in the development of compact and reliable optical radi- the LTB crystal possesses tetragonal symmetry (point ation sources that can operate at a wavelength of about group 4mm) and its lattice parameters are a = b = 1 µm, and high-power UV light sources for use in var- 0.9477 nm and c = 1.0286 nm [9, 10]. Its structural ious areas of science and industry are in ever increasing motif consists essentially of two equivalent tetrahedra demand. The most important components in solid-state and two equivalent trigonal structures, with the polar c UV radiation sources are frequency converters based on axis parallel to the (001) crystallographic direction. nonlinear crystals, in particular, on the borates of alkali LTB single crystals are good ionic conductors due to β and alkaline-earth metals, namely, -BaB2O4 (BBO), the lithium cations they contain; indeed, at 300 K, the LiB3O5 (LBO), and CsLiB6O10 (CLBO). Li2B4O7 lith- one-dimensional ionic conductivity of LTB along the ium tetraborate (LTB) has not been recognized as a pos- polar axis is 3.5 × 10Ð6 (Ω cm)Ð1 [11, 12]. sible candidate for such applications for a long time. A large number of publications on LTB deal with its LTB is traditionally employed in acousto- and opto- optical properties under irradiation. It is well known, electronics and solid-state thermoluminescent dosime- for instance, that LTB crystals remain transparent in the try. Only very recently have its nonlinear properties spectral region 1.2Ð6.7 eV under irradiation by gamma been discovered and studied in the UV region and has rays or electrons (E = 1.25Ð1.3 MeV) up to an absorbed the possibility of its use been demonstrated for second dose of 106 Gy [13]. At higher doses, weak bands (∆k < and ﬁfth YAG:Nd laser harmonic generation [1]. While Ð1 the efﬁciency of the nonlinear conversion in the LTB 1.0 cm ) of stable optical absorption peaking at 4.1 and crystal was found to be slightly lower than that in BBO, 5.2 eV appear [14]. At the same time, data on the tran- LBO, and CLBO, the LTB has an obvious advantage in sient optical absorption and the attendant relaxation of its higher radiation strength. The optical-damage electronic excitations are lacking. The discovery of the threshold for LTB irradiated by high-power pulsed nonlinear properties of LTB has stimulated studies in YAG:Nd laser radiation (1.5 J) (τ = 10 ns, λ = 1064 nm) this direction. is as high as 40 GW cmÐ2 [1], which exceeds the thresh- The present study was aimed at investigating the old for the other borates by a few times. A not less LTB crystal by luminescence and optical spectroscopy important aspect is the availability of well-developed with nanosecond time resolution under electron-beam technologies for growing large bulk LTB crystals of a excitation.

1086 OGORODNIKOV et al. 2. EXPERIMENTAL TECHNIQUE follows: an average electron energy of 250 keV, an exciting-pulse duration of about 20 ns, a pulse current In this work, we used Li2B4O7 single crystals of high density variable from 10 to 1000 A cmÐ2, and a maxi- optical quality Czochralski-grown from a stoichiome- Ð2 3 mum pulse energy of 0.16 J cm . The excitation was try melt in platinum crucibles up to 200 cm in volume. envoked with pulse energies of 12 or 23% of the maxi- After the Grade Ch Li2B4O7 reagent was recrystallized mum level. For decay times up to 20 µs, the probing and maintained in this condition for the required time, light was provided by an INP-5.50 pulsed lamp; for the temperature was lowered to the melting point, longer processes, by KGM12-100 and DDS-30 lamps. 917°C. The crystals were grown on differently oriented seeds at pull rates of up to 3 mm/day and rotation Polarization measurements were performed with speeds of about 20 rpm. Li B O crystals grown for Rochon (TOA) and FrankÐRichter (PCL) prisms. The 2 4 7 degree of polarization was estimated from the relation 15 days typically measured 40 mm in length and P = (I|| Ð I⊥)/(I|| + I⊥), where the indices || and ⊥ refer to 35 mm in diameter and weighed up to 70 g. The results of preliminary studies of the grown crystals are the probing-light electric vector oriented parallel or described in detail in our publications [15, 16]. The perpendicular to the crystal optical axis, respectively, samples were 7 × 7 × 1-mm plane-parallel plates with and I is the luminescence intensity (PCL) or optical polished faces perpendicular or parallel to the c crystal- density (TOA). lographic axis. A detailed description of the experimental setup and 3. EXPERIMENTAL RESULTS of the luminescence and absorption spectroscopy with Figure 1 presents polarized TOA spectra of an LTB nanosecond time resolution employed is given in [17]. crystal measured at 290 K 1.25 µs after an excitation The transient optical absorption (TOA) and pulsed pulse for two different orientations of the probing-light cathodoluminescence (PCL) were measured photoelec- electric vector relative to the crystal optical axis, trically in the spectral region 1.2Ð5.5 eV with an MDR3 namely, for E || C (σ polarization) and E ⊥ C (π polar- monochromator equipped with 1200- and 600-mmÐ1 ization). The spectra of σ- and π-polarized TOA extend interchangeable gratings, an FÉU-97 or FÉU-83 PM over a broad region from 2.0 eV to above 5.0 eV and tube, and an S8-12 storage oscillograph. When measur- represent a superposition of several constituent Gauss- ing the dependence of the PCL and TOA on electron- ian-shaped bands (Table 1). As follows from an analy- beam power, the excitation pulse length was kept con- sis of Table 1, the major part of the optical density is stant. The TOA was studied by the total internal reﬂec- concentrated in a broad, weakly polarized band G4 tion technique. Optical densities greater than unity peaking at 4.52 eV. The observed spectral dependence were measured in the transmission geometry character- of the polarization ratio (Fig. 1) is due to the contribu- ized by a shorter optical path with subsequent reduc- tion of differently polarized G1…G3 bands; more spe- tion. π ciﬁcally, the G2 and G3 bands are responsible for the A nanosecond-range electron accelerator was used negative value of PD in the region 2.3Ð4.5 eV, while the σ as a source of excitation. The beam parameters were as contribution from the weak G1 band results in the sign

1.0 0.50 E⊥C z x hν 1 E||C y – D D 0.5 e 2 0.5 10–1 0.25 G3 G2 0 0 Optical density

G1 Polarization ratio 3 Optical density Optical density

–0.5 0 10–2 1 2 3 4 5 10–6 10–5 10–4 10–3 10–2 10–1 Photon energy, eV Decay time, s

Fig. 1. (1, 2) Polarized TOA spectra of LTB at 290 K, measured in (1) the E ⊥ C and (2) E || C configurations, and (3) the polarization ratio. Points are experimental data, and Fig. 2. TOA kinetics of LTB in the 3.8-eV band measured at solid lines are data fits. G1, G2, and G3 are unfolded constit- 290 K. Points are experimental data, and solid lines are data uent bands. fits.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TRANSIENT OPTICAL ABSORPTION AND LUMINESCENCE 1087

1 1

1 2 D 1 3 4

5 –1 2 –1 10 Optical density 10 Optical density

1 101 102 1 100 101 Electron beam power, % Decay time, µs

Fig. 3. Plots of optical density in the bands at (1) 3.8 and (2) Fig. 4. TOA kinetics of LTB measured in the 3.8-eV band at 2.1 eV measured with a 1-µs delay vs. electron beam power different temperatures (K): (1) 325, (2) 334, (3) 425, (4) obtained on an LTB crystal at 290 K. 461, (5) 538, and (6) 623.

σ reversal of PD in the region 1.5Ð2.3 eV. Below 1.5 eV, neling probability (r), which depends on the distance the π-polarized TOA dominates (Fig. 1), which implies r between defects as π the existence of a TOA band in LTB at longer wave- σ() σ () lengths. r = 0 exp Ðr/a . (3) At 290 K, the TOA decays uniformly over the entire Figure 2 illustrates the fit of experimental data on D(t) spectrum and the sample recovers to the original optical to Eq. (2) made within a decay time region extending transmission level in a time of less than 1 s. No increase over about five decades. in optical density with increasing number of excitation Increasing the power of the exciting electron beam pulses was observed. Figure 2 shows the TOA decay brings about a proportional growth in the optical den- kinetics of LTB measured at 290 K in the 3.8-eV band. sity uniformly over the whole spectrum. Figure 3 shows A characteristic feature of the kinetics is a compara- the dependences of TOA optical density in LTB on tively slow, monotonic relaxation of optical density, excitation power measured in the 2.1- and 3.8-eV which is observed over a decay time range covering bands. As seen from Fig. 3, these relations, when plot- about five decades. As follows from an analysis of ted on a log–log scale, can be fitted within two decades Fig. 2, the decay kinetics is satisfactorily described by with straight lines with a slope of unity. The linear law straight lines within a limited decay time interval of of growth in the optical density and the absence of sat- 2.5Ð3 decades when plotted both on the DÐlog(t) and uration within a broad range of exciting power variation logÐlog scale. In the latter case, the decay kinetics can may suggest that the corresponding color centers are be fitted by a linear relation, associated with intrinsic lattice defects. Note that the () TOA decay kinetics remains virtually unchanged. Only ÐlogDt = Ap+ log t. (1) at a very low excitation power can weak components be µ This implies a power-law dependence of optical density distinguished at decay times of about 100 s; these on time, D(t) ∝ tÐp. In the region of short decay times, the exponent p (the index of asymptotic behavior) is Table 1. Polarized TOA band parameters of LTB measured about 0.04, and it increases slightly for longer decay at 290 K immediately after the pump pulse times. Such characteristics are usually typical of the ∆ kinetics of tunneling recharging (TR); in the case of a Band Em, eV E, eV PD random distribution of fixed defects characterized by a concentration n(t) and kinetics parameter N [n(t) Ӷ N], G1 1.80 0.30 >+0.5 ' the TP kinetics is described by the relation [18] G2 2.25 1.20 Ð0.418 G 2.97 0.64 Ð0.684 4π 3 3 2 nt()= n exp Ð------a N ln ()σ t , (2) 0 3 0 G3 3.70 0.64 Ð0.692 G4 4.52 2.20 Ð0.034 where n0 = n(0), a is one half the Bohr radius of the Note: E , ∆E, and P are the position of the maximum, FWHM, σ m D defect wave function, and 0 is the prefactor of the tun- and the degree of polarization, respectively.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1088 OGORODNIKOV et al.

60 395 K II 1 10–4 N

3 3

a 30 I

Intensity, arb. units 2

10–5 1.5 2.0 2.5 3.0 3.5 0 103/ , K–1 2 3 4 5 T Photon energy, eV

Fig. 6. PCL spectra of LTB measured (1) immediately after Fig. 5. Temperature dependence of the parameter a3N of the pump pulse, (2) with a 100-ns delay, and (3) their differ- TOA kinetics in LTB measured in the 3.8-eV band. ence. components obey a first-order hyperbolic law and respond formally to activation energies of 0.16 eV (I) apparently originate from competing relaxation pro- and 0.10 eV (II). cesses. Figure 6 and Table 2 present PCL spectra of LTB Figure 4 plots TOA kinetics in LTB measured at var- measured at 290 K immediately after an excitation ious temperatures. Throughout the temperature range pulse (GT) and with a 100-ns delay (GS) and their differ- covered (300Ð623 K), the TOA decay curves drawn on ence (GF). The difference spectrum GF, which is prima- a log–log scale can be formally fitted with a linear rela- rily due to the contribution of fast components with tionship. For the microsecond-scale decay time region, time constants shorter than 100 ns, is characterized by the exponent p varies monotonically from 0.038 a long-wavelength shift of 0.26 eV relative to the slow- (300 K) to 0.480 (623 K); however, the initial intensity component spectrum. The fast-component spectrum D(t 0) remains constant within the whole temper- has a wide-band pedestal whose height grows with ature interval studied. Figure 4 illustrates an approxi- excitation power; indeed, for an electron-beam power mation of the experimental data by Eq. (2) for a fixed level of 23%, the pedestal height is about 14% of the GT σ value of the prefactor, 0 = 270 GHz. intensity. The temperature dependence of the kinetics param- The PCL decay kinetics of LTB in the nano- and eter a3N is linearized in the Arrhenius coordinates microsecond ranges measured at 23%-excitation power (Fig. 5) and can be fitted by a linear relation, can be approximated by the sum of two components with an initial intensity ratio of 1 : 1.1. The first of them 3 is an exponential with τ = 30 ns (in some cases, 110 ns), ()3 10 log a N = B0 + B1------, (4) and the second is a hyperbola with an exponent of 1.6 T and a characteristic decay time of about 340 ns (Fig. 7). where B0 and B1 are the fitting parameters. In Fig. 5, one Figure 8 shows the PCL polarization in LTB in the can separate two temperature intervals, 290Ð395 K (I) 3.2-eV band measured with the polarizer rotated in the and 395Ð650 K (II), which differ in the slope of their (100) plane. The angle θ is reckoned from the E || C ori- straight lines; B1 = Ð0.78 (I) and Ð0.52 (II), which cor- entation. The angular dependence of the PCL polarization is approximated by the relation 2 ()θ Table 2. Parameters of the time-resolved PCL spectra in IA= 0 + Acos . (5) LTB at 290 K The degree of polarization PL = A/A0, estimated from ∆ Band Em, eV E, eV Im, arb. units the fitting parameters of this relation for the nanosecond range, increases from 35 to 50%. G 3.36 1.00 24 F Figure 9 plots the decay kinetics of the polarized GS 3.62 0.99 25 luminescence of LTB measured for E || C and E ⊥ C, as GT 3.50 1.05 49 well as the time evolution of the polarization ratio. The ∆ luminescence polarization of the centers usually Note: Em, E, and Im are the position of the maximum, its FWHM, and the intensity at the maximum for the constituent PCL decreases slightly with increasing decay time, whereas spectral bands, respectively. our measurements reveal an opposite tendency. As seen

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TRANSIENT OPTICAL ABSORPTION AND LUMINESCENCE 1089

102 60 1 1 2 101 40 2

100 20 3 Intensity, arb. units Intensity, arb. units 3

10–1 102 103 0 90 180 270 360 Decay time, ns Angle, deg

Fig. 8. LTB PCL polarization in the 3.2-eV band measured Fig. 7. PCL decay kinetics in LTB measured in the 3.5-eV with the analyzer rotated in the (100) plane (1) immediately band. Points are experimental data, solid line (1) is a fit after the excitation pulse and with a delay of (2) 62 and using (2) a hyperbolic and (3) an exponential component. (3) 300 ns. The angle is reckoned from the E || C orientation. from Fig. 9, the characteristic time of growth of the considerably lower temperatures [20]. Experimental polarization ratio is about 30Ð40 ns. This may be a con- data on LTB lattice dynamics show the lithium atoms to sequence of the PCL kinetics following a complex reveal a strong anharmonicity relative to the boronÐ course in LTB. A quantitative analysis of the polarized oxygen framework already at temperatures above PCL kinetics shows that the polarization ratio increases 230 K [21]. Viewed from this standpoint, defects in opposite phase with the decay of the component with involving an interstitial lithium ion and a lithium τ = 30 ns. Unfolding of the curve into its constituents vacancy appear to be the most probable candidates. In showed that the PCL decay components have a degree of polarization PL of about Ð60% (the exponential component) and of +0.47% (the hyperbolic component). 2 100 The variation of their contributions with time accounts (a) for the observed evolution of the polarization ratio 1 (Fig. 9). 1 2 4. DISCUSSION OF RESULTS 50 The first thing that draws one’s attention when analyzing the results is the linear growth (on a logÐlog 3 scale) of the color-center concentration, which occurs within two decades of pump power variation, and the absence of saturation. This gives one sound grounds to 0.1 0 assume that the color centers responsible for the tran- (b) sient optical absorption in LTB are native lattice defects which are formed by the electron beam pulse and then 1 1 decay in a time shorter than 1 s. Intensity, arb. units Polarization ratio 2 There are many publications on the optical and EPR 50 spectroscopy of radiation defects in LTB (see, e.g., [13, 14, 19]). However, all known radiation defects involving boron and oxygen atoms (vacancies, interstitials, 3 etc.) are stable at room temperature. At the same time, at 290 K, no stable color centers involving defects on the lithium sublattice have been observed in LTB. 0.1 0 0 200 400 Moreover, at comparatively low temperatures, crystal- Decay time, ns line LTB is a quasi-one-dimensional Li+ superionic conductor. The onset of efficient ion transport is pre- Fig. 9. Decay kinetics of polarized PCL in LTB in the bands ceded by an increase in the lithium sublattice anharmo- at (a) 3.7 eV and (b) 2.9 eV for the orientations (1) E || C nicity along the polar axis, which becomes manifest at and (2) E ⊥ C, and (3) the polarization ratio.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1090 OGORODNIKOV et al. the first case, trapping of an itinerant electron by a lith- occurs at room temperature already after their TR. ium ion may bring about the formation of an Li0 elec- Because it is the tunneling recharging of defects that tronic center. In the second case, a hole center forms accounts for the TOA in LTB, it appears reasonable to when one of the oxygen ions surrounding a lithium use the theory of TR of randomly distributed fixed vacancy traps an itinerant hole. As shown earlier, it is defects and to describe the TOA kinetics by Eq. (2). such a color center (a hole OÐ center) that is responsible At low temperatures, tunneling recombination of for TOA in the region 2Ð5 eV in LiB3O5 [22]. The opti- radiation defects is the major channel of their recharg- cal transitions in LBO are assigned to hole transfer ing. Indeed, TR has been detected and studied in a large from a local level of the OÐ center, which is split off the number of crystalline nonmetallic solids, as well as in valence-band top into the band gap, to states in the glasses and biological objects. The TR kinetics is usu- upper part of the valence band, which are derived from ally assumed to be temperature-independent. In some the O 2p orbitals. This actually reduces to hole transfer cases, however, the TR kinetics is influenced by other, among the oxygen ions surrounding the defect and is temperature-dependent processes. These processes are treated within the small-polaron theory as interpolaron capable of affecting either the concentration of the absorption. In this connection, it appears reasonable to defects taking part in the TR or the electron tunneling relate the transient optical absorption in LTB in the transfer probability. The first case relates to the condi- region 2Ð5 eV to the color centers representing small tions where, in addition to the two partners involved in polarons, which become optically absorbing when the tunneling recombination, there are defects of a third hole is transferred in a photoinduced process from the kind, whose diffusion brings about destruction of one localized ground state to one of the more or less equiv- of the pair components. A typical illustration of this alent oxygen ions surrounding the lithium vacancy. process is provided by tunneling recombination of the A second point worth noting is that the TOA decay {F, Vk} pair in KCl occurring simultaneously with the kinetics obeys the law of tunneling recombination thermally stimulated migration of the HA(Na) centers, between randomly distributed fixed defects over a time which destroy one of the pair components [27]. The interval covering not less than five decades. As follows second case is realized in wide band-gap dielectrics from the above analysis, the most probable partners in with strong electronÐvibration coupling. It is known tunneling recombination may be native lattice defects, [28] that the probability of the electron tunneling trans- more specifically, the electronic Li0 and the hole OÐ fer can be affected in these conditions by interaction center. At room temperature, the lithium cations are with local and collective vibrational modes, reorienta- mobile, thus suggesting the existence of a diffusion- tion of low-symmetry centers taking part in the TR, controlled reaction (DCR) of annihilation between the slow diffusion of one of the tunneling recombination lithium vacancies and interstitials which is described by partners, and breakdown of the adiabatic electronÐ the relation vibration coupling with increasing separation between the TR partners. In both cases, the defect distribution n nt()= ------0 -, (6) function changes in distance and angle (for low-sym- 1 + t/th metry centers), which affects the exponent p of the defect concentration decay kinetics in the asymptotic where n0 is the initial defect concentration and th is their limit. half-life. As known from the theory of DCR [23], t = h The temperature dependence of the TR kinetics 1/(Kn ), where K is the DCR rate constant: 0 parameter a3N observed by us (Fig. 5) unambiguously R implies the existence of a thermally stimulated process, K = 4πR D 1 + ------, (7) 0 Rπ which affects the tunneling recombination of defects DRt and accounts for the TR kinetics being temperature- 3 where R0 is the radius of the recombination sphere and dependent. The kinetics parameter a N contains a component that influences the electron tunneling transfer DR is the diffusion coefficient. We use now the known data on the ionic electrical conductivity σ(T) of LTB probability (one half the Bohr radius a) and the initial [24–26] and Einstein’s relation to estimate t : concentration N of dominant defects. The initial con- h centration of the color centers produced by an electron 2 e Ð1 beam is known to be usually temperature-independent t ≈ ------()σT , (8) h 4πR k [29]. Indeed, from an examination of Fig. 4, one may 0 b conclude that the TOA decay curves measured in LTB where kb is the Boltzmann constant. Assuming R0 = at various temperatures should converge to the same ≈ σ ≈ × Ð6 Ω Ð1 value of the optical density as t 0. 5 Å, we obtain th 0.03 s for 3.5 10 ( cm) at 300 K. As follows from an analysis of Fig. 2, this time Let us estimate the possible decrease in the concen- corresponds to the extreme points of the time interval of tration of one of the tunneling pair components due to our measurements, where the optical density decreases the diffusion-controlled annihilation with mobile by a factor of roughly 20. It can be maintained that the defects of the third type. Among them are primarily assumed diffusion-controlled annihilation of defects defects on the lithium sublattice. It is known [20] that

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TRANSIENT OPTICAL ABSORPTION AND LUMINESCENCE 1091 ∝ បω in LTB at room temperature, the lithium ions can modes. Assuming a exp(Ð /kbT), we obtain from migrate along the spontaneous-polarization direction (c Eq. (4) an estimate បω = 0.16/3 and 0.10/3 eV. IR opti- axis). The ionic electrical conductivity of the LTB crys- cal and Raman scattering spectroscopy measurements tal was studied in [25, 26], and the activation energies of LTB show [30Ð32] that the vibrational modes of lith- of lithium ion migration along the polar axis were ium ions surrounded by four oxygen ions lie at 35Ð found to be 0.54 eV at 350Ð390 K and 0.39 eV at 390Ð 75 meV. Our estimate of បω is in agreement with this 530 K. These energies exceed, by a few times, the acti- interval. vation energies of the TOA kinetics parameter, 0.16 and A comprehensive comparison of the TOA and PCL 0.10 eV, determined by us. Furthermore, the above esti- decay laws shows that the PCL decay kinetics cannot mates of t show that diffusion-controlled defect anni- h be described by a derivative of the optical-density hilation involving lithium ions can become manifest at decay with respect to time, which, in turn, is propor- room temperature only for comparatively long decay tional to the color-center concentration. This means that times. This is at odds with the results presented in the tunneling transport of electrons among the main Fig. 4. Therefore, we believe that the observed temper- 3 defects governing the TOA is nonradiative and that it ature dependence of the product a N (Fig. 5) is due to does not produce resonance (or close to resonance) its first component (a3), which governs the probability conditions for tunneling electron transport from a of the electron tunneling transfer. defect to the excited state of a self-trapped exciton Let us discuss possible mechanisms which could (STE) located nearby. In this connection, we believe the account for the temperature dependence of the electron PCL and TOA in LTB to be governed by different relax- tunneling transport probability in LTB. The totality of ation processes. the experimental data amassed in this study gives us The main PCL band of LTB at 3.6 eV is close in its grounds to suggest that one of the partners in the tun- characteristics to the fast intrinsic luminescence band neling recombination could be a lithium sublattice of LTB [15], which originates from radiative annihila- defect (for instance, the lithium vacancy). In this case, tion of the STE. The decay kinetics under photoexcita- diffusive transport of lithium cations along the polar tion of this band is described by a component with τ ≈ axis should affect the defect distribution function in dis- 1 ns [15]; however, pumping by unfiltered synchrotron Ӷ tance even for t th. This should result, in particular, in radiation in the x-ray range at 290 K revealed exponen- an increase in the number of closely separated pairs tial components with time constants of 10 and 150 ns and, hence, in an increase in the tunneling transport [33]. This compares favorably with the PCL exponen- probability with increasing temperature. It is evident, tial decay components with τ = 30 and 110 ns observed however, that because of the intricate chain of causal in this study (Figs. 7, 9). relations, the slope of the asymptotic exponent p plot- The slow hyperbolic PCL decay component with a ted in the Arrhenius coordinates can no longer be iden- characteristic time of 340 ns and an asymptotic expo- tified with the activation energy of the original diffusion nent of 1.6 is apparently of another nature. This is sug- process. At the same time, a qualitative relation should gested, in particular, by its degree of polarization hold. Indeed, an analysis of the temperature depen- (Fig. 8). We believe that the major part of the LTB PCL dence of exponent p in Fig. 4 revealed two temperature is due to interdefect tunneling recombination. It is intervals differing in slope and a break at 395 K. The known [34] that one-site small polarons can act as same temperature intervals (I, II) and a trend of short-lived nuclei in the self-trapping of holes, exci- decreasing slope in the crossover from the low-temper- tons, or electronÐhole pairs. Tunneling transport of an ature linear part I for T < 390 K to the high-temperature + electron from a deep electronic level to an excited level part II are characteristic of the ionic (Li ) electrical con- of a hole polaron migrating near this trapping center ductivity of LTB along the polar axis [24Ð26]. A differ- may create an STE in the σ state. Radiative annihilation ence is observed only in the slope of the linear portions of such an STE accounts for the characteristic σ-polar- plotted in the Arrhenius coordinates; namely, the slope ized luminescence of LTB. Indeed, the observed PCL of the linear portions I and II in the temperature depen- of LTB is σ polarized to about 50%. The kinetics of this dence of the electrical conductivity exceeds by about luminescence is rate-limited by the tunneling recharg- fourfold that for the asymptotic exponent p. We note ing in the deep-electronic-center and hole polaron pair. that the nature of this inflection in the temperature We note for comparison that a similar holeÐelectron dependence of electric transport in LTB is of a funda- mechanism of tunneling electron transport from an Ag0 mental character; indeed, above 390 K, the correlation deep trapping center to the excited state of a V center effects acting between mobile lithium ions result in k their cooperative migration along the polar axis of the migrating in its vicinity has been recently detected crystal, thus reducing the activation energy [25, 26]. experimentally in the RbCl : Ag alkali halide crystal [35]. One of the possible candidates for a deep elec- On the other hand, one cannot disregard the possi- tronic center in LTB can be an F center. An indirect bility that the observed temperature dependence of the indication of this is the presence of a rapidly decaying tunneling probability originates from the interaction of band at 3.36 eV in the PCL spectrum. Earlier, we a tunneling electron with local or collective vibrational observed luminescence bands of F-like centers in a

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1092 OGORODNIKOV et al. similar crystal, LBO, at 3.3 and 2.7 eV [36]. It is known 19. E. F. Dolzhenkova, M. F. Dubovik, A. V. Tolmachev, [37, 38], that recombination luminescence, including et al., Pis’ma Zh. Tekh. Fiz. 25 (17), 78 (1999) [Tech. one that is thermally stimulated at 400Ð500 K, in irra- Phys. Lett. 25, 709 (1999)]. diated LTB crystals with impurities which do not pro- 20. S. Matyyasik and Yu. V. Shaldin, Fiz. Tverd. Tela duce activated luminescence bands is represented by (St. Petersburg) 43 (8), 1405 (2001) [Phys. Solid State the same dominant band at 3.38 eV. A conclusion was 43, 1464 (2001)]. drawn that this luminescence center is of intrinsic 21. A. É. Aliev, V. F. Krivorotov, and P. K. Khabibulaev, Fiz. nature and that it is related to F-type centers created in Tverd. Tela (St. Petersburg) 39 (9), 1548 (1997) [Phys. an LTB lattice irradiated by neutrons or electrons with Solid State 39, 1378 (1997)]. energies E = 1.25Ð1.3 MeV [37]. These defects are sta- 22. I. N. Ogorodnikov, A. V. Porotnikov, S. V. Kudyakov, ble at room temperature, and their annealing or carrier et al., Fiz. Tverd. Tela (St. Petersburg) 39 (9), 1535 delocalization are assumed to be responsible for the (1997) [Phys. Solid State 39, 1366 (1997)]. thermally stimulated luminescence peaking at 418 K 23. Yu. R. Zakis, L. N. Kantorovich, E. A. Kotomin, (0.94 eV) and 479 K (1.80 eV) [38]. V. N. Kuzovkov, I. A. Tale, and A. L. Shlyuger, Models of Processes in Wide-Gap Solids with Defects (Zinatne, REFERENCES Riga, 1991). 24. A. É. Aliev and R. R. Valetov, Kristallograﬁya 36 (6), 1. R. Komatsu, T. Sugawara, K. Sassa, et al., Appl. Phys. 1507 (1991) [Sov. Phys. Crystallogr. 36, 855 (1991)]. Lett. 70 (26), 3492 (1997). 2. J. D. Garret, M. Natarajan, and J. E. Greedan, J. Cryst. 25. A. É. Aliev, I. N. Kholmanov, and P. K. Khabibullaev, Growth 41, 225 (1977). Solid State Ionics 118 (1Ð2), 111 (1999). 3. M. Adachi, T. Shiosaki, and A. Kawabata, Jpn. J. Appl. 26. A. É. Aliev, I. N. Kholmanov, and P. K. Khabibullaev, Phys., Part 1 24 (S3), 72 (1985). Dokl. Akad. Nauk 365 (2), 178 (1999) [Dokl. Phys. 44, 4. M. Adachi, S. Yamamichi, M. Ohira, et al., Jpn. J. Appl. 147 (1999)]. Phys., Part 1 28 (S28-2), 111 (1989). 27. Ch. B. Lushchik and A. Ch. Lushchik, Decay of Electron 5. T. Kitagawa, K. Higuchi, and K. Kodaira, J. Ceram. Soc. Excitations with Defect Formation in Solids (Nauka, Jpn. 105, 616 (1997). Moscow, 1989). 6. T. P. Balakireva, V. V. Lebold, V. A. Nefedov, et al., 28. U. T. Rogulis and I. K. Vitol, in Electron Processes and Inorg. Mater. 25 (3), 462 (1989). Defects in Ionic Crystals (Latv. Univ. im. P. Stuchki, 7. S. J. Fan, G. S. Shen, W. Wang, et al., J. Cryst. Growth Riga, 1985), pp. 22Ð23. 99 (1Ð4), 811 (1990). 29. A. K. Pikaev, Modern Radiation Chemistry. Solid and 8. K. Byrappa, V. Rajeev, V. J. Hanumesh, et al., J. Mater. Polymers. Applied Aspects (Nauka, Moscow, 1987). Res. 11 (10), 2616 (1996). 30. Ya. V. Burak, Ya. O. Dovgiœ, and I. V. Kityk, Zh. Prikl. 9. J. Krogh-Moe, Acta Crystallogr. 15 (3), 190 (1962). Spektrosk. 52 (1), 126 (1989). 10. J. Krogh-Moe, Acta Crystallogr. B 24 (2), 179 (1968). 31. V. N. Moiseenko, A. V. Vdovin, and Ya. V. Burak, Opt. 11. A. É. Aliev, Ya. V. Burak, and I. T. Lyseœko, Izv. Akad. Spektrosk. 81 (4), 620 (1996) [Opt. Spectrosc. 81, 565 Nauk SSSR, Neorg. Mater. 26 (9), 1991 (1990). (1996)]. 12. S. F. Radaev, N. I. Sorokin, and V. I. Simonov, Fiz. Tverd. Tela (Leningrad) 33 (12), 3597 (1991) [Sov. 32. A. V. Vdovin, V. N. Moiseenko, and Ya. V. Burak, Opt. Phys. Solid State 33, 2024 (1991)]. Spektrosk. 90 (4), 625 (2001) [Opt. Spectrosc. 90, 555 (2001)]. 13. A. O. Matkovskiœ, D. Yu. Sugak, S. B. Ubizskiœ, O. I. Shpotyuk, E. A. Chernyœ, N. M. Vakiv, and 33. I. N. Ogorodnikov, V. A. Pustovarov, L. I. Isaenko, et al., V. A. Mokritskiœ, Inﬂuence of Ionizing Radiations on Nucl. Instrum. Methods Phys. Res. A 448 (1Ð2), 467 Electronics Materials (Svit, L’vov, 1994). (2000). 14. Ya. V. Burak, B. N. Kopko, I. T. Lyseœko, et al., Izv. 34. S. Iwai, T. Tokizaki, A. Nakamura, et al., Phys. Rev. Lett. Akad. Nauk SSSR, Neorg. Mater. 25 (7), 1226 (1989). 76 (10), 1691 (1996). 15. I. N. Ogorodnikov, V. A. Pustovarov, A. V. Kruzhalov, 35. E. A. Vasil’chenko, I. A. Kudryavtseva, A. Ch. Lushchik, et al., Fiz. Tverd. Tela (St. Petersburg) 42 (3), 454 (2000) et al., Fiz. Tverd. Tela (St. Petersburg) 40 (7), 1238 [Phys. Solid State 42, 464 (2000)]. (1998) [Phys. Solid State 40, 1128 (1998)]. 16. A. Yu. Kuznetsov, L. I. Isaenko, A. V. Kruzhalov, et al., 36. I. N. Ogorodnikov, V. A. Pustovarov, M. Kirm, et al., Fiz. Fiz. Tverd. Tela (St. Petersburg) 41 (1), 57 (1999) [Phys. Tverd. Tela (St. Petersburg) 43 (8), 1396 (2001) [Phys. Solid State 41, 48 (1999)]. Solid State 43, 1454 (2001)]. 17. B. P. Gritsenko, V. Yu. Yakovlev, and Yu. N. Safonov, in Proceedings of All-Union Conference on Modern State 37. M. Martini, C. Furetta, C. Sanipoli, et al., Radiat. Eff. and Advanced Aspects of High-Speed Photography, Cin- Defects Solids 135 (1Ð4), 133 (1995). ematography, and Metrology of Fast Processes, Mos- 38. M. Martini, F. Meinardi, L. Kovács, and K. Polgar, cow, 1978, p. 61. Radiat. Prot. Dosim. 65 (1Ð4), 343 (1996). 18. V. N. Parmon, R. F. Khaœrutdinov, and K. I. Zamaraev, Fiz. Tverd. Tela (Leningrad) 16 (9), 2572 (1974) [Sov. Phys. Solid State 16, 1672 (1974)]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1093Ð1097. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1048Ð1052. Original Russian Text Copyright © 2002 by Bedilov, Beisembaeva, Davletov.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Effect of g-Radiation-Induced Defects in Glass on Laser Destruction M. R. Bedilov, Kh. B. Beisembaeva, and I. Yu. Davletov Research Institute of Applied Physics, National University of Uzbekistan, Tashkent, 700174 Uzbekistan e-mail: [email protected] Received June 7, 2001

Abstract—Processes of degradation and plasma ejection upon exposure of optical materials to a single laser pulse and γ-radiation in the subthreshold, threshold, and far-above-threshold regions are studied using a complex method. © 2002 MAIK “Nauka/Interperiodica”.

The progress in the development of high-power microscopic methods [15]. This allowed us to analyze solid-state lasers, nonlinear optical media, and ele- not only the morphology of laser-induced degradation ments and devices in thermonuclear fusion reactors in the threshold and far-above-threshold regions of operating in extreme conditions is determined to a con- laser power density but also the ion component of the siderable extent by the development of radiation-resis- plasma formed as a result of degradation of glass, as tant optical materials. It has been established that the well as to determine the composition of the target and production of such materials depends mainly on knowl- to find out which elements facilitate the destruction. In edge of the mechanisms of their degradation under the addition, the analysis of the results obtained by using action of a high-power laser beam and nuclear radia- the above methods revealed a correlation between the tion. It is well known that the surface of solids (includ- surface degradation of the solid and the characteristics ing transparent insulators) is destroyed under the action of multiply charged ions emitted from the solid under of laser radiation after the attainment of a certain den- the action of a high-power luminous flux. sity of the incident luminous energy. Despite the large The targets (silicate glasses of the GLS type) were number of publications devoted to the analysis of laser- prepared in the form of pellets ~2 mm thick and 10 mm induced degradation of the surface of optical materials in diameter. Radiation-induced defects were created by [1Ð14], its relation to the imperfection of the structure holding samples in the channel of a γ-radiation source of the solid remains unclear [15]. It has been estab- with a power of 1500 R/s until the samples received a lished that the presence of various dynamic (molecular dose of 109 R. A laser pulse of duration 50 ns and power vibrations, density and concentration fluctuations, etc.) 60 MW was focused on the surface of a target in the and static (foreign impurities and inclusions) optical form of a spot ~250 µm in diameter. In order to deter- inhomogeneities in transparent dielectrics facilitates mine the radiation-damage threshold accompanied in the occurrence of various nonlinear effects (including the given case by degradation of the optical material, self-focusing), which lowers the radiation resistance of luminescence, and ejection of ionized mass, laser flares the optical material. with a successive increase in the incident radiation In the present work, we study the effect of γ-ray- intensity were used. The instant of the beginning of induced defects in glass on laser degradation under the degradation was fixed microscopically from the emer- action of a laser beam with a power density q = 0.1Ð gence of ion peaks. These peaks were registered by a 1000 GW/cm2. Experiments were made in a wide range VEU-1A detector, whose signal was fed to a storage of laser power densities embracing the subthreshold oscilloscope. Mass-, charge-, and energy separation of (108Ð109 W/cm2), threshold (109Ð4 × 109 W/cm2, ion components was carried out by using a mass spec- depending on γ-ray-irradiance), and far-above-thresh- trometer. The relative error of measurements of the old (4 × 109Ð1012 W/cm2) regions of optical-material amplitude of ion signals did not exceed ~8%. degradation. It should be noted that the experiments As a result of the experiments, we obtained informa- were carried out with single laser pulse irradiation of tion on the degradation of silicate glass in the sub- the objects under investigation and, hence, the accumu- threshold, threshold, and far-above-threshold regions in lation effect was not manifested in the subthreshold the case of a single interaction of laser radiation pulse region. In this case, the process of laser-induced degra- with optical materials and on the formation of multiply dation is connected with the formation of a dense high- charged ions of plasma in wide ranges of laser power temperature plasma. For this reason, the experiments densities and γ-radiation doses. The microscopic inves- were made using the mass-spectroscopic and optical tigations revealed that the destruction of an unexposed

1094 BEDILOV et al.

0.6 tion, we investigated the absorption spectra of the given 2 samples in the UV, visible, and IR regions. It was established experimentally that as the γ-radiation dose 0.4 increases, absorption in the regions under investigation becomes stronger. For example, for a dose of 109 R, the

, mm 1 d 0.2 absorption of the given glass increases by 16% in the visible spectral region and by 7% in the region of incident laser radiation. It follows, hence, that the amount 0 of luminous energy absorbed in the sample exposed to 109 1010 1011 γ 2 radiation is larger than that in the sample not sub- q, W/cm jected to irradiation, which increases the thickness of the material being evaporated. On the other hand, the Fig. 1. Dependence of the diameter d of a crater formed on emergence of a huge number of defects in γ-ray-irradi- the surface of (1) the initial and (2) γ-ray-irradiated glass to 9 ated samples absorbing in the UV and visible regions a dose of 10 R on the laser radiation power density. causes an increase in the diameter of the appearing crater due to additional heating by radiation emitted by the target produced in the threshold region has the form of multiply charged laser-produced plasma. a crater with fused brims containing small-sized defects A comparison of the experimental results obtained in the form of indentations having a depth of the order during investigation of various multiply charged ions of a tenth of a micrometer or less. The reason for the emitted by the plasma proved that the number N, charge emergence of such microcraters in the threshold region multiplicity Z, and energy E of the ions increase with is apparently associated with individual impurity inclu- the diameter of the crater formed on the unexposed sur- sions and optical inhomogeneities in the sample, lead- face. For example, in the far-above-threshold region, ing to the absorption of laser radiation at local centers where q = 20 GW/cm2, the crater diameter attains val- [4]. The magnitude of the laser-induced degradation µ threshold of the surface of a glass of the GLS type, ues of ~150 m. It was found that the formed plasma which was determined by using the above technique, contains, in its composition, all the elements constitut- 2 ing glass (Li, O, Na, Si, K, and Nd), as well as uncon- amounted to ~4 GW/cm under the given experimental trollable impurities (H, Be, B, C, etc.). The maximum conditions. The crater diameter d in this case was µ charge multiplicities for ions forming the glass matrix ~50 m. As the laser power density q increases, the cra- obtained in this case are as follows: Z = 4 for the Si ter diameter becomes larger and attains the value max ~300 µm in the far-above-threshold region for q = ions, Zmax = 3 for the Nd and O ions, and Zmax = 2 for 1000 GW/cm2 (Fig. 1). There exists the following the Li, Na, and K ions. All ion components were characterized by a broad energy spectrum with a single dis- dependence between the power density q and the crater +1 +1 0.4 tribution peak. The maximum energy for the Li , O , diameter: d ~ q . The defects induced in the glasses +1 +1 +1 +1 under investigation by γ radiation lower the damage Na , Si , K , and Nd ions was 300, 400, 800, 900, threshold. For example, the degradation threshold for a 950, and 2500 eV, respectively (Fig. 2). As the power 9 density q in the far-above-threshold region increased to GLS-type glass receiving a radiation dose of 10 R 2 2 90 GW/cm (the corresponding crater diameter was amounts to ~1 GW/cm under the given experimental µ conditions; i.e., it was equal to 1/4 of the threshold prior 280 m), Li and K ions with Zmax = 3, Si and Nd ions to irradiation. The diameter of the crater emerging on with Zmax = 4, and O and Na ions with Zmax = 5 were such an irradiated surface for q = 1 GW/cm2 amounts to detected in the plasma formed (see Fig. 3 for ~100 µm. With increasing radiation dose, the diameter 20 GW/cm2 and Table 2 for 90 GW/cm2). The maximal of the degradation crater increases signiﬁcantly and for energy Emax increases in this case by a factor of 3.0Ð3.5 q ≥ 1 GW/cm2, the glass surface experiences cata- for Li+1, O+1, and K1+ ions and by a factor of 2.9Ð2.5 for strophic breakdown. In this case, the size of the crater Na+1, Si+1, and Nd+1 ions. The intensity of the ion on the exposed surface is connected with the power beams also increases considerably. For example, an density q through the relation d ~ q0.55 (Fig. 1). Morpho- increase in the power density q of the laser from 20 to logical analysis of the breakdown pattern proved that, 90 GW/cm2 leads to a 1.5Ð2.0 fold increase in the num- after the action of γ radiation, shallow caverns existing bers of Li, O, Na, and Si ions of all charge multiplici- in the initial crater coalesce into large damaged regions ties, while the intensity of multiply charged ions of K whose size may attain several tens of micrometers. It and Nd increases by a factor of 4Ð5. The increase in the was found that the destruction begins in small isolated values of parameters of multiply charged ions of the regions containing pile-ups of absorbing defects. In our plasma upon an increase in the value of q can be opinion, the role of such defects may be played by for- explained by the fact that, as the diameter of the crater eign impurities and inclusions present in the glass increases, the plasma attains its critical density more before irradiation and the regions of accumulation of rapidly and, hence, a larger part of laser radiation power γ-ray-induced defects. In order to verify this assump- is spent for heating and ionization of the plasma cluster.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

EFFECT OF γ-RADIATION-INDUCED DEFECTS IN GLASS 1095

At the same time, it was established experimentally 10–1 that γ-ray-irradiation of the samples under investigation 2 (a) considerably changes the parameters of multiply charged ions constituting the plasma in the threshold 3 and far-above-threshold regions. It should be noted that γ 1 as the -radiation dose becomes larger, the ion intensity –2 increases with the crater diameter, while the values of 10 4 Zmax and Emax of the multiply charged ions thereby decrease (Figs. 2Ð4 and Tables 1, 2). It should be 6 emphasized that the extent of variation of the character- 5 istics (N, Z, E) of multiply charged ions under investigation are determined to a considerable extent by the 10–3 power density q of the laser. It can be seen from Tables 1 500 1000 1500 2000 2500 , rel. units –1 10 1

and 2 that a less significant change in the values of N dE (b) and E of ions is observed when q ≥ 10 GW/cm2 (which / 2 is an order of magnitude higher than the damage thresh- dN 3 old). This may be due to partial annealing of γ-ray- induced defects (for large values of q). The strong 4 –2 change in the characteristics of multiply charged ions 10 of the plasma formed as a result of degradation of a 5 γ-ray-irradiated target can be explained by the violation of the relation between the initial size d of the plasma 6 and the characteristic recombination length l(Z). This is clearly manifested in an analysis of the dependence 10–3 between the diameter of the crater formed on the sur- 500 1000 1500 2000 face of the γ-ray-irradiated target and the parameters of E, eV multiply charged ions emitted by the plasma. It is found Fig. 2. Typical energy spectra of singly charged ions (1) that an increase in the size of the crater (the value of d) +1 +1 +1 +1 +1 +1 on the given surface upon a transition from the thresh- Li , (2) O , (3) Si , (4) Na , (5) K , and (6) Nd in the plasma of (a) the initial and (b) γ-ray-irradiated GLS-1 old to the far-above-threshold region of degradation of 9 2 the optical material leads to a considerable change in glass with a radiation dose of 10 R for q = 20 GW/cm . the characteristics of the ejected plasma. In this case, the values of Z and E of the plasma ions decrease sharply, while the number of low-charge ions increases. (a) An increase in γ-radiation dose intensifies the degrada- O4+ Si4+ Si2+ tion of the optical material and leads to the ejection of O5+ Li+ Si3+ O+ a low-charge high-density laser plasma. Li3+ Li2+O3+ O2+ Na2+ K2+ Na+Si+ K+ The degradation of the surface of optical materials was analyzed by three independent (microscopic, spectroscopic, and mass-spectrometric) methods for the O3+ Si4+ Na3+ Na2+ (b) 8 12 2 laser power density q = 10 Ð10 W/cm . We experi- Na4+ Si3+ Si2+ O+ Na+ mentally established the subthreshold, threshold, and 4+ 5+ 2+ 3+ 2+ + + 2+ far-above-threshold regions of optical material degra- O Na O K K Si K Nd dation induced by laser and nuclear radiation. Also established were the form of destruction and the defects and elements responsible for the beginning of degradation in the vicinity of the damage threshold for optical materials and, especially, for the formation of a high- density multi-elemental low-charge ion plasma in the Fig. 3. Typical massÐcharge spectra of ions in the plasma of far-above-threshold region of degradation of a γ-ray- GLS-1 glass formed as a result of action of laser radiation irradiated material. The breakdown threshold in exper- with q = 20 GW/cm2. The ion energy E/Z is equal to (a) 200 iments with the unexposed object under investigation and (b) 400 eV. was ~4 × 109 W/cm2; after γ-ray-irradiation, this value decreased by a factor of four. It should be noted that experiments with multiple irradiation of optical materi- ing the degradation of the material under investigation als in the subthreshold laser power region, in which the depending on the number of laser pulses “fired” at the accumulation effect is manifested, are of certain inter- same spot (the laser power density q was 5 × 108 W/cm2, est. We specifically carried out experiments for detect- which corresponds to the subthreshold region). It was

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1096 BEDILOV et al.

Table 1. Effect of γ radiation on the characteristics of multiply charged ions detected for q = 20 GW/cm2 Elements Characteristic Dose, R Li O Na Si K Nd

Zmax 023242 3 109 12122 2

Emax, eV 0 300 400 800 900 950 2500 (Z = 1) 109 150 300 500 550 600 2000 dN ------, rel. units 0 2.6 6.0 0.8 2.8 0.4 0.2 dE max (Z = 1) 109 10.0 6.5 2.0 3.0 1.0 0.1

Table 2. Effect of γ radiation on the characteristics of multiply charged ions detected for q = 90 GW/cm2 Elements Characteristic Dose, R Li O Na Si K Nd

Zmax 0 355434 109 343434

Emax, eV 0 800 1400 2000 2200 3000 3900 (Z = 1) 109 600 1200 2000 2100 3000 3900 dN ------, rel. units 0 3.7 9.0 2.5 10.0 2.0 1.0 dE max (Z = 1) 109 3.9 10.0 4.2 10.0 4.0 1.0 found that the destruction of the surface takes place methods used (especially the microscopic and mass- after 12 laser shots owing to the accumulation effect. spectrometric methods) are sensitive to the accumula- The form and size of the defects observed in the sub- tion effect and provide information on the absorption threshold region are close to the results obtained in the spectra, the pattern of material degradation, and the threshold range of q values. It should be noted that the damage threshold, as well as information on the mass, charge, and energy composition of products after each action of laser and nuclear radiation on the optical- 4+ 3+ 3+ 2+ + (a) material surface. The methods of investigation used by Si Na Si Si O us supplemented one another and allowed us to study O4+ O3+ O2+ Na2+ K2+ Na+Si+ K+ the degradation of the surface of materials not only in the threshold and far-above-threshold regions but also in the subthreshold range of q values. A comparison of the degradation of optical materi- (b) 3+ 3+ 2+ 2+ + + als and the ejected charged particles in the threshold Na Si Na Si O Na and far-above-threshold regions of q values proved that Si4+ O2+ K3+ K2+ Si+ K+ Nd2+ these regions differ in the type and extent of surface degradation of the material and in the character of the charged-particle ejection from the damaged region of the target. Indeed, the character of degradation of optical materials in the subthreshold, the threshold, and far- above-threshold regions is different; the information concerning this difference is essential for establishing Fig. 4. Typical massÐcharge spectra of ions in the plasma of the mechanism of material degradation in a wide range GLS-1 glass subjected to the action of γ radiation with a of power densities during a single act of exposure to the dose of 109 R formed by laser radiation with q = luminous ﬂux. All these processes are interrelated; con- 20 GW/cm2. The ion energy E/Z is equal to (a) 200 and sequently, the optical-material degradation mechanism (b) 400 eV. becomes more complicated as the power density of the

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EFFECT OF γ-RADIATION-INDUCED DEFECTS IN GLASS 1097 laser increases from 108 to 1012 W/cm2. The dominating REFERENCES process in the subthreshold region is associated with 1. R. Abdupataev, M. R. Bedilov, Kh. B. Beœsembaeva, and the accumulation effect, which ultimately leads to the P. K. Khabibullaev, Dokl. Akad. Nauk SSSR 286 (4), degradation of the optical material. In the threshold 857 (1986) [Sov. Phys. Dokl. 31, 130 (1986)]. region, the processes determining the material degrada- 2. N. G. Basov, Yu. A. Zakharenkov, A. A. Rupasov, tion are thermal heating, melting, evaporation, and G. V. Sklizkov, and A. S. Shikanov, Diagnostics of Dense Plasma (Nauka, Moscow, 1989). emission of particles. In the far-above-threshold region, 3. M. R. Bedilov, P. K. Khabibullaev, and Kh. B. Beœsem- the leading role is played by laser-induced thermal baeva, Radiation-enhanced Processes in Solid-State explosion, resulting in the formation of craters and mul- Lasers (Fan, Tashkent, 1988). tielemental plasma containing various ions in a wide 4. Z. T. Azamatov, P. A. Arsen’ev, M. R. Bedilov, range of charge multiplicity and energy. The results of Kh. S. Bogdasarov, A. A. Evdokimov, and V. M. Tsikha- mass-spectroscopic measurements showed that differ- novich, Defects in Materials for Quantum Electronics ent defects and elements of the matrix of the material (Fan, Tashkent, 1991). are responsible for the degradation of the optical mate- 5. N. G. Basov, Yu. A. Zakharenkov, N. N. Zorev, A. A. Rupasov, G. V. Sklizkov, and A. S. Shikanov, rial, depending on the laser power density. In the sub- Advances in Science and Technology (VINITI, Moscow, threshold region, radiation defects and uncontrollable 1982), Vol. 26, Parts 1, 2, p. 304. impurities play the major role when the accumulation 6. M. R. Bedilov and A. N. Ishmuratov, Fiz. Tverd. Tela effect occurs; in the threshold region, radiation defects (St. Petersburg) 38 (6), 1649 (1996) [Phys. Solid State and some elements of the matrix are of importance, 38, 911 (1996)]. while in the far-above-threshold region, the degradation 7. V. V. Artem’ev, A. M. Bonch-Bruevich, I. E. Morigev, is associated with all defects and elements constituting et al., Zh. Tekh. Fiz. 47 (1), 183 (1977) [Sov. Phys. Tech. the optical material. In our opinion, the three stages of Phys. 22, 106 (1977)]. 8. B. M. Ashkinadze, V. I. Vladimirov, V. A. Likhachev, and degradation (subthreshold, threshold, and far-above- I. A. Ivanov, Zh. Éksp. Teor. Fiz. 50 (5), 1187 (1966) threshold) are manifested consecutively in the break- [Sov. Phys. JETP 23, 788 (1966)]. down of a laser system as a result of interaction of a 9. P. I. Bal’kyavichyus, E. K. Kostenko, I. P. Lukoshyus, bell-shaped laser pulse with optical materials. In the and É. K. Maldutis, Kvantovaya Élektron. (Moscow) 5 far-above-threshold region of material degradation, (9), 2032 (1978). various processes occur during a single laser pulse: 10. A. M. Bonch-Bruevich, I. V. Aleshin, Ya. A. Imas, and from the accumulation effect to a laser-induced thermal A.V. Pavshukov, Zh. Tekh. Fiz. 41 (3), 617 (1971) [Sov. explosion with the formation of a crater and the ejection Phys. Tech. Phys. 16, 479 (1971)]. 11. I. M. Buzhinskiœ and A. E. Pozdnyakov, Kvantovaya Éle- of multielemental low-charge ion plasma. It should be ktron. (Moscow) 2 (7), 1550 (1975). noted in conclusion that the results obtained in this 12. Yu. K. Donileœko, A. A. Manenkov, and V. S. Nechitaœlo, work on the type and extent of damage depending on Kvantovaya Élektron. (Moscow) 5 (7), 194 (1978). γ the -radiation dose, on the form of degradation in the 13. Yu. I. Kyzylasov, V. S. Starunov, and I. A. Fabelinskiœ, three regions of the q values, and on the formation of a Fiz. Tverd. Tela (Leningrad) 12 (1), 233 (1970) [Sov. plasma with known mass, charge, and energy parame- Phys. Solid State 12, 186 (1970)]. ters (especially in the threshold and far-above-threshold 14. A. G. Molchanov, Fiz. Tverd. Tela (Leningrad) 12 (3), stages of degradation of the optical material) are useful 954 (1970) [Sov. Phys. Solid State 12, 749 (1970)]. in developing high-power solid-state lasers and in 15. M. R. Bedilov, Kh. B. Beœsembaeva, and M. S. Sabitov, establishing the reasons for the failure of optical laser Kvantovaya Élektron. (Moscow) 30 (2), 201 (2000). systems under the action of high-power laser and nuclear radiation. Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1098Ð1101. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1053Ð1056. Original Russian Text Copyright © 2002 by Oleœnich-Lysyuk. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

The Influence of the Degree of Instability of a System on the Oscillating Behavior of Time Dependences of the Internal Friction A. V. Oleœnich-Lysyuk Chernovtsy National University, Chernovtsy, 58012 Ukraine Received November 13, 2001

Abstract—The speciﬁc features of the oscillations observed in the time dependences of the internal friction of polycrystalline beryllium irradiated with high-energy electrons and aged at T = 300 K within different time intervals are investigated. It is demonstrated that the ageing leads to a gradual decrease in the amplitude and the period of oscillations. The assumption is made that the observed phenomenon has a synergetic nature. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION plates by the electroerosion method with subsequent elimination of the deformed layer. In recent years, considerable research attention has been focused on systems in which a disordered chaotic According to the chemical analysis, the content of motion of the elements involved transforms into an impurities in the condensate was as follows: 0.009Ð ordered motion under the action of external excitations. 0.05 wt % Fe, 0.003Ð0.14 wt % Al, 0.0037Ð0.04 wt % This crossover occurs through self-organization pro- C, 0.06Ð0.15 wt % O, 0.003Ð0.042 wt % Ga, and 0.01−0.017 wt % N. The electrical resistivity ratio cesses of nonequilibrium structures, such as the ρ ρ BelousovÐZhabotinsky reactions in chemistry, the 295 K/ 4.2 K for different samples was in the range 190Ð Benard cells in hydrodynamics, the ordering of mag- 220, which corresponds to an estimated integral purity netic domains in the physics of magnetic phenomena, of better than 99.95 wt % Be. the formation of slip lines and bands and the formation The condensate had a columnar structure with a 40- of cellular and block dislocation structures in the phys- to 70-µm mean cross section of grains. The length of ics of strength and plasticity, etc. [1]. Among similar the grains was equal to (0.2Ð1)d, where d = 1.5Ð3 mm phenomena are oscillating time variations in the inter- is the thickness of the condensate plate. nal friction observed in a number of systems after The internal friction was measured on a low-fre- cyclic deformations [2, 3]. However, the nature of oscil- quency relaxometer of the inverse torsion pendulum lations of the internal friction and the causes of their type at frequencies ~1 sÐ1 according to standard tech- occurrence still remain unclear. The goal of the present niques [2Ð4]. This made it possible to carry out the work was to investigate how the degree of instability of experiment with an accuracy of better than 1% and to the studied system affects the manifestation of the record reliably the oscillating time dependences of the internal friction oscillations. internal friction. The time dependences of the internal friction were measured at strains γ = 9 × 10Ð6 ± 1 × 10Ð7 both after the cyclic deformation of the sample at a con- 2. SAMPLE PREPARATION γ stant strain amplitude 0 for 5 min followed by a sharp AND EXPERIMENTAL TECHNIQUE decrease in the strain to the aforementioned values and The MTBK-1.5 Be samples irradiated with high- without cyclic deformation after the measurement of energy (~18 MeV) electrons and aged within different the amplitude dependence of the internal friction. time intervals at room temperature in air were chosen as the object of investigation, because, as was established 3. RESULTS AND DISCUSSION experimentally in [4], irradiation with high-energy electrons leads to the appearance of clearly deﬁned Figure 1 shows the time dependences of the internal oscillations in the time dependences of the internal fric- friction of the beryllium samples which were irradiated tion. The samples to be studied were prepared from and then aged within six months at room temperature plates produced through the condensation of beryllium both after cyclic deformation for 5 min at stain ampli- γ × Ð5 × Ð5 × Ð5 from the vapor phase onto substrates at temperatures in tudes 0 = 1.4 10 , 2.5 10 , and 6.6 10 and with- the range 500Ð600°C. Samples in the form of parallel- out cyclic deformation after measurement of the ampli- γ × epipeds 1 × 1 × 80 mm in size were cut from these tude dependence of the internal friction to max = 6.6

THE INFLUENCE OF THE DEGREE OF INSTABILITY OF A SYSTEM 1099

10Ð5. A comparison of the data presented in Fig. 1 and 200 (a) the results obtained immediately after irradiation of the 190 sample revealed the following features: (1) natural age- 180 ing of the sample leads to the appearance of clearly 0 10 20 30 deﬁned oscillations with a 10- to 11-min period at 200 strains which did not induce oscillations in the as-irra- (b) diated material (cf. Fig. 1a of this work and Fig. 2a 195 given in [4]), and (2) the time dependences of the inter- 190 γ 185 nal friction at large strain amplitudes 0 exhibit quite a –4 0 10 20 30 different behavior. , 10

An increase in the ageing time to ten months leads to –1 195 (c) γ × Ð5 Q a speciﬁc localization of oscillations at 0 = 2.2 10 . However, their amplitude and period are less than those 175 observed after ageing within six months. At large strain γ 150 amplitudes 0, the oscillations lose their periodicity. For γ × 0 10 20 30 example, after cyclic deformation for 5 min at 0 = 3.5 Ð5 10 , a shadow-type burst appears against the back- 195 (d) ground of weak variations in the internal friction at the 185 17th minute of the measurements, whereas after the γ × Ð5 175 cyclic deformation for 5 min at 0 = 6.6 10 , the oscil- 0 10 20 30 lations arises only after a lapse of some time, more pre- t, min cisely, at the 15thÐ30th minutes of observation. Fig. 1. Time dependences of the internal friction of the In this case, a pronounced temperature hysteresis is MTBK Be samples aged within six months after cyclic observed in the temperature dependences of the internal γ × Ð5 × Ð5 deformation for 5 min at 0 = (a) 1.4 10 , (b) 2.5 10 , friction and the square of the frequency of free torsional Ð5 2 and (c) 6.6 × 10 and (d) after measurement of the ampli- vibrations f (which is proportional to the effective γ × tude dependence of the internal friction up to max = 6.6 shear modulus Geff) of the samples aged within both six Ð5 and ten months: the curves of heating and cooling do 10 . not coincide over a rather wide range of temperatures (Fig. 2). This suggests that a weakly pinned disloca- 300 tion-impurity structure of beryllium undergoes a trans- (a) formation in the course of heating (cooling) due to 1 appreciable internal stresses [3, 4]. –4 200 , 10

Ageing of the samples within 12 months leads to a –1 100 decrease in the amplitude and the period of oscillations Q over virtually the entire range of the studied strains 2 (Fig. 3). In this case, the defect structure of the samples 0 100 200 300 400 is characterized by a higher stability (Fig. 4), as judged T, °C from the absence of speciﬁc anomalies in the tempera- 1.06 2 (b) ture dependences of the internal friction and f 2 [3]: the –2 1.02

absorption of the elastic energy weakly varies with tem- , s 2 perature and corresponds to the background absorption f 0.98 over the entire temperature range covered; moreover, 1 0.94 the curves f 2(T) do not exhibit a temperature hystere- 0 100 200 300 400 sis. Analysis of the internal friction background and the T, °C temperature behavior of the amplitude dependence of the internal friction indicates a substantial stabilization Fig. 2. Temperature dependences of (a) the internal friction of the beryllium defect structure. and (b) f 2 for the MTBK Be samples after ageing within six months: (1) heating and (2) cooling. Therefore, in the course of natural ageing, the stabilization of the dislocation-impurity structure, whose equilibrium was disturbed under irradiation, brings about a gradual decrease and appreciable suppression can be applied to the possible mechanism of this phe- of the oscillating behavior of the time dependence of nomenon. To the best of my knowledge, the periodic the internal friction. This corroborates our assumption increase and decrease in the absorption of the elastic that the oscillations of the internal friction actually energy have deﬁed description in the framework of cur- occur only in a nonequilibrium system under the action rently available models of time dependences of the of external factors. On this basis, some considerations internal friction that are based on simple notions of

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1100 OLEŒNICH-LYSYUK inhomogeneities in dislocation-impurity structures [1]. 100 (a) 90 Under external stresses, the evolution of these structures can occur through the generation of inhomoge- 80 0 10 20 30 neous density fluctuations pinning the dislocation of defects. In this case, proper allowance must be made for 95 the fact that the probability of forming the aforemen- (b) tioned fluctuations along the dislocations is substan- 85 tially larger than the probability of generating similar –4 75 fluctuations in perpendicular directions, because the 0 10 20 30 , 10 mobility of pinning defects along the dislocation pipes –1

Q 95 considerably exceeds the bulk mobility. This is con- (c) 85 firmed by our estimates of the dislocation pipe diffusion coefficients Dd for the basic types of impurities in 75 the MTBK Be material, the comparison of the esti- 0 10 20 30 mated values with the volume diffusion coefficient DV, 95 (d) and the binding energy of the dislocation with pinning 90 defects Eb (for example, Dd for Fe in MTBK Be at the 85 experimental temperatures was estimated at 5.29 × −13 2 × Ð31 2 80 10 m /s, DV = 1.25 10 m /s, and Eb = 0.056 eV). 0 10 20 30 Moreover, the density fluctuations pinning the disloca- t, min tion of defects, according to [1, 5], become possible upon the attainment of a critical density. Reasoning Fig. 3. Time dependences of the internal friction of the from the above considerations, we can explain the MTBK Be samples aged within 12 months at room temper- absence of oscillations at γ = 1.4 × 10Ð5 immediately γ × Ð5 0 ature after cyclic deformation for 5 min at 0 = (a) 2.2 10 , after irradiation and their presence in the time depen- Ð5 Ð5 (b) 3.5 × 10 , and (c) 6.6 × 10 and (d) after measurement dences of the internal friction after ageing within six of the amplitude dependence of the internal friction up to γ × Ð5 months. In actual fact, ageing is attended by the return max = 12 10 . of some part of the impurities to the dislocations. This is supported by the analysis of the amplitude dependences of the internal friction and the calculation of the

–4 200 1 (a) binding energy of the dislocation with pinning defects.

, 10 100 2 As a consequence, clearly deﬁned oscillations appear in

–1 the time dependences of the internal friction of the aged Q γ × 0 100 200 300 400 samples after cyclic deformation for 5 min at 0 = 1.4 T, °C 10Ð5 (Fig. 1a). Further ageing brings about an increase 1.06 (b) in the number of impurities at dislocations and stabili- –2 zation of the dislocation-impurity system. As a result, , s

2 1.02 f 1 the most distinct oscillations manifest themselves only 2 γ × Ð5 0.98 after cyclic deformation for 5 min at 0 = 2.2 10 , i.e., 0 100 200 300 400 after the formation of an optimum amount of impurities T, °C involved in the dislocations. However, their amplitude and period are substantially less than those observed Fig. 4. Temperature dependences of (a) the internal friction 2 after ageing within six months. Hence, the stabilization and (b) f for the MTBK Be samples aged within of the system, as a whole, can lead to the fact that the 12 months: (1) heating and (2) cooling. optimum conditions for the generation of oscillations will not be achieved at any strain. This assumption is interaction between dislocations and the dislocation conﬁrmed to some extent by the ﬁrst publications con- environment. In my opinion, the phenomenon under cerning the oscillations revealed in the time depen- consideration can be adequately described only in the dences of the internal friction [2]. Strongin and Yakov- framework of the concepts regarding the evolution of a ishin [2] observed the oscillations in the AlÐAg system at a content of 0.01 wt % Ag after quenching and did dislocation-impurity ensemble and the development of not reveal oscillations at any other content of impurities collective and cooperative phenomena in this ensemble and in the annealed samples. In my opinion, all the [1, 5]. Indeed, consideration of the kinetic nature of the foregoing counts in favor of the assumption that the interaction between dislocations and the dislocation basic conditions of the occurrence of internal friction environment destabilized under irradiation can provide oscillations are as follows: (i) an optimum amount of an explanation for both the loss of spatial stability by a impurities in the region of dislocations and (ii) a certain dislocation-impurity ensemble and the formation of instability of the system due to either the formation of

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 THE INFLUENCE OF THE DEGREE OF INSTABILITY OF A SYSTEM 1101 a substructure or quenching, irradiation, or some other REFERENCES factor. Therefore, under certain conditions, oscillations of the internal friction can be considered to be an indi- 1. G. A. Malygin, Usp. Fiz. Nauk 169 (9), 979 (1999). cation of the instability of the system. 2. B. G. Strongin and P. A. Yakovishin, Fiz. Khim. Obrab. Mater., No. 4, 123 (1982). 3. A. V. Oleœnich, N. D. Raranskiœ, and B. G. Strongin, 4. CONCLUSION Metalloﬁz. Noveœshie Tekhnol. 16 (4), 47 (1994). Thus, it was demonstrated that the oscillations in 4. A. V. Oleœnich, B. G. Strongin, N. D. Raranskiœ, et al., time dependences of the internal friction can appear Metalloﬁz. Noveœshie Tekhnol. 19 (1), 62 (1997). only under at least two conditions: (1) an optimum 5. G. A. Malygin, Fiz. Tverd. Tela (Leningrad) 33 (11), amount of impurities in the region of dislocations and 3267 (1991) [Sov. Phys. Solid State 33, 1844 (1991)]. (2) an instability of the dislocation-impurity system, as a whole, which allows one to assign this phenomenon to the synergetic type. Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1102Ð1105. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1057Ð1059. Original Russian Text Copyright © 2002 by Fedorov, Kuranova, Tyalin, Pluzhnikov.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Effect of Dislocation Distribution in Twin Boundaries on the Nucleation of Microcracks at the Twin Tip V. A. Fedorov, V. A. Kuranova, Yu. I. Tyalin, and S. N. Pluzhnikov Derzhavin State University, Tambov, 392622 Russia e-mail: [email protected] Received September 4, 2001

Abstract—The effect of dislocation distribution in the boundaries of an arrested twin on the nucleation of microcracks at its tip is investigated. The twin is simulated by a double step pileup (cluster) of twinning dislocations located in adjacent slip planes. The equilibrium equations for dislocations are solved numerically. Clus- ters with different total numbers of dislocations and with different ratios of the numbers of dislocations at the upper and lower twin boundaries are considered. The formation of microcracks as a result of coalescence of head dislocations according to the force and thermally activated mechanisms is analyzed. The equilibrium con- ﬁgurations of a single twin boundary and of the twin are calculated. It is found that the condition for microcrack formation at the twin tip considerably depends on the ratio of the numbers of dislocations in twin boundaries. In the limit, this condition coincides with the condition of crack formation at the tip of a single twin boundary with the same total number of dislocations. It is shown that thermally activated formation of a microrack corresponds to lower values of the critical stress. © 2002 MAIK “Nauka/Interperiodica”.

It was shown in [1, 2] that the actual structure of dis- for each boundary. The equilibrium equations for the location pileups may considerably change the condi- upper boundary with n1 dislocations have the form tions for the formation of microcracks in them. From n this point of view, twins and twin boundaries wherein 1 ()x Ð x 2 Ð ()y Ð y 2 each twinning dislocation moves in its own slip plane ∑ ()x Ð x ------i j i j i j []()2 ()2 2 are of special interest. Such defects are usually simu- j = 1, xi Ð x j + yi Ð y j lated by step dislocation clusters [3, 4], the dislocations ji≠ being distributed in pairs symmetrically relative to the n twinning plane passing through the twin tip. Obviously, 2 ()x Ð x 2 Ð ()y + y 2 in the general case, twin boundaries may contain differ- + ∑ ()x Ð x ------i j i j (1) i j []()2 ()2 2 ent numbers of dislocations. In the present work, we j = 2 xi Ð x j + yi + y j consider the conditions for the initiation of microcracks at the tip of an arrested twin with different ratios of τ Ð ------==0, i 23,,… ,n, twinning dislocations in its boundaries in calcite. Db The boundary of a twin is simulated by a solitary where xi and yi are the coordinates of the ith disloca- step pileup of twinning dislocations, each of which is tion, D = G/[2π(1 Ð ν)], G is the shear modulus (3.2 × displaced relative to a neighboring dislocation by a dis- 1010 N/m2), ν is the Poisson ratio (ν = 0.3), and b is the tance equal to the separation h between the planes Burgers vector for twinning dislocations (1.269 × × Ð10 (3.82 10 m). The twin is represented by a double 10−10 m). The ﬁrst and second terms describe the inter- step pileup of twinning dislocations. We assume that action of the ith dislocation with dislocations from the the head dislocation is ﬁxed at a point with coordinates upper and lower boundary, respectively. x = y = 0 and belongs simultaneously to the upper and lower boundaries. The dislocation pileup is pressed The equations for dislocations in the lower bound- against the head dislocation by an external stress τ. ary have a similar form,

Let us consider twin boundaries containing different n 2 ()2 ()2 numbers (n , n ) of dislocations. Formally, this means ()xi Ð x j Ð yi Ð y j 1 2 ∑ xi Ð x j ------that each boundary must be analyzed separately and the []()2 ()2 2 j = 1, xi Ð x j + yi Ð y j equilibrium equations for dislocations must be written ji≠

EFFECT OF DISLOCATION DISTRIBUTION IN TWIN BOUNDARIES 1103

n1 2 2 20 ()x Ð x Ð ()y + y 2 + ∑ ()x Ð x ------i j i j (2) i j []()2 ()2 2 j = 2 xi Ð x j + yi + y j 10 τ Ð ------==0, i 23,,… ,n. 1

h 0 Db / y Thus, we obtain a system of nonlinear equations (1) and –10 3 (2) with the number of unknowns n1 + n2 Ð 1. The numerical solution of Eqs. (1) and (2) by the method of successive approximations [5] gives the equilibrium –20 coordinates xi of dislocations, which are functions of 0 0.025 0.050 0.075 0.100 the elastic constants G and b, of the numbers of dislo- µ τ x, m cations n1 and n2, and of the external stress . From the standpoint of microcrack initiation, the distance d Fig. 1. Distribution of dislocations at the tip of defects: (1) between the head dislocations whose coalescence leads AT; (2) TB, and (3) ST. to the formation of a crack nucleus is of interest to us. The mechanisms of coalescence of head dislocations can be divided into force mechanisms and thermally aries, an asymmetric twin (AT) with different numbers activated mechanisms. of dislocations in the boundaries, and a single twin In a planar pileup, the coalescence of head disloca- boundary (TB). We considered pileups with different tions occurs when they approach each other to a dis- total numbers n of dislocations and with different ratios tance d = b [6]. In a step pileup, head dislocations expe- of the numbers n1 and n2 of dislocations in the upper rience coalescence after approaching each other to the and lower boundaries of the AT. It should be noted that dislocations in the ST boundaries are arranged in pairs critical distance dc1 = 2.41h. The force with which the first dislocation repels the second one attains its maxi- (Fig. 1). In other words, the values of coordinates xi for mum value in this case. The subsequent approaching of the ith dislocations of the upper and lower twin bound- dislocations until they merge into one occurs without aries coincide, while the coordinates yi are equal in an increase in the external stress. Consequently, we can magnitude but have opposite signs. This result is obvi- τ ous and is a consequence of the interaction of disloca- assume that the critical stresses cr required for the nucleation of cracks are equal to the stresses required tions moving in parallel slip planes. If we assume that for head dislocations approaching each other to a dis- one of the dislocations is stationary, the equilibrium state of the other dislocation corresponds to two posi- tance dcr. This criterion will be referred to as the force criterion. tions, namely, x = h and x = 0, in which the force of interaction is equal to zero, the stable position being the In the case of thermally activated nucleation of one with x = 0. In the case of deviation from this posi- microcracks, we assume that head dislocations experi- tion, a moving dislocation will experience the action of ence coalescence not simultaneously over their entire a force returning it to the equilibrium position. length, but initially only over a short segment as a result of ejection of a double kink by the second dislocation If the equality of the numbers of dislocations in the of the cluster due to thermal fluctuations. A microcrack ST boundaries is violated, i.e., the ST is converted into nucleus of length l that is formed upon the coalescence an AT, the arrangement of dislocations changes consid- of this kink and the first dislocation subsequently erably (Fig. 1). However, pairwise arrangement of dis- expands over the entire dislocation length. It was shown locations in the AT boundaries is also encountered, but in [7] that the energy barrier for the crack nucleation is the total number of such pairs does not exceed a few completely determined by the first stage of the process, percent. We denote by d the distance between the head viz., the formation of a double kink. An expression for and second dislocations in the twin boundary. The val- the energy W of the double-kink formation is given in ues of d for the upper and lower AT boundaries are the [3]. smaller, the larger the number of dislocations in the τ boundary, and the values of d for both AD boundaries In order to determine the critical external stress cr, may differ by a factor of several units for an insignifi- we calculated the dependence of W on τ and determined τ cant difference between n1 and n2. For example, for n1 = the value of for which the value of W coincides with 10 and n = 12, the ratio d /d ≈ 4. Moreover, the a preset value. In the case under investigation, this 2 1 2 smaller of d1 and d2 turns out to be much smaller than energy was chosen equal to 1 eV, which is comparable the value of d for an ST. This can be clearly seen from to the minimum height of the potential barrier for the Fig. 1, which shows the distribution of dislocations formation of a double kink and amounts to a value of 3 directly at the cluster tips. If we take into account the the order of Gb [8]. fact that it is the value of d that determines the crack In our calculations, we analyzed a symmetric twin nucleation stresses, we can expect the crack formation (ST) with equal numbers of dislocations in the bound- conditions to change at the AT tip.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1104 FEDOROV et al.

8 cence). For the remaining dislocations, with larger num- ӷ bers, we have xi Ð xj h. For example, (xi Ð xj)/h > 10 for 7 3 2 n > 10; i.e., in this case, adjacent dislocations and, even more so, dislocations with a larger difference in the indices interact as if they are located in the same plane. 6 Indeed, we can disregard, to a high degree of accuracy, the terms (y Ð y )2 in Eq. (3). In other words, the tail

h i j / 5 d part in step clusters can be replaced by a planar configuration of dislocations and the shape of the defect tip, 4 as well as the coalescence conditions for head dislocations, is determined by the interaction of a small num- 3 ber of head dislocations for which ∆y = |y Ð y | is com- 1 ∆ | | i j parable to x = xi Ð xj . 2 Figure 2 shows the calculated dependences of d on 1.0 1.5 2.0 2.5 3.0 3.5 the external stress τ for three types of clusters: TB, ST, τ, 10–7 N/m2 and AT. It can be seen that the dependences for TB and AT are quite similar. This is a consequence of the Fig. 2. Dependence of the distance between head disloca- above-mentioned coincidence of equilibrium positions tions on the applied stress for various dislocation clusters: of dislocations in TB and in AT boundaries; i.e., TB can (1) TB, n = 50; (2) ST, n1 = n2 = 25; and (3) AT, n1 = 20, ӷ be regarded as a limiting case of AT for which n1 n2 n2 = 30. ӷ (or n2 n1). By comparing the values of τ for AT and ST (with It is worth noting that in the distribution of disloca- equal total numbers of dislocations in the boundaries), tions at the tip of a twin, the values of coordinates of we find that, for d = 2.41h, the crack nucleation accord- dislocations at the AT boundaries are close to the values ing to the force mechanism in AT takes place at much of the dislocation coordinates in a single boundary with lower stresses (by a factor of approximately 1.7). the total number of dislocations n = n1 + n2. This result We tried to find out whether the obtained result will become clear if we consider the expression for the depends on the number of dislocations in the clusters stresses exerted by the jth dislocation on the ith disloca- under investigation. If we plot the dependence d = f(τ) tion, (or of W on τ) in relative units d = f(τn/D), the obtained 2 2 2 2 points corresponding to different values of n for the Gb ()x Ð x []()x Ð x Ð ()y Ð y τ = ------i j i j i j -. (3) same cluster fit into the same curve to a high degree of ij π()ν 2 2 2 2 1 Ð []()x Ð x + ()y Ð y accuracy. By way of example, Fig. 3 shows a curve i j i j describing such a dependence for an AT. Thus, the For adjacent dislocations, we have yi Ð yj = h and xi Ð xj is results presented in Fig. 2 can be generalized to the case comparable with h only for dislocations adjoining the of other values of n by simple renormalization of the head of the pileup (x2 Ð x1 = 2.41h in the case of coales- critical stress.

9 2.2

8 2 1.8 3 7 1 1.4 6 2 h / , eV d 5 W 1.0 4 0.6 1 3

2 0.2 0.08 0.10 0.12 0.14 0.16 1.2 1.6 2.0 2.4 2.8 3.2 3.6 –7 2 τn/D τ, 10 N/m

Fig. 3. Dependence of the distance between head disloca- Fig. 4. Dependence of the double-kink energy W on the tions on the applied stress for an AT: (1) n1 = 10, n2 = 12; external stress: (1) TB, n = 50; (2) ST, n1 = n2 = 25; and and (2) n1 = 20, n2 = 30. (3) AT, n1 = 20, n2 = 30.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EFFECT OF DISLOCATION DISTRIBUTION IN TWIN BOUNDARIES 1105

The calculated activation energy for the crack nucle- 3. V. A. Fedorov and Yu. I. Tyalin, Kristallografiya 26 (4), ation is shown in Fig. 4. In comparing these results with 775 (1981) [Sov. Phys. Crystallogr. 26, 439 (1981)]. the data presented in Fig. 2, we see that the thermally 4. V. A. Fedorov, V. M. Finkel’, V. P. Plotnikov, et al., Kri- activated nucleation corresponds to lower values of stallografiya 33 (5), 1244 (1988) [Sov. Phys. Crystallogr. critical stress, but the difference is not large (~25%). 33, 737 (1988)]. The values of double-kink energy amount to ~0.5 eV 5. J. Ortega and W. Rheinboldt, Iterative Solution of Non- provided that the force criterion d = 2.41h is satisfied. linear Equations in Several Variables (McGraw-Hill, New York, 1970; Mir, Moscow, 1975). REFERENCES 6. A. N. Stroh, Adv. Phys. 6 (24), 418 (1957). 7. V. I. Vladimirov, Physical Nature of Metal Fracture 1. V. N. Rybin and Sh. K. Khannanov, Fiz. Tverd. Tela (Metallurgiya, Moscow, 1984). (Leningrad) 11 (4), 1048 (1969) [Sov. Phys. Solid State 11, 854 (1969)]. 8. A. N. Orlov, Introduction to the Theory of Defects in 2. V. I. Vladimirov and Sh. Kh. Khannanov, Fiz. Met. Met- Crystals (Vysshaya Shkola, Moscow, 1983). alloved. 31 (4), 838 (1971). Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1106Ð1110. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1060Ð1063. Original Russian Text Copyright © 2002 by Mulyukov, Sharipov, Korznikova.

MAGNETISM AND FERROELECTRICITY

The Influence of Crystalline Grain Size on the Thermal Stability of the Er0.45Ho0.55Fe2 Compound Kh. Ya. Mulyukov, I. Z. Sharipov, and G. F. Korznikova Institute of Metal Superplasticity Problems, Russian Academy of Sciences, ul. Khalturina 39, Ufa, 450001 Bashkortostan, Russia e-mail: [email protected] Received July 13, 2001

Abstract—The phase composition and the temperature dependence of the magnetization of the Er0.45Ho0.55Fe2 compound in coarse-grained, microcrystalline, and submicrocrystalline states are investigated experimentally. It is found that, upon heating under vacuum, the Er0.45Ho0.55Fe2 microcrystalline powder with a crystalline grain size of ~1 µm undergoes decomposition into pure iron and rare-earth (erbium and holmium) oxides and nitrides at a temperature of 500 K. The changes observed in the phase composition of the microcrystalline powder due to annealing are conﬁrmed by x-ray diffraction analysis. Heating of the Er0.45Ho0.55Fe2 submicrocrystalline sample leads to a partial change in the phase composition. The phase composition of a large crystal (~1 mm in size) remains unchanged upon heating to 1080 K. It is shown that the thermal stability of the Er0.45Ho0.55Fe2 compound depends on the crystalline grain size. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. SAMPLE PREPARATION AND EXPERIMENTAL TECHNIQUE Earlier investigations into the magnetic properties of FeÐNdÐB submicrocrystalline alloys prepared by crys- A coarse-grained sample in the form of a parallelepiped with a maximum linear size of 1 mm was cut tallization of rapidly quenched amorphous ribbons from an ingot of the Er Ho Fe compound. A sam- revealed that pure iron precipitates in the course of 0.45 0.55 2 ple with a microcrystalline structure was prepared by annealing [1, 2]. The phase instability of multicompo- grinding in an agate mortar because of the high brittle- nent compounds has also been observed in other stud- ness of the chosen material. In order to ensure against ies. In particular, Vereshchagin et al. [3] and Absalya- oxidation, the powder was ground in ethanol. The mov and Mulyukov [4] observed transformations of microcrystalline powder with a particle size of ~1 µm higher metal oxides into lower metal oxides and reduc- was separated through sedimentation. The size of pow- tion of metal oxides to pure metal during vacuum der particles was determined on a JSM-840 scanning annealing of microcrystalline powders. Yermakov [5] electron microscope. Samples with a submicrocrystal- described the phase separation upon treatment of alloy line structure were obtained under the conditions of powders in a ball mill. In these works, the phase insta- severe plastic deformation with the use of Bridgman bility is attributed to the inﬂuence of severe plastic anvils under a pressure of 10 GPa at room temperature. deformation on the crystal structure. The question now arises as to why a similar phenomenon is observed dur- The temperature dependence of the magnetization ing annealing of a multicomponent compound with a of the samples under investigation was measured using submicrocrystalline structure formed through crystalli- a vacuum automated magnetic balance [6] in the tem- zation of the amorphous alloy. perature range from 80 to 1080 K. The phase composition of the samples was analyzed on a DRON-3M auto- With the aim of obtaining additional information on mated x-ray diffractometer with computer processing the phase instability of multicomponent compounds, of the results obtained. we analyzed the phase composition and the tempera- The microstructure of the submicrocrystalline sam- ture dependence of the magnetization of the ples was examined using a JEM 2000EX transmission Er0.45Ho0.55Fe2 compound in coarse-grained, microc- electron microscope. rystalline, and submicrocrystalline states. The choice of the Er0.45Ho0.55Fe2 compound as the object of investigation was made for the following reasons: (i) this com- 3. EXPERIMENTAL RESULTS pound is a ferrimagnet containing iron, and (ii) the tem- Figure 1 shows the temperature dependence of the perature dependence of the magnetization can serve as magnetization σ(T) for a coarse-grained sample. It can a highly sensitive indicator of changes in the phase be seen from Fig. 1 that, upon heating, the magnetiza- composition of the studied sample. tion of the sample decreases drastically and becomes

THE INFLUENCE OF CRYSTALLINE GRAIN SIZE ON THE THERMAL STABILITY 1107

200 200 2

150 150 1 100 100 , arb. units , arb. units σ σ 50 50

0 200 400 600 800 1000 0 200 400 600 800 1000 T, K T, K

Fig. 1. Temperature dependence of the magnetization of a Fig. 2. Temperature dependence of the magnetization of a coarse-grained sample. microcrystalline powder. equal to zero at 570 K. This dependence is character- which was measured during cooling of the submicroc- ized by an abrupt transition from a ferrimagnetic state rystalline sample (see Fig. 3), does not coincide with to a paramagnetic state. The curves σ(T) measured dur- the cooling curve of the microcrystalline powder ing heating and cooling of the sample are virtually iden- (Fig. 2, curve 2) and consists of two portions. The low- tical. This confirms the assumption that the phase com- temperature portion virtually follows the dependence position of the coarse-grained sample remains stable up σ(T) measured during heating of the sample, whereas to 1080 K. the high-temperature portion is similar to the dependence σ(T) for pure 3d ferromagnets. The temperature dependence of the magnetization σ(T) for a microcrystalline sample is depicted in Fig. 2. The x-ray diffraction pattern of the microcrystalline It is seen from this figure that the dependence σ(T) powder immediately after grinding is displayed in obtained upon heating of the microcrystalline sample Fig. 4a. All the peaks revealed in the x-ray diffraction (Fig. 2, curve 1) differs substantially from the heating pattern correspond to the main phase of the curve of the coarse-grained sample (Fig. 1). As the tem- Er0.45Ho0.55Fe2 compound. The x-ray diffraction pattern perature increases, the magnetization first decreases, as of the same powder annealed at 1080 K exhibits many is the case with the coarse-grained sample, but does not additional peaks (Fig. 4b). The new phases formed reach zero and begins to increase at temperatures above upon annealing were identified according to the angular 500 K. With a further increase in the temperature, the positions of additional peaks. It is found that heating of magnetization passes through a maximum at 830 K, the microcrystalline powder of the studied compound decreases, and becomes equal to zero at 1033 K. leads to the formation of pure iron and different phases Another feature in the dependence σ(T) of the microc- of rare-earth compounds. The additional peaks observed rystalline sample is that curve 2, which was measured in the x-ray diffraction pattern of the annealed powder during cooling of the sample, differs noticeably from (Fig. 4b) correspond to the following phases: Ho2O2 curve 1, which was recorded during heating. Note that and Er2O3 (lines 2, 6, 15, 16, and 19Ð21), ErO2 (lines 3, curve 2 is similar to the temperature dependence of the 16, and 21), Ho2N3 (lines 2, 6, 15, 16, and 19Ð21), HoN saturation magnetization of pure 3d ferromagnets. Moreover, the Curie temperature determined by the extrapolation of the steepest portion in curve 2 to the temperature axis coincides with the reverence Curie 200 point of pure iron. Figure 3 shows the dependence σ(T) for a submicro- 150 crystalline sample upon heating and cooling. In this case also, curve 1, which was measured during heating 100 2 of the submicrocrystalline sample to 600 K, coincides , arb. units σ with the heating curve for the coarse-grained sample. 1 At higher temperatures, the magnetization of the sub- 50 microcrystalline sample increases, passes through a maximum at 950 K, and decreases to zero at 1033 K. It 0 200 400 600 800 1000 is worth noting that the increase in the magnetization of T, K the submicrocrystalline sample at temperatures above 600 K (Fig. 3, curve 1) is less pronounced than that of Fig. 3. Temperature dependence of the magnetization of a the microcrystalline powder (Fig. 2, curve 1). Curve 2, submicrocrystalline sample.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1108 MULYUKOV et al.

(a)

(b)

7 11 13 22 23 24 27

1 2 12 25 3 10 20 26 9 45 15 6 8 16 18 14 17 19 21

20 30 40 50 60 7080 90 2θ, deg

Fig. 4. X-ray powder diffraction pattern of a microcrystalline sample (a) prior to and (b) after heating to 1080 K.

(lines 4 and 9), Fe4N (lines 10, 16, 19, and 20), HoFe8 ically, upon heating of the microcrystalline and submi- (line 9), Er2Fe17 (lines 3, 9, 19, and 20), and Fe (lines 14 crocrystalline samples above 500 K, their magnetiza- and 16). tion increases and passes through a maximum. This The electron microscopic investigation revealed that increase in the magnetization can be associated with the the structure of the submicrocrystalline sample consists precipitation of a new ferromagnetic phase with a high of crystalline grains with a mean size of less than Curie temperature. The curve σ(T) measured during 100 nm (Fig. 5). As can be seen from the electron cooling of the microcrystalline sample corresponds to microscope image of the submicrocrystalline sample, the precipitated phase. Judging from this curve, the pre- the crystalline grains have diffuse boundaries and cipitated phase consists of pure iron, because the tem- involve cells with a size of the order of 10 nm. perature at which the magnetization of the sample becomes equal to zero coincides with the Curie point of iron. 4. DISCUSSION In order to conﬁrm the above inference, which was The experimental results demonstrate that the tem- made on the basis of magnetic measurements, the phase perature dependences of the magnetization of samples composition of the microcrystalline sample was addi- in different structural states differ signiﬁcantly. Specif- tionally investigated using x-ray powder diffraction. As

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 THE INFLUENCE OF CRYSTALLINE GRAIN SIZE ON THE THERMAL STABILITY 1109 was noted above, the diffraction pattern in Fig. 4a corresponds to the main phase of the Er0.45Ho0.55Fe2 alloy. Consequently, the phase composition of the studied compound remains unchanged in the course of grinding. The x-ray diffraction pattern of the annealed powder exhibits sharp peaks assigned to the new phases (Fig. 4b). It should be noted that this diffraction pattern also contains maxima associated with the main phase; however, no indications of the main phase are revealed in the temperature dependence of the magnetization measured during cooling of the microcrystalline sample below 1080 K. The matter is that x-ray powder diffraction experiments require a sufficiently large amount of the studied compound. For this reason, the sample was prepared from the whole of the ground powder without separation into fractions with different particle sizes. As a consequence, upon heating of the sample intended for x-ray diffraction measurements, the compound undergoes decomposition within small-sized crystalline grains of the powder, whereas the main phase is retained in large-sized crystalline grains. On the other hand, the presence of the maxima associated with the main phase in the x-ray diffraction pattern is convenient for the following reasons: first, these max- × 200 nm ima serve as reference points, and, second, they clearly 40 K demonstrate that no decomposition of the compound occurs in larger sized particles. Fig. 5. Electron microscope image of the submicrocrystal- Thus, the magnetic measurements made it possible line structure of the studied compound. to reveal only pure iron precipitated upon heating of the microcrystalline sample, whereas x-ray powder diffraction analysis identified both the pure iron phase and residual gas virtually coincides with that of the coarse- other phases formed in the course of decomposition of grained sample. However, no indications of oxidation the studied compound. are observed after heating of the coarse-grained sample This raises the following question: Why does pure to 1070 K (the magnetization was measured accurate to iron precipitate during heating of the microcrystalline within 1%), whereas the heating of the submicrocrys- sample? A close examination of the x-ray diffraction talline sample under the same conditions is accompa- pattern of the annealed microcrystalline sample shows nied by the decomposition of the studied compound that the diffraction maxima revealed after the annealing with the precipitation of pure iron in considerable are attributed to oxides and nitrides of erbium and hol- amounts. mium in different valent states. Consequently, rare- earth metal ions within a crystalline grain exhibit an Analysis of the temperature dependences of the increased reactivity and interact with a residual gas magnetization σ(T) (Figs. 2, 3) indicates that, in the (heating was performed under vacuum at a residual submicrocrystalline sample, unlike the microcrystal- pressure of 1.33 × 10Ð2 Pa). One reason for this reactiv- line sample, the studied compound undergoes partial ity is that the microcrystalline powder possesses a well- decomposition. This fact demonstrates the role played developed surface. Hence, rare-earth metals in contact by the free surface in the interaction with a residual gas. with a residual gas actively interact with oxygen and The partial decomposition of the submicrocrystalline nitrogen. However, under the same conditions of mea- sample can be associated with the specific features of surements, similar interactions in the coarse-grained the submicrocrystalline structure, because this structure sample are not observed. is characterized by a substantial volume fraction of For the purpose of elucidating the influence of the crystalline grain boundaries and a high density of crystalline grain size on the reactivity of rare-earth defects (Fig. 5). It is known that the coefficient of grain metal ions, we carried out a similar investigation of the boundary diffusion in nanocrystalline materials is sev- sample with a submicrocrystalline structure. Since eral orders of magnitude larger than that in coarse- severe plastic deformation of the material with the use grained materials [7]. Most likely, it is this circum- of Bridgman anvils results in the formation of a mono- stance that encourages diffusion of oxygen and nitro- lithic sample, the surface area of its contact with a gen atoms into the submicrocrystalline sample and

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1110 MULYUKOV et al. results in the formation of iron and rare-earth metal REFERENCES compounds with oxygen and nitrogen. 1. T. Miyazaki, H. Takada, and M. Takahashi, Phys. Status Solidi A 99, 611 (1987). 5. CONCLUSION 2. Kh. Ya. Mulyukov, R. Z. Valiev, G. F. Korznikova, and V. V. Stoljarov, Phys. Status Solidi A 112, 137 (1989). Thus, our investigation has demonstrated that the 3. L. F. Vereshchagin, E. V. Zubova, K. P. Budrina, and thermal stability of the Er0.45Ho0.55Fe2 compound G. L. Aparnikov, Dokl. Akad. Nauk SSSR 196 (4), 817 depends on the crystalline grain size. (1971). 4. S. S. Absalyamov and Kh. Ya. Mulyukov, Dokl. Akad. Nauk 375 (4), 469 (2000) [Dokl. Phys. 45, 657 (2000)]. ACKNOWLEDGMENTS 5. A. Ye. Yermakov, Mater. Sci. Forum 179Ð181, 455 This work was supported by the Russian Foundation (1995). for Basic Research (project no. 00-02-17723) and the 6. Kh. Ya. Mulyukov, I. Z. Sharipov, and S. S. Absalyamov, State Program of Support for Leading Scientiﬁc Prib. Tekh. Éksp., No. 3, 149 (1998). Schools of the Russian Federation (project no. 00-15- 7. H. Gleiter, Nanostruct. Mater. 1, 1 (1992). 99093). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1111Ð1116. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1064Ð1069. Original Russian Text Copyright © 2002 by Gulyaev, Zil’berman, Elliott, Épshteœn. MAGNETISM AND FERROELECTRICITY

Magnetostatic Energy and Stripe Domain Structure in a Ferromagnetic Plate of Finite Width with In-Plane Anisotropy Yu. V. Gulyaev*, P. E. Zil’berman*, R. J. Elliott**, and É. M. Épshteœn* * Institute of Radio Engineering and Electronics, Russian Academy of Sciences, pl. Vvedenskogo 1, Fryazino, Moscow oblast, 141120 Russia ** University of Oxford, Department of Physics, 1 Keble Road, Oxford, OX1 3NP United Kingdom Received July 30, 2001

Abstract—The magnetostatic energy and domain structure (DS) in a long ferromagnetic plate of a finite width with in-plane anisotropy are calculated for the case of the domain magnetization vectors lying in the plane of the plate. The situation where the DS period is much shorter than the width but is considerably larger than the thickness of the plate is analyzed in detail. The equilibrium DS period and the width ratio of two adjacent domains are determined as functions of an external magnetic field parallel to the plane of the plate by minimizing the energy. The DS period is found to be proportional to the plate width and the domain wall energy and inversely proportional to the squared saturation magnetization. While the width of the favorable domains (with the magnetization parallel to the field) grows with increasing field, the unfavorable domains, rather than disappearing completely, form relatively narrow transition regions between the favorable domains, i.e., 360° domain walls. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION if it forms, depends to a large extent on the magnetic Calculations of the magnetostatic energy of a charges on the small-area end faces. The DS does form domain structure (DS) in ferromagnets have a long his- in such films, which was confirmed experimentally by tory dating back to the works of Landau and Lifshitz [1] various methods, for instance, by observing the Fara- and Kittel [2]. The most widely used model considered day effect in visualization media applied to a film [8] in many monographs and textbooks (see, e.g., [3Ð5]) is and by transmission electron microscopy [9Ð11]. The a ferromagnetic plate (film) of finite thickness with the most copious information was obtained in studies [9Ð easy axis perpendicular to the large plate face. In this 11], which reported on the observation of a DS with model, the DS is represented by an array of alternating nearly parallel stripes having curved walls. The stripe antiparallel magnetized stripes of equal width whose width was on the order of a few microns and tens of magnetization is perpendicular to the large face. Fol- microns, with the magnetization lying in the film plane. lowing [6], we shall call these formations Faraday The theory of such Cotton domains (using the termi- domains. The width of a Faraday domain is propor- nology of [6]) has not been well elaborated, which tional to the square root of the plate thickness. Applica- impedes experimental interpretation. The experimen- tion of an external magnetic field along the easy axis tally established properties of Cotton domains contrast increases both the domain structure period and the rel- with those of Faraday domains. At the same time, while ative width of the favorable domains with the magneti- the existing theories (see, e.g., [6, 12]) provide a gen- zation parallel to the field. By contrast, the width of the eral basis for analysis, they are not sufficient to interpret unfavorable domains tends to zero. In a sufficiently these differences directly. Their interpretation requires strong magnetic field, the unfavorable domains disap- further progress in theory; this work is an attempt along pear and the favorable ones merge to form a single- these lines. domain structure. The theory of Faraday domains is presently a subject of renewed interest, which was We will show that the dependence of domain width apparently stimulated by the application of bubble on the geometric parameters of the plate, magnetiza- domains in information storage systems. tion, and external magnetic field changes its character The recent development of an area of research called in comparison to the behavior described in [2Ð5]. In spintronics [7] has initiated an intense study of thin particular, unfavorable domains, rather than disappear- metal ferromagnetic films with high magnetization and ing with increasing field, form transition layers in-plane anisotropy (with the easy axis lying in the film between the favorable domains, which may be treated plane). The magnetization vector of such plates and as 360° domain walls. The formation of such walls was films should also lie in the film plane, such that the DS, reported in experimental studies [9, 11].

1112 GULYAEV et al.

y 2. MAGNETOSTATIC ENERGY We assume the plate to occupy the region Ð∞ < x < ∞, L L L L z Ð-----y ≤ y ≤ -----y , Ð-----z ≤ z ≤ -----z in space (Fig. 1). The DS x H 2 2 2 2 is periodic along the x axis, and the magnetization in the domains is parallel or antiparallel to the z axis. The magnetostatic potential ϕ(x, y, z) is described by the Poisson equation + ∂2ϕ ∂2ϕ ∂2ϕ ------++------= Ð 4πρ()xyz,, , (1) ∂x2 ∂y2 ∂z2 where – L L ρ()σxyz,, = ()x fy()δz Ð -----z Ð,δ z + -----z (2) 2 2 L 1, y ≤ -----y +  2 fy()=  (3)  L 0, y > -----y,  2 σ – and (x) is the surface density of magnetic charges on the z = Lz/2 plane, which is a periodic function of period W (we do not specify its explicit form in this stage). It is these charges that generate the magnetostatic field ϕ and increase the magnetostatic energy. The DS forma- + L tion is a consequence of the energy minimization z requirement. To calculate the energy and the DS, one has to solve Eq. (1). In doing this, we do not impose any constraints on Ly and Lz. Ly A solution to Eq. (1) has the form ϕ(),, Fig. 1. Ferromagnetic plate with a domain structure. The xyz arrows specify the direction of the magnetization vectors in ∞ ∞ ∞ domains and the external magnetic field H. On the end face, ρ()x',,y' z' the signs of the surface magnetic charges are specified. Also = ∫ dx' ∫ dy' ∫ dz'------shown are the coordinate axes and the plate width Ly and 2 2 2 ∞ ∞ ∞ ()xxÐ ' ++()yyÐ ' ()zzÐ ' thickness Lz. Ð Ð Ð L -----y ∞ 2 1 = ∫ dx' ∫ dy'σ()x' ------In this work, we make some simplifying assump- 2 ∞ 2 2 Lz Ð Ly ()() tions. In particular, while the plate is of finite thickness, Ð----- xxÐ ' ++yyÐ ' z Ð ----- 2 2 the plate length is assumed to be infinite in this stage of consideration. We also assume that the plate’s easy axis (4) lies in the large face and is perpendicular to the end faces. We neglect the magnetic flux closing in the plate, 1 i.e., the effect of closure domains [1, 12]. This neglect Ð ------. 2 rests on the following grounds. First, as will be shown 2 2 L ()xxÐ ' ++()yyÐ ' z + -----z later, the DS period can be substantially shorter than the 2 plate width, such that the contribution due to the closure domains to the magnetostatic energy will not be Expanding σ(x) in a Fourier series overly large. Second, an external magnetic field ∞ destroys closure domains. In this situation, a periodic π σ() σ inkx 2 DS forms in the plate. The geometry of the plate and of x ==∑ ne , k ------, (5) W the DS is shown in Fig. 1. n = Ð∞

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MAGNETOSTATIC ENERGY AND STRIPE DOMAIN STRUCTURE 1113 substituting Eq. (5) into Eq. (4), and integrating over x', Using the relations we obtain, after some manipulations, W 1 Ðinkx σ() σ σ σ* L  2 -----∫ x e dx ==n, Ðn n , -----y  2 Lz W 2 ()yyÐ ' + z + ----- 0  2 ϕ()xyz,, = dy'σ ln------∫ 0 2 we perform integration over x in Eq. (8) and finally  2 Lz L ()yyÐ ' + z Ð ----- Ð-----y   obtain 2  2 L y  η2 2 2 2 + Lz ∞ η()ση  EM = ------∫ d Ly Ð 0 ln------2 inkx Ðinkx Ly Lz  η +2 ()σ e + σ e 0 ∑ n Ðn (9) n = 1 (6) ∞  σ 2[]()η ()η2 2 2 +4∑ n K0 nk Ð K0 nk + Lz . 2 L × K nk() yÐ y' + z Ð -----z  02 n = 1 This is the equation for the average magnetostatic  energy density we have been looking for; this expres-  sion is valid for arbitrary ratios of the lengths Ly, Lz, 2  π ()2 Lz and W (W = 2 /k). Ð K0nk yÐ y' + z + ----- , 2  First, we consider how this relation behaves in the  ∞ ӷ known limiting cases. The limit of Ly , i.e., Ly  ӷ ӷ Lz and Ly W, is considered. We also assume that Lz W. It is these assumptions that were made in [13] in the where K is the McDonald function (modified Bessel 0 calculation of the energy EM. This limiting case corre- function of the second kind). sponds essentially to a plate in which the domain mag- Knowing the potential ϕ and the surface magnetic- netization vectors are perpendicular to the large face, charge density σ, we can calculate the magnetostatic i.e., to a plate containing Faraday domains. In this case, energy. By definition, this energy per unit volume of the the second of the McDonald functions in Eq. (9) is plate can be written as exponentially small. We obtain

W 2 2 2  2 1 2 L 1 2 L 1 E = ------ σ ---L ln1 + -----z Ð ---L ln-----y + 1 E = ------dxσ()x M 0 y 2 z 2 m ∫ Ly Lz 2 2  2WLy Lz Ly Lz 0 L -----y (7) ∞ L 2 y  Ly 2 Lz L σ ()η ()ηη × dy ϕ xy,,----- Ð.ϕ xy,,Ð-----z +2Ly Lz arctan----- + 4 ∑ n ∫ Ly Ð K0 nk d  ∫  Lz  2 2 n = 1 0 Ly Ð----- ∞ 2 ∞ ≈ πσ 2 8 σ 2 Ly () 2 0 + ------∑ n ------∫K0 x dx Substituting Eqs. (5) and (6) into Eq. (7), we obtain Ly Lz nk for σ and ϕ n = 1 0 (10)

W Ly ∞ ∞ 2 σ() η()η EM = ------∫ dx x ∫ d Ly Ð 1 () ≈ πσ 2 2W 1 σ 2 WLy Lz Ð ------2 -2∫ xK0 x dx 2 0 + ------∑ --- n , 0 0 n k Lz n 0 n = 1 ∞  η2 2 which coincides with [13], Eq. (17). ×σ + Lz ()σ inkx σ Ðinlx  0 ln------+ 2 ∑ ne + Ðne (8)  η2 Now we consider another limiting case, namely, that n = 1 Ӷ Ӷ of a long thin rod (Ly W, Lz W ). In this case,

2 2 2 2  η × []()η ()η 2 2 1 + Lz K0 nk Ð K0 nk + Lz . K ()nkη Ð K ()nk η + L ≈ --- ln------,  0 0 2 η2

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1114 GULYAEV et al. such that Substituting these relations into Eq. (12) for the magnetostatic energy yields ∞ 2 2 2 σ σ  π π EM = ------0 + 2 ∑ n 2Ly 2 2Lz Ly Ly Lz E = ------M ()2ξ Ð 1 ln------+ C Ð ln------n = 1 M L 0 W (11) z  W L y 2 2 (15) η + L ∞ ×ηd ()L Ð η ln------z . 3 4 lnn  ∫ y 2 ()π ξ  η Ð C + --- Ð -----2 ∑ ------2 12Ð cos n . 0 2 π  n = 1 n According to Parseval’s theorem, The value ξ = 1/2 corresponds to the minimum of the ξ ∞ W EM ( , W) function for any W, as should be expected σ 2 σ 2 1 []σ()2 from physical considerations. 0 + 2 ∑ n = -----∫ x dx. W We turn now to calculating the equilibrium DS n = 1 0 period W0. To ﬁnd this period, we have to add to EM the If the plate contains a stripe DS, σ(x) takes on only two domain wall energy values, +M0 and ÐM0, where M0 is the absolute value of 2γ σ 2 2 ED = ------, (16) magnetization in a domain. Therefore, [ (x)] = M0 . W Thus, the presence or absence of a DS does not affect γ the magnetostatic energy in any way. The latter state- where is the energy per unit domain wall area (the fac- ment implies that a DS does not originate at all, because tor 2 appears, because there are two domain walls per its formation would not bring about a decrease in period W). As for the anisotropy energy, it should be energy. noted that the magnetization vectors in domains of both types and the easy axis (z axis) are collinear (Fig. 1). We turn now to the case of most interest to us Therefore, the anisotropy energy does not change in a ӷ ӷ (Fig. 1), namely, Lz W Ly . This relation corre- DS rearrangement and does not affect the domain sponds to a thin ferromagnetic plate in which the parameters; therefore, we disregard this energy in the domain magnetization vectors are parallel to the large further calculation. The equilibrium period W0 is found face. In this case, Eq. (9) assumes the form from the condition ∂()E + E 2L  L M D y σ 2z 3 ------∂ = 0 , (17) EM = ------ 0 ln----- + --- W Lz  Ly 2 (12) where one should substitute the equilibrium value ξ = ∞  1/2. We obtain σ 2 () 3 Ð2∑ n lnnkLy Ð ln 2 Ð --- + C , 2  γ L n = 1 z W0 = ------2 -. (18) M0 Ly where C = 0.5772… is Euler’s constant.

4. EFFECT OF AN EXTERNAL MAGNETIC FIELD 3. DOMAIN STRUCTURE In the presence of an external magnetic ﬁeld H par- We consider the periodic stripe DS with a period W allel to the easy axis, the contributions of Eqs. (15) and shown in Fig. 1; the domain width is w+ and wÐ for the (16) to the magnetic energy should be complemented domains with the magnetization oriented parallel and by the Zeeman energy antiparallel to the z axis, respectively. The relation ()ξ between the widths of these domains can be conve- EH = ÐHM0 2 Ð 1 . (19) ξ ≡ niently characterized by the parameter w+/W with W = w + w . The DS is determined now by minimizing the sum EM + + Ð ξ ED + EH with respect to variations in and W. One For the stripe DS of interest, we can write should bear in mind that E does not depend on ξ and D ξ EH is independent of W. The equilibrium value of the σ 2 2()ξ 2 0 = M0 2 Ð 1 , (13) parameter should no longer be equal to 1/2, because the favorable domains expand and the unfavorable ones M 2 shrink. While Eq. (17) is valid for the calculation of the σ 2 0 ()π ξ equilibrium DS period in the presence of an external n = 2------12Ð cos n . (14) πn magnetic ﬁeld, the ξ parameter in it can now assume

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 MAGNETOSTATIC ENERGY AND STRIPE DOMAIN STRUCTURE 1115

1.0 6 2 3 5 5 0.8 1 4

0 4 W 0.6 W , 0 – 3 ξ w W

0.4 , 0

+ 2 w W 2 3 0.2 1 1 0 0 0 0.2 0.4 0.6 0.8 1.0 –10 –5 0 5 10 ξ f ξ Fig. 2. Relative domain width = w+/W plotted vs. dimensionless magnetic field f = (L /4L )(H/M ) for different val- Fig. 3. Domain widths w and w and domain structure z y 0 + Ð ξ ues of the ratio R = 2Lz/W0: (1) 10, (2) 100, (3) 250, (4) period W = (w+ + wÐ) plotted vs. parameter : (1) w+/W0, (2) 1000, and (5) 10000. wÐ/W0, and (3) W/W0. any value in the interval 0 < ξ < 1. Solving this equation change as the field is further increased in absolute mag- for the DS period as a function of the parameter ξ yields nitude. This saturated state corresponds to the values ξ = 0 and 1 and is reached for the field ()ξ W0 W = ------ξ()ξ-, (20) 2 4 1 Ð 4LyM0 2M0 Ly HH==s ------ln------γ , (23) where W0 is given by Eq. (18). The period grows with- Lz out limit as ξ 0 and ξ 1; however, the condi- Ӷ whence one obtains the effective demagnetization factor tion W Lz we have imposed forces us to restrict the analysis to a region of values not very close to the above 4L 2M2 L boundaries in which the condition ξ(1 Ð ξ) ӷ W /4L is y 0 y 0 z N = ------ln------γ . (24) met. Lz After the substitution of Eq. (20) and summation As follows from Eq. (20), domains of both types persist ∂ EM as one approaches the saturation state (ξ = 1). Indeed, over n [14], the derivative ------∂ξ becomes for zero relative width of the domains with magnetization antiparallel to the applied magnetic field, their ∂ 8L ξ ξ EM y 2 ()ξ absolute width wÐ = (1 Ð )W for 1 tends not to ------∂ξ = ------M0 f , Lz zero but rather to a finite value wÐ = W0/4, which is only one half the value for zero external magnetic field. The 2L same holds for ξ 0. The dependence of the widths f ()ξ = ()2ξ Ð 1 ln ------zξ()1 Ð ξ (21) W of domains with parallel and antiparallel magnetiza- 0 tion, as well as of their sum (the DS period W), on ξ is shown graphically in Fig. 3. 1 πξ πΓ() ξ Ð lnπ--- sin Ð 2 . ∂() 5. DISCUSSION OF RESULTS EM + EH Equation ------∂ξ = 0 yields a relation for determi- According to Eq. (20), the Cotton DS under study nation of the equilibrium value of ξ, differs substantially from the Faraday DS. Indeed, many authors, starting with Kittel [2], estimated the Lz H γ 2 f ()ξ = ------. (22) period of the Faraday structure as W0' ~ 2 Lz/M0 (in 4LyM0 the notation used here). A comparison of this estimate Figure 2 plots the reciprocal function describing the with Eq. (20) shows that in our case, the DS period is dependence of ξ on H for various values of the ratio of proportional to the first power (rather than to the square the plate width to the equilibrium DS period in the root) of Lz and to the first power (rather than the square γ absence of a magnetic field, R = 2Lz/W0. We readily see root) of domain wall energy and is inversely propor- that, as the field increases, one comes to a state which tional to the square (and not to the first power) of the can be treated as saturation, because this state does not magnetization M0.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1116 GULYAEV et al. Ӷ Because the DS period W0 depends on the geometric is not constrained by the condition W Lz, we would parameters of the plate, one has to establish the cases in not, nevertheless, come, within our model, to definite Ӷ Ӷ ξ which the condition Ly W0 Lz imposed by us in conclusions on the behavior of function wÐ( ) for deriving the simplified expression (12) for magneto- ξ 0, 1. The point is that we approximated the γ static energy holds. For this purpose, we use the esti- domain wall energy with the expression ED = 2 /W, γ 2δ δ which does not take into account the internal structure of mate ~ M0 , where stands for the domain wall the wall. Indeed, the question of whether domain walls thickness, and substitute this estimate into Eq. (20) for brought closer to one another with increasing field H W0. Then, this condition will take on the form would annihilate or not should depend on such structural δ Ӷ Ӷ δ characteristics of the wall as, for instance, the chirality, a Ly Ly . (25) point which has been corroborated by direct experimental observations [11]. Theoretical analysis of this impor- According to [1], we have δ ~ α/β , where α is the tant problem requires dedicated consideration. nonuniform exchange constant, which we estimate here as α ~ 10Ð12 cm2, and β is a dimensionless anisotropy parameter which determines the anisotropy field Ha ~ ACKNOWLEDGMENTS β M0. As an illustration, we consider a film of cobalt for The authors are indebted to V.G. Shavrov for valu- which β ~ 4.2 [12]. Putting the geometrical parameters able remarks. L = 50 nm and L = 60 µm, as in [15], we see that con- y z This study was supported by the Russian Founda- dition (25) holds. One can readily verify that condition tion for Basic Research (project no. 00-02-16384) and (25) is also satisfied for the same geometrical parame- ISTC (grant no. 1522). ters for films of iron (β ~ 1.7 [12]) and nickel (β ~ 0.29 [1]). Next, we use inequality (25) to compare the DS REFERENCES γ 2 δ periods W0 and W0' . Because W0 ~ Lz/M0 Ly ~ Lz /Ly 1. L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935); L. D. Landau, Collection of Works γ 2 δ and W0' ~ 2 Lz/M0 ~ Lz , Eq. (25) suggests that (Nauka, Moscow, 1969), Vol. 1, p. 129. ӷ 2. C. Kittel, Phys. Rev. 70, 965 (1946). W0 W0' ; therefore, in plates with the same geometrical parameters, the period of the Cotton domains is sub- 3. S. Chikasumi, The Physics of Ferromagnetism. Mag- stantially larger than that of the Faraday domain. A netic Characteristics and Engineering Applications (Syokabo, Tokyo, 1984; Mir, Moscow, 1987). numerical estimation of the DS period W0 and of the 4. G. S. Krinchik, Physics of Magnetic Phenomena (Mosk. saturation field Hs given by Eq. (23) for a Co film with µ Gos. Univ., Moscow, 1985). parameters M0 ~ 1400 G, Ly = 50 nm, and Lz = 60 m µ 5. M. I. Kaganov and V. M. Tsukernik, Nature of Magne- yields W0 ~ 6 m and Hs ~ 14 G. These estimates agree in order of magnitude with experimental data (see, e.g., tism (Nauka, Moscow, 1982). [9Ð11]). 6. V. D. Buchel’nikov, V. A. Gurevich, and V. G. Shavrov, Another substantial, distinctive feature of the struc- Fiz. Met. Metalloved. 52 (2), 298 (1981). ture under study is the above persistence of the unfavor- 7. L. Mei, Prog. Nat. Sci. 10 (4), 259 (2000). able domains for ξ 1 and ξ 0 (for zero relative 8. V. S. Gornakov, V. I. Nikitenko, L. H. Bennett, et al., J. width, their absolute magnitude tends to a finite limit), Appl. Phys. 81 (8), 5215 (1997). which implies the persistence of domain walls in a 9. X. Portier and A. K. Petford-Long, J. Phys. D 32 (1), 1 fairly strong magnetic field (in absolute magnitude), (1999). | | H > Hs [see Eq. (23)]. As already mentioned, this ° 10. A. K. Petford-Long, X. Portier, E. Yu. Tsymbal, et al., result correlates with the observation of 360 domain IEEE Trans. Magn. 35 (2), 788 (1999). walls in experiments with films containing Cotton domains [9, 11]. 11. X. Portier and A. K. Petford-Long, Appl. Phys. Lett. 76 (6), 754 (2000). We note in conclusion that 360° domain walls sepa- 12. I. A. Privorotskiœ, Usp. Fiz. Nauk 108 (1), 43 (1972) rating favorable domains were obtained by us using an [Sov. Phys. Usp. 15, 555 (1972)]. approximate expression for the energy EM, Eq. (15), Ӷ 13. Yu. V. Gulyaev, P. E. Zil’berman, and É. M. Épshteœn, which is valid for W Lz or, which is the same, for ξ(1 Ð ξ) ӷ W /4L ~ δ/4L [see Eq. (18) and the discus- Radiotekh. Élektron. (Moscow) 46 (7), 862 (2001). 0 z y δ Ӷ sion after Eq. (20)]. Because /Ly 1, according to 14. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Eq. (25), we can approach the saturation limits, i.e., the Integrals and Series (Nauka, Moscow, 1981; Gordon points ξ = 0, 1, fairly closely. However, these points and Breach, New York, 1986). should be excluded from consideration. 15. A. M. Baranov, A. I. Chmil, R. J. Elliott, et al., Europhys. Even if we base our analysis on a more general Lett. 53 (5), 625 (2001). expression for energy given by Eq. (9), whose validity Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1117Ð1121. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1070Ð1074. Original Russian Text Copyright © 2002 by Kveglis, Jarkov, Bondarenko, Yakovchuk, Popel.

MAGNETISM AND FERROELECTRICITY

Formation of Tetrahedrally Close-Packed Structures in TbÐFe and CoÐPd Nanocrystalline Films L. I. Kveglis, S. M. Jarkov, G. V. Bondarenko, V. Yu. Yakovchuk, and E. P. Popel Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received August 10, 2001

Abstract—The crystal structure of Tb30Fe70 and Co50Pd50 nanocrystalline films with strong magnetic anisot- 6 ropy perpendicular to the film plane (K⊥ ~ 10 erg/cm3) is investigated using electron diffraction and transmission electron microscopy. All the studied films in the initial nanocrystalline phase undergo an explosive crystallization with the formation of dendrite structures. It is demonstrated that, after crystallization, the TbÐFe and Co–Pd films exhibit a tetrahedrally close-packed atomic structure that has no analogs among these materials in the equilibrium state. The internal stresses in the films under investigation are estimated from an analysis of the bend extinction contours in the electron microscope images. The inference is made that strong perpendicular magnetic anisotropy can be associated with magnetostriction anisotropy due to the specific features of the film structure. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ropy. Leamy and Dirks [8] proposed one more model, namely, the model of a fractal structure formed perpen- A unique combination of magnetic properties ren- dicular to the film plane. The role played by each factor ders nanocrystalline materials very attractive for practi- in the formation of perpendicular magnetic anisotropy cal applications, in particular, for the design of storage depends on the particular material and technique used media used in magnetic and thermomagnetic data for the preparation of the samples. recording devices. The recording density is the most Earlier [9], we studied Dy–Co films with strong per- important parameter of storage media. Nanocrystalline 5 3 pendicular magnetic anisotropy (K⊥ ~ 10 erg/cm ). It materials with magnetic anisotropy perpendicular to was shown that, in the initial state, these films consist the film plane are materials of the future. The recording of 10- to 15-Å clusters whose structure is similar to a density provided by nanocrystalline materials with per- tetrahedrally close-packed structure of the CaCu type pendicular magnetic anisotropy considerably exceeds 5 [9]. The SmCo5 alloy with the same structure possesses the density achieved in materials with magnetic anisot- the largest crystallographic magnetic anisotropy con- ropy in the film plane. The recording density in materi- stant K ~ 108 erg/cm3 [10]. als with perpendicular magnetic anisotropy can be as 1 high as 1012 bit/cm2. It is common knowledge that rare-earth metalÐtransi- tion metal alloys belong to materials with the strongest At present, materials with a large perpendicular- magnetostriction occurring in nature [10]. However, magnetic-anisotropy energy constant have been studied available data on the contribution of magnetostriction extensively. As a rule, these materials are produced anisotropy to perpendicular magnetic anisotropy in from rare-earth metalÐtransition metal (DyÐCo, TbÐFe, nanocrystalline films of transition metal alloys are very etc.) and 3d metalÐ3d metal (CoÐPd, CoÐCr, etc.) scarce. This is associated with the difficulties encoun- alloys [1Ð6]. However, despite extensive investigations tered in evaluating the magnitude of the magnetostriction into the properties of storage media used in magnetic and its contribution to perpendicular magnetic anisot- and thermomagnetic recording devices, the question as ropy on the basis of experimental data. As is known, to the origin of perpendicular magnetic anisotropy there exist two sources of stresses generated in films: remains open. This can be explained by the fact that the (1) stresses induced by a substrate or multilayers and perpendicular magnetic anisotropy by itself and the (2) internal stresses due to specific features of the atomic perpendicular-magnetic-anisotropy energy depend on structure. Draaisma et al. [2] examined CoÐPd multi- many factors. The basic models thus far applied to the layer films and considered different mechanisms respon- explanation of the origin of perpendicular magnetic sible for the formation of perpendicular magnetic anisot- anisotropy in films are associated with the following ropy. These authors made the inference that magneto- factors [7]: (1) pair atomic interaction, (2) anisotropy of striction anisotropy due to lattice mismatch between the columnar structure, (3) crystallographic anisotropy, cobalt and palladium plays a decisive role in the forma- (4) surface anisotropy, (5) exchange anisotropy tion of perpendicular magnetic anisotropy. Kobayashi et between multilayers, and (6) magnetostriction anisot- al. [11] assumed that strong perpendicular magnetic

1118 KVEGLIS et al. ples were prepared through thermal explosive evaporation under vacuum at a residual pressure of 10Ð5 Torr and magnetron sputtering under vacuum at a residual pressure of 10Ð6 Torr onto different substrates (glass, crystalline and amorphous silicon, fused silica, NaCl, MgO, and LiF). The microstructure and phase composition of the films were analyzed using PRÉM-200 and JEM-100 C transmission electron microscopes. The chemical composition of the films was checked by x- ray fluorescence analysis. The perpendicular magnetic anisotropy constant K⊥ was determined by the torque method at room temperature in magnetic fields with strengths up to 17 kOe. 2 µm 3. RESULTS Fig. 1. Electron microscope image illustrating the onset of In the initial state, the Tb–Fe and Co–Pd films possess dendritic crystallization in the Tb–Fe film. 5 3 perpendicular magnetic anisotropy (K⊥ ~ 10 erg/cm ). The electron diffraction patterns of these films exhibit a 7 3 diffuse halo. The electron microscopic investigation anisotropy in Tb–Fe films (K⊥ = 2 × 10 erg/cm ) is associated with the magnetostriction anisotropy arising from revealed that the Tb–Fe and Co–Pd films consist of the difference between the thermal expansion coeffi- 20- to 30-Å clusters. It is found that dendritic crystal- cients of the film and the substrate. lization occurs in the films under the action of an electron beam in the transmission electron microscope or during In our recent works [12, 13], we investigated CoÐPd annealing under vacuum at a residual pressure of films with a large perpendicular magnetic anisotropy 10−5 Torr and an annealing temperature T = 260Ð 6 3 ann constant (K⊥ ~ 10 erg/cm ). Such a strong perpendicu- 300°C. In the course of crystallization, the perpendic- lar magnetic anisotropy was explained by the self-orga- ular magnetic anisotropy constant increases to ≈5 × nization of crystalline modules through the coalescence 106 erg/cm3. The velocity of the crystallization front was of assemblies of these modules according to the general estimated visually during electron microscopic observa- rules. In this case, the three-dimensional space is filled tions and reached 1 cm/s. After the completion of den- in an imperfect manner. The misorientation angle dritic crystallization, particles forming the film did not between the faces of the adjacent module assemblies increase in size as compared to the initial state. Similar containing tetrahedra and octahedra can be as large as effects were observed earlier for Co–Pd films [12]. several degrees. As a consequence, considerable stresses arising in the material can be partly relieved Figure 1 displays the electron microscope image through displacements and rotations of module assem- illustrating the onset of the dendritic crystallization in blies and the formation of fractures and cracks in the the Tb–Fe film. It can be seen from Fig. 1 that bend material. Making allowance for a giant magnetostric- extinction contours clearly manifest themselves in the tion of CoÐPd alloys, it was assumed that the magneto- crystallized region. After further annealing, a continu- striction anisotropy makes a substantial contribution to ous network of intersecting bend extinction contours is the large perpendicular magnetic anisotropy constant. observed throughout the electron microscope image of However, our attempts to describe the structure of CoÐ the Tb–Fe film (see [13] for Co–Pd films). The electron Pd films adequately were unsuccessful. The purpose of diffraction pattern of the crystallized region of the TbÐ the present work was to elucidate the structure of TbÐ Fe film (Fig. 2a) is identified as the TbFe2 structure Fe and Co–Pd films with strong perpendicular mag- (Fd3m) with the lattice parameter a = 7.10 Å and the netic anisotropy and to evaluate the contribution of the []011 orientation. The electron diffraction pattern of magnetostriction anisotropy to the perpendicular mag- the crystallized region of the Co–Pd film (Fig. 2b) con- netic anisotropy. tains sets of point reflections which disagree with all known structures of CoÐPd alloys. The diffraction reflections observed in the electron diffraction pattern 2. SAMPLE PREPARATION correspond to interplanar distances typical of the (111) AND EXPERIMENTAL TECHNIQUE and (620) atomic planes in a face-centered cubic struc- In this work, we investigated the structure and mag- ture with the lattice parameter a = 3.75 Å. A similar set netic properties of TbÐFe (30 at. % Tb and 70 at. % Fe) of reflections can be observed in the electron diffraction and CoÐPd (50 at. % Co and 50 at. % Pd) nanocrystal- pattern of the face-centered cubic lattice oriented along line films with strong perpendicular magnetic anisot- the [134] zone axis. However, the electron diffraction ropy [12, 13]. The films were examined in the initial pattern of the Co–Pd film (Fig. 2b) exhibit superstructure state and after annealing under vacuum. The film sam- reflections with respect to the CoPd face-centered cubic

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

FORMATION OF TETRAHEDRALLY CLOSE-PACKED STRUCTURES 1119

(111) (111) (400) (422) (311) (311) (200) (222) (3/2 1/2 0) (111) (111) (022) (022) (310) (111) (111) (222) (200) (222) (311) (311) (400)

(a) (b)

(c)

Fig. 2. Electron diffraction patterns of (a) Tb–Fe and (b) Co–Pd films after dendritic crystallization. (c) Schematic representation of a superposition of the diffraction patterns of Tb–Fe and Co–Pd films. Closed circles represent the reflections of TbFe2, and open squares indicate the reflections of CoPd.

° lattice, namely, the (3/21/20), (310), and (9/23/20) Annealing at Tann > 300 C brings about disturbance superstructure reflections. The intensities of the afore- of the dendrite structure in Tb–Fe and Co–Pd films. The mentioned superstructure reflections are substantially electron diffraction patterns of the disturbed dendrite higher than the intensity of the (620) structure reflec- structures in Tb–Fe and Co–Pd films are shown in tion. It is worth nothing that the angle between vectors Figs. 3a and 3b, respectively. These diffraction patterns of the [111] type in the electron diffraction pattern dis- are also not typical and exhibit sets of reflections played in Fig. 2b is equal to ≈54°, whereas this angle which, according to the interplanar distances, corre- for a cubic lattice should be equal to 70.5°. Attempts to spond to the (111) and (200) atomic planes in the TbFe2 identify the electron diffraction pattern under consider- structure (Fd3m) with the lattice parameter a = 7.10 Å ation as a hexagonal close-packed structure showed (Fig. 3a) and in the CoPd face-centered cubic structure that this structure should be described by the ratio c/a ≈ with the lattice parameter a = 3.75 Å (Fig. 3b). How- ever, the [111] and [200] vectors in both structures are 2.18, which is not characteristic of hexagonal close- nearly parallel to each other. This situation is, in princi- packed structures of metallic compounds [13]. ple, impossible for single crystals with a cubic lattice. A comparison of the electron diffraction patterns of ≥ ° After annealing at Tann 400Ð450 C, the structure of Tb–Fe and Co–Pd films (Figs. 2a, 2b) demonstrates that the films under investigation relaxes to the equilibrium the directions of the reciprocal lattice vectors in these state. The films have a fine-grained structure which patterns almost coincide with each other. A schematic manifests itself in diffuse fringes in the electron diffrac- representation of a superposition of the electron diffraction patterns. The electron diffraction pattern of the TbÐ tion patterns is depicted in Fig. 2c. It can be seen that Fe film corresponds to the TbFe2 structure (Fd3m) with the (111)-type reflections of CoPd are superposed on the lattice parameter a = 7.10 Å. The electron diffrac- the (311)-type reflections of TbFe2 and that the tion pattern of the Co–Pd film is identified as a face- (3/21/20)-type superstructure reflections with respect centered cubic structure with the lattice parameter a = to the CoPd face-centered cubic structure are super- 3.75 Å. The perpendicular magnetic anisotropy con- 4 5 3 posed on the (220)-type reflections of TbFe2. stants of these films are equal to ~10 Ð10 erg/cm .

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1120 KVEGLIS et al. structure, the former structure is atomically disordered and imperfect. Note that, within the same film, the packing of tetrahedra can undergo changes according to the basic rule of packing; i.e., tetrahedra are packed in (200) (111) such a face-to-face manner as to provide the highest local density [15]. It is believed that film materials with a similar structure can experience strong internal stresses. The electron diffraction pattern of the Tb–Fe film ° annealed at Tann > 300 C (Fig. 3a) is similar to the diffraction pattern of the Co–Pd film annealed at Tann = 320°C (Fig. 3b). These diffraction patterns have defied (a) interpretation both with the use of a superposition of the electron diffraction patterns obtained for differently oriented crystalline grains and from the viewpoint of a particular single crystal. In our recent work [13], we (200) proposed a scheme for identifying similar electron dif- (111) fraction patterns with the use of crystalline-module assemblies. These assemblies are composed of tetrahedra joined together in the same fashion as a Boerdijk spiral. The three-dimensional space is filled with these module assemblies in a percolation manner to form a structure with strong internal stresses. This is indicated by the bend extinction contours observed in the electron microscope images (see Fig. 1 for the Tb–Fe film (b) and [13] for the Co–Pd film). Analysis of the bend extinction contours was per- Fig. 3. Electron diffraction patterns of (a) TbÐFe and (b) ° formed according to the procedure described in [16]. Co–Pb films after annealing under vacuum at Tann > 300 C. For the studied films, the elastic stress is estimated at ~1011 N/m2. In the case when the stress in the film does 4. DISCUSSION not exceed the elastic limit, the perpendicular magnetic anisotropy constant is approximately equal to The investigation into the magnetic properties and ~106 erg/cm3. However, dark-field electron micro- analysis of the electron diffraction patterns of TbÐFe scopic examinations of these films revealed a plastic films revealed that, in the initial nanocrystalline state, flow. This manifests itself as rotational effects, i.e., these films are characterized by the perpendicular mag- rotations of film regions ~1 µm in size. Consequently, 5 3 netic anisotropy constant K⊥ ≈ 2 × 10 erg/cm . After stresses arising during the formation of the dendrite the dendritic crystallization from the initial nanocrys- structure substantially exceed the elastic limit and can talline state, the films possess the largest perpendicular make a considerable contribution to the perpendicular 6 3 magnetic anisotropy constant (K⊥ ≈ 5 × 10 erg/cm ) magnetic anisotropy due to magnetostriction effects. and have a TbFe2-type structure which corresponds to a Laves phase (the MgCu2 type) [10]. Different polytypes of the Laves phase family belong to the group of FrankÐ 5. CONCLUSION Casper tetrahedrally close-packed structures [14]. In Thus, the above investigation demonstrated that TbÐ the case of a TbFe2-type structure, the tetrahedra involved in the structure form mutually penetrating Fe and Co–Pd films with large perpendicular magnetic FrankÐCasper polyhedra with coordination numbers of anisotropy constants have a tetrahedrally close-packed 12 and 16. In turn, the mutually penetrating FrankÐ structure. The dendritic crystallization of the initial Casper polyhedra consist of close-packed tetrahedra. nanocrystalline phase leads to the formation of a film As is known, crystalline materials described by FrankÐ structure similar to the structure of the Laves phase. Casper polyhedra exhibit a tendency for the unit cell After the dendrite structure is destroyed, the film struc- parameters to decrease to 30% [14]. ture is built up in the same manner as a Boerdijk spiral. The specific features of the dendritic growth in the films In the case of Co–Pd films, we can assume that, after give rise to strong internal stresses. These stresses are the completion of dendritic crystallization (Fig. 2b), the responsible for a dominant contribution of the magne- film structure is described by close-packed tetrahedra, tostriction anisotropy to the perpendicular magnetic as is the case with a TbFe2-type structure. Apparently, anisotropy. It should be emphasized that strong internal the structure formed in the Co–Pd film is similar to the stresses are associated with the specific features in the structure of the Laves phase; however, unlike the latter film structure and do not depend on substrates.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 FORMATION OF TETRAHEDRALLY CLOSE-PACKED STRUCTURES 1121

ACKNOWLEDGMENTS 6. T. Tarahashi, A. Yoshihara, and T. Shimamori, J. Magn. We would like to thank V.N. Matveev for supplying Magn. Mater. 75 (3), 252 (1988). the samples prepared by magnetron sputtering for our 7. W. H. Meiklejohn, Proc. IEEE 74 (11), 1570 (1986). investigations. 8. H. J. Leamy and A. G. Dirks, J. Appl. Phys. 50 (4), 2871 This work was supported by the Russian Foundation (1979). 9. A. S. Avilov, L. I. Vershinina-Kveglis, S. V. Orekhov, for Basic Research (project no. 00-02-17358), the International Association of Assistance for the promo- et al., Izv. Akad. Nauk SSSR, Ser. Fiz. 55 (8), 1609 (1991). tion of cooperation with scientists from the New Inde- pendent States of the former Soviet Union (project 10. K. N. R. Taylor, Adv. Phys. 20, 551 (1971). no. 00-100), the 6th Competition of Research Projects 11. H. Kobayashi, T. Ono, A. Tsushima, and T. Suzuki, Appl. of Young Scientists of the Russian Academy of Sci- Phys. Lett. 43 (4), 389 (1983). ences (1999) (project no. 56), and the Krasnoyarsk 12. L. I. Vershinina-Kveglis, V. S. Zhigalov, and I. V. Staro- Regional Scientiﬁc Foundation. verova, Fiz. Tverd. Tela (Leningrad) 33 (5), 1409 (1991) [Sov. Phys. Solid State 33, 793 (1991)]. 13. L. I. Kveglis, S. M. Jarkov, and I. V. Staroverova, Fiz. REFERENCES Tverd. Tela (St. Petersburg) 43 (8), 1482 (2001) [Phys. 1. Y. Ochiai, S. Hashimoto, and K. Aso, IEEE Trans. Magn. Solid State 43, 1543 (2001)]. 25 (5), 3755 (1989). 14. W. B. Pearson, The Crystal Chemistry and Physics of 2. H. J. G. Draaisma, W. J. M. de Jonge, and F. J. A. den Metals and Alloys (Wiley, New York, 1972; Mir, Mos- Broeder, J. Magn. Magn. Mater. 66 (3), 351 (1987). cow, 1977). 3. P. F. Carcia, J. Appl. Phys. 63 (10), 5066 (1988). 15. N. A. Bulienkov and D. L. Tytik, Izv. Akad. Nauk, Ser. 4. V. H. Harris, K. D. Aylesworth, B. N. Das, et al., Phys. Khim., No. 1, 1 (2001). Rev. Lett. 69 (13), 1939 (1992). 16. I. E. Bolotov and V. Yu. Kolosov, Phys. Status Solidi A 5. T. Miyazaki, K. Hayashi, S. Yamaguchi, et al., J. Magn. 69, 85 (1982). Magn. Mater. 75 (3), 243 (1988). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

MAGNETISM AND FERROELECTRICITY

Two-Dimensional Heisenberg Model with Spin s = 1/2 and Antiferromagnetic Exchange Treated as a Spin Liquid E. V. Kuz’min Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received October 1, 2001

Abstract—The spin system of the Heisenberg model (s = 1/2) on a square lattice with antiferromagnetic (AFM) exchange between nearest neighbors (in which there is no long-range magnetic order at any T ≠ 0) is treated as a spatially homogeneous isotropic spin liquid. The double-time temperature Green’s function method is used in the framework of a second-step decoupling scheme. It is shown that, as T 0, the spin liquid goes over ε (without any change in symmetry) to a singlet state with energy (per bond) 0 = Ð0.352 and the correlation ξ ∝ Ð1 length diverges as T exp(T0/T). The spatial spin correlators oscillate in sign with distance, as in the AFM state. The theory allows one to calculate the main characteristics of the system in all temperature ranges. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION mirror images of each other. It is generally agreed that The objective of this paper is to discuss the problem at T = 0, the two-sublattice AFM state with long-range of the ground state and the main thermodynamic prop- order is the closest approximation to the ground state of erties of the s = 1/2 spin system described by the isotro- the system [3]. However, Anderson [4] assumed that the pic Heisenberg model on a square lattice with antiferro- ground state of the Hamiltonian (1) on a square lattice magnetic (AFM) exchange interaction between nearest can be disordered and described by a wave function neighbors (J > 0). The Hamiltonian H and the total spin with resonant valence bonds (RVBs). Later [5], the S are energy of the disordered (singlet) state was calculated numerically with RVB wave functions on 128 × 128 1 × H ==---J∑s s , S ∑s . (1) and 256 256 lattices and was found to be Ð0.3344 per 2 f fa+ f bond, which is equal, within 0.1%, to the best result for fa f the energy of the ordered AFM state. However, in [6], We will consider the general case of the Hamiltonian using exact diagonalization for a small (4 × 4) cluster, the (1) on an alternant lattice1 of N sites with periodic energy of the singlet state was calculated to be Ð0.3509. boundary conditions. In Eq. (1), f speciﬁes the lattice Different methods for solving this problem are sites, D is the lattice dimensionality, z is the number of reviewed in [3]. nearest neighbors, and a are the vectors connecting nearest neighbors. The ground state and the thermody- The thermodynamic properties are also of fundamental importance. According to the MerminÐWagner namic properties of the system essentially depend on its ≠ dimensionality D. theorem [7], long-range magnetic order can exist at T The problem of the ground state (at T = 0) for the 0 only on three-dimensional (or quasi-two-dimen- two-dimensional model (D = 2, z = 4) still remains sional) lattices (D = 3) up to the critical temperature Tc. unsolved. Marshall [1, 2] argued that the ground state Therefore, the AFM state on a square (D = 2) lattice is represented by the “pricked-out” temperature point and of the Hamiltonian (1) on alternant lattices is a nonde- ≠ generate singlet with total spin S = 0 (this statement has the problem arises of describing the system at T 0. In been rigorously proved only for a one-dimensional a sense, the case of dimensionality D = 2 is critical or chain). On the other hand, on alternant lattices of D = 2 intermediate between a one-dimensional system (in and 3, the spin distribution can have a chessboard pat- which the long-range magnetic order can never occur) tern described by the Néel wave function of an antifer- and three-dimensional systems with a long-range mag- romagnet with two equivalent sublattices which are netic order at temperatures below the critical point. The theory of the thermodynamic properties of this 1 A lattice is termed alternant if it is made up of two interpenetrat- ing equivalent sublattices A and B, such that the nearest neighbors system was developed in [8Ð11], where the long-range of a site of sublattice A are sites of sublattice B alone and vice AFM order of the Néel type was postulated to occur at versa. T = 0.

TWO-DIMENSIONAL HEISENBERG MODEL 1123

In this paper, the two-dimensional system is rier transform of the correlation function assumed to be in a nonmagnetic state with a well-devel- () Ðiqr 〈〉z()z() oped short-range AFM order. This state is referred to as K q ==∑e Kr 4 s q s Ðq the spin liquid (SL). In describing the thermodynamic r properties of the SL, we make some assumptions and = 2〈〉s+()q sÐ()Ðq , (5) follow the method used in [8Ð11]. As T 0, the SL goes over, without any change in symmetry, to a singlet 1 iqr () ε Kr = ----∑e K q state with an energy per bond 0. However, it is not N known in advance whether or not this state is the q ε with evident properties K(q) = K(Ðq) and K = K . Given ground state, because the AFM state with energy AF r Ðr ε ε the Fourier transform K(q), one can calculate any space can also arise. Only a comparison between 0 and AF correlators Kr. We follow the method applied in [8Ð11] will allow one to draw a conclusion as to the type of the and calculate K(q) using the double-time retarded tem- ground state. perature Green’s functions [12]. Below, we calculate the z z function 〈〈s (q)|s (Ðq)〉〉ω ≡ G(q, ω). Since the spin corre- + Ð lators are isotropic, we have 〈〈s (q)|s (Ðq)〉〉ω = 2G(q, ω). 2. SPIN LIQUID: CORRELATION FUNCTIONS AND GREEN’S FUNCTIONS 3. EQUATIONS OF MOTION We deﬁne the spin liquid as a spatially homoge- It is convenient to go over to the dimensionless neous state (with the short-range order symmetry Hamiltonian h = H/z J; all energetic parameters will unbroken) in which (i) the spin correlation functions also be measured in units of z J. The equations of are isotropic, i.e., motion have the form (ប = 1)

+ 1 z + z + --- () 1 〈〉x x 1 〈〉y y isúf = ∑ sf sfa+ Ð sfa+ sf , ----∑ sf sfr+ = ----∑ sf sfr+ z N N a (6) f f (2) z 1 + Ð + Ð ----- () 1 〈〉z z ≡ 1 isúf = ∑ sf sfa+ Ð sfa+ sf , = ----∑ sf sfr+ ---Kr, 2z N 4 a f 2 z ∂ s 1 z z and depend only on the magnitude r = |r| of the space f ------() Ð------2- = 2∑ sf Ð sfa+ + Rf, (7) 2 ∂ 〈〉 t 2z a vector, with K0 = 1 and, hence, sf = 3/4 (here and henceforth, 〈…〉 is a thermodynamic average at T ≠ 0 or where |Ψ 〉 the expectation value in the singlet state 0 at T = 0), 1 z + Ð z 1 + Ð + Ð R = ---- ∑ s s s + s ---()s s + s s and (ii) the following averages are zero: f 2 f fa+ fa+ ' faa+ Ð '2 f fa+ fa+ f z aa≠ ' 〈〉a 〈〉α (3a) sf ==0, S 0, z 1()+ Ð + Ð Ð sfa+ --- sf sfaa+ Ð ' + sfaa+ Ð 'sf (8) 〈〉α β γ ≠≠ 2 sf smsn = 0, fmn; (3b) z 1 + Ð + Ð Ð s ---()s s + s s . the averages of any other odd products of the operators fa+ '2 f fa+ fa+ f at different sites are also zero (α = x, y, z or +, Ð, z). In the second-order differential equation, the kinematic The properties of the SL state are mainly determined properties of spin operators at one site are exactly taken by the temperature dependence of the spin correlation into account, which is of fundamental importance. functions. The SL state energy per bond (in units of J) is The chain of equations is cut off at the second step ≠ by linearizing the operator Rf (a a') according to the 〈〉H 3 ε ==------Ð---K , (4) scheme [8Ð11] ()1/2 zNJ 4 1 z + Ð + Ð z 1 z s s s ≈ α 〈〉s s s = ---α K s , f fa+ fa+ ' 2 fa+ fa+ ' f 2 2 aaÐ ' f where Ka = ÐK1 (K1 > 0) is the correlation function for (9) nearest neighbors (a is the lattice parameter). At T = 0, z + Ð ≈ α 〈〉+ Ð z 1α z ε sfa+ 'sf sfa+ 1 sf sfa+ sfa+ ' = --- 1Kasfa+ ', Eq. (4) gives the energy of the singlet state 0. 2 α To describe the SL state, we go over to the Fourier where 1 is a factor which corrects the decoupling transforms of the spin operators and introduce the Fou- approximation for nearest neighbors (Ka = ÐK1) and

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1124 KUZ’MIN α 2 is an analogous factor for the correlators at distances function can be written as r = d = 2 a (along a diagonal) and r = 2a (in what fol- () Ω lows, we put a = 1). Thus, we have 〈〉z()z()≡ 1 () A q q s q s Ðq ---K q = ------Ω - coth------τ (18) 4 2 q 2 1 1 z z ()R = ------∑ [α K ()s Ð s f lin 2 2 2 aaÐ ' f fa+ z aa≠ ' (10) or α ()z + 1K1 sfa+ ' Ð sfa+ ' Ð a ], K 1 Ð γ λE ()∆ K()q = ------1 ------q coth ------q . (19) λ E ()∆ 2τ 1 1 q K ==----∑K ---[]1 ++()z Ð 2 K K . (11) 2 2 aaÐ ' z d 2a z aa, ' Using the definition of the space correlators Kr of Eq. (5), one can derive the relations Using Eqs. (6) and linearized equation (7), the Fourier transform of the Green’s function in the second-step 1 n K1 decoupling scheme is routinely found to be K ===----∑()Ðγ K()q ------I ()∆τ, , K 1, (20) n N q λ n 0 A()q K q G()q, ω ==------, A()q ------1()1 Ð γ (12) ω2 Ω2 2 q Ð q where with the excitation spectrum n 1 ()γ1 Ð γ ()Ð λE ()∆ Ω λ ()∆ ()∆ ()γ ()γ ∆ I ()∆τ, = ------q q coth ------q , q ==Eq , Eq 1 Ð q 1 ++q , n ∑ ()∆ τ N Eq 2 q (21) 1 γ = ---- ∑ cosq , n = 012.,, q D j j (13) These equations should be solved self-consistently in λ combination with Eq. (15) for the pseudogap and where is the hardness parameter of the excitation Eq. (14). spectrum, α λ2 α As in [8–11], there are five parameters (K1, K2, 1, = 1K1/2, (14) α ∆ 2, ) to be found from three equations (20) (n = 0, 1, ∆ π π γ and the pseudogap at q = Q = ( , ) (where Q = Ð1) 2) and Eq. (15). The needed fifth equation can be cho- is given by sen arbitrarily to some extent. Shimahara and Takada α α λ2 [8] put rα = ( 1 Ð 1)/( 2 Ð 1) = const and found this ∆ 1 12+ α 1 1 + = ------2 ------+.2K2 Ð --- (15) parameter from the condition that the AFM state with 2λ z z sublattice magnetization m = 0.3 exist at T = 0; that is, Ω ≥ The spectrum q 0 describes collective triplet excita- the ground state is postulated to be antiferromagnetic. tions from the singlet state. All parameters of the spec- We will refer to this as the ShimaharaÐTakada (ST) trum are temperature-dependent and should be calcu- condition. lated self-consistently. In this paper, the needed fifth equation for closing the set of equations is chosen on the basis of Eq. (3b). 4. SELF-CONSISTENCY EQUATIONS Using the rules for calculating the products of spin The imaginary part of the Green’s function (12) is operators at one site and Eq. (3b), one can exactly cal- equal to culate the average: 1 (), ω Ðπ---ImG q + i0 〈〉z 1 〈〉+ Ð 1 ˜ (16) Rfsf ==------∑ sfa+ sfa+ ' ---K2, A()q 2 8 []δω()δωΩ Ω ()≡ (), ω 4z aa≠ ' (22) = ------Ω - Ð q Ð + q n q ; 2 q ˜ K2 = K2 Ð 1/z, therefore, the spectral density for the z components is ω/τ which allows one to find (), ω τ e (), ω J q ; = ------ω/τ n q , (17) e Ð 1 2 z ∂ s z 11 τ Ð------f s = --- K + ---K . (23) where = T/zJ is the dimensionless temperature. Using ∂ 2 f 82 z 1 a spectral theorem [12], the equal-time correlation t

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TWO-DIMENSIONAL HEISENBERG MODEL 1125

In the framework of the linear theory used here, we expression (23). Calculating the right-hand side, we have represent Eq. (28) in the form

∂2 z sf z 1()1λ ()∆τ, Ð------s --- K2 + K1/z = --- K1P , ∂ 2 f 8 4 t lin Ω (29) 1 + K ()∆τ, 1 ()γ ()∆ q 11 []α ()α˜ z Ð 1α 2 P = ----∑ 1 Ð q Eq coth------τ . = ---------+ 2 1 + K1 Ð 1K1 K2 Ð ------1K1 . N 2 8 z z q (24) Equation (29) is equivalent to the requirement that From the condition that exact equation (23) and 〈 z 〉 〈( z 〉 approximate equation (24) be the same, we obtain the Rf sf = Rf)lin sf and, therefore, that Eq. (23) be iden- fifth equation needed to close the set of equations. tical to Eq. (24). Thus, instead of the ST condition used Equation (24) is conveniently represented in an in [8], we derived a new self-consistency condition in equivalent form by expressing it in terms of the second the form of Eq. (29). Therefore, the self-consistent sec- moment. Using the spectral theorem, the single-site ond-step decoupling scheme can be based on the average can be written as requirement that the first three moments (M0, M1, M2) ∞ be calculated exactly. 〈〉z()z() Ðiωt ()ωω sf t sf 0 = ∫ e J0 d , Ð∞ (25) 5. SELF-CONSISTENT SOLUTION OF THE EQUATIONS 1 J ()ω = ----∑ J()q, ω; τ , 0 N In what follows, the set of equations (20), (15), and q Ω λ ∆ λ2 α (29) with q = Eq( ) and = 1K1/2 will be solved where J (ω) is the spectral density of the single-site analytically and numerically. For this purpose, we rep- 0 ω ≡ ω Green’s function Gff( ) G0( ). resent Eq. (21) in the form The zeroth moment is defined as the average in ()∆τ, ()∆ ()∆τ, Eq. (25) at t = 0: In = In + Bn , n 1 ()γ1 Ð γ ()Ð (30) z z z z 1 ()∆ q q M ===〈〉s ()0 s ()0 〈〉s s --- In = ----∑------()∆ , 0 f f f f N Eq 4 q ∞ Ω (26) 1 A()q  n = J ()ωω d = ------coth ------q . ()γγ () ∫ 0 ∑ Ω τ ()∆τ, 1 1 Ð q Ð q N 2 q 2 Bn = ----∑------∞ q ()∆ Ð N Eq q (31) This relation is a sum rule and reproduces the expres- × 2 ------()λ ()∆ τ . sion for the correlator K0. Differentiating Eq. (25) with expEq / Ð 1 respect to time and putting t = 0, we obtain the first moment ∆ τ The integrals In( ) correspond to zero temperature ( = ∆ ∆ ∞ 0); therefore, we put = (0) in them. The energy of ∂ z() sf t z() ω ()ωω the system given by Eq. (4) is minimal when K1 reaches M1 ==i------sf 0 ∫ J0 d . (27) ∆ ∂t t = 0 its maximum and, as can be easily shown, when (0) = Ð∞ 0. In this case, the integrals take the form The left-hand side of Eq. (27) can be exactly calculated 1 Ð γ using Eq. (6), which gives K1/4. The same is obtained ()≡ 1 q ()γ n In 0 In = ----∑ ------γ Ð q by calculating the right-hand side; that is, we have an N 1 + q identity. The second moment is defined as q 1 (32) γ ∞ 1 Ð n 2 z = D()γ ------()Ðγ dγ . ∂ s ()t ∫ γ f z() ω2 ()ωω 1 + M2 ==Ð------sf 0 ∫ J0 d . (28) Ð1 ∂t2 t = 0 Ð∞ Here, D(γ) is the density of states (on the square lattice) γ The left-hand side of Eq. (28) is given by exact for the dispersion law q = 0.5(cosqx + cosqy); this den-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1126 KUZ’MIN sity of states is expressed through the complete elliptic The energy of the system per bond (in units of J) is integral of the ﬁrst kind K(x) as ε () 0 ==Ð3/4 K1 Ð 0.352, (39) 2 2 1 1 1 D()γ = -----K()1 Ð γ ≈ --- Ð --- Ð --- ln γ , ε 2 π π which is lower than the energy of the AFM state AF = π 2 Ð0.335 (the best result of the spin-wave theory [3]). An 1 (33) energy very close to this numerical value was found in ε ()γγ [8] ( AF = Ð0.3508) for the sublattice magnetization ∫ D d = 1. ε m = 0.3 and in [9, 10] ( AF = Ð0.345). We will show that Ð1 the solution found is a singlet. A singlet state at τ = 0 A calculation gives the following values: I = 1.396 and must satisfy the condition 〈S2〉 = 0. Let us consider the ∆ τ 0 I1 = I2 = 0.555. The integrals Bn( , ) can also be function expressed through the density of states. 2()τ ≡ 1 〈〉2 1 〈〉 S ---- S = ----∑ sfsm 5.1. Low-Temperature Regime N N fm (40) In the case of τ 0, we calculate the integrals B n 1 3 3 following the procedure employed in [8, 13]. When = ----∑ 〈〉s s ==---∑K ---K()0 . τ N f fr+ 4 r 4 0, the vicinity of the point q = Q is the most fr r important, because this region can lead not only to power-law terms but also to anomalous terms that do From Eq. (19), it follows that not vanish as τ 0. Calculations give K()0 = lim K()q () q → 0 () () D 1 3 B0 t = Ct + ------t , γ Ω (41) 2 K1 1 Ð q q 4τ (34) = ------lim ------coth------= ------, λ q → 0 E ()∆ 2τ α ()2 + ∆ () () () 3 ()3 q 1 B1 t ==B2 t Ct Ð ---D 1 t 2 which gives K(0) = 0 and 〈S2〉 = 0 at τ = 0. Thus, to within terms O(t5). Here, D(1) = 1/π and according to the approximate analytical theories developed to date, the ground state of the spin system on a 2 () 2 t τ λ() square lattice is a singlet. This conclusion is supported Ct ==--- ln------, t / 0 . (35) ε π 2∆ by the result 0 = Ð0.3509 [6] obtained by the exact- diagonalization method for a cluster. We represent the low-temperature dependence of the energy gap parameter in the most general form, 5.3. The Equation for ∆ 2β 2t ∆()t = ∆t exp Ð------0 , (36) ≠ 0 t The solution with C 0 obtained above implies an exponential ∆(τ) dependence, according to Eq. (36), and which allows us to consider two scenarios: we will have allows one to ﬁnd the parameter τ = λt = (πλ/4)C = β 0 0 power-law behavior ( > 0) if t0 = 0 and exponential 0.11. In the case of a pure power-law ∆(τ) dependence ≠ behavior if t0 0. As in [8Ð11], we will say that conden- (for which C = 0), the set of equations has no self-con- sation occurs in the vicinity of the point q = Q = (π, π) sistent solution. It should be noted that in the three- γ (where Q = Ð1) if the function given by Eq. (35) tends dimensional case, the condensate does not occur (C = to a nonzero value as τ 0: 0) because of the square-root dependence of the density of states near the limits of the spectrum. 4t C()0 ==limCt() ------0 ≡ C, (37) In the low-temperature range, Eq. (15) for the t → 0 π energy gap ∆ has a solution which is independent of the preexponential factor in τ 2 Eq. (36). ∆τ()≈  ()τ τ 2 λ--- exp Ð2 0/ . (42)

5.2. The Ground State of the System on a Square Lattice It follows from Eq. (42) that in this temperature range, λ α α The set of equations (20), (15), and (22), which the parameters , 1, and 2 show a pure power-law dependence (as was found in [8]) with the main contri- allows the parameters of the system to be determined ∝τ3 self-consistently, possesses the following solution for bution : τ ∆ = 0, = 0, and P(0, 0) = 0.84: 3 3 λτ()= λÐ ------()τ/λ , λ π ==0.744, K1 0.469, 2 (43) (38) α α α ()τ α τ3 α ()τ α τ3 1 ===2.236, 2 2.65, C 0.189. 1 ==1 Ð A1 , 2 2 Ð A2 .

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TWO-DIMENSIONAL HEISENBERG MODEL 1127

1.0 1.0

0.8

1 0.6

Κ ) τ , 0.5 ( c λ 0.4 λ 0.2 Κ1

0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 τ τ

Fig. 1. Dependence of the hardness parameter of the excitation spectrum λ(τ) (solid curve) and the modulus of the corr- τ τ elator for nearest neighbors K1( ) (dashed curve) on dimen- Fig. 2. Speciﬁc heat c( ) (in dimensionless units) of the spin sionless temperature τ = T/zJ for a square lattice (z = 4). liquid on a square lattice.

τ κτ3 Since K1( ) = K1 Ð in this temperature range, the ceptibility (at ω = 0), in accordance with Eq. (12), has energy of the system behaves as the form

3 ()γ ετ()==Ð()3/4 K ()τ ε + ()κτ3/4 (44) zz K1 1 Ð q K1 1 1 0 χ ()q, 0 ==------2 2 γ ∆ 2 Ω 2λ 1 ++q and the specific heat c(τ) ∝ τ2. q (46) 1 1 = -----α ------γ ∆. 1 1 ++q 5.4. Thermodynamic Properties of the Spin Liquid From Eq. (46), it follows that The replacement of the ST condition by Eq. (29) does not affect the character of the temperature depen- χzz(), α τ dence of the model parameters; however, the numerical q =00 ==1/2 1 0.212, 0, values of the coefficients are changed insignificantly. 2 ()τ τ (47) zz 1 2λ exp 2 0/ For this reason, we will only discuss the main results χ ()qQ=0, ==------, α ∆ α 2 and will not present details concerning self-consistent 1 1 τ calculations of the thermodynamic SL characteristics. with the latter expression diverging as τ 0 (as in the Self-consistent calculations of the hardness parame- AFM state). ter λ(τ) of the excitation spectrum and the modulus of τ the correlation function for nearest neighbors K1( ) are represented in Fig. 1. Their asymptotic temperature dependence (for τ > 2) is described by a power law: λ(τ) ≈ 0.18/τ1/2 and K (τ) ≈ 0.06/τ. 1 0.6 The specific heat (in dimensionless units) is c(τ) = ∂ε(τ)/∂τ; its temperature dependence is shown in Fig. 2.

τ τ ∝ τ2 ) For 0, we have c( ) in accordance with τ 0.4 ( 2

Eq. (44); the maximum is reached at the temperature S τ ≈ ≈ τ * 0.2 2 0, and the asymptotic high-temperature τ ∝ τ2 τ ∝ τ dependence is c( ) 1/ , because K1( ) 1/ . 0.2 The dynamic susceptibility of the spin system (in dimensionless units) is given by [12]

αβ α β 0 0.5 1.0 1.5 2.0 χ ()q, ω = Ð 〈〉〈〉s ()q s ()Ðq ω. (45) τ

In the SL state, the correlation functions are isotropic Fig. 3. Temperature dependence of the mean square of the +Ð zz and, therefore, χ (q, ω) = 2χ (q, ω). The static sus- total spin of the system (per lattice site) S2(τ) = NÐ1〈S2〉.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1128 KUZ’MIN γ λ ()∆ The thermodynamic longitudinal susceptibility, by K1 1 iqr 1 Ð q Eq definition, is [12] = -----λ-----∑e ------()∆ coth------τ , (50) N Eq 2 q zz 1 z 2 z 2 1 χ = --- []〈〉()S ()0 Ð 〈〉S ()0 ≡ --- χ˜ , T T 0 iQr ()xy+ (48) e = Ð1 1 z 2 1 2 1 χ˜ ===---- 〈〉()S ---S ()τ ---K()0 0 N 3 4 and oscillate in sign with distance, as in the AFM state. ( and equals the dynamic susceptibility for ω 0, By definition, K(q) = ∑ exp Ðiqr)Kr; therefore, at q 0. At low temperatures, we have χ˜ zz (τ) = χ˜ zz (0) + r τ3 q = 0 and q = Q, we have a1 , in accordance with the temperature dependence of α τ χ˜ zz α 1( ); the numerical value of (0) = 1/2 1 = 0.212 K()0 = ∑K , K()Q = ∑ K . (51) agrees with the result obtained in [8] if the normaliza- r r tion to z = 4 is taken into account (T/J = zτ, 0.212/4 = r r 0.053, as in [8]). The susceptibility χ˜ zz (τ) reaches a We note that 〈S2〉 = N(3/4)K(0), i.e., 〈S2〉 = 0 at τ = 0 τ ≈ ≈ τ maximum at 0.1 0 and then decreases, with the (the ground state is a singlet); therefore, as can be seen asymptotic high-temperature dependence being 1/4τ. from the expressions for K(0), the correlations of alter- The temperature dependence of the mean square of the nating sign cancel out at τ = 0. It follows from the gen- total spin of the system S2(τ) is shown in Fig. 3. eral expression for K(q) at q = Q that The coefficients characterizing the asymptotic tem- λ ∆ perature dependence of the functions considered above () K1 2 2 K Q = ------coth------. (52) (in the case of τ ∞, where there are no correlations) λ 2∆ 2τ can be found analytically under the following assump- Ω tions: (i) the excitation spectrum is limited, q Substituting the solution for ∆ at low temperatures λ ∆()γ λ ∆ ≡ 2 τ 1 Ð q , i.e., const c ; (ii) S ( ) given by Eq. (42), we obtain α τ 3/4; and (iii) the decoupling parameters 1( ) 1 α τ () τÐ1 ()τ τ and 2( ) 1 and K2 1/4. Condition (ii) is K Q = 2K1 exp 2 0/ , (53) equivalent to K(0) 1. Using the strong inequality ∆ ӷ 2, we find that ∆(τ) 4τ and condition (i) reduces which diverges as τ 0. λ τ τ τ λ∆ to ( ) = c /2 . Asymptotically, I0 2/ , I2 Now, we discuss the behavior of the correlation (1/z)2τ/λ∆, and Eq. (15) for the energy gap takes the functions at large distances r of the order of the linear 2 2 form ∆ = 1/8λ , from which it follows that c = 1/2z = dimensions of the system N (D = 2). At τ ≡ 0, ∆ = 0, 1/8 and c = 0.354. Thus, we have the following asymp- we obtain from Eq. (50) totic (τ ∞) temperature dependence of the parameters of the system: K 1 iqr 1 Ð γ 2 K = ------1 ------e ------q d q. (54) 0.177 0.063 r λ 2∫ 1 + γ ∆τ()==4τ, λτ() ------, K ()τ =------, ()2π q τ 1 τ (49) In this integral, the main contribution comes from the λτ() ∆τ() χ˜ zz()τ 1 region (of radius k Ӷ π) of the point q = Q of the Bril- c ==0.354, =-----τ. 0 4 louin zone. Substituting q = Q Ð k, we obtain These expressions are identical to the numerical calcu- k0r lations to within less than one percent. iQrK 2 1 K ≈ e ------1 ------J ()x dx---, r λ π ∫ 0 r 0 (55) 6. SPACE CORRELATIONS 0.284 K ≈ ------, r ∞, Let r = (x, y) be the vector connecting two arbitrary r r sites of a square lattice (a = 1). According to Eqs. (5) and (19), the space correlators have the form which is indicative of the absence of long-range order in the ground state. Integrating Eq. (55) over the two- 1 iqr K = ----∑e K()q dimensional lattice volume, we find K(Q) ∝ N ; that r N q is, we have a divergence in the thermodynamic limit

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 TWO-DIMENSIONAL HEISENBERG MODEL 1129

(N ∞). The staggered magnetization m is defined state is a singlet (with the total spin S = 0) with an ε as [3] energy per bond 0 = Ð0.352. At T = 0, the AFM state ε with energy AF can compete with the singlet state. For 2 the best result obtained analytically within the spin- 2 1 xy+ m = ----∑()Ð1 s wave theory, we have ε ≤ ε ; that is, the ground state N f 0 AF f is a singlet. It should be noted that the results of numer- (56) ical calculations at T = 0 are contradictory, because the 1 3 1 3 difference in energy of these states is very small. In the = ------∑ K = ------K()Q . N4 r N4 SL state, there is a well-defined short-range AFM order r with spin correlators oscillating in sign according to distance. At high temperatures (T ӷ J), the system In this definition, the factor 1/N is of importance; since asymptotically goes over to the paramagnetic state ∝ τ 2 ∝ Ð1〈 2〉 χ ∝ K(Q) N at = 0, we have m 1/N 0 in the (K1 0, N S 3/4) with susceptibility 1/T. thermodynamic limit. At low temperatures, the dominant contribution in ACKNOWLEDGMENTS Eq. (50) also comes from the vicinity of the point q = Q; this equation can be written as The author is grateful to participants of the Kourovka-2000 School S.V. Maleev, N.M. Plakida, and ikr Yu.G. Rudoœ, as well as to V.V. Val’kov, V.I. Zinenko, iQr4τ 1 e 2 K ()τ ≈ e ------d k. (57) r α 2∫ 2 and A.L. Pantikyurov, for helpful discussions and criti- 1 π k + 4∆ cal comments. The correlation length ξ is commonly defined by the This study was supported by the Russian Founda- equation 4∆(τ) = κ2 = ξÐ2. For large values of r, we find tion for Basic Research, project no. 00-02-16110.

k0r 4τ 1 xJ ()x REFERENCES K ()τ = ------0 dx r α π ∫ 2 2 2 1 2 x + κ r 1. W. Marshall, Proc. R. Soc. London, Ser. A 232, 48 0 (58) (1955). τ ()κ 4 2 exp Ð r 1 … 2. D. C. Mattis, The Theory of Magnetism (Harper and = -----α π------1 Ð ------κ- + . 1 κr 8 r Row, New York, 1965; Mir, Moscow, 1967). 3. E. Manousakis, Rev. Mod. Phys. 63 (1), 1 (1991). ξ ∝ τÐ1 τ τ It follows from Eq. (42) that /a exp( 0/ ), as in 4. P. W. Anderson, Mater. Res. Bull. 8, 153 (1973). [8, 13, 14]. Despite the fact that there are regions of large values of r, such that κr = r/ξ ~ 1, the correlation 5. S. Liang, B. Doucot, and P. W. Anderson, Phys. Rev. | | τ Lett. 61 (10), 365 (1988). function possesses the property Kr 0 as 0 (there is no long-range order). 6. K. Fabricius, U. Low, and K.-H. Mutter, Phys. Rev. B 44 (18), 9981 (1991). It was shown in [14] that the behavior of the two- dimensional Heisenberg model with AFM exchange in 7. N. Mermin and G. Wagner, Phys. Rev. Lett. 17 (22), 1136 (1966). the long-wavelength and low-temperature regions can be described in terms of the quantal nonlinear σ model 8. H. Shimahara and S. Takada, J. Phys. Soc. Jpn. 60, 2394 using the renormalization group method. The results (1991). obtained above correspond to the classical renormaliza- 9. A. F. Barabanov and V. M. Berezovskiœ, Zh. Éksp. Teor. tion regime; however, in a higher order approximation, Fiz. 106, 1156 (1994) [JETP 79, 627 (1994)]. we have ξ/a = Cξexp(2πρ /T), where Cξ ≈ 1; that is, the s 10. A. F. Barabanov and V. M. Beresovsky, Phys. Lett. A preexponential factor is temperature-independent [3, 186, 175 (1994). 14]. Thus, the low-temperature dependence of ξ can be different, but there is always an exponential divergence 11. A. F. Barabanov, V. M. Beresovsky, and E. Zasinas, as T 0 in both AFM and SL singlet states. Phys. Rev. B 52 (14), 10177 (1995). 12. S. V. Tyablikov, Methods in the Quantum Theory of Magnetism (Nauka, Moscow, 1975, 2nd ed.; Plenum, 7. CONCLUSION New York, 1967). 13. M. Takahashi, Phys. Rev. B 40 (4), 2494 (1989). Thus, we have described the Heisenberg model with Hamiltonian (1) on a square lattice as a thermodynam- 14. S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. ically stable homogeneous isotropic nonmagnetic spin Rev. B 39 (4), 2344 (1989). liquid at any temperature T. In the limit as T 0, its Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1130Ð1134. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1082Ð1086. Original Russian Text Copyright © 2002 by Bursian, Zaœtsev. MAGNETISM AND FERROELECTRICITY

Explanation of the Friedel Rule Violations in Electron Diffraction Patterns of Some Metallic Alloys. A Metallic Ferroelectric? É. V. Bursian and A. I. Zaœtsev Russian State Pedagogical University, St. Petersburg, 191186 Russia e-mail: [email protected] Received July 12, 2001; in ﬁnal form, September 17, 2001

Abstract—An assumption of the possible existence of a kind of condensed state possessing metallic conduction while at the same time being made up of electrically polar unit cells is drawn from an analysis of the spe- ciﬁc features of electron diffraction in some metal alloys. Examples of alloys in which such a state may be realized are presented. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In this communication, we show that the diffraction pattern asymmetry observed in some intermetallic non- The diffraction of electrons about 100 keV in energy magnetic compounds is likewise associated with the observed in transmission from thin layers of some asymmetry of the electric potential within each lattice AuCu-type and other binary alloys exhibits an asym- unit cell. metry in the reflection intensities that is sometimes very strong [1Ð3]. The mnk and mnk reflections may differ in brightness so strongly as not to be seen on one side 2. SPECIFIC FEATURES OF ELECTRON of the (000) central reflection at all. This effect could be DIFFRACTION FROM AN due to the crystal being inclined with respect to the ASYMMETRIC CELL POTENTIAL beam; however, for λ ~ 0.04 Å and a lattice constant It appears natural to describe the propagation, ~4 Å, the Ewald sphere becomes virtually plane, thus through a solid, of fast electrons with a de Broglie complicating this trivial interpretation. In addition, wavelength λ Ӷ a, whose kinetic energy is consider- changing the object position does not remove the elec- ably higher than the potential interaction with the crys- tron diffraction asymmetry. In diffraction theory, this is tal lattice, with a wave function in the eikonal approxi- called violation of the Friedel rule [4]. This violation is mation [10], which offers a particularly revealing pat- known to be observed in ferromagnetic materials [5, 6], tern of the wave phase variation inside the medium. We where it is due to the deflecting action of the inherent follow the reasoning outlined in [9]. The wave function magnetic field of the magnetized parts of the sample. ψ of electrons striking a layer of a solid of thickness d When the microscope is adjusted to an image in the at normal incidence is changed after the layer transit by conjugate plane and the bright-field geometry is used, a factor ϕ(x) (see Fig. 1, with only one coordinate dis- this, in accordance with the theory of Abbe, does not played for the sake of brevity): affect the image at all. However, in the case of vignett- ing (asymmetric restriction of inclined beams, dark- d ϕ() i (), ≡ ÐiW() x field geometry), the effect permits the observation of x = exp Ðប------∫dUxzz e , (1) ferromagnetic domains, because the image is produced v z only by the electrons deflected to one side. 0

Some authors have assigned the electron diffraction k asymmetry in nonmagnetic crystals to absorption [7, 8]. z A strong violation of the Friedel rule was also observed in ferroelectrics with spontaneous electric E0 I polarization [9]. An asymmetry in charge density in a unit cell possessing a dipole moment results in an ab k echelle-type effect in optics, where for λ < a (a is the k lattice constant) each lattice unit cell acts simulta- z neously as a small prism by deﬂecting the rays predom- kx inantly to one side of the central maximum (phase-pro- ﬁled diffraction grating). Fig. 1. Wave propagation through an asymmetric potential.

EXPLANATION OF THE FRIEDEL RULE VIOLATIONS 1131 where v is the velocity of an electron moving across z ប the ﬁlm, vz = kz/m. Since the potential energy U(x) is a periodic function, W(x) and ϕ(x) are also periodic [W(x + a) = W(x), ϕ(x + a) = ϕ(x)]. In the absence of U(x), we obviously have ϕ(x) = 1, such that the difference ϕ(x) Ð 1 describes the electron interaction with the solid. Because the interaction is c periodic, the electrons will undergo scattering and π ± acquire wave vectors from the set kxn = 2 n/a, n = 1,

±2, …, with the scattering amplitude being Potential b a/2 ()ϕ[]() a f n = ∫ exp Ðikxnx x Ð 1 dx. (2) Ð/2a | |2 The intensity of the nth reflection is defined as In ~ fn . Here and subsequently, we drop some factors for the sake of clarity. –10 –8 –6 –4 –2 0 2 4 6 8 10 We assume the net interaction of the electron with x the film to be small (W Ӷ 1), a situation where the electron suffers one or only a few scattering events. Then, Fig. 2. Potential U(x) for (a) a symmetric and (b, c) asym- ϕ ≈ metric ion positions. (x) Ð 1 iW(x), such that, as follows from Eq. (2), fn =

Ðf Ð*n and the intensities of the opposite reflections are equal irrespective of the actual form of W: I = I . One can conveniently express W0 through the mutually n Ðn related λ and the accelerating voltage V: The Friedel rule can be violated only in the dynamic approximation, i.e., only in multiple scattering in not U d πE ad W = π------0 --- ≡ ------0 ---. (5) very thin films. Under multiple scattering [W(x) > 1], 0 V λ V λ the term with unity in Eq. (2) for n ≠ 0 virtually does not Ӷ π contribute. As is evident from Eq. (3), for W0 , the intensities For a potential symmetric with respect to some point are symmetric, but as W0 grows, an asymmetry appears. π [U(Ðx) = U(x) and W(Ðx) = W(x), where the origin on the For W0 = 2 , all reflections except that with n = 1 van- π ish. Similarly, for W0 = 2 N, the only reflection retained x axis is at the center of symmetry], we have fn = Ð f Ð*n and the intensities of opposite reflections are equal irre- is that with number n = N. Polarization reversal, i.e., the replacement W0 ÐW0, will also bring about the spective of the actual form of W (In = IÐn). Thus, in the case of symmetric potentials, the Friedel rule also holds replacement In IÐn. in multiple scattering. Thus, an essential point in explaining a violation of This equality does not hold, however, in the general the Friedel rule is the possibility of realization of an case if the following conditions are simultaneously sat- asymmetric potential of the electron interaction with isfied: the lattice. (1) the value of W, which has the meaning of the phase change, is of order unity or larger, and 3. ENERGY RELATIONS (2) the potential within the unit cell is asymmetric, i.e., Such an asymmetry can be obtained if one of the ions in a unit cell is displaced. Consider, for simplicity, Ux()≠≠Ux()Ð ; ϕ()x ϕ()Ðx a one-dimensional problem. Let us assume that the Cu+ for any choice of the origin in x. ion in an AuCu-type alloy does not occupy, for some reason, the central position between the Au+ ions but is To estimate the effect of the asymmetry, we use a displaced from it by ∆x (for instance, ∆x = 10Ð2a). This comb-type approximation for the potential (Fig. 1) and is energetically unfavorable. Denote the corresponding steepen, for simplicity, one side of the comb tooth. loss in energy compared with the original symmetric Then, we have ∆ε arrangement of the ions by +. If, however, the charges sin()W /2 of these ions are not equal because of a large difference f ≈ ------0 . (3) n W Ð 2πn in electronegativities, the resultant distribution of the 0 potential (the pseudopotential, which is usually We have taken into account that the phase incursion is employed in the theory of metals) in such a diatomic lattice will naturally be asymmetric (curve b in Fig. 2). x eU d WW==Ð ---, W ------0 -. (4) In this case, the increase in the ion energy can be com- 0a 0 បv pensated by a decrease of the electronic band in energy

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1132 BURSIAN, ZAŒTSEV combination of the fundamental solutions ϕ (x) and a ϕ 1 2(x) of Eq. (6): b ψ () ()ϕ() ()ϕ() k x = C1 E 1 x + C2 E 2 x . (9) Here, ϕ (x) and ϕ (x) are solutions to Eq. (6) for Ða/2 ≤ c 1 2 x ≤ a/2 that satisfy the boundary conditions ϕ (), ϕ' (), Energy 1 EaÐ /2 ==1, 1 EaÐ /2 0, (10) ϕ (), ϕ (), 2 EaÐ /2 ==0, 2' EaÐ /2 1. The requirement of Bloch’s theorem for the linear combination in Eq. (9) ψ ()λ()ψ() –0.6 –0.4 –0.2 0 0.2 0.4 0.6 k a/2 = k k Ða/2 , (11) k ψ ()λ()ψ() k' a/2 = k k' Ð/2a Fig. 3. E(k) calculated numerically for nondisplaced and yields a characteristic equation displaced ion positions (curves aÐc correspond to Fig. 2). λ2 ()λϕ ()ϕ, (), () +1 EaÐ /2 + 2' EaÐ /2 +wa/2 = 0 (12) ∆ε ( Ð). Let us estimate the possibility of such a compen- with real coefficients. The Wronskian of the equation ϕ ϕ ϕ ϕ sation. W = 1(E, x)(2' E, x) Ð 2(E, x)(1' E, x) is a constant ∆ε The energy + can be estimated from the relation [W(x) = W(0)] equal, according to Eq. (10), to unity. λ ∆ε ~ mxω2 2, where ω is the frequency of longitu- By solving this equation, one can find as a func- + LO LO tion of energy and, thus, determine E(k). For this energy dinal optical vibrations. For x ~ 10Ð2a, the cost in to belong to an allowed band (for k to be real), the con- energy is of the order of 10Ð2 eV. dition Now, we estimate ∆ε . The interaction potential of a ϕ ()ϕ, (), ≤ Ð 1 Ea/2 +22' Ea/2 (13) free electron with the ion core can be obtained as a sum of effective interaction potentials with the ions U(x) = has to be met. Figure 3 presents the energies of the lower band E(k) Uα (|x Ð x α|), where x α is the position of the ion of ∑ i i obtained by numerical calculation of the lower-band i, α α energy E(k) for several ion displacements. The lower species in the ith unit cell and Uα(x) is the effective valence band is seen to lower to yield a gain in electron electron interaction potential with an ion of species α. ∆ε energy Ð. Considered from the physical standpoint, We shall choose a sufficiently “shallow” effective this means that ion asymmetry makes one of the poten- pseudopotential of the electron interaction with an ion tial boxes slightly wider. This conclusion depends only such that it will accommodate one atomic level only. weakly on the actual form of the effective potential. This is needed to ensure filling of the valence band For displacements of the order of 10Ð2a, the values ∆ε states with a sufficiently large group velocity (i.e., with of Ð are also about a few hundredths of an electron- a good conductivity). volt, such that, on the whole, the gain in energy for some displacements may, in principle, be real in some The electron wave functions ψ (x) in the periodic k cases, thus making a state with displaced core ions potential are solutions to the Schrödinger equation energetically favorable. ψ ()ψ() () k" +00.26 Ek Ð Ux k = (6) 4. ELECTRON DENSITY DISTRIBUTION which satisfy, according to Bloch’s theorem, the AND DIPOLE MOMENT OF THE UNIT CELL requirement The electronic state thus produced is of particular ψ ()λxa+ = ()ψk ()x , (7) interest. The electron density distribution (the probabil- k k ity density of an electron being at point x) can be writ- where ten as ρ() ψ()2 λ()k = exp()ikx (8) x = k x k, (x is measured in Å; E and U, in eV). where 〈 f (k)〉 = ∫ f (k(E))g(E)dE. Here, g(E) is the den- ψ The wave function k(x) for a given energy E was sity of states in the band and integration is performed looked for within a unit cell Ða/2 ≤ x ≤ a/2 as a linear over the band energies. The dipole moment is calculated

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 EXPLANATION OF THE FRIEDEL RULE VIOLATIONS 1133 as Ð∫exρdx = Ð∫Px()dx . The corresponding curves 0.7 are displayed in Fig. 4. The electron density is naturally 0.6 the highest at the atom with a deeper potential well. 0.5 The total dipole moment of the unit cell is the sum of the ion and electron contributions. We note that the 0.4

total charge of the unit cell is zero and the ion dipole ρ moment of the cell should not depend on the method 0.3 used in the calculations. As follows from the calculations, adding these contributions in the case of dis- 0.2 placed ions gives a state possessing a nonzero total 0.1 dipole moment of the unit cell. 0 5. DISCUSSION 1.5 Thus, the asymmetry in reflections reveals an asym- 1.0 metry in the internal electric field, i.e., the cell polarization, and theoretical estimates support the possibility of realization of this state. Any total dipole moment in a 0.5 metal crystal becomes naturally screened completely in a time less than 10Ð16 s, and macroscopic polarization is P 0 always exactly zero. This does not imply, however, that a unit cell cannot have a dipole moment. –0.5 As for matching these moments in direction in adjacent cells, such a coherence between charge displace- –1.0 ments is not caused by the dipoleÐdipole interaction, as –1.5 is the case with ferroelectrics. Thus, extrapolation of –4 –2 0 2 4 the dielectric approach used when considering conduct- x ing ferroelectrics would be invalid. In this case, the long-range interaction is due to a purely quantum Fig. 4. Distribution of band-averaged electron density ρ and effect, namely, the need of joining the electron wave dipole moment P (in arbitrary units) over the unit cell. One functions (with Bloch’s theorem not violated). of the atoms (with a shallower potential well) is at the center (x = 0), and the other is displaced from the unit-cell edge Because the potential distribution within a unit cell (x = +d/2 = +3) to the center by 2/15 (dashed curve) and by and within one domain of macroscopic size differ little 3/15 (dotted line). from the situation in poorly conducting pyroelectrics and ferroelectrics (although for another reason), there are grounds to use, in this case, the term metallic ferro- diffraction measurements. We have not succeeded in electric. To be more rigorous, this is one more kind of finding published data on the refinement of point condensed state. groups of metallic alloys in this context. Revealing this state in a macroscopic experiment A study of the literature on intermetallic alloys [12, (pyroelectric or piezoelectric effect, etc.) appears diffi- 13] shows that, in addition to the ordering phase transi- cult if at all possible because almost instantaneous screening sets in. It may turn out that electron scattering λ Ӷ for a is the only method by which one can detect Phase transition temperatures in some binary metal alloys spontaneous electrical polarization in strongly conduct- Melting Other phase ing crystals, naturally provided that one does not reveal Alloy effects where such polarization noticeably affects the temperature, °C transitions, °C macroscopic properties (see, e.g., [11]). AuMn 1260 600, 232, 124 In accordance with the Neumann principle, in order AuZn 600 515, 225 to learn whether a metallic compound can be classed 3 among pyroelectrics, one has to prove that the point Au3Zn 700 425, 230 group of the crystal does not contain certain symmetry AuCu 910 410, 385 elements. One has to establish that this point group is a CuZn 1100 450, 200, Ð100 ∞ subgroup of the limiting group mm, i.e., that it Cu Zn 1000 500, 210 belongs to one of the following classes: 1, 2, 3, 4, 6, m, 3 mm2, 3m, 4mm, and 6mm. However, metallographic lit- NiTi 1310 800, 100 erature and reference books present, as a rule, only the Note: The temperatures of the ordering phase transitions are type of Bravais lattice inferred from x-ray and neutron underlined.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1134 BURSIAN, ZAŒTSEV [14]. These domains were called antiphase [6], and they L were given another interpretation. The possibility of the unit cells making up a domain having ordered polarity 910 °C was not considered. At the same time, the presence of such domains suggests the existence of a nontrivial mechanism of long-range interaction and possibly α bears on the considerations outlined here.

ACKNOWLEDGMENTS 410 °C The authors are indebted to the Chair of Theoretical Physics, RSPU, for fruitful discussions. This study was supported by the Russian Founda- II tion for Basic Research (project no. 00-02-16735). 385 °C REFERENCES 1. J. B. Coreia, H. A. Davies, and C. M. Cellars, Acta I Mater. 45 (1), 177 (1997). 2. M. J. Whiting and P. Tsakiropoulos, Acta Mater. 45 (5), 2037 (1997). 3. N. D. Zemtsova, Fiz. Met. Metalloved. 89 (3), 75 (2000). 4. J. M. Cowley, Diffraction Physics (American Elsevier, New York, 1975; Mir, Moscow, 1979). 5. R. Heidenreich, Fundamentals of Transmission Electron Au 50 at. % Cu Microscopy (Interscience, New York, 1964; Mir, Mos- cow, 1966). Fig. 5. Part of the phase diagram of AuCu. L is liquid, α is 6. P. B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, the disordered phase, II is the orthorhombic phase with and M.J. Whelan, Electron Microscopy of Thin Crystals superstructure (a ~ 3.96, b ~ 43Ð44, c ~ 3.85 Å), and I is the (Butterworths, London, 1965; Mir, Moscow, 1968). face-centered tetragonal phase (a ~ 3.95, c ~ 3.68 Å). 7. S. Miyake and R. Uyeda, Acta Crystallogr. 8, 335 (1955). 8. G. Kästner, Acta Crystallogr. 43 (5), 683 (1987). tions investigated comprehensively in metallography, one or more structural transitions are observed in many 9. É. V. Bursian, A. B. Vall, and N. N. Trunov, Izv. Akad. cases at lower temperatures; those transitions involve a Nauk, Ser. Fiz. 58 (5), 219 (1994). change in lattice symmetry and superficially resemble 10. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic the transitions that occur in ferroelectrics. They are Theory (Nauka, Moscow, 1974; Pergamon, New York, exemplified in the table. 1977). The largest difference between the electronegativi- 11. É. V. Bursian and Ya. G. Girshberg, Coherent Effects in ties is found in the AuCu alloy, which has been well Ferroelectrics (Prometeœ, Moscow, 1989). studied in metallography and can serve as a model com- 12. M. Hansen and K. Anderko, Constitution of Binary pound. Figure 5 shows a part of its phase diagram. Alloys (McGraw-Hill, New York, 1958; Metallurgizdat, Phases existing below 410°C are suggestive of having Moscow, 1962). ordered electrical unit-cell polarization (the onset of 13. A. E. Vol, Structure and Properties of Binary Metallic quantum phase coherence). Systems (Nauka, Moscow, 1960), Vol. 3. It can be added that there are numerous accounts 14. D. W. Pashley and A. E. B. Presland, J. Inst. Met. 87, 419 even of extrareflections, superstructure, and the obser- (1959). vation of a fine but distinct domain pattern in AuCu Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1135Ð1144. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1087Ð1095. Original Russian Text Copyright © 2002 by Kvyatkovskiœ. MAGNETISM AND FERROELECTRICITY

On the Nature of Ferroelectricity in Sr1 Ð xAxTiO3 and KTa1 Ð xNbxO3 Solid Solutions O. E. Kvyatkovskiœ Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received October 10, 2001

Abstract—Cluster calculations of the local adiabatic potential for an impurity atom in position A in Sr1 Ð xAxTiO3 (A = Mg, Ca, Ba, Pb, Cd, Zn), as well as for the Nb and O atoms in the TaÐOÐNb chain in KTa1 Ð xNbxO3, were carried out in the nonempirical HartreeÐFockÐRoothaan MO-LCAO formalism. For comparison, similar calculations of the local adiabatic potential were performed for a sublattice-A atom in the ATiO 3 cubic perovskites (A = Ca, Sr, Ba, Pb), for K and Ta atoms in KTaO3, and for Li in K1 Ð xLixTaO3. The calculations revealed that in all the cases considered, except the Zn, Mg, and Li impurities, the impurity atoms move in single-well potentials and that the corresponding solid solutions are displacive ferroelectrics. Zn in Sr1 Ð xZnxTiO3 and Mg in Sr1 Ð xMgxTiO3 were found to occupy off-center positions, as does the Li atom in K1 Ð xLixTaO3; i.e., they move in a multiwell local potential. An explanation is proposed for the ﬁrst-order Raman scattering observed in the paraelectric phase of the above solid solutions with central impurities. The critical concentration xc for the displacive KTa1 Ð xNbxO3 and Sr1 − xAxTiO3 solid solutions was calculated in the virtual-crystal approximation within the soft ferroelectric mode theory. The values of xc thus obtained agree with the available experimental data. © 2002 MAIK “Nauka/Interpe- riodica”.

1. INTRODUCTION appears at a temperature Tm identiﬁed with the phase- transition temperature TC. In accordance with the theo- The SrTiO3 (STO) and KTaO3 (KTO) quantum paraelectrics [1Ð4] and the related solid solutions [1Ð6] retical predictions made for displacive low-temperature have recently attracted considerable interest. Under ferroelectrics [10, 11, 18, 28Ð30], in KTa1 Ð xNbxO3 (KTN) [20], Sr1 Ð xAxTiO3 (A = Ca, Ba, Pb, Cd) [5, 6, normal conditions, STO and KTO are incipient ferro- 16 18 electrics. Having anomalous dielectric properties at low 21Ð27], and SrTi ( O1 Ð x Ox)3 [17], the transition tem- temperatures, these materials remain paraelectrics perature near xc scales with concentration as TC(x) = 1/2 down to 0 K [7, 8] because of the quantum effects [9Ð A(x Ð xc) . 11]; in STO, this is likewise due to the cubic structure For isotopic impurities, both the nature of the becoming distorted below the point of the structural ordered state in SrTi(16O 18O ) [16, 17] and the phase transformation (105 K) [12]. STO and KTO are 1 Ð x x 3 similar in dielectric properties to the paraelectric phase phase-transition mechanism [18, 19] are presently understood quite well. However, the mechanism of the of real perovskite ferroelectrics (BaTiO3, PbTiO3, KNbO ); in fact, there is a soft IR-active transverse induced phase transition and the nature of the ordered 3 polar state in quantum paraelectrics doped by noniso- optical (TO) mode whose frequency ω tends to zero f topic impurities still remain unclear. Some authors with decreasing temperature [12, 13] and the dielectric identify the phase transition as displacive ferroelectric permittivity ε grows anomalously for T 0 K [7, 8]. [20, 21, 31], while others [32, 33] believe the low-tem- ε ω2 However, the growth of (T) and the decrease in f (T) perature phase in KTN to have a glasslike character. saturate below 40 K for STO and 30 K in KTO. As a Ordering in KTN was shown in [34] to exhibit long- result, the soft mode remains stable down to 0 K and range order. The phase transformation in weakly doped ε ε ε (T) reaches the values a = 41900 and c = 9380 in Sr1 Ð xCaxTiO3 is interpreted in [35Ð37] as a diffuse, per- STO [14] and ε = 3800 in KTO [15], while varying only colation-type ferroelectric transition which is associ- weakly in the immediate vicinity of 0 K [7, 8]. ated with the presence of ferroelectric microregions Because STO and KTO are close to the ferroelectric (FMR) induced by Ca impurities in a strongly polariz- state, even a weak doping by isoelectronic impurities able host [38]. The ordered state in Sr1 Ð xBaxTiO3 was drives them to a low-temperature ferroelectric state or a found [23, 26] to be glasslike for xg = 0.0027 < x < xc = glass-type polar phase [1Ð6, 16Ð19]. There is a critical 0.035 and ferroelectric (with long-range order) for concentration xc above which a maximum in the tem- x > xc = 0.035. A similar situation is observed to exist in perature dependence of the dielectric permittivity Sr1 Ð xCdxTiO3 [27].

1063-7834/02/4406-1135 $22.00 © 2002 MAIK “Nauka/Interperiodica” 1136 KVYATKOVSKIŒ To explain the nature of the FMRs, the specific fea- data suggests that Li in KTL is an off-center ion and tures in the dielectric properties [32], light scattering moves in a multiwell potential [48]. experiments [39], and the fine structure in the x-ray The above subject of controversy is connected inti- absorption spectra (XAFS) [40], the model of an off- mately with the problem of the existence of structural center impurity ion moving in a multiwell potential is disorder in pure perovskites. To explain their experi- frequently used. Interaction between the impurity-ion mental results, some authors [49Ð53] suggest that the Ti dipole moments via the soft polar TO mode was shown and Nb atoms in the cubic phase of BaTiO3 (BTO), in [41, 42] to play an important role in the low-temper- PbTiO3 (PTO), and KNbO3 (KNO) are off-center and ature state of the system in doped incipient ferroelec- move in a multiwell local potential. Although the trics with off-center impurities. This interaction can ini- experimentally observed anomalous diffuse x-ray scat- tiate either a dipole-glass phase or the transition to a tering in perovskites [49] can be attributed to specific long-range order ferroelectric state, depending on the features in the perovskite phonon spectrum [54Ð57], actual impurity concentration [2, 42]. The latter will be one should not disregard the results of other experi- an orderÐdisorder phase transition occurring in the sys- ments, for instance, of those on inelastic light scattering tem of impurity ions in a strongly polarizable medium. [51] and XAFS [53], which defy, at present, all The existence, in the local impurity potential, of several attempts to make interpretations in terms of delocalized equivalent minima that are displaced from lattice sites collective excitations (phonons). and separated by energy barriers suggests an obvious We present here a solution to the problem of the relaxation mechanism capable of accounting for the local structure of doped and pure perovskites based on glasslike behavior that has been observed in many cases direct nonempirical calculations of the local potential [23, 27, 32, 33, 43]. Finally, the existence of off-center for impurity ions in STO : X (X = Ca, Ba, Pb, Cd, Mg, impurities may obviously explain the FMR formation. Zn) and in KTO : X (X = Li, Nb), as well as for atoms If an impurity ion occupies, on average, the central at the A and B sites in the pure perovskites BTO, STO, position at a lattice site and moves in a single-well CaTiO3 (CTO), PTO, KNO, and KTO. We discuss the potential, the transformation will more likely be a dis- origin of first-order Raman scattering in the paraelec- placive phase transition in a random field of impurity tric phase of STO : Ca and KTN. In Section 3, we also centers. Thus, the shape of the local potential acting on consider an approach to describing the static properties an impurity atom is a problem of crucial importance in of the Sr1 Ð xAxTiO3 and KTa1 Ð xNbxO3 solid solutions understanding the nature of the phenomena observed in with central impurities which is made in the virtual- doped quantum paraelectrics. crystal approximation in terms of the theory of the soft ferroelectric mode. The existence itself of off-center impurities in the compounds under consideration is believed in [43] to have been established reliably. However, the only thing 2. LOCAL POTENTIALS proven to date is the off-center position of the Li+ ion in FOR ATOMS IN PEROVSKITES K1 Ð xLixTaO3 (KTL) [1, 2, 44]. The off-center position As already pointed out, adequate description of of Li+ in KTL can be assumed based on the consider- some properties of perovskite compounds and of the able difference in size between Li+ and K+ [45]. In all related solid solutions requires knowledge of the shape other cases, the off-center position of an impurity may of the local adiabatic potential for the host and impurity be questioned [46]. For instance, the Ba2+ and Pb2+ ions atoms. Cluster ab initio calculations for B-type and substituting for the Sr2+ ion in STO are larger than Sr2+ oxygen atoms in pure perovskites were reported in [58, and they move, most likely, in a single-well potential 59]. Ab initio local-potential calculations for the impu- 2+ 2+ 2+ 2+ + [45]. The Ca , Mg , Cd , and Zn ions are smaller rity atom Li (Li ion) in KTaO3 : Li (KTL) were made than Sr2+ [45]. It would be difficult, however, to make a in [44]. We report here on cluster ab initio calculations sound judgement of the shape of the local potential for of the local adiabatic potential for the K, Ta, and oxy- an impurity ion drawing only from empirical estimates gen atoms in KTO and for A-type atoms in CTO, STO, of the ion size. As for KTN, the Nb5+ and Ta5+ ions have BTO, and KNO, as well as for the impurity atoms Ba, equal ionic radii within the accuracy with which these Ca, Mg, Cd, and Zn in STO, Li in KTL, and Nb in empirical quantities were determined [45]. The inter- KTN. pretation of experimental data obtained on this compound was based in many studies, starting with the work of Yacoby [39], on the assumption of Nb being 2.1. Cluster Model and Method of Calculation off-center. However, the conclusion that Nb atoms of the Total Energy occupy off-center positions is based on indirect data, in To find the local adiabatic potential for an atom sit- the sense that one can put forward another explanation ting at a lattice site, one has to calculate the total crystal of the observed phenomenon. A direct study of the local energy as a function of atomic displacement from the potential for the Nb atom in KTN based on 93Nb NMR equilibrium position at the site, ∆E(η), assuming the measurements [47] shows Nb to move in a single-well remaining atoms to occupy their equilibrium positions local potential [47], whereas an analysis of 7Li NMR in the perovskite cubic lattice. The supercell [44, 60,

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 ON THE NATURE OF FERROELECTRICITY 1137

61] and quasimolecular cluster [58, 59] methods are the (a) most suitable approaches to nonempirical calculation of the properties of point defects in crystals.1 We use here the approach described in [58, 59]; this method is based essentially on isolating the fragment of the crystal of interest, which ensures correct description of the chemical bonding and point symmetry. Thereafter, this fragment is simulated by a quasimolecular cluster with the geometry of the crystal fragment. B The smallest cluster that can reproduce the chemical A bonding and the local properties (determined by this (b) O bonding) for an atom at site B is the octahedral cluster nÐ H (BO6) (cluster I). The cluster second in complexity that is appropriate in symmetry to a B atom consists of seven octahedra, more specifically, of a central (B'O6) and six neighboring (BO6) octahedra each sharing a common oxygen atom with the central octahedron. This cluster also contains eight A-type atoms occupying the corners of the cubic primitive cell. To reduce the cluster charge, hydrogen atoms are attached to the broken OÐB bonds at a distance of 1 Å. All this adds up to (c) n+ [B'O6 A8B6(OH)30] positively charged clusters, where n = 2 for BTO, STO, CTO, and PTO and n = 1 for KNO and KTO (cluster VII). The smallest possible cluster for an A atom is for- nÐ mally ((AO12) ). However, this cluster cannot provide an adequate description of the local potential for the A atom. Therefore, we shall consider as the minimum cluster for atom A the cluster which is next in complexity and contains seven A atoms (A' at the cluster center Fig. 1. Main and additional fragments of clusters (a) VII, (b) and six A atoms at the fourfold axes) and eight octahe- VIII, and (c) II. dra (BO6) with B-type atoms lying on the threefold axes at the corners of the cubic cell. To reduce the cluster charge, hydrogen atoms were added to the broken OÐB Each cluster also contains additional fragments; one of bonds at a distance of 1 Å. This results in negatively them is shown in Fig. 1 for each cluster. The total num- nÐ ber of additional fragments is determined by the cluster charged [A'O12 A6B8(OH)24] clusters, where n = 2 for symmetry: six Ti(OH)5 groups in cluster VII, eight BTO, STO, CTO, and PTO and n = 1 for KNO and (OH) groups and six A atoms in cluster VII, and two KTO (cluster VIII). In the case of a perovskite solid 3 H4OH groups in cluster II. solution AB B' O , the smallest cluster for the oxy- 1 Ð x x 3 By definition, the local potential is the difference gen atom consists of two octahedra, (BO6) and (B'O6), between the total energies of the crystal in the distorted sharing a common oxygen atom. This cluster also con- and undistorted configurations: tains four A atoms (Ca, Sr, Ba, Pb, K) at the corners of ∆ ()η ()η the cubic-cell face perpendicular to the BÐOÐB' chain; E = E Ð E0, (1) in order to reduce the cluster charge, hydrogen atoms were added to the broken OÐB outer bonds at a distance where η is the displacement of the central atom from of 1 Å from the oxygen atom. This yields a positively the lattice site. The calculations were made using the n+ ∆ η ≈ ∆ cl η cl η charged cluster [OBB'A4(OH)10] , where n = 2 for approximation E( ) E ( ), where E ( ) is the KNO and KTO (cluster II). total energy of the corresponding cluster. The total energies and the one-electron cluster properties were We shall describe now the way in which the clusters calculated by the HartreeÐFockÐRoothaan nonempiri- were constructed. The main fragments are the primitive cal MO LCAO SCF formalism [63] using the PC cubic cells for the ABO3 perovskite structure that are GAMESS version [64] of the GAMESS (US) QC pack- centered on the corresponding atom, namely, on a B age [65]. The correlation effects were taken into atom for cluster VII (Fig. 1a), an A atom for cluster VIII account (for simple octahedral clusters) within the (Fig. 1b), and on an oxygen atom for cluster II (Fig. 1c). Möller–Plescet perturbation theory (MP2) [65]. The 1 There also exists a method of quantum clusters embedded in a calculations were made using the following sets of crystal lattice [62]. Strictly speaking, however, this method is atomic basis functions: for the oxygen atoms, the basis semiempirical. sets were TZV (10s6p)/[5s3p] [65, 66] with d-type

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1138 KVYATKOVSKIŒ

Table 1. Local force constants kloc (determined by short-range for the Ta atom at the center of clusters I and VII, were interaction) for atoms at the A and B sites in ABO3 perovskite WTBS [69]. For the Li, Mg, Ca, and Zn atoms in the A' compounds obtained by cluster ab initio calculations position, we also used the Roos ADZ ANO sets [70]. loc 2 For the remaining K, Ca, Sr, Ba, and Pb atoms in clus- Compound Atom Cluster k , eV/Å a0, Å ters VIII, as well as for Ta in clusters VII, we used the CaTiO Ca VIII 7.6 3.8367 effective core potential with the corresponding basis 3 sets for valence orbitals [71]. The WTBS and Roos Ti I 32.2 (29.4) ADZ ANO basis sets and the relevant information were VII 32.0 obtained from [72].

SrTiO3 Sr VIII 13.5 3.905 Ti I 28.6 (26.0) 2.2. Pure Perovskites VII 28.5 We consider a crystal with only one atom belonging, for example, to sublattice s and displaced from the cen- BaTiO3 Ba VIII 15.9 3.996 tral equilibrium position (at the lattice site). We deﬁne loc Ti I 24.5 (22.2) local force constants k (s) as coefﬁcients of the quadratic terms in the expansion of the total crystal energy VII 23.9 η in powers of displacements i(s) of this atom: PbTiO3 Pb VIII 6.8 3.970 1 loc 4 ∆E()η ==E()η Ð E()0 ---k ()ηs η + ᏻ()η , (2) Ti I 25.6 (23.2) 2 i j VII 20.1 where the second term on the right-hand side includes 2 KTaO3 K VIII 7.0 3.9845 the anharmonic terms of expansion ∆E. According to loc ∆ η η2| Ta I 43.7 (43.4) Eq. (2), k = 2 E( )/ η → 0 [for symmetry considerations, Eq. (2) does not contain linear terms]. VII 39.0 Table 1 presents the results of cluster ab initio calcu- loc KNbO3 K VIII 6.4 4.0214 lations of the local force constants k (s) performed for Nb I 35.7 (35.7) the A- and B-type atoms in the cubic perovskites CTO, STO, BTO, PTO, KNO, and KTO. We readily see that all VII 30.6 the kloc constants are large compared to the atomic force Note: The values shown in parentheses include the correlation 2 3 ≈ 2 loc loc constant for the perovskites kat = e /r0 2 eV/Å , where corrections to k . The values of k (B) for STO, BTO, ≈ PTO, and KNO were taken from [59]. The values of the lat- r0 2 Å is the BÐO bond length, as well as in compari- loc tice constant a0 were taken from [73] (for KTO, from [15]). son to the k for cations in typical ionic crystals [58]. The values of kloc are the largest for the oxides of tanta- loc lum, niobium, and titanium, which is indicative of the Table 2. Force constants k for A' atoms in the ABO3 : A' stabilizing effect the BÐO bond covalency exerts on the perovskite compounds and for Nb atoms in KTaO3 : Nb obtained from nonempirical cluster calculations stability of the central position of these atoms in the nÐ (BO6 ) oxygen octahedron [59]. These results sug- Compound Cluster kloc, eV/Å2 gest that the assumption of the B atoms in the paraelectric phase of perovskite ferroelectrics being STO : Ca VIII 6.4 off-center, which is frequently used in interpreting STO : Ba VIII 19.4 XAFS [40, 53] and in inelastic light scattering [51] experiments, is apparently erroneous. STO : Pb VIII 8.6 STO : Cd VIII 3.6 2.3. Doped Perovskites STO : Mg VIII Շ0 Table 2 presents the results of cluster ab initio cal- STO : Zn VIII Շ0 culations of the force constants kloc for the impurity KTO : Li VIII Շ0 atoms Ca, Ba, Pb, and Cd substituting for Sr in STO and

KTO : Nb VII 35.3 2 In contrast to the optical force constant k(s), which corresponds to the displacement of sublattice s as a whole and whose magnitude in the perovskites is affected noticeably by the long-range polarization functions [67]; for the hydrogen atoms, the dipoleÐdipole interaction [59], the local force constant kloc(s) DZV of Dunning and Hay [65]; and for Ti and Nb, contains contributions due to short-range interactions only, i.e., loc sr (13s7p5d) and (14s8p7d), respectively, taken from k (s) = k (s). The contribution of the dipoleÐdipole interaction to the restoring force in a lattice with one displaced atom falls off [68]. The basis sets used for the Li, K, Mg, Ca, Sr, Ba, with distance R as RÐ6. The contribution due to the Madelung Pb, Cd, and Zn at the center of cluster VIII, as well as energy to kloc(s) for the A and B sites in cubic perovskites is zero.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 ON THE NATURE OF FERROELECTRICITY 1139

loc for the Nb impurity atom substituting for Ta in KTO. Table 3. Local force constants k for A' atoms in ABO3 : A' Let us compare kloc from Table 2 for the atom acting as compounds derived from nonempirical cluster calculations an impurity with that from Table 1 for the same atom, (cluster VIII) for two basis sets for the A' impurity atom but in the dissolved component of the solid solution, (WTBS [69] and Roos ADZ ANO [70]) namely, Ca in STO : Ca and CTO, Ba in STO : Ba and kloc, eV/Å2 BTO, Pb in STO : Pb and PTO, and Nb in KTN and loc Compound KNO. We readily see that the corresponding k con- WTBS Roos ADZ ANO stants have close values and that the differences are caused primarily by the difference in the lattice con- STO : Ca 6.4 5.4 stants between the host and the dissolved component. STO : Mg 0.25 Ð0.31 According to Table 2, the impurity atoms in these solid solutions move in a single-well potential; i.e., they are STO : Zn Ð0.16 Ð0.56 central impurities. The situation with the Nb impurity KTO : Li 0.004 Ð0.24 in KTaO3 is particularly interesting. Contrary to the fairly common idea of the Nb impurity in KTaO3 being off-center [39, 40], calculations show (Tables 1, 2) that 3 As already mentioned, the paraelectric phase of the KTaO3 lattice is too tight for the Nb impurity atom. STO : Ca and KTN exhibit ﬁrst-order Raman scatter- Thus, STO : Ca, STO : Ba, STO : Pb, STO : Cd, and KTN can be considered to be displacive ferroelectrics ing, which is forbidden in the cubic phase of pure per- (we shall call them, for brevity, displacive solid solu- ovskites. To explain this phenomenon, the assumption tions). of the impurity atom (Nb) in the cubic cell being off- center was put forward [39]. As shown above, the Nb The situation with materials doped by Mg, Zn, and atom in KTN and the Ca atom in STO : Ca occupy cen- Li is radically different (Tables 2, 3). The harmonic tral positions with symmetry O in the paraelectric force constants kloc for the Mg, Zn, and Li impurity h atoms are small in magnitude (|kloc| Շ 0.5 eV/Å2) and phase. At the same time, the existence of such central negative, as shown by exact calculations making use of impurity atoms brings about a lowering of symmetry 4 large basis sets [70] for impurity atoms. This means for the neighboring oxygen atoms from D4h to C2v in that the Mg and Zn impurities in STO and Li in KTO the CaO12 complexes in STO : Ca and down to C4v in are off-center and move in a multiwell local potential, the NbO6 complexes in KTN. In view of the fact that whose shape in the [001] direction is displayed in oxygen atoms take part in all optical lattice vibrational Figs. 2 and 3. The displacement from the central posi- modes in perovskites, this lifts the forbiddenness from η ∆ tion min and the corresponding gain in energy Emin ﬁrst-order Raman scattering. This reasoning also holds calculated in the HartreeÐFock approximation are for other solid solutions with central impurity atoms. 0.32 Å and 7 meV for STO : Mg, 0.41 Å and 24 meV for STO : Zn, and 0.63 Å and 22 meV for KTL, respectively. 2 0.04 1 3 5 6 These results, demonstrating the off-center position 4 of the Mg and Zn impurities in STO and of Li in KTO, are primarily of a qualitative character. Because of the smallness of kloc Շ 1 eV/Å2, the values of the local 1 Ba potential parameters can change markedly toward 0.02 2 Pb increasing |kloc| when either the electronic correlations 3 Ca 4 Cd , eV or the structural relaxation of the impurity atom neigh- 0 5 Mg η ∝ | loc|1/2 E borhood are taken into account; we have min k 6 Zn ∆ ∝ | loc|2 loc ӷ 2 – and Emin k . At the same time, for k 1 eV/Å , E

the effect of the electronic correlations is small (Table 1) = 0 E

and their inclusion, as well as taking into account the ∆ structural relaxation of the impurity environment, cannot affect the conclusion that the Ca, Ba, Pb, and Cd impurities in STO and Nb in KTO occupy the central positions. –0.02

3 As follows from calculations, the anharmonic contribution to the local potential is positive and virtually does not affect the shape –0.8 –0.4 0 0.4 0.8 of the potential down to η ≈ 0.2 Å. Impurity atom displacement along [001], Å 4 As evident from the data for kloc in STO : Ca (Table 3), the basis loc loc 2 corrections to k are signiﬁcant only for k Շ 1 eV/Å and Fig. 2. Local adiabatic potential for the Ca, Ba, Pb, Cd, Mg, small for |kloc| ӷ 1 eV/Å2. and Zn impurity atoms in STO.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1140 KVYATKOVSKIŒ

1 2 The lowering of symmetry for the oxygen atoms surrounding impurity atoms results in oxygen atom dis- 1 Nb placements directed either inward or outward, depending 2 K on the size of the impurity ion. These considerations are, 3 Li however, insufﬁcient for determination of the character 0.02 of the Nb environment relaxation in KTN, because, according to [45], the octahedrally coordinated Nb5+ and Ta 5+ ions have the same radii of 0.6 Å. To establish the character of the relaxation the nearest environment of a Nb atom undergoes in KTN, we carried out a cluster , eV 0 ab initio calculation of the local potential for the oxygen E atom in the TaÐOÐNb chain in the longitudinal direction –

E which was based on cluster II (Fig. 1c). The results of the

= 0 calculation are shown graphically in Fig. 4. A calculation E

∆ made in a nonrelativistic approximation with the WTBS basis set [69] for the Nb and Ta atoms shows that the oxy- 3 gen atom is displaced from the center of the cubic-cell face toward the Nb atom by 0.018 Å (the attendant energy gain is 10 meV). Taking into account the smallness of this effect, as well as the need of including the inﬂuence of the relativistic effects on the electronic struc- –0.02 ture of Ta, we also performed a calculation using the effective core potentials (ECP) for the Nb and Ta atoms, –0.8 –0.4 0 0.4 0.8 with the corresponding valence-orbital basis sets Impurity atom displacement along [001], Å (SBKJC), which took into account the relativistic effects Fig. 3. Local adiabatic potential for the K atom and the Nb in the DiracÐHartreeÐFock equation [65, 71]. In this and Li impurity atoms in KTO. case, the oxygen atom is displaced from the center of the cubic-cell face toward the Ta atom by 0.01 Å (the gain in energy is 1 meV). Treated in the language of ionic radii, WTBS this means that if the relativistic effects are neglected, the VI 5+ VI 5+ 0.10 SBKJC tantalum ion is larger, rion (Ta ) Ð rion (Nb ) = 0.036 Å. At the same time, if the relativistic effects are included in the calculation of the electronic structure of the Nb and VI 5+ Ta atoms, the niobium ion is larger, rion (Nb ) Ð VI 5+ rion (Ta ) = 0.02 Å. The latter result conforms to the lat- , eV

0 tice constant of the cubic phase of KNbO3 being larger E by about 0.04 Å than that of KTaO3 (Table 1). –

E 0.05 =

E 3. VIRTUAL-CRYSTAL MODEL ∆ As follows from the preceding section, dilute solid solutions STO : Ba, STO : Ca (SCT), STO : Pb, STO : Cd, and KTN belong to the displacive type. This conclusion correlates with the observation of a clearly pronounced soft ferroelectric TO mode in SCT [35, 36] 0 and KTN [31, 74Ð76] (single-mode behavior in the terminology introduced in [77]). According to [31], the frequency of the soft TO mode decreases at a given –0.04 0 0.04 0.08 temperature with increasing x and no new low-fre- Oxygen atom shift along Ta–O–Nb chain, Å quency modes associated with Nb are observed. The temperature dependence of the soft TO mode exhibits a Fig. 4. Local adiabatic potential for the oxygen atom along minimum at a temperature close to Tm [31, 35, 36, 75, ω ≈ Ð1 the TaÐOÐNb chain in KTN calculated in the nonrelativistic 76]. In KTN, at the minimum f 7 cm [31, 75], approximation (WTBS) and including the relativistic effects which is substantially lower than the zero-temperature in the electronic structure of Nb and Ta (SBKJC). Open sym- ω soft TO mode frequency for pure KTO, f(T = 0 K) = bols refer to calculations made in the HartreeÐFock (HF) Ð1 approximation, and ﬁlled symbols are calculations with due 20 cm [13]. This suggests the existence, in displacive account of the correlation corrections (HF + MP2). solid solutions, of a soft ferroelectric mode dressed by

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 ON THE NATURE OF FERROELECTRICITY 1141 interactions with impurity atoms, i.e., with parameters Table 4. Critical concentration xc for the solid solutions depending on the impurity concentration. Sr1 Ð xAxTiO3 (A = Ca, Ba, Pb) and KTa1 Ð xNbxO3 This suggests that the free energy of the above dis- xc placive solid solutions can be expressed in terms of the Compound generalized force constant kf, ss(T, x) corresponding to experiment theory the soft TO mode of a mixed compound. In considering Sr Ca TiO 0.0018a 0.0026Ð0.0032 the soft TO mode amplitude yf now to be an order 1 Ð x x 3 b parameter, we can present the free energy Fss of a dis- Sr1 Ð xBaxTiO3 0.035 (0.0027) 0.0013Ð0.012 placive ferroelectric solid solution, by analogy with a c pure displacive ferroelectric [4, 78, 79], in the form of Sr1 Ð xPbxTiO3 0.0015 0.0029Ð0.0039 d a Landau expansion: KTa1 Ð xNbxO3 0.008 0.01Ð0.030 a b c d 1 2 ah —[21], —[22, 23, 26], —[5, 25], —[20]. See text for explana- F ()y ,,Tx = ---k , ()T y + F (){}y Ð v PE⋅ , (3) tion. ss f 2 fss f f 0 ω ah Table 5. Harmonic force constants kh and frequencies h where Fss includes the anharmonic expansion terms. for the cubic phase of perovskite compounds obtained from Treated within this approach, the ferroelectric ab initio calculations phase-transition temperature TC(x), which is a function Compound k , eV/Å2 ω , cmÐ1 of the impurity concentration, can be found by solving h h 5 a c the equation CaTiO3 Ð2.23 153i (), d kfss, TC x = 0 (4) 140i SrTiO Ð0.175a 41ic and the critical concentration xc, defined by the condi- 3 a e tion TC(xc) = 0, is a solution to the equation BaTiO3 Ð3.401 72i (), c kfss, T C = 0 xc = 0. (5) 178i 219if To make a quantitative estimation of xc, we invoke the a g virtual-crystal (VC) approximation. The compounds PbTiO3 Ð2.507 125i under study are solid solutions of two isomorphous 144ic compounds. For instance, Sr1 Ð x AxTiO3 can be repre- KTaO 0.48b 80ib sented in the form (1 Ð x)SrTiO3 + xATiO3, etc. In the 3 VC approximation, a solid solution is treated as a per- 61ih fect crystal with the average values of the parameters a j determined by Vegard’s rule. The VC approximation KNbO3 Ð2.993 115i provides a simple expression for the critical concentra- 143ic tion xc through the parameters of the matrix (solvent) 203ih and solute which automatically takes the effect of quan- 147ik tum fluctuations (zero-point atomic vibrations) on xc l into account. Introducing the notation km and ki for the 197i force constants of the matrix and impurity (dissolved) a b c d e f g h components, respectively, we write the constant k = —[79], —[80], —[83], —[81], —[82], —[84], —[85], — 0, ss [86], j—[87], k—[88], l—[89]. kss(T = 0) in the form () () k0ss, x = 1 Ð x k0, m + xk0, i. (6) Equation (8) is more convenient to use in calculations ω From Eqs. (5) and (6), we can find x : than Eq. (7), because the frequencies 0, m, unlike the c corresponding force constants, are experimentally mea- k , 0 m surable quantities. This is essential for SrTiO3 and xc = ------. (7) k0, m Ð k0, i KTaO3 because of the need to take into account, in these compounds, the zero-point vibration contribu- One can also write an expression for xc which is sim- tion, nonempirical calculations for which (in contrast to ilar to Eq. (7) but is written in terms of the frequencies the contribution calculated in the harmonic approxima- rather than of the force constants: tion) are still lacking. 2 In the solid solutions under study, x Ӷ 1 (Table 4). ω , c x = ------0 m -. (8) This imposes constraints on the parameters of the host c ω2 ω2 0, m Ð 0, i matrix and of the impurity component: < ω2 < 5 The generalized force constant is a static (thermodynamic) quan- k0, i 0, 0, i 0 tity which can vanish even if the frequency of the corresponding Ӷ ω2 Ӷ ω 2 lattice excitation (in our case, of the soft TO mode) remains finite. and k0, m k0, i , 0, m 0, i .

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1142 KVYATKOVSKIŒ

For our solid solutions, these conditions are met satis- 2. B. E. Vugmeister and M. D. Glinchuk, Rev. Mod. Phys. factorily. Table 4 lists the theoretical values of xc calcu- 62 (4), 993 (1990). lated from Eq. (8). The calculations were made using 3. W. Kleemann, Int. J. Mod. Phys. 7 (13), 2469 (1993). ω the experimental values of 0, m for STO and KTO 4. O. E. Kvyatkovskiœ, Fiz. Tverd. Tela (St. Petersburg) 43 ω available in [12, 13]. To ﬁnd the frequencies 0, i for the (8), 1345 (2001) [Phys. Solid State 43, 1401 (2001)]. impurity component of a solid solution, we assumed 5. V. V. Lemanov, Ferroelectrics 226, 133 (1999). the zero-point vibration contributions to ω in all per- f 6. V. V. Lemanov, in Defects and Surface-Induced Effects ovskites to be of about the same order of magnitude. ω in Advanced Perovskites, G. Borstel, A. Krumins, and The zp contribution can be estimated using the values D. Millers (Kluwer, Dordrecht, 2000), p. 329. quoted in [4] for STO (ω ≈ 35.4 cmÐ1) and KTO (ω ≈ zp zp 7. K. A. Müller and H. Burkard, Phys. Rev. B 19 (7), 3593 Ð1 ω 20 cm ). Taking into account the values of h given in (1979). ω2 ω2 ω2 Table 5 and the equality 0 = h + zp [4], we obtain 8. S. H. Wemple, Phys. Rev. 137 (5A), 1575 (1965). ω2 ≈ ω2 9. J. H. Barrett, Phys. Rev. 86 (1), 118 (1952). 0 h for the cubic phase of CTO, BTO, PTO, and KNO. Thus, one can recast equality (8) for the solid 10. A. B. Rechester, Zh. Éksp. Teor. Fiz. 60 (2), 782 (1971) solutions under study in the approximate form [Sov. Phys. JETP 33, 423 (1971)]. 11. D. E. Khmel’nitskiœ and V. L. Shneerson, Fiz. Tverd. 2 ω , Tela (Leningrad) 13 (3), 832 (1971) [Sov. Phys. Solid x ≈ ------0 m- (9) c ω 2 State 13, 687 (1971)]. hi, 12. A. Yamanaka, M. Kataoka, Y. Inaba, et al., Europhys. ω Lett. 50 (5), 688 (2000). and use the results of the ab initio calculations of h, i presented in Table 5. The last column of Table 4 con- 13. H. Vogt, Phys. Rev. B 51 (13), 8046 (1995). tains the ranges of theoretical estimates of xc obtained 14. H. Uwe and T. Sakudo, Phys. Rev. B 13 (1), 271 (1976). from Eq. (9) with due account of the scatter in the val- ω 15. G. A. Samara and B. Morosin, Phys. Rev. B 8 (3), 1256 ues of h in Table 5. (1973). In view of the fact that the theoretical values of the 16. M. Itoh, R. Wang, Y. Inaguma, et al., Phys. Rev. Lett. 82 critical concentration thus found were obtained within (17), 3540 (1999). such a simple model, they describe fairly well, even though they are slightly high, the available experimen- 17. M. Itoh, R. Wang, and T. Nakamura, Appl. Phys. Lett. 76 tal pattern for all the solid solutions except STO : Ba. (2), 221 (2000). This overestimation is possibly due to the neglect of the 18. O. E. Kvyatkovskii, Solid State Commun. 117 (8), 455 host matrix and impurity lattices undergoing mutual (2001). relaxation in the solid solution. 19. A. Bussmann-Holder, H. Büttner, and A. R. Bishop, J. It should be stressed that description of the phenom- Phys.: Condens. Matter 12, L115 (2000). ena accompanying phase transitions in solid solutions 20. U. T. Höchli, H. E. Weibel, and L. A. Boatner, Phys. Rev. is beyond the scope of the present work and, accord- Lett. 39 (18), 1158 (1977). ingly, outside the limits of applicability of the VC 21. J. G. Bednorz and K. A. Müller, Phys. Rev. Lett. 52 (25), approximation. Such a description would require the 2289 (1984). invocation of phenomenological theory or a theory 22. V. V. Lemanov, E. P. Smirnova, and E. A. Tarakanov, Fiz. based on a model lattice Hamiltonian [90]. The results Tverd. Tela (St. Petersburg) 37 (8), 2476 (1995) [Phys. obtained in this study provide a basis for selecting the Solid State 37, 1356 (1995)]. appropriate model description. 23. V. V. Lemanov, E. P. Smirnova, and E. A. Tarakanov, Phys. Rev. B 54 (5), 3151 (1996). ACKNOWLEDGMENTS 24. P. A. Markovin, V. V. Lemanov, O. Yu. Korshunov, et al., Ferroelectrics 184, 269 (1996). I am indebted to V.V. Lemanov and P.A. Markovin for numerous discussions of the work and to 25. V. V. Lemanov, E. P. Smirnova, and E. A. Tarakanov, Fiz. S.B. Vakhrushev, T.R. Volk, and V.A. Trepakov for Tverd. Tela (St. Petersburg) 39 (4), 714 (1997) [Phys. valuable comments. Solid State 39, 628 (1997)]. This study was supported by the Russian Founda- 26. P. A. Markovin, V. V. Lemanov, M. E. Guzhva, and tion for Basic Research (project nos. 00-02-16919, 01- W. Kleemann, Ferroelectrics 199, 121 (1997). 02-17801) and NWO (grant no. 16-04-1999). 27. M. E. Guzhva, V. V. Lemanov, P. A. Markovin, and T. A. Shuplygina, Ferroelectrics 218, 93 (1998). 28. R. Oppermann and H. Thomas, Z. Phys. B 22 (4), 387 REFERENCES (1975). 1. U. T. Höchli, K. Knorr, and A. Loidl, Adv. Phys. 39 (5), 29. T. Schneider, H. Beck, and E. Stoll, Phys. Rev. B 13 (3), 405 (1990). 1123 (1976).

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 ON THE NATURE OF FERROELECTRICITY 1143

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82. R. E. Cohen and H. Krakauer, Phys. Rev. B 42 (10), 6416 87. D. J. Singh and L. L. Boyer, Ferroelectrics 136, 95 (1990). (1992). 83. W. Zhong, R. D. King-Smith, and D. Vanderbilt, Phys. 88. R. Yu and H. Krakauer, Phys. Rev. Lett. 74 (20), 4067 Rev. Lett. 72 (22), 3618 (1994). (1995). 84. P. H. Ghosez, X. Gonze, and J.-P. Michenaud, Ferroelec- 89. C.-Z. Wang, R. Yu, and H. Krakauer, Phys. Rev. B 54 trics 206, 205 (1998). (16), 11161 (1996). 85. R. E. Cohen and H. Krakauer, Ferroelectrics 136, 65 (1992). 90. A.P. Levanyuk and A. S. Sigov, Defects and Structural Phase Transitions (Gordon & Breach, New York, 1988). 86. A. V. Postnikov, T. Newmann, and G. Borstel, Phys. Rev. B 50 (2), 758 (1994). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1145Ð1151. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1096Ð1101. Original Russian Text Copyright © 2002 by Vazhenin, Guseva, Artemov.

MAGNETISM AND FERROELECTRICITY

Selective Averaging of EPR Transitions of a High-Spin Center in the Vicinity of Their Accidental Coincidence V. A. Vazhenin, V. B. Guseva, and M. Yu. Artemov Research Institute of Physics and Applied Mathematics, Ural State University, Yekaterinburg, 620083 Russia e-mail: [email protected] Received June 26, 2001

Abstract—The averaging of a part of spin packets for two transitions of the Gd3+ trigonal center in ferroelectric lead germanite in the vicinity of coincidence of their positions leading to the emergence of an additional EPR signal is investigated. A computer simulation of the experimental spectrum gives the temperature dependence of the spinÐlattice relaxation time connecting the doublets in which resonance transitions occur and leading to selective averaging of the packets. An increase in the relaxation rate for intra- and interdoublet transitions is observed in the vicinity of the structural ferroelectric transition. © 2002 MAIK “Nauka/Interperiodica”.

Electron paramagnetic resonance is successfully In order to explain the emergence of an additional used to detect structural phase transitions, to determine signal, we assumed [5Ð7] the existence of rapid relax- the magnitude and temperature variation of the local ation transitions between the 3 4 and 5 6 dou- order parameter, and to analyze relaxation characteris- blets, which led to averaging of a part of the spin pack- tics of paramagnetic defects that reflect the critical ets belonging to the above EPR transitions and corre- dynamics of the lattice [1, 2]. In the case of high-tem- sponding to the same local crystal field (isofield perature structural transformations, because of the high transitions). rates of energy transfer from the spin system to the lattice, only the relaxation parameters obtained from an The necessary condition of averaging of these packets is the inequality W > ∆ν, where W is the probability analysis of the line shape are used. This analysis is ∆ν extremely complicated due to the existence of alterna- of relaxation transitions between doublets and is the tive temperature-dependent mechanisms of broadening [1, 3]. The effects analyzed in the present work may facilitate the analysis of spin relaxation in the vicinity 500 2 3 of structural transformations (at least, in some materials). 3 4 450 1. In gadolinium-doped lead germanate (LG) 1 ()1 Pb5Ge3O11 (ferroelectric transition P3(C3 ) P 6 C3h 400 1 2 3+ at Tc = 450 K [4]), an EPR spectrum of the Gd trigonal center is observed for which the strong magnetic field 4 5 approximation is satisfied to a high degree of accuracy 350

(Fig. 1). For an arbitrary orientation of the magnetic , mT B ﬁeld, the degeneracy of the EPR spectrum for Gd3+ ions located in the opposite domains is removed. In our pre- 300 vious works [5Ð7], we observed an additional signal 7 8 whose intensity increased upon a decrease in the dis- 250 tance between the initial signals 3 4 and 5 6 of the same type of domains in the vicinity of coinci- 5 6 dence of their resonance positions (for the polar angle 200 6 7 θ ≈ ° || of magnetic ﬁeld 0 41 , z C3) (Fig. 2). It was proved that this signal cannot be due to a multiquantum transi- 0 15 30 45 60 75 90 tion, although two-quantum transitions can be clearly θ, deg observed in the vicinity of the intersection of the angular dependences corresponding to transitions 4 5 Fig. 1. Polar angular dependence of the positions of transi- and 3 4, as well as 4 5 and 5 6, 3 3+ tions in the Gd trigonal center in the zy plane. Energy lev- 4 and 2 3, and 5 6 and 6 7. els are labelled in ascending order.

∆ 348 Bpp = 2.7 mT 5

+ 5 6 4 346

3 – 344 2 – +

, mT + B 342 – –

1 3 4 340 ∆ Bpp = 1.4 mT + 338 40 41 42 θ, deg 335 340 345 B, mT Fig. 3. Fragment of the polar angular dependence of transitions 3 4 and 5 6 (Fig. 1) illustrating the forma- Fig. 2. EPR spectrum in the region of coincidence of position of the quasi-symmetric distribution of spin packets. tions of the 3 4 and 5 6 transitions at 458 K Solid lines correspond to b = 0; dashed lines, to b = ∆θ ° ° ° ° 21 21 (0.05% Gd2O3) for (1) = 0 , (2) 0.5 , (3) 0.63 , (4) 0.75 , +100 MHz; and dotted lines, to b21 = Ð100 MHz. Plus and and (5) 1.75°. The dashed curve is described by Eq. (4); minus signs on the right of the absorption lines indicate the ∆ Bpp is the spacing between the extremum points of the positions of “isofield” packets corresponding to the maxi- absorption signal derivative. mum and minimum values of b21. distance (in frequency units) between packets in the trum of Gd3+ trigonal centers in LG for an arbitrary ori- spectrum. The key point in the proposed model is the entation (θ ≠ 0°, 90°) of the polarizing magnetic field is quasi-symmetric arrangement (see below and Fig. 3) of static modulation of the spin Hamiltonian parameter b21 isofield spin packets in a pair of initial EPR signals. In (c21), which increases as we approach the structural- the case of antisymmetric arrangement of isofield pack- transition temperature. The spread in the values of b43 ets, the value of ∆ν is approximately the same for all makes a noticeable but not dominating contribution for pairs and no selective averaging takes place. It was azimuthal angles ϕ close to zero. assumed in [5Ð7] that this form of inhomogeneous According to our calculations, the arrangement of broadening is due to the spread in the value of sponta- spin packets for the transitions 3 4 and 5 6 neous polarization and, hence, in the value of the spin due to the fluctuations in b21 (Fig. 3) is similar to the Hamiltonian parameter b proportional to it. This 43 packet distribution as a result of modulation of b43; con- enabled us [5Ð7] to qualitatively describe the main fea- sequently, the approach developed in [5Ð7] to explain tures of the behavior of the additional EPR signal. We the emergence of the additional signal remains valid. considered cross-relaxation and spinÐlattice interaction The strong dependence of the magnitude of the as the mechanisms triggering transitions between initial effect on the extent of inhomogeneous broadening of doublets. Thus, the specific feature of the EPR spec- the initial lines is indicated by the change in the form of trum observed for LG is analogous to the effects emerg- the EPR spectrum in the region of coincidence of the ing when relaxation without spin flop between Kramers resonance positions upon the application of an external doublets originating from a vibronic doublet split as a electric field (Fig. 4). The observed decrease in the result of the interaction with random deformations is additional-signal intensity is due to a decrease in the taken into account [8, 9]; this phenomenon is also close inhomogeneous width of the initial signals in the elec- in nature to cross-singular effects in the NMR of poly- tric field, which was discovered in [11] and is appar- crystals [10]. ently associated with the saturation of polarizability. It was shown in [11] that (at least in the ~±50 K The present work aims to quantitatively describe the neighborhood of the ferroelectric transition) the main spectrum with an extra signal in the region of coinci- mechanism of inhomogeneous broadening of the spec- dence of the resonance positions of EPR transitions and

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 SELECTIVE AVERAGING OF EPR TRANSITIONS 1147 to determine the temperature variation of the rates of intra- and interdoublet relaxation in the vicinity of the ferroelectric phase transition point. 2. Measurements were made with the help of a 3-cm EPR spectrometer on LG single crystals grown using the Czochralski method in air or nitrogen from a melt 1 having stoichiometric composition and containing 2 0.0075 and 0.05 mol % Gd2O3 impurity. The specimens were transferred to the single-domain state prior to each cycle of temperature measurement in order to avoid overlapping of the transitions under investigation with the signals from the centers in oppositely magnetized domains. At each temperature, the EPR spectrum was recorded in the following cases: at the point of coinci- 340 345 dence of resonant positions of the initial transitions B, mT ∆θ θ θ ∆θ ( = Ð 0 = 0); for “strong” disorientation ( = 1.5°Ð2°), where there is no extra signal and the distor- Fig. 4. EPR spectra in the region of coincidence of positions of the 3 4 and 5 6 transitions at 454 K for ∆θ = tion of the initial signals due to the interdoublet relax- 0.5° for a weakly doped sample (1) in zero field and (2) for ation is small; and for “weak” disorientation (∆θ = E = 7 kV/cm. 0.3°Ð0.6°), when the averaging of spin packets is most noticeable (Fig. 2). At the first stage of measurements (the magnetic field lies in the zx plane corresponding to Ω the maximum splitting between the same transitions in is the transition probability from one frequency to the Gd3+ centers in oppositely magnetized domains), it another; and ±δ are the initial frequencies. was found that, in the vicinity of Tc the sample is trans- We assumed that the initial lines consist, in contrast formed into a multidomain state leading to uncontrollable distortion of the form of the signals being detected, to Eq. (1), of Lorentzian packets with intensities having which is especially undesirable for ∆θ = 0.3°Ð0.6°. For a Gaussian distribution this reason, the subsequent measurements were made in m the zy plane, in which the signals from the centers in I exp()Ðn/σ 2 () ------0 - different domains coincide. IB = ∑ 2, (3) 1 + []αT ()BBÐ Ð n 3. In order to determine the probability of interdou- nm= Ð eff 0 blet transitions, a computer simulation of the EPR spec- where B is the magnetic induction, B0 is the resonance trum with an extra signal was carried out using an α expression [12] derived to describe the spectrum of a position, 2m + 1 is the number of spin packets, = π β spin system with several close frequencies character- 2 geff /h, Teff is the effective relaxation time associated ized by infinitely narrow lines and performing transi- with the homogeneous line broadening, and σ is the tions (motion) between the energy levels corresponding parameter characterizing inhomogeneous broadening. to these frequencies: In this case, the expression describing the shape of I()ω = Re{}WA⋅⋅()ω Ð1 1 , (1) the EPR spectrum in the region of accidental coincidence of transitions has the following form (taking into where W is a vector with components equal to the ini- account the above-mentioned quasi-symmetric distri- tial-transition probabilities and 1 is a unit vector; in the case of two close frequencies, the matrix A is given by bution of the packets):

m i()ΩÐωδ+ Ð Ω Ð1 2 A()ω = ; (2) IB()= Ð∑ Re()WA⋅⋅()B 1 exp()Ðn/σ , (4) Ω ()Ωωδ Ði + Ð nm= Ð

1 1 1 iα()anB+ Ð Ð ------Ð ------T 2τ 2τ A()B = eff , (5) 1 α()1 1 -----τ i bndÐ Ð B Ð ------Ð -----τ 2 T eff 2

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1148 VAZHENIN et al.

1.5 2 6 , mT pp

B 1.0 , mT ∆ σ

0.5 1 300 400 500 T, K 300 400 500 T, K Fig. 5. Temperature dependence of the line width (the dis-

tance between the extrema of the first derivative of the Fig. 6. Temperature dependence of σ for (1 ) 0.0075% || absorption signal) of the 4 5 transition for B C3. Gd2O3 and (2) 0.05% Gd2O3. where a and b are the resonance positions of the initial for the initial signals 3 4 and 5 6 in the region signals, 1/2τ is the interdoublet transition probability, of coincidence were close to unity. σ σ σ σ and d = b / = b / a is the ratio of the inhomogeneous- In the case of a weak disorientation ∆θ, the initial broadening parameters of the initial lines. parameters could be varied in the course of simulation only to within the error in their magnitudes. After dif- The EPR spectrum can also be simulated using the ferentiation, the obtained spectrum was compared with formulas derived in [13] as a result of simultaneous the experimental spectrum which presents the first solution of the Bloch equations for two low-spin sys- derivative of the absorption spectrum. Figure 2 shows tems coupled by exchange interaction. However, the an example of the description of the observed spectrum lower flexibility of these systems (e.g., the impossibil- on the basis of the proposed model (curve 3). It can be ity of correctly taking into account the difference in the seen that the stronger discrepancy is observed for the probabilities of the initial transitions) noticeably wors- wings of the three-component EPR signal, which fall ens the description of the experimental spectrum. off at a slower rate than those of the calculated curve. In In order to avoid ambiguity in determining the our opinion, this is due to the existence of a quasi-con- parameters of the EPR spectrum being constructed, we tinuous spectrum of triclinic centers (leading to signals must accurately determine their initial values. The resin the wings of the trigonal spectrum and arising as a onance positions measured for ∆θ = 0 and 1.5°Ð2° were result of local compensation of the excess charge of Gd3+ [14]) and due to the disregard of signals from odd used to determine the positions of the initial transitions 155 157 (a, b) in the absence of the effects of spin packet aver- isotopes Gd and Gd (the natural abundance of each of these isotopes is ≈15%). The inclusion of the signal aging. The relaxation time Teff was estimated from the temperature dependence of the width of the 4 5 from the odd isotopes in the computational model is || hampered in view of the complex hyperfine structure transition for B C3 (Fig. 5), which has an extremely θ ≠ ° ° θ ≈ ° small width at low temperatures and does not experi- for 0 , 90 (the splitting for 41 amounts to tens ence broadening due to the spread in the fine-structure of megahertz) associated with the strong quadrupole parameters. In other words, we assumed that the depen- interaction [15]. dence of the relaxation time on the type of transition The temperature dependences of the inhomoge- and the orientation of the polarizing field is extremely neous and homogeneous broadening parameters and of weak. This assumption was made in connection with the interdoublet relaxation time for two paramagnetic- the complexity of separating the relaxation contribution impurity concentrations, which are obtained as a result to the line widths of the transitions 3 4 and 5 of computer simulation of the EPR spectrum, are pre- 6 due to the strong temperature dependence of inhomo- sented in Figs. 6Ð8. The σ(T) dependences (Fig. 6) are geneous broadening [11] (see also Fig. 6). The values in qualitative agreement with the results [11] on the σ σ 3+ of b, a, and the ratio of the initial-transition probabil- behavior of inhomogeneous line broadening for Gd in ities were estimated from the synthesis of the shape of the vicinity of the ferroelectric phase transition. the initial lines for a strong disorientation ∆θ with the The strong temperature dependence and the weak help of Eq. (3) (for a given value of Teff). The ratios of concentration dependence of Teff (Fig. 7) indicate that intensities and inhomogeneous broadening parameters the homogeneous broadening of signals is controlled

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 SELECTIVE AVERAGING OF EPR TRANSITIONS 1149

2.5 6 1 4

5 2.0 2 3

s 1 4 s s –8 1.5 –8 2 –8 , 10 , 10 , 10 eff τ τ T 3

2 1 1.0 2

0.5 1 300 400 500 300 400 500 T, K T, K

τ Fig. 7. Temperature dependence of Teff for (1) 0.0075% Fig. 8. Temperature dependence of for (1) 0.0075% Gd2O3 and (2) 0.05% Gd2O3. Gd2O3 and (2) 0.05% Gd2O3. by the spinÐlattice relaxation. A slight deviation from 4. The observation and investigation of relaxation the monotonic temperature dependence of the 4 5 averaging at other accidental coincidences of resonance || transition line width for B C3 in the vicinity of the positions of EPR transitions (Fig. 1) are undoubtedly of structural phase transformation (Fig. 5) might be attrib- considerable interest. uted to the measurement error; however, the simulation The effects under investigation cannot take place in of a spectrum with an extra signal which would be ade- the vicinity of intersection of the angular dependences quate for the experimental spectrum proved to be of transitions having a common energy level, while impossible under the assumption of a monotonic Teff(T) two-quantum transitions can be observed clearly (see dependence. Section 1). In the regions of coincidence of the resonance positions of transitions occurring in the doublets The temperature and concentration dependences of separated by a single energy interval (like the transi- the time τ (Fig. 8) are similar, on the whole, to the tions 3 4 and 5 6), a quasi-symmetric respective dependences of Teff (the dependence for a arrangement of isoﬁeld spin packets is realized only for sample with a low Gd concentration is represented by a the transitions 2 3 and 4 5. An analysis of straight line as a result of the considerable errors Eqs. (4) and (5) in the case of antisymmetric distribu- encountered in determining τ). This fact conﬁrms the tion of the spin packets revealed that the effect of conclusion [7] that interdoublet relaxation is due to motion (relaxation transitions) on the line shape and τ spinÐlattice interaction. Like Teff(T), the (T) curve has width is minimal in this case. a narrow minimum in the region of the ferroelectric A detailed analysis of the spectrum in the region of transition. Such a feature in the behavior of the spinÐ coincidence of the transitions 2 3 and 4 5 θ ≈ ° lattice relaxation time determined from the measure- ( 0 42 ) did not reveal any traces of an extra signal, ments of the EPR line width in the vicinity of Tc was whose observation is complicated by the strong differ- 2+ observed in BaTiO3 : Mn [16] and was attributed to ence in the intensities and widths of the initial lines, this the anomalous interaction of the spin system with trans- difference increasing as the temperature approaches Tc. verse optical phonons [17Ð19]. An increase in the spinÐ It was proved in [7], however, that the EPR spectrum lattice relaxation rate in the region of the structural is perturbed by the averaging effects even in the case of phase transformation was also observed in NMR of exact coincidence of the resonance positions (∆θ = 0) 23 3+ NaNbO3 at Na nuclei [20] and, probably, at Fe of the initial transitions. Indeed, it can be clearly seen impurity ions in PbTiO3 [21]. On the contrary, an from Fig. 2 that the width of the resultant line for ∆θ = increase in the spinÐlattice relaxation time was 0 (curve 1) is noticeably smaller than the width of the observed in [2] at the Mn2+ centers in the vicinity of the initial signals (curve 5). The spectrum of the transitions phase transition in calcium tris-sarcosine chloride. 2 3 and 4 5 for ∆θ = 0 presented in Fig. 9 also

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1150 VAZHENIN et al.

the 3 4 and 5 6 transitions) of the angular 1 dependences of positions of signals with quasi-sym- ∆ metric arrangement of the spin packets. In addition, a 2 Bpp = 0.52 mT ∆ three-component spectrum can be observed only when Bpp = 0.44 mT τ σ definite relations are observed between Teff, , and . It follows from the results presented in Figs. 6Ð8 that an extra signal is observed for close values of the quanti- βσ π πτ ties geff , 1/2 Teff, and 1/2 . The changes in the spectrum associated with interdoublet relaxation become more pronounced upon an increase in the relaxation rate; however, the interdoublet relaxation rate in our model cannot exceed the relaxation rate within a doublet and, hence, the optimal relation connecting τ with T has the τ ≈ eff 337 338 339 form Teff. Since an extra signal cannot emerge in the B, mT absence of inhomogeneous broadening, another condition for the observation of a three-component spectrum Fig. 9. EPR spectrum in the case of exact coincidence of has the form 1/2πT ≤ g βσ. positions of the 2 3 and 4 5 transitions at room eff eff temperature: (1) experiment and (2) the sum of the initial The shape of the EPR spectrum of a large number of signals in the absence of interdoublet relaxation. high-spin paramagnetic centers with axial symmetry is mainly determined by the axial parameter of the spin Hamiltonian b20, which fluctuates due to the defect demonstrates a slight narrowing of the resultant line nature of the crystal [22]. If there exists a spread in the relative to the sum of the initial lines, indicating the values of b20 (at least, when the strong magnetic field averaging of spin packets in this case also. Unfortu- approximation holds), the polar angular dependence of nately, the error in determining the relaxation time for the positions of transitions contains the points of coin- exactly coinciding positions of the initial packets is cidence of signals with quasi-symmetric distribution of very high. the isofield spin packets. However, the probability of In the region of the coincidence of resonance posi- detecting the selective averaging of the EPR spectrum tions of transitions occurring in doublets separated by is low in view of the requirement that the above rela- τ σ two energy intervals, the spin packets associated with tions between Teff, , and must be satisfied, as well as modulation of b21 are arranged quasi-symmetrically for of the requirements imposed on the intensity and width the transitions 3 4 and 6 7, 2 3 and of the initial signals. 5 6, and 4 5 and 7 8. However, the region of intersection of the angular dependences for the transitions 2 3 and 5 6 is covered by the ACKNOWLEDGMENTS intense signal of the 4 5 transition, while the inten- The authors express their sincere gratitude to sities of the transitions 4 5 and 7 8 are abso- A.N. Legkikh for his assistance in measuring and pro- lutely incomparable. An analysis of the spectrum in the cessing spectra and to N.A. Legkikh for valuable dis- region of coincidence of the 3 4 and 6 7 sig- cussions of the results. nals, which also have noticeably different peak intensi- ≈ ∆ ∆ ≈ This research was partly supported by the American ties (I34/I67 3 for Bpp67/ Bpp34 1.5), did not reveal Foundation for Civilian Research and Development for any interaction of the spin packets. CIS countries, grant no. REC-005. No averaging effects were also detected in the region of intersection of the angular dependences for transitions separated by three energy intervals. In this REFERENCES group, the intersection of the dependences for the tran- 1. G. F. Reiter, W. Brelinger, K. A. Müller, and P. Heller, sitions 2 3 and 6 7 occurring between two Phys. Rev. B 21 (1), 1 (1980). high-intensity EPR signals are the most promising for 2. W. Windsch and G. Volkel, Ferroelectrics 24, 195 the observation of an extra signal as regards the ratio of (1980). ≈ intensities (I23/I67 2.5). 3. B. E. Vugmeœster, M. D. Glinchuk, A. A. Karmazin, and Computer simulation of the three-component spec- I. V. Kondakova, Fiz. Tverd. Tela (Leningrad) 23 (5), trum by Eqs. (4) and (5) proved that increasing the ratio 1380 (1981) [Sov. Phys. Solid State 23, 806 (1981)]. of the intensities of the initial signals up to four or 4. Y. Iwata, J. Phys. Jpn. 43, 961 (1977). σ σ increasing the ratio b/ a up to two for conserved val- 5. V. A. Vazhenin and K. M. Starichenko, Pis’ma Zh. Éksp. ues of relaxation parameters leads to a spectrum with- Teor. Fiz. 51 (8), 406 (1990) [JETP Lett. 51, 461 (1990)]. out any noticeable evidence of an extra signal. In our 6. V. A. Vazhenin and K. M. Starichenko, Fiz. Tverd. Tela opinion, this fact completely explains the absence of an (St. Petersburg) 34 (1), 172 (1992) [Sov. Phys. Solid additional signal for intersections (other than those for State 34, 90 (1992)].

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 SELECTIVE AVERAGING OF EPR TRANSITIONS 1151

7. V. A. Vazhenin, K. M. Starichenko, and A. D. Gorlov, 15. A. D. Gorlov, A. P. Potapov, and Yu. A. Sherstkov, Fiz. Fiz. Tverd. Tela (St. Petersburg) 35 (9), 2449 (1993) Tverd. Tela (Leningrad) 27 (3), 625 (1985) [Sov. Phys. [Phys. Solid State 35, 1214 (1993)]. Solid State 27, 388 (1985)]. 8. J. R. Herrington, T. L. Estle, and L. A. Boatner, Phys. 16. V. V. Shapkin, B. A. Gromov, G. T. Petrov, et al., Fiz. Rev. B 3 (9), 2933 (1971). Tverd. Tela (Leningrad) 15 (5), 1401 (1973) [Sov. Phys. 9. I. B. Bersuker and V. Z. Polinger, Vibronic Interactions Solid State 15, 947 (1973)]. in Molecules and Crystals (Nauka, Moscow, 1983; 17. B. I. Kochelaev, Zh. Éksp. Teor. Fiz. 37, 242 (1959) [Sov. Springer, New York, 1989). Phys. JETP 10, 171 (1960)]. 10. É. P. Zeer, V. E. Zobov, and O. V. Falaleev, Novel Effects in NMR of Polycrystals (Nauka, Novosibirsk, 1991). 18. N. Kumar and K. P. Sinha, Physica (Amsterdam) 34, 387 (1967). 11. V. A. Vazhenin, E. L. Rumyantsev, M. Yu. Artemov, and K. M. Starichenko, Fiz. Tverd. Tela (St. Petersburg) 40 19. C. Y. Huang, Phys. Rev. 161 (2), 272 (1967). (2), 321 (1998) [Phys. Solid State 40, 293 (1998)]. 20. G. Bonera, F. Borsa, and A. Rigamonti, J. Phys. (Paris) 12. A. Abragam, The Principles of Nuclear Magnetism 33 (C2), 195 (1972). (Clarendon, Oxford, 1961; Inostrannaya Literatura, 21. S. T. Kirillov and Yu. G. Plakhotnikov, Pis’ma Zh. Éksp. Moscow, 1963). Teor. Fiz. 34 (11), 572 (1981) [JETP Lett. 34, 548 13. H. S. Gutowsky and S. H. Holm, J. Chem. Phys. 25 (6), (1981)]. 1228 (1956). 22. N. V. Karlov and A. A. Manenkov, Quantum Ampliﬁers 14. V. A. Vazhenin, K. M. Starichenko, and A. V. Gur’ev, Fiz. (VINITI, Moscow, 1966). Tverd. Tela (Leningrad) 30 (5), 1443 (1988) [Sov. Phys. Solid State 30, 832 (1988)]. Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

LATTICE DYNAMICS AND PHASE TRANSITIONS

Specific Features of Low-Temperature Phonon Scattering in Materials with Tilted Disclination Loops S. E. Krasavin and V. A. Osipov Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia e-mail: [email protected] Received August 7, 2001

Abstract—Phonon scattering by static stress ﬁelds of circular wedge disclination loops is investigated in the framework of the deformation potential approach. Numerical calculations of the mean free path l and thermal conductivity κ demonstrate that the temperature dependence of κ exhibits a minimum at a certain temperature T* in the low-temperature range. The thermal conductivity κ sharply increases as TÐ3 with a decrease in temperature (T < T*) and exhibits a dislocation behavior (κ ~ T2) with an increase in temperature (T > T*). The results obtained for the wedge disclination loop are compared with the available data for uniaxial disclination dipoles. It is shown that the properties of uniaxial disclination dipoles serving as sources of phonon scattering are similar to those of wedge disclination loops. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION condition λ ~ L, which, in the approximation of thermal phonons, leads to the following estimate of the transi- ≈ ប It is well known that dislocations, together with tion temperature: T* vs/2LkB (see, for example, [6]). other structural defects and chemical impurities, are the It was also demonstrated that the uniaxial and biaxial main sources of phonon scattering at temperatures sub- Θ wedge disclination dipoles with in-place axes of rota- stantially below the Debye temperature D [1, 2]. In the tion, which are the two limiting cases of an arbitrary case of phonon scattering by immobile dislocations, the κ wedge disclination dipole, are distinctly different dependence of the thermal conductivity on the tem- objects from the point of view of phonon scattering. perature T is described by a quadratic relationship. The uniaxial wedge disclination dipole is a highly However, the role played by linear rotational defects screened system in contrast with the biaxial wedge dis- (disclinations) in phonon scattering is as yet poorly clination dipole [6]. This manifests itself in a strong understood despite the fact that these defects are of dependence of the mean free path l on the wave vector great importance in nanocrystalline materials [3], com- k in the long-wavelength limit [l(k) ~ kÐ5 at λ Ӷ L]. For posites [4], and topologically disordered systems [5]. The problem of heat transfer in a wide variety of mate- a uniaxial wedge disclination dipole in the short-wavelength limit, the phonon scattering exhibits a disloca- rials with rotational plastic strains remains unsolved. In Ð1 our recent works [6, 7], we investigated the temperature tion nature: l(k) ~ k . At the same time, the biaxial behavior of the thermal conduction associated with the wedge disclination dipole with in-place axes of rotation contribution of static disclination dipoles of the tilted obeys the relationships l(k) ~ kÐ1 at λ Ӷ L and l(k) type (wedge disclination dipoles) to heat transfer. const in the opposite limit. This behavior sets the biax- Moreover, we examined all types of wedge disclination ial wedge disclination dipole apart from other similar dipoles, namely, uniaxial dipoles, biaxial dipoles with defects. out-of-place axes of rotation, and biaxial dipoles with in-place axes of rotation. Exact expressions were In the present work, we examined another stable dis- obtained for the mean free path of phonons due to static clination object, namely, a circular wedge disclination stress fields for each type of dipoles in the Born approx- loop [8, 9]. Considerable interest expressed in the prob- imation. lem of phonon scattering by static strain fields of circular wedge disclination loops stems from the fact that It has been found that the wedge disclination dipoles disclination loops seem to be the most commonly serving as effective centers of phonon scattering are encountered elements of the three-dimensional discli- characterized by a specific linear parameter, namely, nation structure in the majority of real media. The the dipole arm L, and that the mode of phonon scatter- results obtained for the circular wedge disclination loop ing depends on the ratio between the wavelength λ of are compared with the available data for another discli- the incident phonon and the dipole arm L. In particular, nation defect, namely, a uniaxial wedge disclination we revealed that the scattering mode changes under the dipole.

SPECIFIC FEATURES OF LOW-TEMPERATURE PHONON SCATTERING 1153

102 100 Disclination dipole Disclination loop Dislocation Point impurity 10–1

~ ~

), cm ~ ~ k ( l –1 10 , W/(K cm) κ 10–2 10–2 10–3 10–4 10–5 10–3 –3 –2 –1 2 0.1 1 10 10 10 / 10 10 k kD T, K Fig. 1. Dependence of the phonon mean free path [calculated from formula (6)] on the reduced wave vector k/kD Fig. 2. Temperature dependence of the thermal conductivity ω × (kD = D/vs) at the following parameters: R = L = 2 calculated from formula (7) for the phonon mean free path −6 ν × 5 determined from formula (6) (for the wedge disclination 10 cm, = 0.1, vs = 4 10 cm/s, B = 0.01, and nd = 15 Ð3 dipole, the mean free path is taken from [6]). The parame- 10 cm . The dashed line represents the dependence ters used in the calculation are the same as those in Fig. 1 l(k/k ) for a uniaxial wedge disclination dipole at the same Θ D ( D = 300 K). The curves calculated for a dislocation and a × 9 Ð3 parameters, except for nd = 6 10 cm . point impurity are depicted for comparison.

∞ 2. THEORETICAL BACKGROUND J (K)J (Kr/R)exp(ÐK|z|/R)KdK is the LifshitzÐ ∫0 2 1 Let us consider the problem of elastic phonon scat- Hankel integral, and Jm(x) is the Bessel function of the tering by static stress fields of a wedge disclination loop first kind. in the framework of the deformation potential approach [1, 10]. Within this approach, the energy of phonon per- In the Born approximation, the matrix element turbation is associated with the relative change in the describing the transition of a phonon from the state k to volume due to strains induced by a circular wedge dis- the state k' has the form clination loop placed in the medium. In this case, the 1 3 〈〉k Ur()k' = --- d rU()r exp()iqr , (3) deformation potential has the following form [1]: V∫ () បωγ U r = SpEij, (1) where q = k Ð k''. Taking into account the axial symme- បω try of the defect under consideration, it is expedient to where is the energy of a phonon with the wave vec- introduce the cylindrical coordinate system into rela- tor k, ω = kv , v is the mean velocity of sound in the γ s s tionship (3). In this case, the matrix element with due medium, is the Grüneisen constant, and SpEij is the regard for relationship (2) has the following form: trace of the deformation tensor due to the circular wedge disclination loop. 2 A q⊥ 〈〉k Ur()k' = 4πiR cosα------J ()q⊥ R , (4) V 2 2 2 It is assumed that a loop of radius R is located in the q⊥ + qz plane z = 0 of the cylindrical coordinate system (r, ϕ, z) α with the rotation axis passing through the origin of the where is the angle between q⊥ = (qx, qy) and the x coordinates and the Frank vector specified by the coor- axis. In order to simplify the subsequent calculations, dinates (0, Ω, 0) with respect to the chosen basis (the we consider an incident phonon whose momentum is in-place circular wedge disclination loop). By using the directed along the kx axis and choose the cylindrical φ explicit expression for the deformation tensor of the cir- coordinate system (k⊥, , kz) in the k space. In this case, cular wedge disclination loop with the in-place axis of we obtain the following expression for the mean free path: rotation Eij (see [9, 11]), we obtain the following relationship for the perturbation energy represented by Ð1 VNk l ()k = ------expression (1): ()πប 2 2 v s U()r = AcosϕJ()2, 1; 1 . (2) 2π k (5) ប πγν σ σ ν Ω π ×φ 〈〉() 2()θ Here, A = kvs (1 Ð 2 )/( Ð 1), = /2 is the ∫ d ∫dkz' k Ur k' 1 Ð cos , σ Frank index, is the Poisson ratio, J(2, 1; 1) = 0 0

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1154 KRASAVIN, OSIPOV where N is the number of defects and θ is the angle of can be estimated by the standard kinetic formula writ- scattering (for elastic scattering, q = |q| = |k Ð k'| = ten in the dimensionless form: θ 2ksin( /2)). The bar above the square of the matrix ele- Θ /T 4 3 D ment signiﬁes averaging over the angle α. The angle θ k T x4ex φ θ κ = ------B - ∫ ------lx()dx, (7) can be expressed through as follows: 1 Ð cos = 1 Ð π2ប3 2 x 2 2 v s ()e Ð 1 ()' 2 φ 0 1 Ð kz/k cos . From relationships (4) and (5), we ប បω Θ បω where x = kvs/kBT = /kBT, D = D/kB, and l(x) is obtain the ﬁnal expression for the mean free path determined by relationship (6).

2π 1 The results of numerical calculations of the temper- 2 φ 2 κ Ð1() 4 2 φ 11Ð Ð z cos Ð z /2 ature dependence of the thermal conductivity due to l k = nd R k B ∫ d ∫dz------phonon scattering by static stress fields of a circular 2 φ 0 0 11Ð Ð z cos (6) wedge disclination loop are represented in Fig. 2. This figure also depicts the dependences κ(T) for a uniaxial × 22 2 φ wedge disclination dipole, a dislocation, and a point J Rk 2 Ð21z Ð Ð z cos , impurity. The behavior of κ(T) for the circular wedge disclination loop is explained in terms of the depen- where z = kz' /k, nd = N/V is the concentration of defects, dence l(k) (see above). It is clearly seen from Fig. 2 that, and B = (πγν(1 Ð 2σ)/(1 Ð σ))2. for the chosen model parameters in relationships (6) and (7), the dependence κ(T) exhibits a minimum at T* ≈ 2 K for both the circular wedge disclination loop and the uniaxial wedge disclination dipole. The thermal 3. RESULTS AND DISCUSSION conductivity κ increases drastically in the temperature 3.1. Figure 1 shows the dependences l(k) calculated range below T*. This increase corresponds to the increase in the mean free path l(x) [see relationship (7)] numerically for a circular wedge disclination loop at T 0 in accordance with the law l(k) ~ kÐ6 for the according to formula (6) and for a uniaxial wedge dis- Ð5 clination dipole (the data are taken from [6]) at R = L (L circular wedge disclination loop and l(k) ~ k for the is the arm of the dipole) under the assumption that the wedge disclination dipole in the long-wavelength limit. capacities of defects specified by the Frank index ν are As a result, when T < T*, the contribution to the thermal conductivity due to phonon scattering by static stress equal to each other. It is seen from Fig. 1 that both fields of the circular wedge disclination loop is propor- defects are characterized by two different scattering Ð3 Ð2 modes at kR Շ 1 and kR տ 1 (k* ~ 1/R is the point on the tional to T (T in the case of the wedge disclination dipole). Note that, below T*, the thermal conductivity κ curve which corresponds to the change in the scattering Շ of both the circular wedge disclination loop and the mode). In the long-wavelength limit kR 1, the mean uniaxial wedge disclination dipole increases more rap- free path l(k) sharply increases in phonon scattering by idly than that of a point impurity for which κ ~ TÐ1 a stress field of the circular wedge disclination loop (dashed line in Fig. 2). Near T*, phonons whose wave- with a decrease in k (Fig. 1). This increase is even more length is comparable to the characteristic size of the pronounced than in the case of the uniaxial wedge dis- defect (kR ~ 1) are predominantly excited, in contrast clination dipole for which l(k) ~ kÐ5 [6]. From relation- with the case kR Շ 1, where the defect is a weak scat- ship (6) at k 0, we obtain the estimate l(k) ~ kÐ6. Such terer of thermal phonons. For kR ~ 1, strong scattering a strong k-dependence for l(k) is specific to defects of arises at the boundary of the defect. In turn, this brings finite size and can be explained by the lack of interfer- about the suppression of heat transfer. Above T*, the ence in scattering from different segments of the defect thermal conductivity for both the circular wedge discli- in the long-wavelength limit (the wavelength exceeds nation loop and the wedge disclination dipole has a dis- the size of the defect) [12]. Therefore, the circular location behavior: κ ~ T2 [l(k) ~ 1/k] (Fig. 2). An wedge disclination loop is all the more a self-screened increase in the thermal conductivity κ above the mini- system compared to the uniaxial disclination dipole. It mum point can be explained by the increase in the num- should be noted that, in both cases, the dependence l(k) ber of short-wave excitations for which the local heat in the long-wavelength limit is stronger than in the case transfer from one phonon to another proceeds more of a point impurity for which l(k) ~ k4 (Rayleigh scatter- rapidly than the phonon scattering by circular wedge ing) [2]. In the opposite limit of short waves (kR տ 1), disclination loops (wedge disclination dipoles), which the dependence l(k) for the circular wedge disclination exhibits a dislocation nature and leads to the suppres- loop exhibits a dislocation behavior similar to that of sion of the thermal conduction. the uniaxial wedge disclination dipole: l(k) ~ 1/k (see 3.2. In this work, we analyzed the contribution to the [6, 7]). thermal conductivity due to phonon scattering by static stress fields of the circular wedge disclination loop in Θ The contribution to the thermal conductivity from the low-temperature range (T < D). It was demon- phonon scattering by circular wedge disclination loops strated that the properties of the circular wedge discli-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 SPECIFIC FEATURES OF LOW-TEMPERATURE PHONON SCATTERING 1155 nation loops serving as sources of phonon scattering are can serve as indirect evidence of the existence of circu- similar to those of disclination dipoles with a common lar wedge disclination loops and wedge disclination axis of rotation (uniaxial wedge disclination dipoles). dipoles in the studied material. Phonon scattering by these defects at different temper- The calculation of the temperature dependence of atures is characterized by a nontrivial dependence κ(T). the thermal conductivity κ with due regard for other In a certain range of temperatures near T*, the thermal sources of scattering, apart from scattering by wedge conductivity κ is strongly suppressed and increases disclination loops and uniaxial dipoles, will be per- above and below the temperature T* (Fig. 2) as TÐ3 (TÐ2 formed in the immediate future. The results obtained for the wedge disclination dipole) with a decrease in the will be reported in a separate paper. temperature when T < T* and as T2 with an increase in the temperature when T > T*. Thus, the circular wedge disclination loop and the uniaxial wedge disclination REFERENCES dipole are specific defects from the point of view of 1. J. M. Ziman, Electrons and Phonons (Clarendon, phonon scattering and exhibit quite a different behavior Oxford, 1960; Inostrannaya Literatura, Moscow, 1962). depending on the phonon wavelength. In the long- 2. P. Carruthers, Rev. Mod. Phys. 33 (1), 92 (1961). wavelength range, the properties of these defects are 3. S. G. Zaœchenko and A. M. Glezer, Fiz. Tverd. Tela similar to those of the point impurity (strong screening (St. Petersburg) 39 (11), 2023 (1997) [Phys. Solid State of stress fields). In the short-wavelength limit, the 39, 1810 (1997)]. phonon scattering has a dislocation nature. 4. I. A. Ovid’ko, Fiz. Tverd. Tela (St. Petersburg) 41 (9), It is evident that, for real materials containing these 1637 (1999) [Phys. Solid State 41, 1500 (1999)]. defects, the situation can be somewhat different due to 5. A. Richter, A. E. Romanov, W. Pompe, and the existence of other scattering sources. For example, V. I. Vladimirov, Phys. Status Solidi B 122 (1), 35 in the low-temperature range, phonon scattering should (1984). arise at the crystal grain boundaries inside the sample 6. V. A. Osipov and S. E. Krasavin, J. Phys.: Condens. Mat- (the sample boundaries correspond to the limiting ter 10, L639 (1998). case), thus limiting an infinite increase in κ below the 7. S. E. Krasavin and V. A. Osipov, J. Phys.: Condens. Mat- minimum conductivity. Our calculations demonstrate ter 13, 1023 (2001). that the inclusion of this channel of scattering gives rise 8. H. H. Kuo and T. Mura, J. Appl. Phys. 43 (4), 1454 to an additional maximum in the range of very low tem- (1972). peratures. In the materials under investigation, defects 9. V. A. Likhachev and R. Yu. Khaœrov, Introduction to the of another type can also exist and contribute to the total Theory of Disclinations (Leningr. Gos. Univ., Leningrad, thermal conductivity κ. It should be emphasized that 1975). the minimum thermal conductivity κ obtained in the 10. V. F. Gantmakher and Y. B. Levinson, Carrier Scattering numerical calculations is observed in the temperature in Metals and Semiconductors (Nauka, Moscow, 1984; range 0.1 < T* < 10 K, in which other main mecha- North-Holland, New York, 1987). nisms of scattering (for example, umklapp processes) 11. V. I. Vladimirov and A. E. Romanov, Disclinations in do not contribute appreciably to scattering. For this rea- Crystals (Nauka, Leningrad, 1986). son, the effects associated with the suppression of the 12. P. Clemence, in Low Temperatures Physics (Springer- thermal conduction in the low-temperature range can Verlag, Berlin, 1956; Inostrannaya Literatura, Moscow, be revealed experimentally. Moreover, the observed 1959). behavior of the thermal conductivity κ is unique and Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1156Ð1165. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1106Ð1115. Original Russian Text Copyright © 2002 by Lemanov, Gridnev, Ukhin.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Low-Frequency Elastic Properties, Domain Dynamics, and Spontaneous Twisting of SrTiO3 near the Ferroelastic Phase Transition V. V. Lemanov*, S. A. Gridnev**, and E. V. Ukhin** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] ** Voronezh State Technical University, Moskovskiœ pr. 14, Voronezh, 394026 Russia Received November 26, 2001

Abstract—The low-frequency elastic properties of strontium titanate near the ferroelastic phase transition were studied by the torsional-vibration technique. Domain wall motion was shown to contribute noticeably to the anomalies in the shear modulus and internal friction. It was established that the wall motion under varying elastic stresses is an unactivated process corresponding to viscous ﬂow with a relaxation time inversely proportional to temperature. Spontaneous twisting of samples at the phase transition was revealed, and a model is proposed to account for the sample chirality and the spontaneous twisting effect. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. MEASUREMENT TECHNIQUE AND EXPERIMENTAL RESULTS Strontium titanate SrTiO3 is an incipient ferroelectric and a quantum paraelectric [1, 2] in which quantum The measurements were carried out on a setup fluctuations suppress the ferroelectric phase transition. (Fig. 1) representing an inverse torsion pendulum It has been recently shown [3] that the improper fer- described in detail in [9]. A long twisting rod (1) with a rigidly fixed cross and inertia weights is connected roelastic phase transition in SrTiO3, which occurs at T ≈ 110 K with the symmetry changing from cubic, through a grip to sample 2, whose lower end is clamped a in a fixed grip. The sample is twisted through the rod by 1 18 Pm3m Ð Oh (for T > Ta), to tetragonal, 14/mcm Ð D4h , means of a differential electromagnetic system consist- plays a dominant role in inhibiting the ferroelectric ing of two current-carrying coils 3 arranged diametri- phase transition (if the strontium titanate remains cubic, cally relative to the pendulum axis and interacting with the ferroelectric phase transition can take place, despite two permanent magnets provided by soft-iron pole the quantum corrections [2]). While the anomalies pieces. Photoelectric sensors 5 with lamps 6 served to Ð5 Ð3 observed in the elastic properties of SrTiO3 at the measure small torsional strains (10 Ð10 ). Large improper ferroelastic phase transition have been stud- strains were measured with capacitance sensors 7. ied already for more than a quarter of a century, the When a periodically varying current of low frequency Ω is passed through coils 3, the sample is twisted peri- experiments were performed, as a rule, at frequencies ϕ ϕ Ω ≥ 108 sÐ1 (see monograph [4], review [5], and the ref- odically to angles from + 0 to Ð 0 under the action of a torque M = M sinΩt which varies from +M to ÐM and erences therein). Fossheim and Berre [6] were the first 0 σ 0 σ 0 to consider the contribution due to ferroelastic produces shear strains varying from + 0 to Ð 0. The domains, which form at the transition temperature T ≈ measurements were performed at the natural resonance a frequency of composite oscillator 1Ð2 (twisting rod 110 K, to the anomalies of elastic properties of SrTiO3. with the inertial system and the sample), which in our This contribution turns out to be particularly large at case was 25 Hz. The internal friction QÐ1 (inverse Q fac- low frequencies, Ω ≈ 101Ð102 sÐ1 [7]. tor) was derived from the decrement of freely decaying It has appeared of interest to study the anomalies in vibrations, for which purpose the sample was twisted to the elastic properties and the domain contribution to a given amplitude by the electromagnetic system and these anomalies using the low-frequency technique of the number of vibrations between two preset vibrational strain amplitudes was counted. The shear modu- torsional vibrations. It is the results of these studies that lus G was inferred from the period (frequency) of the are dealt with in this paper. Just as we were coming to resonance vibrations. the end of our experiments, a paper appeared [8] which reported on a study of SrTiO3 by a similar technique. In The maximum shear stresses in the sample were discussing our results, we shall compare them with lit- about 10 MPa. Such stresses corresponded to shear erature data, including those from [7, 8]. strains on the order of 10Ð4 and to sample twist angles

LOW-FREQUENCY ELASTIC PROPERTIES 1157

(the angle of turn of the upper sample cross section relative to the lower, rigidly fixed cross section) ϕ ≈ 0.1° = Ð3 1.7 × 10 rad. We note that the shear modulus G and 1 internal friction QÐ1 were measured at strain amplitudes 6 substantially below their maximum values. 7 The experiments were carried out on single-crystal 4 samples of nominally pure strontium titanate. The sam- 3 ples measured typically 2 × 2 × 10 mm. Samples of two 5 orientations were used (Fig. 2). In the [100] samples, the long axis coincided with the [100] direction and the side faces coincided with the (010) and (001) planes. In the [110] samples, the long axis was aligned with [110] and the (110 ) and (001) planes were parallel to the side faces. The temperature dependences of the shear modulus To vacuum pump G and internal friction QÐ1 are shown in Figs. 3 and 4. When the [100] sample undergoes a phase transition from the cubic to tetragonal phase, the shear modulus G decreases abruptly by about 3%, with no anomalies in QÐ1 seen to occur at the phase transition. In the [110] sample, the modulus G decreases by about 40%, while 2 the internal friction QÐ1 grows strongly upon the transition. This anisotropy can be assigned to the ferroelastic domains contributing to the elastic properties. As will be shown in the next section, the elastic stresses generated in a twisted sample do not initiate domain wall motion in the [100] sample but activate it in the [110] sample. This implies that the static shear stresses created under static sample twisting should not affect the domain structure in the [100] sample but should make the [110] sample single-domain. The experiment confirmed this expectation. The results presented in Fig. 5 indicate that the change in the shear modulus G and the internal friction QÐ1 occurring at the phase transition in the [110] sample depend strongly on the static shear Fig. 1. Experimental setup. (1) Twisting rod with cross, (2) σ sample in clamps, (3) electromagnetic coils producing stress 00 and decrease as it increases, i.e., as the sample torque, (4) permanent magnets with annular soft-iron pole becomes increasingly single-domain. The dependence pieces, (5) photoelectric sensors for small strains, (6) elec- of the shear modulus G and of the internal friction QÐ1 tric lamps, and (7) capacitance sensors for large strains. σ on static stress 00 measured in the low-symmetry phase at 80 K is displayed in Fig. 6. The relative changes in the shear modulus G and internal friction QÐ1 decrease with increasing absolute magnitude of the static shear stress σ 00 as a result of the sample becoming single-domain. [100] [110] Such processes should, in principle, give rise to a hysteresis dependence of the shear modulus G and internal Ð1 σ friction Q on the shear stress 00 (the butterfly pattern). While hysteresis was indeed seen in the experiment (not shown in Fig. 6), it was very weak and occurred in an irregular manner. A similar result was obtained in [8], where hysteresis loops were success- – fully observed in ϕ(σ) only after etching off a surface [001] [110] layer about 5 µm thick from the sample. Ð1 In addition to measurements of G and Q in sam- [010] [001] ples twisted by external forces, we also observed spontaneous twisting of the [110] sample to an angle ϕ in sϕ the absence of external forces (Fig. 7). The angle s, which is zero in the symmetric phase, becomes nonzero 3/4 at T = Ta, to grow subsequently as (Ta Ð T) as the temperature is further lowered. Fig. 2. Sample orientation.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1158 LEMANOV et al.

1.1 6

1.0 1 4 –2 m G , 10 / –1 G

0.9 2 Q

0.8 2 0 Shear modulus Internal friction 0.7

0.6 90 120 150 180 210 240 T, K

Ð1 Fig. 3. Temperature dependences of (1) shear modulus G/Gm and (2) internal friction Q in the [100] sample.

10 1.0 1 8

0.9 –2 m G , 10 / –1 G 6 Q 0.8

4 0.7 Shear modulus Internal friction 0.6 2 2

0.5 0 90 120 150 180 210 240 T, K

Ð1 Fig. 4. Temperature dependences of (1) shear modulus G/Gm and (2) internal friction Q in the [110] sample.

3. DISCUSSION OF RESULTS applied to the sample (Fig. 1) and the elastic strains created in it. The results presented in Figs. 3Ð6 indicate that the ferroelastic domains forming in strontium titanate in The problem of thin-rod twisting has been consid- ≈ the improper ferroelastic phase transition at Ta 110 K ered in the theory of elasticity [10, 11]. Rather than contribute appreciably to the temperature dependences dealing with exact solutions to this problem, we shall of the shear modulus G and internal friction QÐ1. consider only its principal qualitative aspects. Domains can obviously affect the shear modulus and internal friction in two ways, namely, through variation Twisting a thin straight rod with the axis z along the of the crystallographic orientation from one domain to rod axis produces pure shear strains uxz and uyz. The another and through domain wall motion under elastic twist angle γ (the angle of turn per unit rod length) is γ = stresses. To establish the mechanism of this effect, one M/C, where M is the torque and C is the so-called tor- should ﬁnd, ﬁrst of all, the relation between the torque sional stiffness of the rod.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 LOW-FREQUENCY ELASTIC PROPERTIES 1159

(a) 1.0 4.5 MPa

0.9 3.6 MPa m 2.6 MPa G / G 0.8

1.5 MPa 0.7 Shear modulus

0.6 0.52 MPa

0.5 8 (b) –2 6 0.52 MPa , 10 –1 Q 4

2 1.5 MPa Internal friction 2.6 MPa 3.6 MPa 4.5 MPa 0 100 110 120 T, K

Fig. 5. Temperature dependences of (1) shear modulus G/G and (2) internal friction QÐ1 in the [110] sample for various static shear σ m stresses 00.

Approximating our sample with a rod of circular possibly should also be taken into account when ana- cross section with a radius R = 10Ð3 m, we have C = lyzing ﬁner effects. (πR4/2)G, where G is the shear modulus; i.e., the rod’s We consider now the domain structure of samples torsional stiffness is C ≈ 2 × 10Ð12 G ≈ 0.2 J m. This yields with the long axis aligned with the [100] and [110] the value of the twist angle γ (rad/m) = 5 M (J), and for directions (Fig. 2). the stresses and strains, we obtain zero at the center and The canonical form of the thermodynamic potential σ maximum values at the sample surface: xz(Pa) = for strontium titanate is well known [12]: − γ Ð3 γ γ × Ð4γ σ G R = Ð10 G ; uxz = Ð0.5 R = Ð5 10 ; yz(Pa) = Fb= ()q2 ++q2 q2 + b ()q4 ++q4 q4 +GγR = +10Ð3Gγ; and u = +0.5γR = +5 × 10Ð4γ. 1 1 2 3 11 1 2 3 yz ()2 2 2 2 2 2 ()2 2 2 + b12 q1q2 ++q1q3 q2q3 + 1/2c11 u1 ++u2 u3 Thus, shear strains in the sample turn out to be ()()2 2 2 + c12 u1u2 ++u1u3 u2u3 + 1/2c44 u4 ++u5 u6 essentially nonuniform along the rod radius. To obtain (1) an exact quantitative solution to the problem, one has to Ð g ()u q2 ++u q2 u q2 Ð g {u ()q2 + q2 take into account the sample’s shape and the crystallo- 11 1 1 2 2 3 3 12 1 2 3 ()2 2 ()2 2 } graphic anisotropy by invoking the concept of the shear + u2 q1 + q3 + u3 q1 + q2 modulus of an anisotropic material for twisting [11]. () Signiﬁcantly, twisting will also bend the rod, which Ð g44 u4q2q3 ++u5q1q3 u6q1q2 .

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1160 LEMANOV et al.

T = 80 K 6.0 3.6 MPa 1.0 10

–2 4.8 deg

m 2.6 MPa –2 , 10 G 2 1

/ 0.9 8 –1

G 3.6 , 10 Q s

ϕ 1 MPa 0.8 6 2.4 0.5 MPa 0 MPa 0.7 4 1.2 Twist angle Shear modulus

Internal friction 0 0.6 2 85 95 105 115 T, K –15 –10 –5 0 5 10 15 σ Static stress 00, MPa Ð1 Fig. 7. Temperature dependences of the spontaneous twist Fig. 6. (1) Shear modulus G/Gm and (2) internal friction Q angle in the [110] sample plotted for various static shear σ σ in the [110] sample plotted vs. static shear stress 00. stresses 00.

30 5 The elastic moduli, quoted in various papers, are c11 = 10 J/m ) are 11 3 3.30, c12 = 1.05, and c44 = 1.27 (in units of 10 J/cm ). [] g11 ==1.8, g12 Ð1.1, g44 = 2.3 6 , (3) Of all the terms in the thermodynamic potential (1), g ==1.3, g Ð2.5, g = 2.3[] 14 . we are primarily interested in the coupling coefficients 11 12 44 gik of the order parameter with elastic strain. It should We note that the condition g11 = Ð2g12 accepted in [12] be borne in mind that the numerical values of the coef- is in sharp contrast with the real situation, because this ficients in Eq. (1) depend on the quantity used as the condition implies that the phase-transition temperature order parameter. The order parameter may have the Ta is independent of hydrostatic pressure, which dis- meaning of the angle of turn of the oxygen octahedra θ agrees with experiment [6, 15]. We also note that the (in radians) or oxygen displacements q(m). In this case, coupling coefficients gik are equal, in order of magni- the coefficients in Eq. (1) will have naturally different tude, to the elastic moduli cik, as should be. dimensionality and different magnitude and their rela- Let us now consider the elastic strains generated in θ ≈ θ tion will be determined by tan = (2/a0)q, where domains when a [100]- or [110]-oriented sample is θ × a0 = 3.905 Å is the lattice parameter. Then, = 5.1 twisted. It is known that strontium titanate has three 9 types of tetragonal domains, with the tetragonal axis c 10 q and the coupling coefficients gik will have the 3 5 (z) along [100] (domain 1), along [010] (domain 2), and dimensions J/m and J/m , respectively, and will be along [001] (domain 3). related through the coefficient 2.62 × 1019 mÐ2.

The numerical values of the coupling coefficients gik 3.1. Sample [100] quoted in different publications [6, 7, 12Ð14] differ strongly both in magnitude (by a factor of six to eight) We introduce the reference frame with the axes and in sign. These differences are particularly large in coinciding with the coordinate axes of domain 1; i.e., [7]. We note that the scatter of literature data for the the x, y, and z axes point along [010], [001], and [100], coefficient g is substantially smaller than that for g respectively, with the z axis ([100]) aligned with the 44 11 long axis of the sample. Then, twisting the sample cre- and g12. We believe the values presented in [6, 14] to be the most reasonable. We present these values for both ates the following shear strain components in the order parameters. domains: ±u and +−u ()domain 1 , In the case where the order parameter is the angle of zy zx turn of the oxygen octahedra θ (in radians), the cou- ±u and +−u ()domain 2 , (4) pling coefficients (in units of 1011 J/m3) are yz yx ± − () uxz and +uxy domain 3 . [] g11 ==0.69, g12 Ð0.42, g44 = 0.88 6 , (2) Using Eq. (1) for the energy and taking into account [] ≠ g11 ==0.5, g12 Ð0.95, g44 = 0.88 14 . that in each domain q1 = q2 = 0, q3 0, we obtain for the domain energy F1 = F2 = F3 = 0. If the order parameter is the oxygen ion displacement q Thus, all domains are energetically equivalent under (in meters), the coupling coefficients (in units of twisting and twisting should not entail domain wall dis-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 LOW-FREQUENCY ELASTIC PROPERTIES 1161 placement; hence, domains contribute neither to the to Eq. (6), is energetically preferable on a sample face, shear modulus G nor to the internal friction QÐ1, which then domain 2 is energetically favorable on the opposite is exactly what is observed in the experiment (Fig. 3). face. Furthermore, because the strain in the sample is As for the change in the shear modulus at the phase nonuniform, the single-domain state does not set in at transition, Eq. (1) shows [12] that the phase transition the sample center at all and the central part of the sam- should produce jumps in all the shear moduli except c ple retains its polydomain pattern, while on the surface, ∆ 66 ( c66 = 0). Whence it follows that the magnitude of the the single-domain state is most pronounced. experimentally observed jump in a shear modulus depends on the relative numbers of differently oriented At the same time, if considered from the viewpoint domains in the sample. of the properties studied (the shear modulus G and the internal friction QÐ1), the sample becomes fully single- domain (for a large enough torque and, accordingly, a 3.2. Sample [110] large enough shear stress) in the sense that it is the Ð1 The sample axes (Fig. 2) coincide with the tetrago- domains contributing to G and Q that disappear. nal axes of domain 3; namely, the X, Y, and Z axes are Therefore, as seen from Fig. 5, the anomalies in the shear modulus and internal friction decrease with σ along the [110 ], [110], and [001] directions, respec- increasing static shear stress 00. tively, with the Y axis ([110]) aligned with the long axis of the sample. Twisting creates strains ±u and +−u We note that, as for the [100] sample, the effective YZ YX shear modulus and effective coupling coefﬁcient are along these axes. determined, for the given domain structure and ﬁxed The strains in the domains can be decomposed into domain walls, by the domain orientation relative to the the following components (along the cube axes, with sample axes and by the fractions of oppositely oriented Z || z in domain 3): domains.

uxx ==Ðuzz Ð0,a/2, uyy = In view of the shear modulus G and internal friction QÐ1 being dependent on static shear stress σ at a con- () 00 uyz ==uxy 2a, uxz =Ða/2 domain 1 , stant temperature (Fig. 6), we note that, in experiments of this kind, one should observe butterfly-type patterns, uxx ===0, uyy Ðuzz a/2, because G and QÐ1 cannot reverse sign as σ reverses (5) 00 () sign, and the domain structure rearrangement with vari- uyz ==Ð,a/2 uxz =2uxy a domain 2 , σ ation of 00 should be accompanied by hysteresis. As u ==Ðu a/2, u =0, already mentioned, the hysteresis in the dependence of xx yy zz Ð1 σ G and Q on 00 observed in our experiments was u ==u 2a, u =Ða/2() domain 3 , extremely weak and irregular and the butterfly pattern yz xz xy was effectively absent. The absence of this relation | | | | where a = uYZ = uYX . could be assigned only to the fact that the ferroelastic In this case, the domain energies are hysteresis loop in the dependence of strain u on stress σ is very narrow, i.e., that the coercive stress σ is small ()2 c F1 = Ð g11 Ð g12 q3uYX, and, hence, the domain wall mobility is high. As already mentioned, ferroelastic hysteresis in ϕ(σ) [or ()2 σ µ F2 = g11 Ð g12 q3uYX, (6) u( )] only after a surface layer 5 m thick had been etched off the sample was successfully observed in [8]. F3 = 0. This result comes somewhat as a surprise; indeed, the domain wall mobility turns out to be higher in the dam- Taking the values of coefficients g11 and g12 given in Eqs. (2) and (3), we obtain (g Ð g ) > 0. This means aged surface layer than in the damage-free layer. This 11 12 problem requires further comprehensive investigation. that for uYX > 0, domain 1 is energetically preferable and domain 2 is the least favorable. The situation Regrettably, our experiments do not permit us to reverses for uYX < 0. determine the coercive stress σ and thus it can only be c σ Thus, varying the shear stress and, accordingly, estimated. As follows from Fig. 6, c is considerably varying the strain uYX gives rise to domain wall motion less than 10 MPa and, judging from the resolution of (vibrations). As a result, domain contributions to the the setup, does not apparently exceed 1 MPa. Using the Ð1 σ σ shear modulus G and internal friction Q form in the concept of a strong stress ( > h), at which the strain [110] sample (Fig. 4). When a constant torque is induced by it exceeds the spontaneous strain [16], we σ ≈ applied to the sample together with a variable torque, obtain the estimate h = Gus 50 MPa. It may appar- the sample becomes single-domain gradually because ently be accepted that the thermodynamic (or intrinsic) of the increasing fraction of energetically favorable coercive stress should be smaller by about an order of σ σ ≈ domains. This transition to the single-domain pattern is magnitude than h [16], i.e., that c 5 MPa. This very specific. The strains on the opposite sample faces means that the experimentally obtained coercive stress σ ≤ have opposite signs; therefore, if domain 1, according c 1 MPa is substantially smaller than the thermody-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1162 LEMANOV et al. namic coercive stress, exactly as is the case with the quencies Ω ≤ 10 sÐ1, which agrees in order-of-magni- coercive fields in ferroelectrics [16]. tude with our experiment. It is at low measuring fre- Now, we consider the information on domain quencies that large variations in the shear modulus were dynamics that can be extracted from the data in Fig. 4. observed to occur near the phase transition (Fig. 4) We The large domain contribution to the shear modulus note that our data on the temperature dependence of the and internal friction at frequencies near 10 Hz, which shear modulus G at 25 Hz correlate with the data of [7], was observed earlier in [7] and in this work, combined where the measurements were performed at 10 Hz with the considerably smaller high-frequency contribu- using another technique. tion, suggests that elastic vibrations interact with It is of interest to compare our results with the avail- domain walls through a relaxation mechanism with suf- able data. ficiently long relaxation times τ . Therefore, at high fre- Peaks of 30Ð300 MHz ultrasound wave damping Ω Ωτ ӷ d quencies , such that d 1, the domain contribution were observed earlier [6] at temperatures of 105.5Ð103 becomes suppressed, as follows from the expressions K slightly below the phase-transition point. The damp- for ∆G/G and ∆α for the relaxation mechanism of ing maximum shifted toward lower temperatures with damping: increasing frequency. The observed damping anomalies were attributed to a relaxation mechanism of elastic ∆ ()Ω2τ2 G/GK= d/1+ d , wave interaction with domain walls, with the relaxation (7) τ × Ð9 ∆α ()Ω2τ ()Ω2τ2 time d = 1.6 10 /(Ta Ð T) (s). Another temperature = Kd/2 d/1+ d , dependence of the relaxation time was established where Kd is the effective coupling coefficient for elastic when using 25Ð85 MHz ultrasonic waves, with the vibrations with domain walls proportional to the relaxation-enhanced damping peaks lying in the tem- squared coupling coefficient g in Eq. (1). perature region of 65Ð85 K [17]. As the frequency Turning now from damping to the internal friction increased, the damping peaks shifted toward higher (QÐ1 = 2α/Ω), we obtain temperatures, with the relaxation time varying as τ = τ exp()U/kT for τ = 1.5× 10Ð11 s and U = ∆ Ð1 Ωτ ()Ω2τ2 d 0 0 Q = Kd d/1+ d ; (8) 0.04 eV = 464 K. whence it follows that In contrast to [6], the measurements in [17] were performed on thin plates of strontium titanate 1.5 × 10 × ∆ Ð1 ∆ Ð1 Ωτ ()Ω2τ2 Q / Qm = 2 d/1+ d , (9) 30 mm in size with the edges aligned with the 〈110〉, 〈100〉, and 〈110〉 directions, respectively, and the ultra- ∆ Ð1 Ωτ where Qm is the maximum internal friction for d = 1, sonic waves propagating perpendicular to the plate plane, i.e., along the 〈110〉 1.5-mm edge. While this ∆ Ð1 Qm = Kd/2. (10) sample differs strongly in size from the single-domain plate used in [18] (0.3 × 3 × 7 mm along 〈110〉, 〈110〉, Equation (9) and the data in Fig. 4 now permit us to and 〈100〉, respectively), it apparently supports a suffi- estimate the domain-wall relaxation times and their ciently simple domain structure compared to that temperature dependence. obtained in samples with dimensions typically Equation (9) yields two values of the domain-wall employed in ultrasound measurements [6]. τ τ relaxation time d. One solution yields a value of d that Thus, at a temperature, for example, of 100 K, the decreases with decreasing temperature, while for the τ × Ð10 τ domain-wall relaxation times d turn out to be 3 10 other, d increases with decreasing temperature. In the [6], 1.5 × 10Ð9 [17], and 10Ð2 s (this work). It follows first case, the relaxation time scales with temperature as τ ≈ that the domain-wall relaxation times in strontium d 0.1/(Ta Ð T) (s) (within at least the 95Ð105 K inter- titanate can vary over a broad range extending at least val), and in the second case, the relaxation time follows Ð2 Ð10 τ τ from 10 to 10 s. The mechanism of domain wall the relation d = 0exp(U/kT), but with a physically τ ≈ Ð22 motion driven by elastic strains can be both thermally meaningless value 0 10 s and with an activation activated [17] (with the domain walls breaking away energy for domain wall motion U = 0.4 eV, which is from the pinning centers) and, as follows from [6] and obviously too high. We come thus to τ ≈ 0.1/(T Ð T) d a τ this work, activation-free, i.e., associated with the vis- (s) as the choice for the temperature dependence for d. cous flow of domain walls. Thus, in our experiments, domain motion under the Damping peaks associated with domains were also action of elastic stresses is activation-free and corre- reported in other studies [8, 19] (which did not deal sponds to viscous flow with a relaxation time inversely with the temperature dependence of the relaxation time proportional to temperature. For such relaxation times, τ ). the change in the shear modulus G at the phase transi- d tion is suppressed at high frequencies, as follows from Compare the maximum value of internal friction ∆ Ð1 Eq. (7). Strong variations in G will be observed only near the phase transition Qm (or of the attenuation Ωτ ≤ ∆α under the condition d 1. For example, at the tem- m) from literature data [6, 8, 17, 19] with our present perature T = (Ta Ð 1) K, this condition is met for fre- results. It was found that over a broad frequency range,

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 LOW-FREQUENCY ELASTIC PROPERTIES 1163

≈ Ð11 extending from 3.5 kHz [19] to 300 MHz [6], internal The coefficient Cq is found to be very small, Cq 10 s K ∆ Ð1 ≈ Ð13 Ω ≈ friction Qm is almost independent (to within a factor [6] or even Cq 10 s K [5]. For our frequencies 2 Ð1 Ωτ Ð9 Ð11 two) of frequency and is approximately 2 × 10Ð3 (a 10 s and for (Ta Ð T) = 1 K, we have q = 10 Ð10 . remarkable result, considering that the above studies Because the coupling constant of elastic vibrations with used different techniques and different samples). Thus, the order parameter Kq < 1, the change in the internal ∆α attenuation m turns out to vary in proportion to the friction through the LandauÐKhalatnikov mechanism at ∆ Ð1 Ωτ ≤ frequency. It is such a behavior of the internal friction the phase transition should constitute Q = Kq q and attenuation that should be observed, according to 10Ð9Ð10Ð11, which is too small a value to be measurable. Eq. (10), provided that the coupling coefficient of elas- Now, let us discuss the results presented graphically tic vibrations with domain walls is the same, irrespec- in Fig. 7. The sample to which no external torque was tive of whether the domain wall motion is thermally applied undergoes spontaneous twisting below the activated or viscous. ϕ phase-transition point, with the twist angle s increas- ∆ Ð1 ing with decreasing temperature. If now a static shear The value of Qm obtained in this work at the fre- σ quency 25 Hz is 9 × 10Ð2, which is about two orders of stress 00 is applied to the sample, the spontaneous twist angle increases substantially (Fig. 7). We note that magnitude larger than is reported in the literature. This σ ≠ ϕ σ means that at such low frequencies, the coupling coef- for 00 0, the contributions to the ( ) dependence ficient for elastic waves interacting with domain walls come not only from spontaneous twisting but also from the twisting associated with the variation of G(T) and is substantially larger than that in the kilohertz or mega- σ σ hertz ranges. It also appears essential that the elastic an increase in 00, which is proportional to 00/G(T). vibrations interact with domain walls primarily in a sur- The first contribution is, however, dominant. face layer. Spontaneous twisting upon phase transition was observed earlier to occur in crystals of KH (SeO ) , α 3 3 2 Interestingly, the background attenuation , as K2ZnCl4, KH2PO4, etc. (see [21] and references reported in the same publications [6, 8, 17, 19] and therein), as well as in Hg2Cl2 crystals [22]. The nature measured over as broad a frequency range, from 35 kHz of this spontaneous twisting phenomenon has not been [19] to 300 MHz [6], scales approximately as α ∝ Ω1.5 clarified. and again the attenuation measured by us exceeds by We consider possible reasons for the spontaneous two orders of magnitude the attenuation that should be twisting. This twisting appears only in the [110] sample α ∝ Ω1.5 observed at this frequency if it behaves as . It (Fig. 2). Retaining the sample frame introduced earlier, is known that, at frequencies above approximately 100 MHz, the attenuation of elastic waves is described namely, with the X, Y, and Z axes lying along [110 ], by the Akhiezer mechanism with the scaling α ∝ Ω2 [110], and [001], respectively, we come to the conclu- and that the attenuation of transverse waves in cubic sion that such a sample will undergo spontaneous twist- dielectrics, measured at 1000 MHz and temperatures of ing if spontaneous pure shear strains uYZ and uYX are 100Ð300 K, is more or less typically 1Ð10 dB/µs [20]. created in it at T < Ta. It is also known that as the frequency is lowered and, Furthermore, for the sample to know the sense in accordingly, the attenuation decreases strongly, the lat- which it should twist, it must possess a definite chiral- ter becomes dominated by other, zero-phonon (non- ity. Then, the “right-handed” sample will twist to the Akhiezer) processes, with the frequency dependence of right; the “left-handed” one, to the left. There is no such the attenuation becoming progressively weaker and chirality in the sample symmetry groups either above or finally disappearing. Such zero-phonon attenuation below the phase transition. This chirality may appear, mechanisms can be associated with the crystallite in principle, in a unipolar sample with different num- structure, dislocations, macrodefects, sample surface, bers of different domains. etc. It is the sample surface that may play the most sub- The spontaneous strains forming in strontium titan- stantial role in our technique. s ate along the cubic axes at the phase transition are uxx = Turning now back to the anomalies in elastic prop- s s erties near the phase transition, we note the absence of uyy = u1s and uzz = u3s. Going over to the sample axes, the internal friction (attenuation) peak (Fig. 3) gov- we find spontaneous shear strains us = (1/2)(u Ð u ) erned by the relation of the elastic vibrations with the YX 3s 1s s order parameter (the LandauÐKhalatnikov mecha- and uYX = Ð(1/2)(u3s Ð u1s) to occur in domains 1 and 2, nism). The LandauÐKhalatnikov attenuation is also respectively. It would seem that such strains forming described by the relaxation equation (7), where the even in a unipolar sample should not bring about sam- relaxation time has the meaning of the order-parameter s τ ple twisting because there are no u strains. Let us relaxation time q. This relaxation time depends on YZ temperature as consider this point in more detail. As follows from Fig. 7, the spontaneous twist angle τ ()() ϕ q = Cq/ T a Ð T s. (11) s is particularly large in the single-domain sample at

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1164 LEMANOV et al.

[110] [110] tially above room temperature. We note that when the sign of the static torque M is reversed, thus reversing 00 σ the sign of the static shear stress 00, the sample starts to twist in the opposite sense, which argues for the twist model proposed. ϕ The twist angle s as a function of temperature should be proportional to the spontaneous strain us(T). Then, for the second-order phase transition, we obtain ϕ ∝ 2 ∝ s qs (Ta Ð T). In the case of the tricritical point, ϕ ∝ 2 ∝ 1/2 s qs (Ta Ð T) . The experimental relation plotted σ 3/4 in Fig. 7 for 00 = 0 is the best fit to the (Ta Ð T) relation. Although a more accurate determination of the critical index would require a more comprehensive study, it is already possible to conclude that the improper ferroelastic phase transition in strontium titanate is a second-order transformation close to the Fig. 8. Left- and right-handed samples of strontium titanate. The filled and open arrows specify the direction of the tet- tricritical point. ragonal axis c on the front and rear faces, respectively. We note in conclusion that spontaneous twisting of samples at a phase transition should also occur in other proper and improper ferroelastic and ferroelectric σ 00 = 3.6 MPa (the sample is treated as single-domain in phase transitions. For instance, in the case of the m3mÐ the sense specified above). In the single-domain sample, 4mm ferroelectric phase transition (BaTiO3, PbTiO3), the only domain left on the front face is the favorable the inclusion of the electrostriction coupling gP2u in the domain 1 with its tetragonal axis c making an angle of thermodynamic potential results obviously in the same 45° with the [110] sample axis and directed to the left of situation as in strontium titanate. Thus, BaTiO and 〈 〉 3 this axis; the domain left on the rear face is the favorable PbTiO3 samples cut along the 110 direction should domain 2 with its c axis at 45° to the right of the sample twist spontaneously at the phase transition and the axis. A sample with such a domain structure is chiral; spontaneous twist angle in BaTiO3 and PbTiO3 should namely, reflection of the sample in the mirror plane be considerably larger than that in SrTiO3 because of reverses the sign of chirality, such that, for example, the the larger spontaneous strain. left-handed sample becomes right-handed (Fig. 8). Can such a sample twist spontaneously? It has been shown ACKNOWLEDGMENTS that a spontaneous shear strain us = (1/2)(u Ð u ) YX 3s 1s One of the authors (V.V.L.) is indebted to O.E. Kvyat- forms at the front face and that the strain at the rear face kovski for valuable discussions. s œ has the opposite sign, uYX = Ð(1/2)(u3s Ð u1s). As a This study was partially supported by NWO, grant result, the side faces tilt, as it were, in opposite direc- no. 16-04-1999. tions. It is this system of spontaneous strains that apparently brings about spontaneous twisting of the sample. σ REFERENCES For 00 = 0, the sample is in a multidomain state and, for equal numbers of different domains, is no longer 1. K. A. Müller and H. Burkard, Phys. Rev. B 19, 3593 chiral and stops twisting. To explain the spontaneous (1979). twisting of such a sample, one has to assume that the 2. O. E. Kvyatkovskiœ, Fiz. Tverd. Tela (St. Petersburg) 43 sample is unipolar (more specifically, that the surface (8), 1345 (2001) [Phys. Solid State 43, 1401 (2001)]. layers on the front and rear faces are unipolar). Such a 3. A. Yamanaka, M. Kataoka, Y. Inaba, et al., Europhys. unipolarity may be induced by internal stresses, whose Lett. 50, 688 (2000). field makes favorable domains dominant over unfavor- 4. M. E. Lines and A. M. Glass, Principles and Applica- able ones. If, for instance, domain 1 is dominant on the tions of Ferroelectrics and Related Materials (Claren- front face and domain 2 is dominant on the rear face, don, Oxford, 1977; Mir, Moscow, 1981). the situation will not differ from that in a single-domain 5. V. V. Lemanov, Ferroelectrics 265, 1 (2001). sample, but the twist angle in the multidomain sample 6. K. Fossheim and B. Berre, Phys. Rev. B 5, 3292 (1972). will be smaller (Fig. 7). In our case, the sample always 7. A. V. Kityk, W. Schranz, P. Sondergeld, et al., Phys. Rev. twisted in the same direction and even when warmed to B 61, 946 (2000). room temperature it did not forget the sense of the twist. 8. A. Binder and K. Knorr, Phys. Rev. B 63, 094106 This means that, in order to remove the internal stresses (2001). determining the sign of chirality and, hence, the sign of 9. S. A. Gridnev, V. I. Kudryash, and L. A. Shuvalov, Izv. the twisting, the sample has to be annealed substan- Akad. Nauk SSSR, Ser. Fiz. 43 (8), 1718 (1979).

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10. L. D. Landau and E. M. Lifshitz, Course of Theoretical 17. E. V. Balashova, V. V. Lemanov, R. Kunze, et al., Ferro- Physics, Vol. 7: Theory of Elasticity (Nauka, Moscow, electrics 183, 75 (1996). 1987; Pergamon, New York, 1986). 18. K. A. Müller, W. Berlinger, M. Capizzi, and H. Gran- 11. Yu. I. Sirotin and M. P. Shaskolskaya, Fundamentals of icher, Solid State Commun. 8, 549 (1970). Crystal Physics (Nauka, Moscow, 1975; Mir, Moscow, 19. O. M. Nes, K. A. Müller, T. Suzuki, and F. Fossheim, 1982). Europhys. Lett. 19, 397 (1992). 12. J. C. Slonczewski and H. Thomas, Phys. Rev. B 1, 3599 (1970). 20. V. Ya. Avdonin, V. V. Lemanov, I. A. Smirnov, and V. V. Tikhonov, Fiz. Tverd. Tela (Leningrad) 14 (3), 877 13. W. Rehwald, Solid State Commun. 8, 1483 (1970). (1972) [Sov. Phys. Solid State 14, 747 (1972)]. 14. H. Uwe and T. Sakudo, Phys. Rev. B 13, 271 (1976). 21. S. A. Gridnev, O. N. Ivanov, L. P. Mikhaœlova, and 15. G. Sorge and E. Hegenbarth, Phys. Status Solidi 33, K79 T. N. Davydova, Fiz. Tverd. Tela (St. Petersburg) 43 (4), (1969); G. Sorge, E. Hegenbarth, and G. Schmidt, Phys. 693 (2001) [Phys. Solid State 43, 722 (2001)]. Status Solidi 37, 599 (1970). 16. L. D. Landau and E. M. Lifshitz, Course of Theoretical 22. A. Binder, K. Knorr, and Yu. F. Markov, Phys. Rev. B 61, Physics, Vol. 8: Electrodynamics of Continuous Media 190 (2000). (Nauka, Moscow, 1982; Pergamon, New York, 1984). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1166Ð1170. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1116Ð1120. Original Russian Text Copyright © 2002 by Sinyavskiœ, Brusenskaya. LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Light Magnetoabsorption in Systems with Quantum Confinement in the Field of Resonant Laser Radiation É. P. Sinyavskiœ and E. I. Brusenskaya Institute of Applied Physics, Academy of Sciences of Moldova, Chisinau, MD-2028 Moldova Received April 12, 2001; in ﬁnal form, October 9, 2001

Abstract—Near-gap light magnetoabsorption is investigated in systems with quantum confinement in the presence of IR laser radiation. It is shown that if the frequency of laser radiation is equal to the cyclotron frequency, the magnetoabsorption line shape can be fully determined by the IR radiation intensity. The frequency dependence of interband absorption of a weak electromagnetic wave can be changed significantly in the case where the laser radiation frequency is equal to the quantum-confinement frequency in parabolic quantum wells and the polarization vector is parallel to the confinement axis. © 2002 MAIK “Nauka/Interperiodica”.

1. Let us consider a quantum-confined system in a (ω ≠ ωv = eH/mvc). The laser radiation intensity is quantizing magnetic field H parallel to the quantum assumed to be low; therefore, we can neglect multipho- confinement axis z. In this system, the energy of an ton transitions between fully quantized hole states and electron (hole) in the quantum well (QW) is fully quan- discrete electron states. tized. The frequency dependence of the light absorption 2. Under the assumption made above, the Hamilto- associated with transitions between hole and electron nian of the system understudy has the form states (in the absence of a magnetic field, this is interband absorption) is characterized by the number of HH= 0 + W, peaks with half-widths determined by the interaction of c + v + បω + carriers with lattice vibrations [1]. At the present time, H0 = ∑Eαaαaα + ∑Eα a˜ αa˜ α + b b, (1) perfect quantum-confined structures can be fabricated α α with a small quantity of lattice defects and impurities. + + It is this factor that determines the presence of high- WV= ∑ αβaαaβ()b + b . ≥ 5 2 mobility ( 10 cm /V s) carriers at low temperatures [2, α 3] and narrow luminescence peaks (~3 meV [4]). We + + + will consider the case of strong magnetic fields where Here, aα (aα), a˜ α (a˜ α ), and b (b) are the creation (anni- the Coulomb electronÐhole interaction is small in com- hilation) operators of electrons, holes, and photons, parison with the distance between Landau quantization respectively; α(N, n, k ) is a set of quantum numbers levels and exciton effects in the light absorption are x negligible. A detailed discussion and the criteria of the characterizing the state of a confined carrier in a quan- validity of such an approximation can be found in [5]. tizing magnetic field; N is the Landau level index; n is When an electronÐhole pair is excited by light, the the quantum-well level index; and kx is the projection of quasi-momentum of the created exciton is equal to the the electron quasi-momentum. momentum of the absorbed photon and, hence, is very For a square QW of width a, we have small [6]; therefore, we can neglect the exciton bands 2 2 appearing in two-dimensional systems in a strong mag- c 1 2 ប π Eα ==បω N + --- + ε n , ε ------, netic field [7]. c2 c c 2 2mca (2) Significant influence of IR laser radiation of fre- 2 2 v 1 2 ប π quency ω upon low-power light absorption occurs Eα ==បω N + --- + ε n , ε ------. ω ω v 2 v v 2 when is equal to the cyclotron frequency c (IR mag- 2mv a netic resonance). In this case, the resonant laser radiation causes the electron states to be nonstationary and, The matrix element Vαβ of the electronÐphoton interac- hence, can control the shape of electromagnetic-wave tion operator between electron wave functions in the absorption peaks. QW in a longitudinal magnetic field is

ω ω V αβ = idδ , ' δ , ()Nδ , Ð N + 1δ , , We will consider the resonance case of ~ c = kx kx nn' NN1 + 1 NN1 Ð 1 eH/mcc (where mc(v) is the effective electron (hole) πω 1/2 (3) mass); therefore, for the sake of simplicity, the interac- ប c de= ------ε ω- , tion of the laser radiation with holes can be neglected V 0mc

ε () where 0 is the permittivity and V is the volume of the where operators l N are deﬁned as system under study. m ()N () () In Eq. (3), it is assumed that the electric ﬁeld of the lm fN = fN+ m, (9) linearly polarized laser radiation is directed along the y axis; i.e., the laser beam falls normal to the surface of with f(N) being an arbitrary function of the Landau the system under study. level index N. The weak-light absorption factor K(Ω) can be Naturally, we have expressed through the dipole-moment correlation func- ()N ()N ˆ tion [8]. Within the approximations made, we can write lm lÐm = I, (10)

2πe2 P e 2 where Î is the unit operator. From Eqs. (8) and (10), it K()Ω = ------cv 0 2 ε បΩ follows that a a+ = δ and that the operators a+ Vm0c 0 N1 N2 N1 N2 N ∞ (4) and aN can be considered boson operators. v + × dtitexp{}[]Ω Ð ()Eα + ε /ប 〈〉aα()t aα . We will solve Eq. (7) in the resonance approxima- ∑ ∫ g |ω ω | |ε| Ӷ ω α tion Ð c = c. In neglecting nonresonant terms Ð∞ (substantiation of this approximation and the corre- Here, P v is the momentum matrix element between the sponding criteria can be found in [9]), Eq. (7) takes the c ε Bloch states, g is the band width, m0 is the mass of a form free electron, e0 is the polarization vector of the weak d + + electromagnetic wave, and c is the speed of light. ξú α() ()ξ()ε ()ε () t = ប--- aNbiexp Ð t Ð aNb exp i t α t , (11)   ε ω ω |ε| Ӷ ω it it where = Ð c and c. aα()t = exp--- H aα expÐ--- H . (5) ប ប A solution to Eq. (11) with the initial condition ξα(0) = aα(0) has the form Averaging in Eq. (4) is performed using the density ξ () {}ε + matrix of the electronÐphoton system. According to α t = exp Ðit aNaN Eq. (5), the operator (12) + id + + × ε -----()ξ ()  exp it aNaN + ប aNb Ð aN b α 0 . it c  ξα()t = exp--- Eα aα()t (6) ប We substitute Eq. (12) into Eq. (6) and find aα(t). satisfies the equation of motion Thereafter, Eq. (4) for the light absorption factor is represented in the form (for nondegenerated semiconduc- d + ξú () []δ δ 〈 α 〉 α ≈ Nnkx t = ---∑ N , Ð N + 1 , tor systems, a aα = 1 Ð n 1) ប NN1 + 1 NN1 Ð 1 N1 ∞ 2π ∞

+ ()Ω ϕ () × ()biexp{}Ð ωt + b exp{}iωt (7) K = K0∑ ∫ dt ∫ d ∫rrPrd Nn, Ð∞ 0 0 × exp{}ξitω ()NNÐ ()t . c 1 N1nkx × {}[]Ω ()ε ប {}ε + exp it Ð Eα* + g / exp Ði aNaNt (13) In deriving Eq. (7), we neglected the inﬂuence of carriers on the photon spectrum; i.e., we put  × ε + id {}iϕ + Ðiϕ expit aNaN + -----raNe Ð aNe .   ប it ()+ it exp---ប H bb+ expÐ---ប H   Here, the following notation is introduced: ≈ biexp{}Ð ωt + b+ exp{}iωt . 2πe2 P e 2 K = ------cv 0 -, 0 2 ε បΩ 2 This approximation is quite natural since all correc- am0c 0 R tions to the photon spectrum are small (on the order of π2ប2 2 ប the inverse volume of the system under study). បω 1 n បω eH Eα* ==c*N + --- + ------, c* ------, For further calculations, it is convenient to introduce 2 2µa2 µc the operators Ð1 Ð1 Ð1 2 cប () () µ N + N ==mc + mm , R ------. aN ==N + 1l1 , aN NlÐ1 , (8) eH

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1168 SINYAVSKIŒ, BRUSENSKAYA The averaging over the free-photon system was fulfilled We note that, for a parabolic QW, the absorption fac- in the usual way [10, 11]; in the averaged operator tor for a weak electromagnetic wave is similar to ϕ expression, the operator b is replaced with z = reÐi , the 1 1 ϕ បω   បω   operator b+ is replaced with z* = rei , and integration is Eq. (16), but Eα* = c*-N + --- + 0*-n + --- and, 2 2 2 performed with respect to d z over the complex plane under the summation sign over n, we should introduce with a weight function P(r). | |2 the factor Vn : + Using the algebra of boson operators for a and a ∞ N N ()λ λ 1/4 v 1 2 [12], we find c ()λ λ V n = ------n - ∫ dzexp Ð--- c + v z + π 2 {}ε 2 n! ∞ exp Ði aNaNt Ð × ()λ ()λ Hn cz Hn v z , + id iϕ + Ðiϕ × ε () () () expit aNaN + -----raNe Ð aNe λ ω i ប បω i ប where i = mi 0 / (i = c, v), 0 is the spacing ω between QW levels of electrons (c) and holes (v), 0* = + {}() {}() {}() () ()v = exp CtaN exp ÐC* t aN exp At ; (14) ω c ω 0 + 0 , and Hn(z) are Hermite polynomials. ε () ird Ðiϕ[]()ε In the case of exact resonance ( = 0), Eq. (16) takes Ct = Ð------បε e expit Ð 1 , the form ∞ 2 2  ir d exp()Ðiεt Ð 1 ()Ω it[]បΩ ()* ε At()= ------t + ------. K = K0∑ ∫ dtexp--- Ð Eα + g 2 ε ប ប ε i Nn, ∞ Ð (17) + 2 2 2 γ t 2 e E Using the definition and properties of the operators aN × expÐ------L []γ t , γ = ------. 2 N 8m បω and aN in Eqs. (8) and (9), one can easily show that c exp()Ct()a+ exp()ÐC*()t a From Eq. (17), it immediately follows that in the field N N ω ω of resonant laser radiation ( c = ), damping of the N ∞ m ()Ct()m()ÐC*()t 1 γ t2 Gaussian type proportional to the factor exp Ð------= ∑ ∑------() (15)  m!m1! NmÐ ! 2 m = 0 m1 takes place; i.e., the laser radiation produces nonsta- () × []N!()Nmm+ Ð ! 1/2l N . tionary electron states. 1 m1 Ð m By integrating over t in Eq. (17), one gets For stationary laser radiation, we have π 2 ()Ω 2 1 2 qnN qnN () 1 δ() K = K0 ------∑------Hn ------expÐ------, Pr = ------π rNÐ 0 , γ 2nn! 2γ 2γ 2 N0 Nn, (18) បΩε where N is the average number of photons in the radi- Ð g Ð Eα* 0 ------ating mode. qnN = ប . Using Eq. (15), we represent Eq. (13) for the light Therefore, the frequency dependence of the light absorption factor in the final form absorption due to electron transitions from the lowest ∞ hole state (N = 0, n = 0) to the electron state (N = 0, n =  ()Ω it[]បΩ ()ε 0) has a Gaussian profile with half-width ∆ = K = K0∑ ∫ dtexp--- Ð Eα* + g ប ប γ Nn, Ð∞ 2 2 ln2 . When the intensity of the resonance laser radiation increases, the heights of absorption peaks × exp{}iζ[]tε + i()1 Ð exp()Ðiεt (16) decrease and their half-widths increase (Fig. 1). If the transition occurs from the hole state (N = 1, n = 0), the ε × ζ 2 t magnetoabsorption peak splits into two (Fig. 2), with LN 4 sin ---- , 2 the amount of splitting δ being 2ប 2γ . For a transition from the hole state N to the Nth Landau level of elec- e2E2ω ζ = ------c -, trons, the absorption peak splits into N + 1 peaks. This 8បm ω2ε2 splitting corresponds to different quantum numbers for c the angular momentum of electron states in the mag- where LN(z) are Laguerre polynomials. netic field.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 LIGHT MAGNETOABSORPTION IN SYSTEMS WITH QUANTUM CONFINEMENT 1169

The absorption line shape of quantum-conﬁned systems in a longitudinal magnetic ﬁeld is determined by multiphonon processes [1]. The half-width of the mag- 1 ∆ netoabsorption line associated with the interaction γ 0 1 of carriers with acoustic lattice vibration modes is /2 3()E2 + E2 m ω 00 c v c c ()

∆ Iq 2 0 ==2 k0Ta0 ln2 , a0 ------, បπρv 2a where Ec(Ev) is the deformation potential constant for an electron (hole), ρ is the density of QWs of width a, 0 5 10 v is the velocity of sound in the crystal, and T is the q /2γ temperature. 00 The resonance laser radiation electric ﬁeld Ecr for Fig. 1. Frequency dependence of the ﬁrst magnetoabsorp- which the absorption line half-width determined by IR tion peak (in relative units). Curves 1 and 2 are drawn for radiation is equal to the line half-width due to the inter- E = 1500 and 3000 V/cm, respectively. action of carriers with long-wavelength acoustic ∆ ∆ phonons ( = 0) can be found to be ω 0.4

4m a k T γ E2 = ------c 0 0 . 1 cr ប 2 e /2 For typical parameters of a GaAsÐAlGaAs square QW 10 0.2 ()

ρ 3 × 5 Iq (Ec = 9 eV, Ev = 7 eV, = 5.4 g/cm , v = 2 10 cm/s, 2 បω Ð2 mc = 0.06m0) and c = 10 eV, T = 10 K, and a = 50 Å, 3 0 we have Ecr = 10 V/cm. For a parabolic QW with the 10 15 20 25 × above parameters and a = 1000 Å, we have Ecr = 4 q /2γ 102 V/cm. 10 3. Let us consider a parabolic QW and let the elec- Fig. 2. Frequency dependence of the second magnetoab- tric field of the laser radiation be directed along the con- sorption peak (in relative units). Curves 1 and 2 are drawn finement axis. For this configuration, the matrix ele- for E = 1500 and 3000 V/cm, respectively. ment of the electronÐphoton interaction operator is δ δ ()δ δ γ V αβ = id0 k , k k , k n nn, + 1 Ð n + 1 nn, Ð 1 , where 0 is the electron-phonon scattering probability x x1 y y1 1 1 បω per unit time in the conduction band and 2 is the where spacing of the QW levels in the valence band. πω d = eប ------1 - 1/2, 0 Vε m ω In the case of scattering on acoustic phonons in the 0 c elastic-scattering approximation, it is easy to show that α (n, kx, ky) are quantum numbers of an electron in a 2 parabolic QW, n is the QW level index, kx and ky are the k TE m λ 1/2 m ω electron wave vector projections, and បω is the spac- បγ = ------0 1 c ------c λ = ------c 1 . (20) 1 0 2 2 2π c ប ing between the QW levels. ប ρv A calculation of the light absorption induced by We note that for the typical parabolic-QW parameters transitions from the valence band to the conduction at T = 10 K and បω = បω = 0.01 eV, we have b = 1 if band is made in the same way as above. As a result, in 1 × ω ω the laser radiation electric field is Ecr = 3.7 10 V/cm. the case of exact resonance ( = 1), one gets Therefore, if the laser radiation electric field is such that E ӷ E (b ӷ 1), the frequency dependence of the inter- 2µe2 P e 2 cr K()Ω = ------cv 0 ∑ V 2F()δ , b , band light absorption is fully determined by IR laser 2 ε ប2Ω n n radiation: am0c 0 n ∞ ∞ ()δ 2 2 π sin nt bt 2 π sin()q x  ()δ , ------() ()δ , n x ()2 F n b = --- + ∫dt exp Ðt Ð ------Ln bt , F n b = --- + ∫dx------expÐ----- Ln x 2 t 2 2 x 2 0 (19) 0 បΩ Ð E˜ Ð nបω γ δ ==------g -0, b -----, ()b ӷ 1 , (21) n បγ 2 0 γ 0 Λ បω n ∆ បΩ ˜ បω ˜ 0 qn ==------, n Ð Eg Ð n 0. បω ==បω + បω , Eg ε + ------, ប γ 0 1 2 g 2 2

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1170 SINYAVSKIŒ, BRUSENSKAYA

3 At N = 0 (transitions to the zero Landau level), we have γ 2 2 2 2 ()Ω π  ប K = K0 ---∑ exp Ð q0n --- ; (24) / γ γ

1  n ∆

() 1

K at N =1, 0  0 1 2 ()Ω π2 2 K = K0 ---∑ q1n expÐ q1n --- . (25) ∆ /ប 2γ γ γ 1 n As immediately follows from Eqs. (24) and (25), the Fig. 3. Frequency dependence of the interband light absorp- half-width of the ﬁrst magnetoabsorption peak is ∆ = tion factor (in relative units) for a parabolic QW under resonant IR laser radiation. The dashed line corresponds to the ប 2γ ln2 and the second peak splits into two peaks, case where there is no laser radiation. with the distance between them being δ = ប 2γ . We note that in this case, ∆ and δ are two times smaller than At a given n, the integral over x in Eq. (21) can be in the case of stationary laser radiation. exactly calculated to give π ∆ REFERENCES ()δ , Φ0 F 0 b ==--- 1 +,------n 0, 2 ប 2γ 1. É. P. Sinyavskiœ and E. I. Grebenshchikova, Zh. Éksp. Teor. Fiz. 116 (6), 2069 (1999) [JETP 89, 1120 (1999)]. π ∆ ()δ , Φ1 2. M. Shayegan, T. Sajoto, M. Santos, and C. Silvestre, F 1 b = --- 1 + ------(22) Appl. Phys. Lett. 53 (9), 791 (1988). 2 ប 2γ 3. T. Sajoto, M. Santos, and M. Shayegan, Appl. Phys. Lett. 55 (14), 1430 (1989). 2 π q1 4. H. Buhmann, W. Joss, K. V. Klitzing, et al., Phys. Rev. Ð q1 --- expÐ----- , n = 1, 2 2 Lett. 66 (7), 926 (1991). 5. W. Edelstem, H. N. Spector, and R. Marasas, Phys. Rev. where Φ(z) is the probability function. B 39 (11), 7697 (1989). Figure 3 shows the frequency dependence of K(Ω) 6. R. J. Elliot and R. London, Phys. Chem. Solids 15 (1), (in relative units) calculated from Eqs. (19) and (22). 196 (1960). The dashed line represents the frequency dependence 7. I. V. Lerner and Yu. E. Lozovik, Zh. Éksp. Teor. Fiz. 78 of interband light absorption in the absence of resonant (3), 1167 (1980) [Sov. Phys. JETP 51, 588 (1980)]. laser radiation; this dependence has a typical stepwise 8. R. Kubo, J. Phys. Soc. Jpn. 12 (6), 570 (1957). shape [13, 14]. 9. E. Yu. Perlin and V. A. Kovarskiœ, Fiz. Tverd. Tela (Le- ningrad) 12 (11), 3105 (1970) [Sov. Phys. Solid State 12, In conclusion, we note that for a chaotic laser radia- 2512 (1970)]. tion for which 10. R. J. Glauber, in Quantum Optics and Electronics (Les 2 Houches Summer School Lectures) (Gordon and Breach, () 1 r New York, 1965; Mir, Moscow, 1967). Pr = ------π - expÐ------, N0 N0 11. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968; Mir, Mos- the features of the interband absorption of a weak elec- cow, 1970). tromagnetic wave speciﬁed above remain the same. For 12. W. H. Louisell, Radiation and Noise in Quantum Elec- instance, according to Eq. (13), K(Ω) for chaotic laser tronics (McGraw-Hill, New York. 1964; Nauka, Mos- radiation takes the form cow, 1972). 13. D. A. Miller, D. S. Chemla, and S. Schmitt-Rink, Phys. 2 2 N Rev. B 33 (10), 6976 (1986). ∞ cosq ---τ () 1 Ð τ Nn γ 14. D. A. Miller, J. S. Weiner, and D. S. Chemla, IEEE J. ()Ω 2 τ K = K0∑2 ---∫d ------. (23) Quantum Electron. 22 (9), 1815 (1986). γ ()τ2 N + 1 Nn, 0 1 + Translated by D. Bayuk

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

The Temperature Dependence of Eu Atom Yield Resonances in Electron-Stimulated Desorption from Oxidized Tungsten V. N. Ageev and Yu. A. Kuznetsov Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received August 22, 2001

Abstract—This paper reports on the ﬁrst measurement of the intensity of the resonances in the yield of europium neutrals as a function of temperature observed in electron-stimulated desorption from tungsten surfaces oxidized to different degrees and having different europium coverages. The measurements were carried out by the time-of-ﬂight method with a surface ionization detector. The temperature dependences obtained for resonances due to europium and tungsten core level ionization differ qualitatively. The relation is reversible for temperatures below the onset of europium thermal desorption. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION for 3 h. The ribbon cleanness was checked by Auger Electron-stimulated desorption (ESD) is widely electron spectroscopy. The oxygen monolayer was prepared by exposing a ribbon heated to T = 1600 K to employed in the analysis and modification of adsorbed × Ð6 layers. However, reliable measurements of the fluxes of oxygen at a pressure of 1 10 Torr for 300 s. The tungsten oxide film was grown on a ribbon heated to neutrals produced in ESD are lacking, and this hampers × Ð6 the development of elaborated models of the process T = 1100 K at an oxygen pressure of 1 10 Torr for involved [1]. 600 s [8]. We measured the yield and energy distributions of The europium was deposited on a ribbon at T = 300 K alkali metal atoms emitted in ESD from layers of these from a direct-heated evaporator made of a tantalum metals adsorbed on the surface of oxidized tungsten [2] tube into which metallic europium was placed. The and molybdenum [3], as well as the yield and energy tube was equipped with several holes for uniform distribution of barium [4] and the yield of europium [5] europium deposition along the ribbon. The concentra- atoms observed in the ESD from layers adsorbed on tion of deposited europium was determined by ther- oxidized tungsten. It was found that the mechanism of modesorption spectroscopy and monitored by the sur- neutral ESD depends substantially on the structure of face ionization current from the heated ribbon. The the atomic valence shells; in particular, the dependence monolayer europium coverage was identified with the of the Eu ESD yield from europium layers adsorbed on maximum ESD yield of Eu atoms after the tungsten 5p oxidized tungsten on the incident electron energy has a and 5s core-level excitation [6]. One europium mono- resonance character, with the electron energies corre- layer (ML) on a tungsten surface coated by an oxygen sponding to the resonance peaks that correlate with the monolayer corresponds to a europium concentration of × 14 2 europium 5p and 5s and tungsten 5p3/2, 5p1/2, and 5s 7.5 10 atoms/cm , and one europium monolayer on core-level excitation energies [6]. a tungsten oxide surface corresponds to a europium 15 2 This paper reports on a study of the temperature concentration of 1 × 10 atoms/cm . dependence of the resonances in the Eu ESD yield from The temperature of the ribbon was lowered by pass- oxidized tungsten. ing gaseous nitrogen, which was precooled in a coiled copper tube immersed in liquid nitrogen, through its 2. EXPERIMENTAL TECHNIQUE hollow fixtures. The ribbon temperature could be varied from 160 to 300 K by properly controlling the nitrogen The instrumentation used and the technique flow. The temperature was derived from the tempera- employed in sample preparation for measurements ture dependence of the ribbon electrical resistance, were described in detail in [7]. The measurements were which was found by placing the ribbon into a cooler performed in a high-vacuum chamber at a base pressure maintained at a known temperature. of less than 5 × 10Ð10 Torr. The targets were textured tungsten ribbons with preferential (100) orientation The ribbon was heated by passing an electric current measuring 70 × 2 × 0.01 mm. The ribbons were cleaned though it. In this case, its temperature was also inferred of carbon by heating in an oxygen environment at a from the temperature dependence of the electrical resis- pressure of 1 × 10Ð6 Torr and a temperature T = 1800 K tance, which was calculated by linear extrapolation of

1172 AGEEV, KUZNETSOV

3000 the relation from the pyrometric region to room temperature. 2600 1 2 The ribbon was irradiated by electrons of energies 2200 Ee = 0Ð300 eV in stationary conditions at an electron current density of 5 × 10Ð6 A/cm2; this a regime did not 1800 produce noticeable ribbon overheating. The desorbing 1400 Eu atoms were ionized in a surface ionization detector 3

, arb. units in which a tungsten ribbon heated to T = 2000 K served q 1000 as the ion emitter. The ion current was measured in the 600 4 particle counting mode. 200 0 3. RESULTS 180 220 260 300 340 360 T, K The temperature dependences of the Eu ESD yield from an oxidized tungsten surface coated by an Fig. 1. Eu ESD yield q vs. substrate temperature plots mea- adsorbed europium ﬁlm are governed by the energy Ee sured at an incident electron energy Ee = 32 eV for different of the electrons striking the surface, the surface cover- europium coverages Θ of tungsten coated by an oxygen Θ Θ age by europium , and the degree of tungsten surface monolayer. : (1) 0.05, (2) 0.07, (3) 0.18, and (4) 0.25. oxidation. Figure 1 displays the ESD yield q of Eu atoms from a tungsten surface coated by an oxygen monolayer plot- 2000 ted vs. the substrate temperature for four values of the 1 1600 europium coverage and an incident electron energy 2 Ee = 32 eV. As the substrate temperature is increased, 1200 the yield q is seen to pass through a broad maximum 3 800 which decreases with increasing europium coverage , arb. units

q and shifts toward higher temperatures. The europium 400 appearance threshold temperature increases with Θ, 0 while the temperature at which the Eu ESD stops is vir- 180 220 260 300 340 360 tually independent of the europium coverage. T, K Figure 2 shows similar plots obtained for the ESD of Eu neutrals from a tungsten oxide surface coated by Fig. 2. Eu ESD yield q vs. substrate temperature plots mea- europium to various coverages. These curves resemble sured at an incident electron energy Ee = 32 eV for different the temperature dependence of the Eu yield from the europium coverages Θ of tungsten coated by an oxide ﬁlm. tungsten surface coated by an oxygen monolayer. How- Θ: (1) 0.05, (2) 0.15, and (3) 0.25. ever, at low europium coverages, Eu atoms start to des- orb at substantially higher temperatures. This effect is illustrated by the curves in Fig. 3, which were obtained Θ at = 0.05 and an incident electron energy Ee = 32 eV 3000 for a tungsten surface coated by an oxygen monolayer 2600 (curve 1) and by a tungsten oxide ﬁlm (curve 2). These graphs provide supportive evidence for the Eu ESD dis- 2200 appearance temperature being independent of the 1 extent of tungsten oxidation at E = 32 eV. 1800 e Figure 4 presents, in graphical form, the EuO mole- 1400 2 cule ESD yield vs. substrate temperature plot for a , arb. units

q 1000 europium layer adsorbed on a tungsten surface coated by an oxygen monolayer and obtained at Θ = 0.70 and 600 Ee = 50 and 80 eV. The EuO yield grows slowly with 200 temperature within the temperature interval T = 180Ð 0 460 K, after which it drops to zero at T = 500 K, with 180 220 260 300 340 360 the ESD disappearance temperature being independent T, K of the incident electron energy. Figure 5 displays similar curves for the ESD of EuO molecules from an adsorbed europium layer on a tung- Fig. 3. Eu ESD yield q vs. substrate temperature plots mea- sten surface coated by an oxygen monolayer obtained sured at an incident electron energy Ee = 32 eV for a europium coverage Θ = 0.05 of tungsten coated by (1) an with electrons of energy Ee = 50 eV for several values oxygen monolayer and (2) an oxide ﬁlm. of the europium coverage. The curves are similar in

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

THE TEMPERATURE DEPENDENCE OF EU ATOM YIELD RESONANCES 1173

2400 2000 1 6 1600 2000 5 1200 1600 4 3 800 1200 , arb. units 2 q 2 , arb. units 800 400 q 1 0 400 160 240 320 400 480 560 0 T, K 160 240 320 400 480 560 640 T, K

Fig. 5. EuO ESD yield q vs. substrate temperature plots Fig. 4. EuO ESD yield q vs. substrate temperature plots measured at an incident electron energy E = 50 eV for dif- Θ e measured at incident electron energies Ee of (1) 50 and (2) ferent europium coverages of tungsten coated by an oxy- 80 eV for a europium coverage Θ = 0.70 of tungsten coated gen monolayer. Θ: (1) 0.18, (2) 0.35, (3) 0.53, (4) 0.63, (5) by an oxygen monolayer. 0.70, and (6) 0.85. shape to the relations presented in Fig. 4. However, the tion and the yield reached a maximum for Θ = 0.03. The temperature at which the EuO yield begins to decrease Eu yield from the ribbon coated by an oxygen mono- changes abruptly with increasing europium coverage layer with 0.05 ML barium decreases with increasing Θ Θ. For Θ > 0.60, the EuO ESD disappearance tempera- almost in the same way as for a ribbon coated by an ture is 500 K, while for Θ < 0.60, it increases to 580 K. oxygen monolayer, while for the oxide-coated ribbon, We note that, at temperatures below the point at which the yield q falls off more slowly with increasing Θ, to the EuO yield starts to decrease, the yield grows almost disappear for Θ = 0.40. in proportion to the europium coverage, which agrees with the relations plotted in Fig. 3 in [9]. For electron energies Ee = 80 eV, the EuO disappearance temperature also drops in a jump from 570 to 500 K within the 1200 europium coverage interval of 0.60 to 0.70 (Fig. 6). The 4 ESD disappearance temperature of EuO molecules 800 2 increases from 500 to 560 K as one crosses over from 3 tungsten coated by an oxygen monolayer to a tungsten 400 , arb. units

Θ q 1 oxide film for a europium coverage = 0.70 and an 0 electron energy Ee = 50 eV (Fig. 7). Moreover, the EuO 160 240 320 400 480 560 640 ESD disappearance temperature for a tungsten oxide T, K film decreases smoothly with the europium coverage increasing to one monolayer. Fig. 6. EuO ESD yield q vs. substrate temperature plots measured at an incident electron energy Ee = 80 eV for dif- It should be pointed out that all the relations describ- ferent europium coverages Θ of tungsten coated by an oxy- ing the ESD yield of Eu atoms and EuO molecules are gen monolayer. Θ: (1) 0.53, (2) 0.63, (3) 0.70, and (4) 0.85. reversible within the temperature variation region where these particles are detected; in other words, europium thermodesorption in this region can be 2200 neglected. 1 1800 Figure 8 plots the Eu ESD yield q for Ee = 32 eV vs. europium coverage of a tungsten ribbon with an oxygen 1400 2 monolayer (curve 1), with an oxygen monolayer over- laid by a 0.05 barium monolayer (curve 2), and with an 1000 , arb. units oxide film (curve 3) for T = 300 K. We readily see that q 600 the yield q of Eu neutrals from an oxygen monolayer- coated ribbon becomes noticeable only after the depo- 200 0 sition of 0.03 ML europium and reaches a maximum at 160 240 320 400 480 560 640 0.05 ML europium, after which it decreases with T, K increasing europium coverage, to disappear altogether for Θ > 0.35. In the case of the ribbon coated by an oxide film or an oxygen monolayer with 0.05 ML bar- Fig. 7. EuO ESD yield q vs. substrate temperature plots measured at an incident electron energy Ee = 50 eV for a ium, no appearance threshold was observed in the europium coverage Θ = 0.70 of tungsten coated by (1) an dependence of q on the deposited europium concentra- oxygen monolayer and (2) an oxide film.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1174 AGEEV, KUZNETSOV 4. DISCUSSION OF RESULTS a substrate with an oxygen monolayer for the same europium coverages. Europium adsorbed on oxidized tungsten lowers the The increase in the substrate temperature entails an work function of the surface and, hence, resides par- increase in the exciton hole and electron recombination tially in ionic form [10]. As established by Auger elec- rate [15], which brings about a decrease in the Eu2+ ion tron spectroscopy and thermodesorption measure- flux to the surface and, accordingly, in the flux of Eu ments, europium is distributed uniformly over the sur- neutrals reflected from the surface. We note that the face of oxidized tungsten even at below monolayer yield disappearance temperature of the Eu atoms asso- coverages, while at above monolayer coverages, it ciated with excitation of the europium core levels does forms three-dimensional islands even at T = 300 K [11]. not depend on the concentration of deposited europium The temperature dependence of the Eu yield from a and the substrate oxidation. It is apparently determined europium layer adsorbed on oxidized tungsten can by the properties of the europium core exciton only. apparently be interpreted in terms of the model pro- The Eu atoms passing through the film of adsorbed posed by us earlier [5, 9]. According to this model, europium ions undergo resonance charge exchange, excitation of the europium 5p and 5s core levels is whose probability grows with decreasing distance accompanied by the formation of core excitons, which 2+ 2+ between the Eu atoms and the adsorbed Eu ions. decay partially with the formation of Eu ions. These Accordingly, the yield of Eu atoms decreases with ions start to move toward the surface as a result of the increasing europium coverage. increasing image potential and of the decreasing repulsion forces which appear when the electronic shells of The emergence of neutrals at electron energies cor- the europium and oxygen ions touch [12]. The Eu2+ responding to the tungsten core-level excitation can be ions become neutralized by the conduction band elec- interpreted as desorption of EuO molecules. At least, trons if the energy of the core holes of the ions moving the surface ionization method does not discriminate the toward the surface drops below the substrate Fermi desorption of the latter from that of Eu atoms [9]. More- level [13]. The electron density in the conduction band over, it is hard to conceive how excitation could be increases with temperature as the electrons leave the transferred from a tungsten core level through oxygen to initiate desorption of Eu atoms. The formation of the donor levels formed in europium adsorption, and, localized tungsten core exciton breaks the bond accordingly, the probability of europium ion neutraliza- between the tungsten and oxygen and favors EuO des- tion also increases. Estimation of the activation energy orption. The yield of EuO molecules grows sublinearly of this process yields ~0.8 eV, a figure substantially less with the substrate temperature increasing to T ~ 460 K; than the band gap width of the WO3 oxide, which is 2+ this is apparently caused by the increase with tempera- about 3.0 eV [14]. The probability of Eu neutraliza- ture in the amplitude of tungsten ion vibrations, which tion decreases as one goes over from a substrate coated produce additional repulsion of the EuO molecules by an oxygen monolayer to that with an oxide film, [16]. For T > 470 K, the tungsten core exciton begins to which may be associated with the increase in the poten- break up, thus giving rise to a sharp decrease in the Eu tial barrier for electrons tunneling from the substrate to yield. The exciton breakup temperature depends on the 2+ the Eu ions. This conclusion is supported by the nature of both the exciton and the surrounding host increase in the temperature of Eu ESD appearance from matrix [15]; therefore, there is nothing strange in that a substrate coated by an oxide film compared to that for the yield of Eu atoms and the yield of EuO molecules start to decrease at different temperatures (Figs. 1, 4). 1600 The abrupt decrease in the ESD yield disappearance 1 temperature of EuO molecules from an oxygen monolayer with the europium coverage increasing from 0.60 1200 2 to 0.70 and the smooth falloff of this temperature with 3 increasing europium coverage on the tungsten oxide 800 film are accounted for by the different patterns of lateral interaction between the europium particles adsorbed on , arb. units

q these surfaces. This conclusion is supported by the fact 400 that one europium monolayer on these surfaces corresponds to different europium concentrations. 0 0.1 0.2 0.3 0.4 The existence of a concentration threshold of Eu Θ, arb. units ESD appearance from a tungsten surface coated by an oxygen monolayer (Fig. 8) suggests that this surface Fig. 8. Eu ESD yield q vs. Eu coverage Θ plots measured at an incident electron energy E = 32 eV at temperature T = has a limited number of sites on which europium can e adsorb to Θ < 0.03 and from which the Eu ESD yield is 300 K for (1) tungsten coated by an oxygen monolayer, (2) + tungsten coated by an oxygen monolayer and a 0.05-ML negligible. Ba ions adsorb by substituting for Ba+ ﬁlm deposited after europium adsorption, and (3) tung- europium at these sites, and the concentration threshold sten coated by an oxide ﬁlm. of Eu ESD appearance vanishes. The oxide-coated

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 THE TEMPERATURE DEPENDENCE OF EU ATOM YIELD RESONANCES 1175 tungsten surface does not have sites with low Eu yield. 5. V. N. Ageev and Yu. A. Kuznetsov, Pis’ma Zh. Tekh. Fiz. These sites are probably associated with direct contact 26 (13), 86 (2000) [Tech. Phys. Lett. 26, 579 (2000)]. between the Eu and tungsten atoms, where the elec- 6. V. N. Ageev, Yu. A. Kuznetsov, and T. E. Madey, J. Vac. tronic excitations of Eu2+ ions relax before the latter Sci. Technol. A 19 (4), 1481 (2001). reach the surface. 7. V. N. Ageev, O. P. Burmistrova, and Yu. A. Kuznetsov, Thus, we have presented the ﬁrst measurement of Fiz. Tverd. Tela (Leningrad) 29 (6), 1740 (1987) [Sov. the temperature dependence of the ESD yield from a Phys. Solid State 29, 1000 (1987)]. europium adatom layer on the surface of oxidized tung- 8. V. N. Ageev and N. I. Ionov, Fiz. Tverd. Tela (Leningrad) sten. This dependence is reversible in temperature. The 11, 3200 (1969) [Sov. Phys. Solid State 11, 2593 yield of Eu atoms and EuO molecules passes through a (1970)]. maximum as the temperature increases and drops to 9. V. N. Ageev, Yu. A. Kuznetsov, and N. D. Potekhina, Fiz. zero at 310 and 500 K, respectively, which is associated Tverd. Tela (St. Petersburg) 43 (10), 1894 (2001) [Phys. with different thermal stability of the europium and Solid State 43, 1972 (2001)]. tungsten core excitons. 10. S. Yu. Davydov, Pis’ma Zh. Tekh. Fiz. 27 (7), 68 (2001) [Tech. Phys. Lett. 27, 295 (2001)]. 11. V. N. Ageev and A. Yu. Afanas’eva, Fiz. Tverd. Tela ACKNOWLEDGMENTS (St. Petersburg) 43 (4), 739 (2001) [Phys. Solid State 43, This study was partially supported by the Russian 772 (2001)]. Foundation for Basic Research (project no. 99-02- 12. P. R. Antoniewicz, Phys. Rev. B 21 (9), 3811 (1980). 17972) and the State Program of the RF “Surface 13. A. G. Borisov and J. P. Ganyacq, Surf. Sci. 445, 430 Atomic Structures” (grant no. 4.5.99). (2000). 14. C. G. Granqvist, Handbook of Inorganic Electrochromic Materials (Elsevier, Amsterdam, 1995). REFERENCES 15. Ch. B. Lushchik and A. Ch. Lushchik, Decay of Electron 1. V. N. Ageev, Prog. Surf. Sci. 47 (1Ð2), 55 (1994). Excitations with Defect Formation in Solids (Nauka, 2. V. N. Ageev, Yu. A. Kuznetsov, and N. D. Potekhina, Fiz. Moscow, 1989), p. 183. Tverd. Tela (St. Petersburg) 38 (2), 609 (1996) [Phys. 16. V. F. Kuleshov, Yu. A. Kukharenko, S. A. Fridrikhov, Solid State 38, 335 (1996)]. V. I. Zaporzhchenko, V. I. Rakhovskiœ, A. G. Naumovets, 3. V. N. Ageev and Yu. A. Kuznetsov, Phys. Low-Dimens. and A. E. Gorodetskiœ, Spectroscopy and Diffraction of Struct. 1/2, 113 (1999). Electrons when Used to Study Solid Surfaces (Nauka, 4. V. N. Ageev, Yu. A. Kuznetsov, and N. D. Potekhina, Moscow, 1985), p. 211. Surf. Sci. 367, 113 (1996). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1176Ð1180. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1126Ð1130. Original Russian Text Copyright © 2002 by Gomoyunova, Pronin, Valdaœtsev, Faradzhev. LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Reactive Epitaxy of Cobalt Disilicide on Si(100) M. V. Gomoyunova, I. I. Pronin, D. A. Valdaœtsev, and N. S. Faradzhev Ioffe Physicotechnical Institute, Russian Academy of Sciences, ul. Politekhnicheskaya 26, St. Petersburg, 194021 Russia Received September 13, 2001

Abstract—The growth of cobalt disilicide on the Si(100) surface by reactive epitaxy at T = 350°C was studied within the 10Ð40 ML cobalt coverage range. A new method of mapping the atomic structure of the surface layer by inelastically scattered medium-energy electrons was employed. The ﬁlms thus formed were shown to consist ° of CoSi2(221) grains of four azimuthal orientations turned by 90 with respect to one another. This domain structure originates from substrate surface faceting by (111) planes, a process occurring during silicide formation. B-oriented CoSi2(111) layers grow epitaxially on (111) facets. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION medium-energy electrons. It has been already used to Thin cobalt disilicide films grown on the surface of advantage in studying the epitaxial growth of ultrathin silicon single crystals enjoy broad application in solid- CoSi2 films on Si(111) [22, 23]. state microelectronics for the production of atomically sharp metal/semiconductor interfaces and are particu- 2. EXPERIMENTAL TECHNIQUE larly promising for use in ultralarge-scale integration purposes. The formation of CoSi2/Si contacts is pres- The instrument used in this study is described in ently best studied for the Si(111) surface. At the same considerable detail elsewhere [24]. We note here only time, of prime interest for silicon technology is the its most important features. The sample is irradiated by Si(100) face. However, the formation of a phase bound- a 2-keV electron beam striking the surface under a ary in this system is a substantially more complex prob- grazing incidence (~10°). The beam current is ~10Ð7 A, lem and recent publications have even reported on self- and its cross-sectional area does not exceed 0.1 mm. We organization of the cobalt disilicide clusters, thus open- study the spatial distributions of electrons that are scat- ing the possibility of using not only planar tered in the surface layer of the sample and lose no CoSi2/Si(100) contacts but also CoSi2 quantum dot sys- more than 10% of their initial energy in reflection. The tems [1, 2]. electrons are detected by a small retarding-field energy Numerous studies have demonstrated that the prop- analyzer with two spherical grids. The electron flux erties of cobalt disilicide films grown on Si(100) are passing through the analyzer grids is amplified by a very sensitive both to the initial state of the substrate microchannel plate and forms (on the luminescent and to the silicide formation regime [1Ð9]. In the con- screen) a diffraction pattern, which is recorded through ditions favoring growth of epitaxial CoSi (111) layers the optical window of the vacuum chamber by a com- 2 puter-interfaced videocamera. The pattern is observed on Si(111), one observes the growth on this surface of ° cobalt disilicide grains oriented differently relative to within a cone of half-angle 57 . the substrate [3, 9Ð13]. To overcome the difficulties The measurements were performed in ultrahigh vac- associated with the problem of nucleation, the tech- uum (~10Ð8 Pa). The single-crystal silicon substrates on niques based on the use of the so-called templates, rep- which the cobalt disilicide layers were to be formed were × × resenting preformed ultrathin CoSi2(100) layers, were cut of KÉF-1 plates and measured 22 14 0.25 mm. developed [9, 10, 14, 15]. The growth of silicide films The crystal surface was aligned with the (100) face to on the templates reduces to upgrowing these perfect no worse than 20′. The samples were pretreated by the layers. The best results are obtained under codeposition method of Shiraki [25] and subsequently cleaned by of Co and Si, a process which removes the problem of heating in vacuum [22], the two procedures culminat- mass transport of components in the system [9, 16, 17]. ing in the obtainment of an atomically clean silicon sur- × In real practice, however, molecular beam epitaxy is face with a (2 1)-type reconstruction. not always appropriate [18]. This makes investigation The cobalt was deposited from a source in which a of the growth of CoSi2 layers on Si(100) by reactive and wire of 99.99%-pure material was heated by electron solid-state epitaxy a topical problem. In this work, we bombardment. The cobalt deposition rate was about used for this purpose a new method which permits one 2 ML/min. We accepted one Co monolayer to be equal to visualize the atomic structure of a nanometer-scale to 6.78 × 1014 atoms/cm2, which corresponds to the sur- surface layer [19Ð21]. The method is based on analyz- face concentration of Si atoms on the (100) face. The ing spatial distributions of inelastically scattered elemental composition of the sample surface was mon-

REACTIVE EPITAXY OF COBALT DISILICIDE ON Si(100) 1177

(a) (b) (c)

Fig. 1. Diffraction patterns observed in the deposition of cobalt on a single-crystal silicon surface heated to 350°C. (a) Initial pattern from the Si(100) facet, (b) after deposition of 10 ML Co and heating of the sample to 600°C, and (c) after repeated deposition of 10 ML Co and heating of the sample to 600°C.

itored by Auger electron spectroscopy. All the measure- ratio between the Co(M2, 3VV) and Si(LVV) low-energy ments were carried out at room temperature after the Auger electron peaks of 53 and 92 eV, respectively, is samples had cooled down. approximately 0.1, which is characteristic of a Si-rich CoSi2 surface [14, 26]. 3. RESULTS OF MEASUREMENTS Further increase in the deposited dose to ~20 ML AND DISCUSSION results in a gradual weakening of the diffraction pattern and the appearance of a diffuse background. This is Because the data relating to the earliest stages in accompanied by a substantial decrease in the silicon silicide formation are controversial, it appeared reason- Auger electron signal and an increase in the Co Auger able to start the work with a study of CoSi2 films of electron peak, which indicates the growth of a disor- thickness in excess of the depth probed by the method dered cobalt film with a silicon admixture. Whence it to be employed, i.e., in the conditions where the sub- follows that the cobalt disilicide layer produced after strate does not contribute directly to the formation of the deposition of the first 10 ML Co already forms a the diffraction patterns observed. Therefore, the mea- noticeable diffusion barrier, which suppresses further surements were conducted on films prepared by depo- ° CoSi2 growth at 350 C. Annealing the sample to T = sition of ten or more cobalt monolayers on Si(100). The 600°C for ≈1 min restores the diffraction pattern substrate temperature chosen was 350°C, i.e., where, (Fig. 1b), as well as the magnitude of the cobalt to sili- according to [14], the silicide formation already takes con Auger signal ratio observed after the deposition of place. 10 ML Co. Further growth of the silicide film in subse- Figure 1a shows the starting diffraction pattern of quent depositions of cobalt and sample heatings to Si(100). The data are presented in the form of two- 600°C follows the above scenario. This can be inferred, dimensional maps of electron reflection intensity in the for instance, from the fact that all the features in the dif- polar and azimuthal emergence angles which are con- fraction patterns obtained after the first cobalt deposi- structed in stereographic projection in the linear gray- tion (Fig. 1b) and on deposition of 20 ML (Fig. 1c) scale contrast code. The bright regions relate in this almost coincide. The only difference between the pat- case to maxima in the angular distributions; the dark terns consists in their contrast, which increases with the ones, conversely, to the minima. The center of the pat- thickness of the silicide layer. tern corresponds to electrons emerging along the sur- We consider now the diffraction pattern typical of face normal; the outer circle, to electrons leaving the CoSi films grown on Si(100). Because differently ori- sample at a polar angle of 60°. 2 ented CoSi2 layers can form on this substrate, we shall Deposition of cobalt changes the observed pattern first discuss the diffraction patterns produced by differ- dramatically. As the adsorbate dose increases in the 1Ð ent facets of cobalt disilicide. Such patterns can be 10 ML range, the diffraction maxima associated with obtained by computer processing of the experimental the substrate smear out gradually and new features data presented in [22] for CoSi2(111). This procedure is appear. The strengthening of their development brings based [27] on the experimental observation that when a about a new diffraction pattern, shown in Fig. 1b, which crystal is turned, the corresponding diffraction pattern shows the structure of the grown cobalt disilicide lay- turns as a whole by the same angle. Therefore, if the ers. While this pattern retains the symmetry of the sub- pattern due to CoSi2(111) is turned so as to bring an strate, its general appearance changes appreciably. As 〈hkl 〉 crystallographic direction to the center of the for the elemental composition of the grown layer, the screen (at zero polar angle), the diffraction pattern

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1178 GOMOYUNOVA et al.

(a) (b) (c)

Fig. 2. Diffraction patterns of different CoSi2 facets obtained by computer treatment of experimental data on CoSi2(111) for (a) CoSi2(100) and (b) CoSi2(221). Pattern (c) was obtained by summing the patterns of four (221) domains turned azimuthally relative to one another by 90°. observed will be that of the (hkl) face. The part of the that the (110) planes of cobalt disilicide are parallel to pattern which is absent can be filled using the symmetry the silicon (110) planes. elements of the (hkl) face by extrapolating the part of The reason for the growth of CoSi2(221) grains on the image thus obtained. Si(100) under reactive epitaxy can be readily under- The relevant publications primarily discuss three stood if we take into account the instability of the (100) face in the course of silicide formation and the possibil- possible orientations of the CoSi2 epitaxial layers which correspond to the (100), (110), and (221) faces. ity of its faceting [10, 28]. Facet growth can be initiated by intense mass transport of silicon occurring in the Only the CoSi2(100) face possesses the fourfold mirror- rotational symmetry seen in the pattern of Fig. 1c. Fig- course of cobalt disilicide formation at an elevated tem- ure 2a displays the diffraction pattern of this face. As is perature, which brings about rearrangement of the crys- obvious from a comparison of the patterns in Figs. 1c tal surface and faceting by the energetically preferable and 2a, this pattern does not agree with the experiment; (111) planes. Further deposition of cobalt onto the hence, the film grown has another orientation. We also Si(111) facets results in epitaxial growth of CoSi2(111) analyzed the possibility of the (110) orientation. domains, which have a B-type azimuthal orientation Because this face is characterized by a twofold rotation antiparallel to the substrate under the conditions of axis, the fourfold symmetry observed in the experiment reactive epitaxy [22]. The driving force of the process can be obtained if we accept the possibility of forma- is the formation of the most energetically favorable tion, on the (100) substrate facet, of two types of phys- CoSi2(111)/Si(111) interface. Because the growing CoSi (111) domains are B-oriented, it is the (221) ically equivalent (110) domains turned by 90° with 2 planes that turn out to be parallel to the (100) planes of respect to one another [9] and add their diffraction pat- the original substrate surface; this exactly is observed terns. The pattern thus obtained was also found to dis- experimentally. Thus, the observed growth of agree with experiment. This means that no CoSi2(110) CoSi2(221) grains is initiated by the surface of the start- grains form in our case. ing silicon crystal becoming faceted by the (111) planes Now, we consider the third possible orientation of in the course of the silicide formation. We note that the CoSi /Si interface forming in these conditions is not the CoSi2 layers. The diffraction pattern corresponding 2 to the (221) face is shown in Fig. 2b. It has the lowest plane and its roughness governs the relief of the grow- symmetry and possesses only one mirror-reflection ing film to a considerable extent. plane. Therefore, the observed fourfold symmetry of Of particular importance for application purposes of this image can arise only if four types of physically thin silicide films is the establishment of the region equivalent CoSi2(221) domains, turned azimuthally by within which these films are thermally stable. It is well- 90° relative to one another, coexist on the substrate sur- known that high-temperature annealing of cobalt disili- face. The net pattern obtained by superposition of the cide films results first in the formation of pinholes and, contributions due to such domains is presented in thereafter, of CoSi2 islands [14, 29]. We studied the Fig. 2c. We readily see that it almost coincides with the thermal destruction of a CoSi2 layer on silicon using a experimental results (Fig. 1c). This implies that the film obtained by depositing 40 ML of cobalt on the CoSi2 film grown by us consists of grains whose (221) crystal. To do this, the sample was subjected to stepped planes are oriented parallel to the substrate surface. As anneals at increasing temperatures (up to T = 1200°C), for the azimuthal orientation of these grains, a compar- with diffraction patterns measured in between. These ison of the patterns shown in Figs. 1a and 2b suggests patterns revealed noticeable changes. Figure 3 presents

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

REACTIVE EPITAXY OF COBALT DISILICIDE ON Si(100) 1179

(a) (b)

Fig. 3. Diffraction patterns of a cobalt disilicide film on silicon annealed to temperatures of (a) 1000 and (b) 1200°C. patterns obtained after two such anneals. Computer ACKNOWLEDGMENTS simulation was used to assist in interpreting the experi- This study was supported by the Russian Founda- mental data. The measured patterns were compared tion for Basic Research (project no. 01-02-17288) and with calculations obtained by summing up the patterns the Ministry of Science, Industry, and Technology of due to CoSi2(221) islands and Si(100) pinholes under the RF (“Surface Atomic Structures” Program, project variation of the statistical weights of the two contribu- no. 5.10.99). tions. The experimental data presented in Figs. 1a and 1c, respectively, were used as reference patterns of the phases. The calculations were compared with experi- REFERENCES ment values using reliability factors, as was done in our 1. I. Goldfarb and G. A. D. Briggs, Phys. Rev. B 60 (7), study of silicide formation on Si(111) [22]. 4800 (1999). As follows from our analysis, an increase in the 2. H. L. Meyerheim, U. Dobler, and A. Puschmann, Phys. annealing temperature is indeed accompanied by a Rev. B 44 (11), 5738 (1991). change in the film morphology, with the substrate 3. D. D. Chambliss, T. N. Phodin, and J. E. Rowe, Phys. becoming increasingly bare. In particular, after the Rev. B 45 (3), 1193 (1992). sample was annealed to 1000, 1120, and 1200°C, the 4. P. A. Bennett, S. A. Parikh, and D. G. Cahill, J. Vac. Sci. regions of the substrate free of the silicide were found Technol. A 11, 1680 (1993). to occupy 8, 34, and 50% of its area, respectively. Bare 5. M. Sosnowski, S. Ramae, W. L. Broun, and Y. O. Kim, substrate regions were also observed in the case of the Appl. Phys. Lett. 65, 2943 (1994). films forming in the initial stages of reactive epitaxy. 6. G. Rangelov, P. Augustin, J. Stober, and Th. Fauster, For instance, after deposition of 10 ML of cobalt Phys. Rev. B 49 (11), 7535 (1994). (Fig. 1b), bare regions occupied about 10% of the sur- 7. O. P. Karpenko and S. M. Yalisove, J. Appl. Phys. 80 face area, and after 10 ML more were deposited and the (11), 6211 (1996). ° sample was annealed to 600 C, their fraction decreased 8. H. Ikegami, H. Ikeda, S. Zaima, and Y. Yasuda, Appl. to 4%. Surf. Sci. 117/118, 275 (1997). Thus, application of the new structural method to 9. S. M. Yalisove, R. T. Tung, and D. Lorentto, J. Vac. Sci. the investigation of cobalt disilicide formation on the Technol. A 7 (3), 1472 (1989). Si(100)Ð(2 × 1) surface revealed that a film consisting 10. J. R. Jiménez, L. J. Schowalter, L. M. Hsiung, et al., J. ° of CoSi2(221) grains turned azimuthally by 90 with Vac. Technol. A 8 (3), 3014 (1990). respect to one another grows on it under reactive epit- 11. L. J. Schowalter, J. R. Jiménez, L. M. Hsiung, et al., J. axy performed at T = 350°C. This domain structure is Cryst. Growth 111, 948 (1991). produced by the substrate surface being faceted by 12. C. W. T. Bulle-Lieuwma, A. H. van Ommen, J. Hornstra, (111) planes in the course of the silicide formation. The and C. N. A. M. Aussems, J. Appl. Phys. 71 (5), 2211 CoSi2(111) epitaxial layers growing on the (111) facets (1992). are B-oriented, and this accounts for the (221) planes of 13. V. Scheuch, B. Voigtlander, and H. P. Bonzel, Surf. Sci. cobalt disilicide becoming parallel to the substrate 372, 71 (1997). plane. In the initial stage of the process, the film is not 14. J. M. Gallego, R. Miranda, S. Molodsov, et al., Surf. Sci. continuous, leaving about 10% of the surface area bare. 239 (3), 203 (1990). Preparation of continuous CoSi2 films with a better 15. V. Buschmann, M. Rodewald, H. Fuess, and G. van Ten- structure requires that after the cobalt deposition, the deloo, J. Cryst. Growth 191, 430 (1998). sample be annealed additionally to higher tempera- 16. R. Tung, F. Schrey, and S. M. Yalisove, Appl. Phys. Lett. tures. 55 (1), 2005 (1989).

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17. R. Stadler, C. Schwarz, H. Sirringhaus, and H. von Kanel, 24. I. I. Pronin, D. A. Valdaœtsev, M. V. Gomoyunova, et al., Surf. Sci. 271, 355 (1992). Zh. Tekh. Fiz. 68 (12), 80 (1998) [Tech. Phys. 43, 1475 18. B. Ilge, G. Palasantzas, J. de Nijis, and L. J. Geerlings, (1998)]. Surf. Sci. 414, 279 (1998). 25. A. Ishizaka and Y. Shiraki, J. Electrochem. Soc. 133, 666 19. N. S. Faradzhev, M. V. Gomoyunova, and I. I. Pronin, (1986). Phys. Low-Dimens. Struct. 3/4, 93 (1997). 26. U. Starke, W. Weiss, K. Heinz, et al., Surf. Sci. 352Ð354, 20. I. I. Pronin and M. V. Gomoyunova, Prog. Surf. Sci. 59 89 (1996). (1Ð4), 53 (1998). 27. I. I. Pronin, N. S. Faradzhev, and M. V. Gomoyunova, 21. M. V. Gomoyunova and I. I. Pronin, Zavod. Lab. 67 (4), Fiz. Tverd. Tela (St. Petersburg) 40 (7), 1364 (1998) 24 (2001). [Phys. Solid State 40, 1241 (1998)]. 22. M. V. Gomoyunova, I. I. Pronin, D. A. Valdaœtsev, and 28. K. Rajan, L. M. Hsiung, J. R. Jiménez, et al., J. Appl. N. S. Faradzhev, Fiz. Tverd. Tela (St. Petersburg) 43 (3), Phys. 70 (9), 4853 (1991). 549 (2001) [Phys. Solid State 43, 569 (2001)]. 29. W. Weiss, U. Starke, K. Heinz, et al., Surf. Sci. 347, 117 23. I. I. Pronin, D. A. Valdaitsev, N. S. Faradzhev, et al., (1996). Appl. Surf. Sci. 175/176, 83 (2001). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1181Ð1187. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1131Ð1136. Original Russian Text Copyright © 2002 by Perekrestov, Koropov, Kravchenko.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Formation of Island Structures in the Course of Deposition of Weakly Supersaturated Aluminum Vapors V. I. Perekrestov*, A. V. Koropov**, and S. N. Kravchenko* * Sumy State University, Sumy, 40007 Ukraine ** Sumy State Agricultural University, Sumy, 40021 Ukraine e-mail: [email protected] Received May 18, 2001; in ﬁnal form, November 1, 2001

Abstract—The formation of island structures in the course of deposition of weakly supersaturated aluminum vapors is investigated by transmission electron (TEM) and scanning electron (SEM) microscopies and electron microdiffraction. The aluminum layers are prepared by dc magnetron sputtering in a high-purity Ar atmosphere. The conditions of formation of statistically homogeneous nanocrystalline layers depend on the deposition temperature and the partial pressures of Ar and reactive gases. The wetting angle between islands and the substrate is calculated from the derived relationships as a function of the condensation temperature. © 2002 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The purpose of the present work is to investigate the formation of nanocrystalline layers upon deposition of The particular interest expressed by researchers in aluminum vapors with a weak supersaturation. The island structures is associated with their unique electri- choice of aluminum as a model material for the forma- cal properties. Nanocrystalline layers can be produced tion of island structures is motivated by its relatively using different techniques. It is well known that, at the low melting temperature at which vapor condensation early stage of vapor deposition of metals, the formation can proceed by the following mechanisms: vapor of a condensate is accompanied by the formation of liquid phase crystal [1] and vapor crystal. In island films in the form of a layer consisting of super- turn, this extends the range of technological procedures critical nuclei [1Ð3]. Further vapor condensation leads used for producing island structures. to the Ostwald ripening of islands and to the formation of a channel structure and a continuous film. On this basis, we consider three possible variants of the prepa- 2. EXPERIMENTAL TECHNIQUE ration of island structures. The first variant consists in interrupting the technological process prior to the for- Experimentally, the growth of island structures is mation of a channel structure. impeded by the diffusion of adatoms, because low rates The second variant is based on the suppression of of layer formation favor the interaction between the the Ostwald ripening and intergrowth of islands during condensate and reactive residual gases. For this reason, the deposition of metals in an atmosphere of reactive a stationary aluminum vapor flux was produced by dc magnetron sputtering in an atmosphere of argon sub- gases (O2, N2, CO2, etc.) [4]. However, this brings about the formation of continuous getter films that are com- jected to deep purification. The partial pressure of reactive residual gases during the growth of island struc- posed of metallic crystals joined through interlayers of × Ð8 oxides, nitrides, and other compounds. These structures tures did not exceed 8 10 Pa, as was required. are referred to as granular films [2]. The structure and phase composition of the conden- Koropov et al. [5, 6] theoretically established that sates were examined by scanning electron (SEM) and the third variant of the formation of impurity-free transmission electron (TEM) microscopies and elec- island structures can be accomplished through deposi- tron microdiffraction. The TEM investigation of the tion of vapors with an extremely weak supersaturation. structure was performed using layers deposited onto In this case, the growth of island structures is associated KCl cleavages. Moreover, island structures were pre- with the resorption of subcritical nuclei due to diffusion pared on glass-ceramic and glass substrates. processes which encourage the formation of individual, The mechanisms of nucleation and growth are gov- relatively large islands, because they exhibit a higher erned primarily by technological parameters, such as degree of equilibrium. Therefore, the formation of the pressure of a working gas (for example, argon) PAr, large-sized island structures upon vapor condensation the partial pressure of reactive residual gases Pg, the can be treated as a qualitative criterion for weak super- condensation temperature Tc, and the growth rate of a saturation of vapors. layer Rc. In turn, the parameters Tc and Rc determine the

1182 PEREKRESTOV et al.

83 nm

° × Ð8 Fig. 1. Structure of an Al layer formed at Tc ~ 500 C, Pg ~ 8 10 Pa, and PAr = 9 Pa. degree of vapor supersaturation, whereas a decrease in i.e., when the wetting angle θ is considerably larger π the partial pressure PAr leads to an increase in the than /2. energy of atoms at the instant of deposition [7, 8]. A change in the condensation mechanism vapor Unlike thermal evaporation, ion sputtering brings about liquid phase crystal can occur for several reasons. the condensate nucleation at any weak supersaturation The main reason is a decrease in the condensation tem- of deposited vapors [7]. In other words, ion sputtering ° perature Tc to 400Ð420 C without variations in the provides extremely low rates Rc and, thus, encourages aforementioned technological parameters. In this case, diffusion processes on the deposition surface. Accord- the crossover to the nucleation mechanism vapor ing to the classical theory of capillarity [9], the nucle- crystal [1] is characterized by the following features in ation rate is determined by the following relationship: the formation of the condensate structure. []()∆ ∆ ∆ ICP= exp Gdes Ð Gsd Ð G0 /kT , (1) At the initial stage, islands predominantly grow ∆ ∆ in the substrate plane so that the growth rate is propor- where G0 is the free energy of a critical nucleus, Gsd tional to the area of the diffusion zone adjacent to the is the free activation energy of surface diffusion of ada- ∆ islands. Consequently, prior to their intergrowth, the toms, Gdes is the free activation energy of desorption, islands are equally spaced (Fig. 2a), in contrast with the P is the vapor pressure over the deposition surface at structure shown in Fig. 1. temperature T, and C ≈ 1022 NÐ1 sÐ1. (2) Upon the crossover to the nucleation mechanism Consequently, the concentration of critical nuclei vapor crystal, the wetting angle θ decreases to at increases with an increase in the adatomÐsubstrate least π/2. As a result, the KCl(001) substrate exhibits an bonding energy, a decrease in the diffusion rate, and orienting effect, which leads to the formation of a struc- lowering of the deposition temperature. Undeniably, in ture with preferred orientation, i.e., texture (see the this situation, a continuous film can be formed even at electron diffraction pattern in Fig. 2a). the early growth stage. In order to decrease the adatomÐ substrate bonding energy upon ion sputtering, it is nec- (3) The growth of islands to the point where they essary to decrease the energy of atoms at the instant of come into contact is accompanied by the formation of a condensation. For this reason, the experiments were layer that does not break into pieces after separation performed at minimum discharge voltages (~60Ð35 V) from the substrate. The time it takes for this layer to be formed is referred to as the overgrowth time t [6]. It is and, if required, at sufficiently high pressures PAr 0 (~9 Pa). With the aim of maintaining a stationary worth noting that a decrease in the condensation tem- ° growth rate R in the range 0.03Ð0.05 nm/s irrespective perature Tc from 500 to 400 C and the corresponding c change in the mechanism of the layer formation result of the partial pressure PAr, the power supplied to the sputtering apparatus was made equal to 3 W. in an increase in the time t0 from 20 to 110 min. More- ° over, the deposition at Tc = 400 C for 4 h does not lead to the formation of a continuous crystalline layer 3. RESULTS AND DISCUSSION (Fig. 2b). In this case, the SEM data indicate a substan- ° tial increase in the island volume (Fig. 2c). The structure of a layer formed at Tc ~ 500 C, Pg ~ × Ð8 8 10 Pa, and PAr = 9 Pa is shown in Fig. 1. Since the (4) A further decrease in the condensation tempera- condensation temperature Tc is higher than (2/3)Tm ture Tc is attended by a gradual decrease in the over- (where Tm is the melting temperature of aluminum), the growth time t0 and an enhancement in the tendency condensate nucleation proceeds through the following toward the formation of a nearly continuous crystalline mechanism: vapor liquid phase crystal [1]. layer. For example, an increase in the rate of crystal This is confirmed by the presence of bulk crystals and growth in the substrate plane at relatively low tempera- ° the absence of any orienting effect of the KCl(001) sub- tures Tc (~120 C) results in the formation of a continu- strate (see the electron diffraction pattern in Fig. 1). ous crystalline layer within 6Ð8 min after the onset of This variant of the growth becomes possible in the case permanent deposition (Fig. 3). Moreover, the orienting of a weak interaction between islands and the substrate; effect of the KCl(001) substrate becomes more pro-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

FORMATION OF ISLAND STRUCTURES 1183

125 nm (a)

417 nm (b)

456 nm (c)

Fig. 2. Structures of Al layers formed upon condensation through the nucleation mechanism vapor crystal at (a) Tc ~ 400Ð ° × Ð8 ° × Ð8 420 C, Pg ~ 8 10 Pa, and PAr = 9 Pa and (b) Tc = 400 C, Pg ~ 8 10 Pa, and PAr = 9 Pa for 4 h. (c) SEM image of the surface of an Al island ﬁlm.

111 nm

° Fig. 3. Structure of an Al ﬁlm prepared at Tc ~ 120 C (the time of continuous deposition is 6Ð8 min).

nounced and brings about the formation of a texture For the above experiments, the overgrowth time t0 with the relationship Al(001) || KCl(001); in this case, can be estimated by ignoring the evaporation of ada- τ ∞ some islands are rotated in the azimuthal plane through toms from the substrate (under the condition s , π τ an angle of /4 with respect to the principal direction of where s is the lifetime of adatoms prior to evaporation) growth (see the electron diffraction pattern in Fig. 3). and atoms from the island surface. For this purpose, we We did not reveal a crossover from the VolmerÐWeber will use the notion of a domain of island inﬂuence mechanism of condensation to the StranskiÐKrastanov whose radius is designated as R0 [5]. If the reevapora- τ ∞ mechanism with a decrease in the condensation tem- tion is insigniﬁcant ( s ), the radius R0 can be perature Tc. determined from the following condition: all the atoms

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1184 PEREKRESTOV et al. From relationships (1)Ð(3), we obtain the equation

2dR 2 3κR ------= πR Kω, dt 0 θ in which the variables R and t can be separated as follows: R 3κR2dR dt = ------. (4) Fig. 4. An island in the form of a spherical segment with a π 2 ω wetting angle θ < π/2. R0K θ ≤ π In order to estimate t0 at /2, we integrate Eq. (4) with respect to the time t from 0 to t0 and over the radius R from R (R is the mean radius of the island base at the

instant t = 0) to R0(R ). When integrating, we take into 2 account that the quantity R0 weakly (logarithmically) depends on R and, to a first approximation, can be treated as a constant [5]. Then, we obtain R sinθ () t0 R0 R 3κ 2 dt = ------R dR, (5) θ ∫ π 2 ω ∫ R0K 0 R from whence it follows that R κ t = ------()R3()R Ð R3 . 0 π 2 ω 0 Fig. 5. An island in the form of a spherical segment with a R0k wetting angle θ > π/2. 1 Making allowance for the fact that R ()R = ------0 π introduced by an external source into the domain of Ni influence of a particular island are adsorbed by it. (where Ni is the island density) [5] and by assuming that Ð1 Hence, the total flux I (s ) of atoms caught in this 3()ӷ 3 island can be calculated from the relationship R0 R R , we finally derive the relationship π 2 κR ()R κ I = R0K, (1) 0 t0 ==------. (6) πKω ππN Kω where K is the flux of atoms condensed on the substrate i Ð2 Ð1 (m s ), R0 = R0(R), and R is the radius of the island At θ = π/2, we have base. The rate of change in the island volume V is given by 2 t0 = ------. 3 πN Kω dV i ------= Iω, (2) dt If θ > π/2 (Fig. 5), relationship (5) takes the form

() θ where ω is the volume occupied by an atom in the t0 R0 R sin 3κ 2 island. For islands in the form of a spherical segment dt = ------R dR, (Figs. 4, 5), we have ∫ π 2 ω ∫ R0K 0 R 3 3 π23Ð cosθ + cos θ V ==κR ; κ ------. because, in this case, the condition of joining islands 3 sin3 θ (the overgrowth of the substrate with islands) is deﬁned θ The rate dV/dt is related to the rate of change in the by the expression R(t0)/sin = R0(R ). As a result, we radius R of the island base through the expression have

κ 3 3 3 dV 2dR ()() θ κ t0 = ------R0 R sin Ð R . ------3= R ------. (3) π 2 ω dt dt R0K

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 FORMATION OF ISLAND STRUCTURES 1185

3() 3θ ӷ 3 120 Under the assumption that R0 R sin R , we obtain

3 3 κR ()R sin θ κθsin 100 t ==------0 ------. (7) 0 πKω ππ ω NiK Here, the multiplier κsin3θ ≡ κ is determined by the 1 , deg formula θ 80 π κ ()θ 3 θ 1 = --- 23Ð cos + cos . 3 60 The flux K can be estimated from the film growth rate: K ≈ 2 × 1018 mÐ2 sÐ1. By using relationships (6) or (7) and the experimental values of t and N , the angle θ can 0 i 40 be easily determined by a graphical method. Figure 6 100 200 300 400 500 θ depicts the obtained dependence (Tc), which confirms Tc, °C the aforementioned regularities in the formation of island structures. θ The crossover of the condensate nucleation mech- Fig. 6. Dependence (Tc). anism (vapor liquid phase crystal) can also be caused by a decrease in the partial pressure PAr to ° × Ð8 Ð6 3Ð0.8 Pa (at Tc = 500 C and Pg = 8 10 Pa). This An increase in the partial pressure Pg to 10 Pa ° decrease in the pressure PAr favors the formation of the (PAr ~ 7Ð9 Pa and Tc = 500 C) results in the formation amorphous phase at the stage of condensate nucleation. of granular films within 30 min after the onset of per- The mechanism of nucleation of an amorphous phase manent deposition (Fig. 8a). In this case, the electron was considered in our recent works [8, 10]. It was dem- diffraction patterns contain additional diffraction max- onstrated that the nucleation of an amorphous phase is ima attributed to the impurity phase and the intensity governed primarily by high energies of atoms at the ratios of the diffraction lines change significantly. It can instant of their deposition and depends on the impuri- be assumed that the microstructure shown in Fig. 8a ties adsorbed on the substrate surface. An increase in contains dark spherical aluminum inclusions sur- the thickness of the continuous amorphous film rounded by the binding impurity phase. The inference (approximately to 3Ð5 nm) leads to an increase in the regarding the formation of granular films is confirmed degree of its nonequilibrium. In turn, this brings about to some extent by the SEM investigation (Fig. 8b). In the formation of crystal inclusions on the film surface actual fact, the contrast observed in the SEM images (Fig. 7a). It should be noted that these crystals grow only stems from phase inhomogeneities of the condensate. in the course of aluminum vapor deposition. The conden- Brighter regions correspond to the reflection of elec- sation mechanism vapor amorphous phase trons from denser aluminum inclusions, and the dark crystal substantially affects the shape and size of the background surrounding the inclusions is formed islands and the texture of their growth. Indeed, unlike through weaker reflections from the impurity phase. the island structures growing through the mechanism vapor liquid phase crystal, thin crystals (up to The mechanisms responsible for the structure for- 10 nm thick) grow on the amorphous phase even at T = mation of island films can be conveniently interpreted 500°C. Consequently, the formation of a continuous in terms of the energies of bonding between an adatom crystalline layer occurs within 7Ð10 min after the onset and the substrate surface (Es), an adatom and the amor- of permanent deposition. Note that, prior to the forma- phous phase (Ea), and an adatom and the crystal (Ecr). tion of the continuous crystalline layer, intercrystalline It can be assumed that aluminum vapor deposition onto regions are filled with the binding amorphous phase. glass, glass-ceramic, and KCl substrates satisfies the Analysis of the bright-field (Fig. 7b) and dark-field following inequality: Es < Ea < Ecr. It is reasonable that (Fig. 7c) images indicates that the contrast observed in thermal accommodation of atoms deposited onto the the intercrystalline regions stems from diffuse electron surface proceeds more efficiently at high bonding ener- scattering. This is the reason why the intercrystalline gies. On the other hand, low energies of bonding regions in dark-field images are universally darkened. between adatoms and the deposition surface encourage At the same time, examinations of dark-field images their reevaporation and migration. The inequality Es < and microdiffraction investigations demonstrate that Ea also accounts for the crossover from the bulk growth the same texture is formed only in local regions of the to the surface growth of crystals upon formation of the layer. This can be associated with a misorienting effect amorphous phase (Figs. 1, 7b). The diffusion Ostwald of the amorphous interlayers. ripening and intergrowth of islands are efficiently sup-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1186 PEREKRESTOV et al.

111 nm (a)

222 nm (b)

222 nm (c)

° × Ð8 Fig. 7. Structures of Al layers formed at Tc = 500 C, Pg = 8 10 Pa, and PAr = 3Ð0.8 Pa: (a) initial stage of crystal nucleation on the surface of the amorphous phase, (b) bright-ﬁeld image of the structure, and (c) dark-ﬁeld image of the structure.

3 µm (a)

6 µm (b)

° Ð6 Fig. 8. (a) Microstructure of an Al granular ﬁlm prepared at Tc = 500 C, Pg = 10 Pa, and PAr ~ 7Ð9 Pa (the deposition time is 30 min) and (b) SEM image of the Al granular ﬁlm.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 FORMATION OF ISLAND STRUCTURES 1187 pressed in the case when the diffusion flux of adatoms growth followed by a high-rate overgrowth of channels from the intercrystalline region onto the crystal and the with the formation of bridges [7]. flux of atoms reevaporated from the intercrystalline (2) The results of calculations of the wetting angle θ region add up to the flux of atoms deposited onto the between islands and the substrate on the basis of the same intercrystalline region. However, in this situation, theoretical concepts advanced in this work are quite ° at Tc < 420 C, the crystals should grow in the substrate consistent with the physics of processes responsible for plane, which must necessarily result in the formation of the formation of island structures. a continuous crystalline layer. The tendency toward the (3) The condensation of weakly supersaturated formation of this layer can be considerably depressed in vapors can be used to control technological processes the case when the energy of bonding between the atoms of producing nanocrystalline layers with a specified and the lateral face of the crystal is less than the energy structure by varying the parameters P , P , and T . of bonding between the atoms and the crystal surface g Ar c oriented nearly parallel to the front of the deposited flux. This ratio of the bonding energies can actually ACKNOWLEDGMENTS hold, because, according to experimental data [4], the This work was supported by the Ministry of Educa- lateral faces of crystals are usually enriched with impu- tion and Science of Ukraine, project no. 84.02.04.00-02. rities. Most likely, this is the reason why the increase in the island volume rather than the formation of a continuous crystalline layer is observed even after aluminum REFERENCES deposition for 4 h (Fig. 2c). 1. L. S. Palatnik, M. Ya. Fuks, and V. M. Kosevich, Mecha- It is known that, in the course of magnetron sputter- nism of Formation and the Structure of Condensed Films ing, the condensation surface is irradiated by a second- (Nauka, Moscow, 1972). ary-electron flux [11], thus increasing the condensation 2. Yu. F. Komnik, Physics of Metal Films: Size and Struc- temperature Tc. Therefore, the thermal conductivity of tural Effects (Atomizdat, Moscow, 1979). the substrate material also plays an important role in the 3. S. A. Kukushkin and A. V. Osipov, Usp. Fiz. Nauk 168 formation of island structures. Since the thermal con- (10), 1083 (1998) [Phys. Usp. 41, 983 (1998)]. ductivity of the glass is less than those of the glass- 4. V. S. Kogan, A. A. Sokol, and V. M. Shulaev, Influence of ceramic and KCl substrates, the growth of statistically Vacuum Conditions on the Formation of Condensate homogeneous island structures on the glass substrate is Structure. Structure of Metal Films Doped with Gases: A more pronounced. In particular, it is revealed that the Review (TsNIIatominform, Moscow, 1987), Part 2. growth of island structures on the glass substrate is 5. A. V. Koropov, P. N. Ostapchuk, and V. V. Slezov, Fiz. accompanied by a change in the layer color from blue Tverd. Tela (Leningrad) 33 (10), 2835 (1991) [Sov. to pink and then to white. This regularity of the change Phys. Solid State 33, 1602 (1991)]. in color is explained by the corresponding increase in 6. A. V. Koropov and V. V. Sagalovich, Poverkhnost, No. 2, the degree of periodicity of the statistically homoge- 17 (1990); No. 5, 55 (1989); No. 6, 50 (1987). neous island structure on which the diffraction of light 7. Handbook of Thin-Film Technology, Ed. by L. I. Maissel occurs. The change in color to white is observed upon and R. Glang (McGraw-Hill, New York, 1970; Sov. disturbance of this periodicity due to the intergrowth of Radio, Moscow, 1977), Vol. 2. some islands. Note that the change in color and its 8. V. I. Perekrestov and S. N. Kravchenko, Vopr. At. Nauki brightness upon condensation on the KCl and glass- Tekh., No. 4, 80 (2001). ceramic substrates are less pronounced. 9. J. P. Hirth, S. J. Hruska, and G. M. Pound, in Single- Crystal Films: Proceedings of the International Confer- ence, Blue Bell, 1963, Ed. by M. H. Francombe and 4. CONCLUSIONS H. Sato (Pergamon, Oxford, 1964; Mir, Moscow, 1966). 10. V. I. Perekrestov, S. N. Kravchenko, and A. V. Pavlov, The results obtained in this work allowed us to make Fiz. Met. Metallogr. 88 (5), 72 (1999). the following inferences. 11. B. S. Danilin, Application of Low-Temperature Plasma (1) The deposition of weakly supersaturated alumi- for Deposition of Thin Films (Énergoatomizdat, Mos- num vapors does not result in the classical variant of the cow, 1989). formation of a channel structure through island inter- Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1188Ð1195. Translated from Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1137Ð1144. Original Russian Text Copyright © 2002 by Marikhin, Novak, Kulik, Myasnikova, Radovanova, Belov.

POLYMERS AND LIQUID CRYSTALS

Resonant Raman Spectroscopy of EthyleneÐAcetylene Copolymers Doped with Iodine V. A. Marikhin*, I. I. Novak*, V. B. Kulik*, L. P. Myasnikova*, E. I. Radovanova*, and G. P. Belov** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia ** Institute of Chemical Physics in Chernogolovka, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected] e-mail: [email protected] Received August 2, 2001; in ﬁnal form, September 10, 2001

Abstract—The relative contents of short and long conjugated chains in ethyleneÐacetylene copolymers (EAC) are determined by varying the lasing wavelength, as was done earlier for pure poly(acetylene). The Raman spec- Ð Ð tra of the copolymer samples doped with iodine contain the bands attributed to the iodine polyions I3 and I5 . Unlike poly(acetylene), the ethyleneÐacetylene copolymer is characterized by the Raman spectra in which the Ð Ð intensities of the bands assigned to the I5 polyions are higher than those of the I3 polyions even at low degrees of doping. The structural features responsible for this difference are discussed. © 2002 MAIK “Nauka/Interpe- riodica”.

1. INTRODUCTION It can be assumed that such an essential difference is associated with appreciable differences in both the Among the polymers with conjugated chains, length and the length distribution of macromolecular poly(acetylene) (PA) possesses the highest conductiv- fragments with conjugated bonds in PA and EAC. 5 ΩÐ1 Ð1 ity (up to 10 cm ) comparable to theoretical esti- However, our investigations with the use of resonant mates [1, 2]. However, poly(acetylene) is highly sus- Raman scattering have revealed that the size distribu- ceptible to oxidative destruction in air and, conse- tion of conjugated regions in EAC is similar to that quently, has not found practical application. According observed in PA synthesized by a conventional method to Pochan [3], the oxidation is initiated by molecular [8]. defects formed in the course of synthesis and (or) cisÐ trans isomerization of the synthesis products upon heat It is known that ethyleneÐacetylene copolymers treatment. contain conjugated chains tens and hundreds of ang- stroms in length. These chains are separated from one Over the last decade, considerable efforts have been another by extended dielectric PE regions. In this made to search for methods of improving synthesis pro- respect, it is of interest to elucidate whether the low cedures [1, 2]. This has allowed one to decrease sub- conductivity of EAC is associated with an insufficient stantially the concentration of molecular defects (and, percolation of conducting regions or the doping of the hence, to increase the conductivity). However, the oxi- copolymer occurs through a mechanism differing from dation resistance of the high-conductivity PA samples that observed in PA. thus far synthesized remains extremely low. Resonant Raman scattering investigations have It has been found that the introduction of sufficiently demonstrated that, upon doping of PA with iodine, Ð Ð flexible poly(ethylene) (PE) spacers into PA molecules charge-transfer complexes of the I3 and I5 types are through the synthesis of ethyleneÐacetylene block formed in conjugated fragments. These complexes are copolymers (EAC) is an effective method of increasing responsible for the generation of free charge carriers in the oxidation resistance of poly(acetylene) [4, 5]. the conjugated chains and conduction in PA in electric Nascent EAC powders can be kept in air over many fields. According to the results obtained in [9Ð12], the years without indications of destruction. Earlier [6], we relative content of these complexes depends on the revealed that doping of copolymers with iodine leads to molar concentration of introduced iodine and the a substantial increase in dc conductivity. However, the degree of orientation of PA samples. Wang et al. [12] Ð maximum conductivity achieved in EAC is estimated to showed that the concentration of the I3 complexes in be σ ≤ 10Ð2 ΩÐ1 cmÐ1 [7], which is seven orders of mag- oriented high-conductivity PA samples is one order of Ð nitude less than the maximum conductivity of PA [1, 2]. magnitude higher than that of the I5 complexes. More-

RESONANT RAMAN SPECTROSCOPY 1189 over, the doping of PA samples leads to a considerable 1500 change in the distribution of conjugated regions over the length toward an increase in the percentage of short fragments [13]. 1000 To the best of our knowledge, the specific features of the doping mechanism in EAC have never been analyzed. In this respect, the main objective of the present work was to perform this investigation. 500 1 Intensity, arb. units 2 2. SAMPLES AND EXPERIMENTAL TECHNIQUE 0 1000 1500 2000 2500 3000 The experiments were carried out using unoriented Frequency, cm–1 copolymer films 100–150 µm thick. The films were prepared by pressing powders at room temperature. The Fig. 1. Raman spectra of (1) undoped and (2) iodine-doped λ content of double bonds in the copolymer was deter- (y = 0.16) EAC samples at L = 632.8 nm. mined by the ozonization technique and was estimated at 20 mol %. The iodine diffusion was performed at a ° temperature of 25 C either from a gas phase or from an 1800 iodine solution in heptane. After doping, the samples (a) were placed in an evacuated tube in order to remove the iodine which was not incorporated into the charge- 1400 transfer complexes. The molar concentration y of iodine in the copolymer samples was calculated from 1000 the relative increment of the mass of the studied sample 600 [(m Ð m0)/m0] according to the formula 2 mmÐ 1M 3 y = ------0 ------CH, (1) 200 1 m n M 0 I 1060 1080 1100 1120 1140 1160 1200 where MI is the molecular mass of iodine, MCH is the molecular mass of the acetylene unit (CH), m is the 0 1000 (b) mass of the sample prior to diffusion, m is the mass of the sample after diffusion, and n is the fraction of the Intensity, arb. units 800 acetylene moiety in the mass of the copolymer. The Raman spectra were recorded on a SPEX Indus- 600 tries spectrometer with holographic gratings at a linear 400 ruling density of 1800 mmÐ1 at room temperature in air. 3 The measurements were performed with radiation at 200 2 wavelengths λ = 632.8 nm (heliumÐneon laser) and 1 λ L 0 L = 476.5, 488.0, and 514.5 nm (argon laser). The 1440 1460 1480 1500 1520 1540 power of a light beam incident on the sample did not Frequency, cm–1 exceed 20 mW. The diameter of an illuminated spot on the sample was approximately equal to 100 µm. The Fig. 2. Profiles of the fundamental bands in the Raman λ backscattered light in a 180° scattering geometry was spectra of EAC at different lasing wavelengths L (nm): (1) recorded using an electronic signal-processing circuit 632.8, (2) 514.5, and (3) 476.5. based on a PC486DX computer. chains. A broader band centered at about 2170 cmÐ1 corresponds to the overtone of the band at 1085 cmÐ1, 3. RESULTS AND DISCUSSION and the band at about 2550 cmÐ1 results from a mixing Figure 1 shows the overall Raman spectra of the of vibrations at frequencies of 1085 and 1470 cmÐ1. undoped and doped copolymer samples. Earlier [13Ð15], it was demonstrated that the consid- According to the analysis performed in our recent erable half-width and asymmetry of the bands at about work [8], all the intense bands observed in the Raman 1100 and 1500 cmÐ1 are due to a superposition of a set spectrum of the copolymer are assigned to the conju- of elementary resonant Raman bands whose frequen- gated PA fragments in a trans conformation. The band Ð1 cies are close to one another and depend on the length at 1085 cm is associated with a mixing of the stretch- of conjugated chains. ing vibrations of the single CÐC and CÐH bonds. The band in the range of 1470 cmÐ1 is attributed to the As is known [13Ð15], the incident light in the red λ stretching vibrations of the double C=C bonds in trans lasing wavelength range ( L > 600 nm), for the most

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1190 MARIKHIN et al.

Sizes of conjugated regions in undoped and doped ethyleneÐacetylene block copolymer samples Band in the range of 1100 cmÐ1 Band in the range of 1500 cmÐ1 size of the conjugated size of the conjugated region NC=C region NC=C

λ Ð1 Ð1 L, nm Ð1 Ð1 frequency of elementary components, cm half-width, cm fraction of the elementary component spectrum, % in the overall according to the method described in [16] according to the method described in [17] frequency of elementary components, cm half-width, cm fraction of the elementary component spectrum, % in the overall according to the method described in [16] according to the method described in [17] Undoped sample 632.8 1080.25 15.46 23.25 28 47 1471.0 8.2 22.8 26 40 1087.0 18.55 20.04 24 34 1474.7 14.7 1.3 25 33 1096.6 17.04 28.93 21 24 1476.2 7.9 31.3 24 32 1100.84 24.22 9.21 20 21 1479.4 14.2 10.1 22 26 1114.62 24.75 0.11 17 15 1482.2 10.8 29.5 21.5 25 1145.72 7.67 0.52 Ð Ð 1497.5 24.5 4.75 16 15 1162.76 18.4 0.29 Ð Ð 1514.8 12.1 0.21 15 12 476.5 1080.55 12.5 7.25 27 46 1474.2 11.5 16.9 25 35 1087.44 12.2 6.12 23 33 1480.7 10.9 18.6 2 26 1095.4 22.0 26.9 20 25 1487.6 11.7 17.8 19 21 1105.85 19.9 27.5 19 17 1494.9 13.0 18.3 18 17 1113.42 15.1 9.5 17 13 1502.9 7.0 4.6 17 15 1120.1 15.8 14.0 16 10 1507.0 7.6 4.5 16.5 13 1129.0 18.3 8.65 15 Ð 1513.1 17.0 10.9 16.0 Ð 1170.55 7.6 0.15 Ð Ð 1527.8 20.0 8.44 13 Ð 488.0 1080.84 13.32 9.7 26 45 1473.3 9.9 9.7 25 36 1087.23 12.35 10.0 23 32 1478.9 11.7 20.4 23 28 1095.23 17.72 22.0 20 25 1486.4 13.7 27.4 20 21 1105.54 18.75 28.0 19 18 1494.4 12.0 10.5 19 17 1111.21 16.83 11.1 18 15 1500.5 15.0 9.8 18 15 1119.22 14.71 12.3 16 13 1500.2 18.8 10.5 17 13 1130.0 16.59 5.9 14.5 10 1521.9 25.8 10.5 15 10 1170.0 14.45 1.1 Ð Ð 1563.1 28.7 1.1 Ð Ð Doped sample (y = 0.016) 488.0 1079.23 16.6 5.8 27 50 1475.4 10.8 9.2 24 32 1086.5 14.6 4.2 24 35 1481.8 12.8 10.2 22 25 1994.5 19.2 18.9 21 25 1487.0 9.8 5.4 20 20 1105.6 17.2 18.0 19 18 1496.0 21.2 33.2 18 16 1115.0 17.6 17.2 17 15 1504.6 8.0 2.6 17 15 1122.6 16.8 14.8 16 12 1513.6 21.4 22.8 16 12 1129.5 16.2 15.6 15 10 1523.5 12.4 2.9 15 10 1141.0 20.4 5.4 Ð Ð 1531.7 28.4 13.7 14 8

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

RESONANT RAMAN SPECTROSCOPY 1191 part, resonantly excites long chains, whereas short 1.0 (a) chains, in this case, are excited nonresonantly; i.e., the intensity of the corresponding bands in the Raman spec- 0.8 trum is underestimated out of proportion. By contrast, 2 λ ≥ 0.6 the incident light at shorter wavelengths ( L 350 nm) gives rise primarily to resonant excitation of short 0.4 chains involving only a few double C=C bonds. 0.2 1

The profiles of the fundamental Raman bands at dif- Intensity, arb. units 0 ferent lasing wavelengths are depicted in Figs. 2a and 1060 1080 1100 1120 1140 1160 2b. As can be seen, the expected effect of the wave- –1 λ Frequency, cm length L most clearly manifests itself in the scattering range corresponding to short conjugated chains. This 1.0 effect can be determined quantitatively from the data (b) (see table) obtained by computer decomposition of the 0.8 complex Raman band profiles at frequencies of 1100 and 1500 cmÐ1 with the use of the standard Pick fit pro- 0.6 gram (by analogy with the decomposition performed 0.4 2 earlier in [8]). 0.2 1

It is seen from Fig. 1 that the doping leads to a dras- Intensity, arb. units tic decrease in the intensity of all the observed bands. 0 1440 1480 1520 1560 This is especially pronounced for the bands at 2170 and –1 2550 cmÐ1, which become almost indistinguishable Frequency, cm even at y = 0.16. As regards the fundamental bands, the 1400 intensity of the band at about 1470 cmÐ1 decreases more (c) Ð1 steeply compared to that of the band at 1085 cm . In 1000 our opinion, this result is quite reasonable because the charge-transfer complexes formed in the vicinity of the 600 1 double bonds should bring about considerable distortions of local vibrations near these bonds. 200 2 The profiles of the fundamental bands (normalized to Intensity, arb. units the same intensity of the band at a maximum) in the 0 Raman spectra of the undoped and doped copolymer 1000 1200 1400 1600 λ Frequency, cm–1 samples at the lasing wavelength L = 488.0 nm are displayed in Figs. 3a and 3b. The actual intensity ratio of Fig. 3. Profiles of the fundamental bands in the Raman these bands is illustrated in Fig. 3c. As can be seen from spectra of (1) undoped and (2) iodine-doped (y = 0.016) λ Fig. 3, the doping results in a significant asymmetrical EAC samples at L = 488.0 nm. (a, b) The bands of the change in the shape of the bands at about 1100 and undoped and doped samples are normalized to the same 1500 cmÐ1 due to a sharp decrease in the scattering inten- intensity of the band at a maximum. (c) The bands illustrat- sity in the wavelength range corresponding to the longest ing the actual intensity ratio for the undoped and doped samples. chains. This effect is caused by the separation of long chains into shorter fragments during the formation of charge-transfer complexes upon doping [13Ð15]. [18, 19]. In the simplest case, the structure of these poly- The results of computer spectral decomposition (see Ð Ð ions is considered a combination of IÐ, I , and I ions. table) clearly demonstrate that, upon doping, the frac- 2 3 tion of short conjugated fragments considerably For each type of ions, there are particular frequencies in increases compared to that of longer fragments. the Raman spectra upon excitation at lasing wave- Ð Thus, analysis of the fundamental Raman bands lengths in the visible spectral range. Ions I do not man- attributed to the conjugated PA fragments in EAC ifest themselves in the Raman spectra but can be 129 shows that the regularities observed in changes of these revealed by I Mössbauer spectroscopy [20]. fragments upon doping are similar to those found for In the Raman spectra of iodine solutions, molecules PA [13Ð15]. I2 can be identified from the broad band which is It should be noted that the type of charge-transfer observed at about 209 cmÐ1 and assigned to the stretch- complexes formed upon iodination of conjugated chains ing vibrations of iodine molecules [19]. The polyions IÐ can also be determined by Raman spectroscopy. The for- Ð 3 and I5 are most frequently formed upon doping of con- mation of different complexes was first observed in jugated polymers. As follows from the calculations per- methanol solutions of iodine and in a number of com- formed by Marks and Kalina [19], the Raman spectrum pounds (with a known chemical structure) containing of the IÐ polyion substantially depends on its geometric Ð Ð Ð 2Ð 3 iodine polyions of specific types: I3 , I5 , I7 , I8 , etc. structure. In the case when the ion has a symmetric lin-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1192 MARIKHIN et al.

Ð Ð ear shape, the Raman spectrum, according to the selec- I5 polyion can be treated both as the combination I2 + I3 tion rules, should exhibit only one active polarized and as the combination of the IÐ ion with two symmet- ν Ð1 mode at the frequency 1 = 118 cm , which corre- rically added units I2. For a symmetric linear chain of sponds to the symmetric stretching vibrations involving the type IÐIÐIÐÐIÐI, the resonant Raman spectrum the motion of only terminal atoms with respect to the exhibits an intense band at ν = 160 cmÐ1. This band is central atom. The in-phase composition of the stretch- attributed to the IÐI valence transition and is most fre- Ð ing vibrations of two iodine atoms IÐI in the I3 ion quently observed upon doping of conjugated polymers. Ð1 Ð results in the band at 118 cm . As regards the I2ÐI ÐI2 chain, the symmetric stretching ν Ð1 vibrations can be responsible for the appearance of a The other two modes at frequencies 3 = 145 cm (asymmetric vibrations involving the motion of the weak band at 107 cmÐ1. ν Ð1 Ð central atom) and 2 = 69 cm (bending vibrations) are For the other combination (I2 + I3 ), the changes in active only in the IR spectrum. However, if the symme- the interatomic distances in the I2 units result in the try of the polyion structure is violated (for example, due appearance of the band at ν = 185 cmÐ1 instead of the to the difference in the distances between terminal band at 209 cmÐ1 [19]. atoms and the central atom or the deviation of the ion Ð ν ν For the I5 polyions with symmetrically bent and zig- structure from the linear geometry), the 2 and 3 zag geometries, the Raman spectra exhibit a more com- modes can manifest themselves in the Raman spectrum plex structure. These spectra are presented in [19]. due to the violation of the selection rules. This would Ð Ð It is quite reasonable that linear polyions of the I3 indicate that the geometry of the I3 polyion had Ð changed in response to force fields induced by sur- and I5 types are most stable. These are singly charged Ð ions whose stability and size are important factors. rounding atoms or molecules. The high lability of the I3 structure can be explained by the fact that the halogenÐ Upon heating of iodine-doped PA, the charge-trans- Ð Ð Ð halogen bonds in the iodine polyion are substantially fer complexes I3 and I5 decompose to form the I2 ions. weakened as the result of an appreciable increase (by These processes were observed by mass spectrometry 0.2 Å) in the interatomic distances. In this case, the ten- in the course of temperature measurements. The forma- Ð sile force constants are less than one-half the corre- tion of the I2 ions was judged from the appearance of sponding constant for the I2 molecule [21]. Conse- broad peaks in the mass spectra with variations in the quently, the location of the central atom can easily temperature [24]. The intensity and location of these change under the action of sufficiently weak forces peaks depend on the degree of doping. At y = 0.07, only induced, for example, by neighboring molecules or one peak is observed at a temperature of 70°C. For this chemical and conformational defects. This feature per- concentration of iodine in PA, the Raman scattering Ð mits the polyions to be readily incorporated into data indicate the occurrence of only the I3 ions, which, paracrystal lattices formed by conjugated macromole- most likely, are thermally stable up to 70°C. At y = 0.22, Ð cules. In the case when the I3 ions are distributed in a the mass spectra exhibit two peaks: the first peak at T = nonrandom manner but form a linear regular chain (this 50°C and the second (more intense) peak at T = 100°C. situation can occur in well-oriented PA samples [22, By comparing these findings with the Raman scattering 23]), the Raman spectrum contains a very intense data, Saalfeld et al. [24] attributed the low-temperature ν Ð1 Ð (slightly shifted in frequency) band at 1 = 110 cm . peak to the formation of I2 upon decomposition of the This band is accompanied by a clearly defined progres- IÐ complexes and assigned the high-temperature peak 5 Ð sion of overtones up to the ninth overtone. For asymmet- to the decomposition of the I3 complexes. From the Ð ν Ð1 ric I3 ions, an intense band at 3 = 150 cm can be foregoing it follows that, upon the formation of chains −1 Ð Ð Ð observed in addition to the intense band at ν = 105 cm . from the I3 and I5 anions, the stability of the I3 anions ν 1 Ð The aforementioned frequencies 1 for the I3 polyions increases by 20Ð30 K. The predominant formation of vary significantly (from 118 to 105 cmÐ1) for different these polyanions upon doping of partly crystalline con- compounds and PA in the works of different authors jugated polymers should be expected for the following [18, 19]. This implies that, in the studied compounds, reasons. In PA crystals, linear segments of macromole- the distances between iodine atoms should differ cules are separated by channels whose walls are formed ν noticeably. As is known, a decrease in the 1 frequency by hydrogen atoms of the CH groups. These channels suggests an increase in the halogenÐhalogen distance. can involve molecules I2. Coppens [18] proved that the As was noted above, this seems to be quite reasonable potential of the covalent interaction between hydrogen because the interatomic distances in the polyions can and iodine atoms has a minimum at r = 3.36 Å; hence, readily change as the result of an appreciable decrease the diameter of the channels should be no less than in the force constants. 6−7 Å. It is evident that channels of similar sizes should Ð encourage the predominant formation of linear iodine In the case of the I5 polyion, there exists a large number of geometric structures, including a symmetric polyions. linear chain, a chain symmetrically bent with respect to Among the currently known polyanions of iodine, the Ð the central atom, and a zigzag chain. It has been shown I3 polyion has the largest constant of formation, which is that the low-frequency Raman spectrum is very sensi- almost two orders of magnitude greater than that for the Ð Ð tive to the configuration of the I5 polyion. Note that the I5 polyion [19]. This accounts for the predominant for-

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 RESONANT RAMAN SPECTROSCOPY 1193

Ð mation of I3 polyions upon doping of PA, which has been 500 observed by many authors [9Ð15, 18Ð25]. From the aforesaid it is clear that a similar analysis 400 of the low-frequency Raman spectra can provide reliable and valuable information regarding the speciﬁc features of the doping of conducting polymers and the 300 real structure of iodine polyanions. 2 Figure 4 displays the Raman spectra of lightly doped 200

(y = 0.04) EAC samples in the range ∆ν < 200 cmÐ1, Intensity, arb. units which we obtained upon excitation with red laser light 1 λ 100 ( L = 632.8 nm). For the iodine-doped sample, the spec- 50 100 150 200 250 trum exhibits two rather broad (with a half-width of Frequency, cm–1 approximately 20 cmÐ1), well-pronounced bands with maxima near 107 and 157 cmÐ1. According to the afore- Fig. 4. Raman spectra of (1) undoped and (2) iodine-doped (y = 0.04) EAC samples in the short-wavelength range at mentioned data available in the literature, we can state λ that, upon doping of EAC, charge-transfer complexes L = 632.8 nm. Ð Ð of two types, namely, I3 and I5 , are formed in poly(acetylene) fragments, as was observed earlier for pure librium toward the right. And vice versa, the removal of poly(acetylene). At the same time, the doping kinetics Ð I2 leads to the shift of the equilibrium toward the left. in the copolymer differs in that the concentration of I5 Ð We believed that the specific features in the supramo- anions is three or four times higher than that of I3 lecular structure of polymers should substantially affect anions even at initial low degrees of doping (y < 0.04, the character and the rate of diffusion into PA crystals. Fig. 4). With a further increase in the degree of doping, Ð As is known [1, 2], high conductivities have been the intensity of the band associated with the I5 anions achieved upon doping of specially synthesized low- becomes progressively higher than that of the band Ð defect PA samples with an increased content of long attributed to the I3 anions. As a consequence, the latter conjugated chains. Electron microscopic and diffrac- band completely disappears at y = 0.2Ð0.3. However, tion investigations have revealed that these samples the authors of all published works dealing with the dop- contain more perfect and extended PA microfibrils [26, ing of PA have argued that, at the early stages of dop- Ð 27]. Furthermore, these samples can be oriented to ing, either only the band of the I3 anions is observed in higher degrees of extension. This results in a very close the spectrum or the intensity of the band attributed to Ð packing of microfibrils with respect to each other and a the I3 anions always exceeds the intensity of the band Ð considerable decrease in the porosity of the samples. In assigned to the I5 anions. The intensities of these bands Ð our opinion, all the aforementioned factors hinder the level off at y = 0.25, and the band of the I5 anions begins diffusion of I inside the fibrils into the crystal lattice of Ð 2 Ð to dominate over the band of the I3 anions only at PA and bring about the predominant formation of I3 higher (limiting) iodine concentrations. In the spectra anions and, also, column structures composed of these of high-conductivity oriented poly(ethylene) samples anions. By contrast, if the PA crystals have a more loose Ð with different degrees of extension, the band of the I3 structure and are surrounded by a free volume, the Ð anions invariably dominates over the band of the I5 amount of penetrating iodine I2 can be appreciably anions even at the highest degrees of doping [12]. Fur- larger, which results in a shift of reaction (2) toward the thermore, at the maximum extension (L/L0 = 8), the right. spectra contain only the band at ν = 107 cmÐ1, which cor- This mechanism is confirmed by the x-ray photo- Ð responds to the I3 complex. The highest conductivity electron spectroscopic data obtained by Ikemoto et al. observed in these samples was attributed by Wang et al. [28], who showed that only the surface layers of PA Ð [12] not only to the absence of the I5 polyions but also fibrils are doped at the initial stages (y < 0.05) and then to the formation of a specific linear lattice composed of dopants are more homogeneously distributed over the Ð I3 polyanions and aligned along the direction of macro- bulk of the sample. molecule orientation. Similar conclusions were drawn Note that, in this case, a substantial amount of in [26]. iodine in the surface region is in a weakly bound state The formation of particular complexes with an and can be readily removed by evacuation. Ð Ð increase in the degree of doping can be represented in It is significant that the I5 / I3 ratio for the surface the form of a dynamic equilibrium according to the layers in PA (determined from the x-ray spectroscopic Ð Ð scheme data) proves to be considerably larger than the I5 / I3 ratio averaged over the sample thickness (obtained ()I2 ()+ Ð I2 ()+ Ð CH x CH x I3 CH x I5. (2) from the Raman spectroscopic data) [28].

It is clear that the larger the amount of iodine I2 intro- We believe that the speciﬁc structure of EAC favors duced into a polymer, the greater the shift of the equi- the diffusion of iodine into crystalline regions of PA

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1194 MARIKHIN et al.

Ð – – I I (0.131 e/Å) for a one-dimensional lattice of I3 anions 3 3 and the lowest density (0.073 e/Å) for a one-dimen- L = 15.24 Å Ð 3.56 sional lattice of I5 anions. The intermediate densities 5.84 5.84 should be observed for combinations of the IÐIÐIÐ Ð Ð Ð 3 5 3 – – (0.099 e/Å) and I5I3I5 (0.082 e/Å) types. The former I5 I5 combination is most frequently encountered in PA L = 27.50 Å (according to x-ray diffraction and Raman spectros- 12 3.50 12 copy). Judging from the intensity ratio of the low-fre- – – – quency bands in the Raman spectrum of the copolymer I3 I5 I3 (Fig. 4), it can be assumed that our samples contain Ð Ð Ð L = 30.28 Å combinations of the I5I3I5 type. However, the ultimate 5.84 3.3012 3.30 5.84 answer to this question calls for large-angle x-ray dif- – – – fraction investigation. I I I 5 3 5 It can be seen from Fig. 5 that, upon formation of the L = 36.44 Å Ð Ð Ð Ð Ð Ð 12 3.30 3.30 12 I3I5I3 and I5I3I5 combinations, the minimum lengths of 5.84 conducting PA regions should be equal to 30 and Ð Ð 36.5 Å, respectively. This implies that shorter conju- Fig. 5. Linear sizes of the I3 and I5 polyions and their combinations according to x-ray diffraction data [23]. gated chains (for the given frequently realized type of doping) do not participate in the generation of free charge carriers, because they cannot combine with the fragments. In our earlier works [6Ð8], we made the dopant in the given form. As a consequence, the attain- inference that, in the structure of copolymers, well- able macroscopic conductivities decrease appreciably. ordered (from the standpoint of x-ray diffraction) PA Unfortunately, the copolymer samples studied in the crystalline grains involving extended conjugated present work are characterized not only by an increased regions alternate with crystalline grains consisting of content of short conjugated chains (see table) but also by PE trans chains. The size of crystalline grains of both a considerable number of isolated double bonds [6Ð8]. It types is estimated at 100 Å [6]. Moreover, the copoly- seems likely that all these factors are responsible for the mers involve rather narrow, strongly defective layers sufﬁciently low electrical conductivities thus far between crystalline grains of both types. These inter- attained in EAC [6Ð8]. layers accumulate conformational, disclination, and chemical defects, including isolated double bonds and The analysis performed also demonstrates that, in short conjugated chains. It can be assumed that a more order to improve the electrical properties of poly(acet- loose (compared to crystalline grains) structure of these ylene) and its copolymers with ethylene, special atten- interlayers facilitates access for iodine to end faces of tion should be focused on the doping process. In particular, the doping needs to be carried out under the mild- crystalline grains composed of conjugated chains, Ð which is unexpected for PA. est conditions with the aim of forming only the I3 anions and the column structures composed of these Furthermore, our x-ray structural investigations anions. demonstrated that the crystalline grains of poly(acetylene) in copolymers are less perfect than those in pure It is in this case that the number of generated charge poly(acetylene). carriers can be increased at the expense of short conjugated chains. Moreover, the high conductivity can be Thus, the above arguments and easier diffusion indi- achieved only with a strict order in the arrangement of cate that equilibrium (2) for the copolymer should more the IÐ anions in order to prevent disturbance of the readily shift toward the right. As a result, the concentra- 3 Ð Ð potential distribution along the molecular axis of PA tion of I5 anions exceeds the concentration of I3 anions. Ð [27]. The incorporation of anions of other types (I5 ) This can also mean that the actual local degrees of dop- leads to a decrease in the conductivity along the chain. ing y can be higher than those determined from the As is known [29], hopping conductivity—a character- increment of the sample mass due to the diffusion. istic property of polymers—is determined not only by Analysis of the data available in the literature shows the transfer of carriers along the chain but also by hop- that an increased concentration of IÐ anions as com- pings from one conjugated chain to another chain both Ð 5 pared to that of I3 anions is an undesirable factor as within a particular crystalline domain and between regards the possibility of achieving high conductivities poly(acetylene) domains. Using poly(acetylene) as an [9Ð12, 16Ð26]. First and foremost, making allowance example, Murthy et al. [23] showed that the efﬁciency for the actual geometric sizes obtained by the diffrac- of the transverse carrier transfer is substantially tion methods for these singly charged anions and their affected by the type of more complex structures con- combinations (Fig. 5) [23], it becomes evident that dif- sisting of column conjugated chains formed in the ferent numbers of free carriers are generated in conju- course of doping. These can be layers of alternating PA gated chains with the same length upon doping. It is molecules and incorporated iodine structures in differ- easy to determine that the generated charge has the ent molar ratios or even continuous layers and iodine highest density per unit length of the conjugated region anions. For the purpose of elucidating the real structure

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 RESONANT RAMAN SPECTROSCOPY 1195 of doped copolymer samples, we are, at present, per- 14. J. Kurty and H. Kuzmany, Phys. Rev. B 44 (2), 597 forming large-angle x-ray diffraction investigations. (1991). 15. G. P. Brivio and E. Mulazzi, Phys. Rev. B 30 (2), 876 (1984). ACKNOWLEDGMENTS 16. E. Mulazzi, G. P. Brivio, E. Faulqnes, and S. Lefrant, This work was supported by the Russian Foundation Solid State Commun. 46 (12), 851 (1983). for Basic Research, project no. 98-03-33264. 17. Y. Furakawa, T. Arakawa, H. Takeushi, et al., J. Chem. Phys. 81 (7), 2907 (1984). 18. P. Coppens, in Extended Linear Chain Compounds, Ed. REFERENCES by J. S. Miller (Plenum, New York, 1982), Vol. 1, p. 333. 1. H. Naarmann, Synth. Met. 17 (1), 223 (1987). 19. T. J. Marks and D. W. Kalina, in Extended Linear Chain 2. J. Tsukamoto, A. Takahashi, and K. Kovasaki, Jpn. J. Compounds, Ed. by J. S. Miller (Plenum, New York, Appl. Phys. 92 (1), 125 (1990). 1982), Vol. 1, p. 197. 3. J.M. Pochan, in Handbook of Conducting Polymers, Ed. 20. W. Kiefer and H. J. Bernstein, Appl. Spectrosc. 25 (4), by T. A. Skotheim (Marcel Dekker, New York, 1986), 500 (1971). Vol. 2, p. 1383. 21. P. Klaboe, J. Am. Chem. Soc. 89 (15), 3667 (1967). 4. L. N. Russiyan, P. E. Matkovskiœ, V. N. Noskova, et al., 22. R. H. Baughman, N. S. Murthy, G. G. Miller, and Vysokomol. Soedin., Ser. A 33 (2), 280 (1991). L. W. Shacklette, J. Chem. Phys. 79 (2), 1065 (1983). 5. L. N. Raspopov, P. E. Matkovskiœ, G. P. Belov, et al., 23. N. S. Murthy, G. G. Miller, and R. H. Baughman, J. Vysokomol. Soedin., Ser. A 33 (2), 425 (1991). Chem. Phys. 89 (4), 2523 (1988). 6. A. N. Aleshin, E. G. Guk, V. A. Marikhin, et al., Vysoko- 24. F. E. Saalfeld, J. J. De Corpo, and J. R. Wyatt, in mol. Soedin., Ser. A 37 (7), 1179 (1995). Extended Linear Chain Compounds, Ed. by J. S. Miller 7. A. N. Aleshin, E. G. Guk, V. A. Marikhin, and L. P. Myas- (Plenum, New York, 1982), Vol. 1, p. 33. nikova, Macromol. Symp. 102, 309 (1966). 25. Handbook of Conducting Polymers, Ed. by T. A. Skot- 8. V. A. Marikhin, I. N. Novak, E. G. Guk, et al., Fiz. Tverd. heim (Marcel Dekker, New York, 1986), Vol. 1, Chap. 2. Tela (St. Petersburg) 39 (11), 2106 (1997) [Phys. Solid 26. A. Montaner, M. Rolland, J. L. Souvajol, et al., Polymer State 39, 1885 (1997)]. 29 (6), 1101 (1988). 9. S. Lefrant, L. S. Lichtmann, H. Temkin, et al., Solid 27. K. Shimamura, Y. Yamashita, H. Kasahara, and State Commun. 29 (3), 191 (1979). K. Monobe, Synth. Met. 17 (1), 485 (1987). 10. K. Kamija and J. Tanaka, Synth. Met. 25 (1), 59 (1988). 28. I. Ikemoto, M. Sakairi, T. Tsutsumi, et al., Chem. Lett. 11. L. S. Hsu, A. J. Signorelli, G. P. Pez, and R. H. Baugh- 1189 (1979). man, J. Chem. Phys. 69 (1), 106 (1978). 29. S. Curran, A. Stark-Hauser, and S. Roth, in Handbook of 12. D. Wang, J. Tsukamoto, A. Takahashi, et al., Synth. Met. Organic Conductive Molecules and Polymers, Ed. by 65 (1), 117 (1994). H. S. Nalwa (Wiley, Chichester, 1997), Vol. 2. 13. H. Kuzmany, Phys. Status Solidi B 97 (2), 521 (1980). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

Physics of the Solid State, Vol. 44, No. 6, 2002, pp. 1196Ð1202. From Fizika Tverdogo Tela, Vol. 44, No. 6, 2002, pp. 1145Ð1150. Original English Text Copyright © 2002 by Perova, Astrova, Tsvetkov, Tkachenko, Vij, Kumar.

POLYMERS AND LIQUID CRYSTALS

Orientation of Discotic and Ferroelectric Liquid Crystals in a Macroporous Silicon Matrix1 T. S. Perova*, E. V. Astrova**, S. E. Tsvetkov*, A. G. Tkachenko**, J. K. Vij*, and S. Kumar*** * Department of Electronic & Electrical Engineering, Trinity College, Dublin, 2 Ireland ** Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia *** Institute for Liquid Crystal Research, Bangalor, India e-mail: [email protected] Received April 10, 2001; in ﬁnal form, October 18, 2001

Abstract—Macroporous silicon with deep regular channels 3Ð4.5 µm in diameter was inﬁltrated with discotic and ferroelectric liquid crystals (LCs) at the temperature of the isotropic phase, and then, the system was slowly cooled to room temperature, with the liquid crystalline mesophase formed. The orientation of the LC molecules in the porous matrix was studied by FTIR spectroscopy. The alignment of LCs was ascertained by comparing the behavior of various vibrational bands of a liquid crystal introduced into the porous matrix with that for LC inside the bulk cells of planar and homeotropic alignment. The molecules of the discotic LC show a planar orientation of their column’s axis with respect to the surface of the macroporous silicon wafer; i.e., they are perpendicular to the channel axis. The long molecular axis of the ferroelectric LC is aligned with the pore walls, having homeotropic orientation with respect to the wafer surface. In a macroporous silicon matrix, both kinds of LCs show unexpected enhancement of the low-frequency vibrational bands. © 2002 MAIK “Nauka/Inter- periodica”.

1 Over the last decade, considerable scientific effort Regular porous structures, such as microporous has been focused on studying liquid crystals within a superlattices and macroporous silicon, have attracted confined geometry. Liquid crystals are a crucially considerable scientific interest because of their having important enabling technology for the manufacture of a photonic band gap [9–11]. Infiltration of porous sys- displays. The understanding of the layer structure, tems of this kind with liquid crystals gives control over phase transitions, order parameter fluctuations, liquid- the position of their photonic band gap, if one relies on solid interface, and dynamics of collective modes and the fact that the refractive index of LC changes with molecular motions will be significantly advanced if temperature [12] or external electric field [13]. It is results on samples restricted to confined geometries are important to find out how different kinds of liquid crys- compared with those in the bulk [1Ð4]. tal molecules will behave in a macroporous silicon (ma- PS) matrix. In this study, Fourier Transform Infrared Porous systems are suitable hosts for liquid crystal Spectroscopy (FTIR) was used to investigate the align- molecules. These systems are characterized by a high ment of liquid crystals, both discotic and ferroelectric, surface-to-volume ratio and, therefore, are very sensitive infiltrated into ma-PS. to any interactions between the infiltrated molecules and the surrounding walls. A possible interaction of this kind is the aligning power of the walls [5], which can give rise 1. EXPERIMENTAL to such subphases as a quasi-nematic layer [6]. Two different types of liquid crystal materials were Polarized infrared spectroscopy with normal and used in this study: (i) a commercial ferroelectric liquid oblique incidence of light is one of the most powerful crystal (FLC) mixture SCE8 (ii) and a triphenylene- techniques that can be used to investigate the orienta- based discotic liquid crystal (DLC) H5TÐNO2 [14]. tion of liquid crystal (LC) molecules with respect to a These LCs were chosen for experimental convenience, substrate [7, 8]. The application of this technique to as they both have a mesophase at room temperature. confined LCs is frequently restricted by the nature of The structural formulae and phase sequences of these the host material (e.g., Vocor glass), which is usually compounds are shown in Fig. 1a. These liquid crystals opaque to light in the IR region. However, this problem were infiltrated into macroporous silicon (shown in Fig. 1b) by means of the capillary effect at tempera- can be overcome by using porous silicon for LC infil- ° tration. In this case, the size and depth of pores and the tures approximately 10 C above the temperature of porosity can be varied widely. transition to the isotropic phase. The macroporous silicon used in this study (Fig. 1b) 1 This article was submitted by the authors in English. has a system of regular cylindrical pores of micrometer

ORIENTATION OF DISCOTIC AND FERROELECTRIC LIQUID CRYSTALS 1197

(a) F R R RRCOO NO2 F R R R COO R' RR SCE8 H5T–NO2 SmC*–59°C–SmA–79°C–N*–100°C–I Cr–(–5°C)–D–134°C–I (b)

–+ (c) α'

IR beam

0° x 90° y z

(d)

Planar Homeotropic

(e)

Planar Homeotropic

Fig. 1. Material properties and experimental setup: (a) structural formulas and phase sequences of ferroelectric liquid crystalline mixture SCE8 and tryphenylene-based discotic liquid crystal H5TÐNO2 (R = OC5H11); (b) SEM image of the macroporous silicon matrix (cross section side and top view) used to inﬁltrate with the liquid crystals; (c) the schematic view of the FTIR experiment shows the sample rotation angle α' and macroporous Si sample coordinate frame; (d) planar and homeotropic alignment of the discotic liquid crystal in a ZnSe cell; (e) planar and homeotropic alignment of the ferroelectric liquid crystal in a ZnSe cell.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

1198 PEROVA et al. diameter and high aspect ratio. The starting material samples. This makes the analysis of the IR spectra of was single-crystal (100)-oriented Czochralski-grown the liquid crystal itself impossible for oblique incidence n-type silicon with resistivity ρ = 15 Ω cm. A standard of the light. Therefore, the alignment of LCs was deter- photolithographic process was employed to form pits mined by comparing the behavior of various vibrational spaced 12 µm apart on the polished surface of the sili- bands in the spectra obtained at normal incidence of the con wafer. Deep pores were etched electrochemically infrared beam on liquid crystals contained in a porous in a 2.5% aqueous-ethanol solution of HF for 300 to matrix with that in spectra of planar and homeotropic 450 min under backside illumination [15] at a voltage bulk LC cells. 2 of 5 V and a constant current density j = 3 mA/cm . The alignment of liquid crystals is usually consid- The pore depth and diameter d were, respectively, µ µ ered with respect to the orientation of the long molecu- 200Ð250 m and d = 3Ð4.5 m, with these parameters lar axis (in the case of rodlike molecules) and with corresponding to a porosity of 5.7Ð12.8% for our trian- respect to the column axis n (which is, in general, per- gular lattice. pendicular to the core plane) for discotic liquid crystals. FTIR measurements were performed using a Biorad The alignment will be planar if the long molecular axis FTS60A spectrometer fitted with a liquid-nitrogen- (column axis) is oriented parallel to the substrate plane cooled MCT detector. A schematic of the FTIR experi- and is homeotropic in the case of perpendicular orien- ment is shown in Fig. 1c. Scans were performed tation. Therefore, conclusions about the alignment can between 450Ð4000 cmÐ1 with a resolution ranging from be drawn from the behavior of bands associated with Ð1 vibrations exhibiting the transition dipole moment par- 2 to 8 cm . A total of 64 scans were coadded to || ⊥ improve the signal-to-noise ratio. IR spectra were ini- allel ( ) and perpendicular ( ) to the rigid part or core tially recorded for an empty macroporous silicon wafer of the molecules. These bands are listed in the table for and then for a matrix infiltrated with the liquid crystal. both kinds of liquid crystals under study. It should be The difference between these two spectra was taken to noted that the spectral position of the majority of the be the spectrum of a liquid crystal infiltrated into the vibrational bands of liquid crystals in a porous matrix is porous matrix. The alignment of the LCs within the the same as that in the bulk LC cells. porous silicon matrix was deduced from a comparison of the relative intensities and positions of the different 2.1. Discotic Liquid Crystal vibrational bands in the obtained spectra with those of the bulk liquid crystal. For this purpose, a number of There are two methods for determining the type and liquid-crystal cells with different types of alignment extent of alignment for discotic liquid crystals: (i) from (homeotropic and planar, Figs. 1d, 1e) were prepared as the ratio of the peak intensities for a particular vibra- follows. tional band in the isotropic and in the discotic phase (RDI = AiD/AiI) and (ii) from the ratio of the peak inten- A planar cell with H5TÐNO2 was fabricated with ZnSe windows (with a spacer of thickness ~12 µm). A sities for the same band in the discotic phase at normal homeotropic cell was obtained when two ZnSe win- and oblique incidence of light (R = Ai ||/Ai⊥), which dows were coated with nylon 6/6 (see [7] for more allows the intensity of the vibrational band to be deter- details). A planarly aligned cell with SCE8 was mined for vibrations with transition dipole moments obtained using two ZnSe windows coated with com- parallel (A||) and perpendicular (A⊥) to the substrate mercial (Nissan Chemical Industries, Ltd.) orientant plane (see [7] for more details). Neither of these tech- RN-1266 (a 0.4% solution in a mixture of 10% butyl niques is suitable for our purposes for the technical rea- cellosolve and 90% N-methyl pyrrolidone) and rubbed sons described above. with velvet in one direction. The thickness of this cell, We introduce here a third method for determining found from infrared fringes, was ~9 µm. A homeotropi- the type of alignment by measuring the ratio intensities cally aligned SCE8 cell, of approximate thickness 9 of two different vibrational bands, i and j, observed in µm, was prepared using ZnSe windows coated with a the mesophase (a type of dichroic ratio): carboxylato chromium complex (chromolane) (see [16] for more details). All these cells were filled by means of Rij = Ai/A j, (1) ° the capillary effect at a temperature of ~10 C above the where A is the intensity of the band associated with transition to the isotropic phase. i vibrations parallel to the LC molecule core and Aj is that associated with vibrations perpendicular to the 2. RESULTS AND DISCUSSION core. Particular care should be taken in choosing the appropriate set of bands for calculating the dichroic Complete information on the structure and orienta- ratio by means of Eq. (1). These bands should corre- tion of LC molecules can be obtained by studying a spond to vibrations with transition dipole moments at a combination of polarized and oblique infrared trans- right angle (~90°) with respect to each other. Our pre- missions [7, 8]. However, our investigations show that vious investigation [7] performed for a number of triph- additional interference effects appear in the frequency enylene derivatives (H5T, H7T, and H7TÐNO2 discotic range 700Ð1500 cmÐ1 for tilted macroporous silicon liquid crystals) demonstrated that the aromatic CÐC

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002

ORIENTATION OF DISCOTIC AND FERROELECTRIC LIQUID CRYSTALS 1199 stretching vibrations at ~1510 and ~1610 cmÐ1 are par- Assignment of infrared vibrational bands allel to the core (i-band) while the CÐH aromatic out- Ð1 Band assignment and orientation of-plane deformation at ~830 cm is purely perpendic- H5TÐNO2 SCE8 of the transition dipole moment Ð1 Ð1 ular to the core (j-band). These vibrational bands are with respect to the core cm cm shown in Figs. 2a and 2b for the H5TÐNO2 discotic liq- ⊥ uid crystal in the porous matrix and for both homeotro- CH2 symmetric stretching 2861 2856 pic and planar alignment of the molecules in a liquid- asymmetric stretching 2934 2926 crystal cell. ⊥ CH3 symmetric stretching 2873 2873 Qualitative information on the type of alignment in asymmetric stretching 2957 2955 the porous matrix can be obtained by comparing the ⊥ spectra of different samples in the region 800Ð890 cmÐ1. N=O asymmetric stretching 1530 Ð Although the vibrational band in this region is rather CÐC aromatic stretching || 1507 1513 complicated, it can be clearly seen that the main maxi- 1521 Ð1 mum of this band is shifted to ~830 cm for the cell 1616 1606 with planar alignment and for the porous matrix, ⊥ whereas for the cell with homeotropic alignment, the NO bending 845 Ð peak lies at ~820 cmÐ1. CÐH aromatic out-of-plane ⊥ 890 846 In order to obtain more precise information on the deformation 831 829 818 alignment of H5TÐNO2 molecules in a porous silicon matrix, we estimated the dichroic ratio for three sam- 791 ples using Eq. (1). For this purpose, we fitted the com- CÐC aromatic out-of-plane Ð1 posite band in the region 800Ð890 cm with four (for ⊥ the bulk planar cell) or five (for the bulk homeotropic deformation 765 || cell) Voight functions in order to determine the inten- NO2 wagging 750 Ð sity of the band at ~830 cmÐ1. The results of this fit are CÐH rocking ⊥ 733 722 shown in Figs. 3a, 3b. It can be seen that the CÐH aromatic out-of-plane deformation occurs at 831 cmÐ1 for both liquid crystal cells and for the porous matrix. The cells depends only on thickness, being independent of dichroic ratio Rij is 31 for the cell with homeotropic the alignment. alignment, 7 for the cell with planar alignment, and is Therefore, we compared the spectra of different only 0.4 for H5TÐNO2 infiltrated into the porous samples by reducing their intensity in the range matrix. At first glance, it seems that the alignment in the 2600 cmÐ1 to the same value. This normalization proce- porous matrix is even better than that for the bulk cell dure allowed us to conclude that there is a strong inten- with planar alignment, but a more detailed analysis of sity enhancement (~15 times) for the low-frequency both types of alignment for a number of triphenylene vibrational bands. derivatives shows that this is not the case. The best pla- Finally, it is obvious from our results that the align- nar alignment is observed for hexapentyltryphenylene ment of H5TÐNO2 in ma-PS is planar with respect to (H5T) [7]. Rij was found to be in the range from 3 to 5 the wafer surface. However, as seen from Fig. 4b, the for H5T. alignment of discotic LCs is homeotropic with respect We conclude that the obtained small value of the to the pore walls. This is in agreement with the results dichroic ratio is due either to the enhancement of the of a previously published work [7], where a high stabil- intensity of the low-frequency vibrational bands or to ity of homeotropic alignment was found for discotic the dampening of the intensity of the high-frequency liquid crystals deposited on untreated substrates. vibrational bands. In order to obtain more precise data on this issue, we compared the results obtained with all three samples in terms of absorption per unit volume. 2.2. Ferroelectric Liquid Crystals This was particularly important for the porous matrix in It should be noted that for this type of molecular which the liquid crystal occupied only a part (VLC) of shape, we have to take into account the polarization of the volume probed by the IR beam. This fraction was light when measuring the spectrum of the cell with pla- evaluated by multiplying the total volume (V) by the nar alignment. We use the following notation: P = 0° for porosity (p = 9.5%). This comparison enabled us to the electric vector of the incident light coinciding with conclude that strong intensity enhancement is observed the long molecular axis, and P = 90° otherwise. The IR for the low-frequency vibrational bands. This conclu- spectra of SCE8 introduced into a porous silicon matrix sion is further supported by a comparison of the inten- show that the relative intensities of (i) “parallel” and (j) sities of the vibrational bands for the alkyl chain in the “perpendicular” bands are, in this case, close to that region 2800Ð3200 cmÐ1. It is accepted that the alkyl observed for the bulk LC cell with homeotropic align- chain is rather disordered for discotic liquid crystals ment (Fig. 5). In particular, as seen from Figs. 5a and 5b, and its absorption intensity in this range for different the intensity of the CÐC aromatic stretching vibration

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 1200 PEROVA et al.

(a) ma-PS + LC (a) 0.10 0.6 Homeotropic 831 Planar 0.08 Planar cell 0.4 0.06

0.2 0.04

0.02 0 0 1580 1600 16200 1640

0.12 Planar (b) Homeotropic 1.5 Absorbance 831 (b) ma-PS + LC 0.08 Absorbance 0.25 1.0 0.20 Homeotropic cell 0.04 0.5 0.15

0.10 0 0 831 900 880 860 840 820 800 0.05 Wavenumber, cm–1 0 Fig. 2. FTIR spectra of the H5TÐNO2 discotic liquid crystal inﬁltrated into porous silicon (solid line) and embedded in 900 880 860 840 820 800 ZnSe cells with planar (dotted line) and homeotropic –1 (dashed line) alignments. (a) CÐC stretching band, parallel Wavenumber, cm to the core; (b) CÐH aromatic out-of-plane deformation band, perpendicular to the core. (Note the different scales Fig. 3. Example of the ﬁtting procedure for the low-fre- for the heavy line in (b) and the fact that the absorption of quency band for H5TÐNO2 contained in (a) planar and all the samples is reduced in accordance with the intensity (b) homeotropic cells. Note that the thickness of the homeo- of the alkyl chain vibrations in the region 2800Ð3200 cmÐ1.) tropic cell is 2.45 times that of the planar cell.

(i-band at ~1513 cmÐ1) in the case of the planar cell is surfaces of Anapore membranes [17] and infiltrated much higher with a polarizer angle of 0° (II polariza- into both microporous [18] and macroporous [12] sili- tion) when the electric vector of the incident light coin- con. Such an orientation is expected if one considers cides with the orientation of the transition dipole the alignment of rodlike molecules with respect to the moment for this molecular unit. At the same time, the surface of the pore, in which case the alignment is pla- intensity of the j-bands perpendicular to the core vibra- nar. The planar alignment is typically observed for rod- tions (e.g., CÐH out-of-plane deformation at ~828, CÐ like molecules on various untreated surfaces, including C aromatic out-of-plane deformation at ~765, and CH2 crystalline silicon [19]. Only a special surface treat- rocking vibrations at ~722 cmÐ1) is smaller. The inten- ment or coating of the substrate surface by a surfactant sity ratio found for these bands in the case of a planar cell at a polarizer angle of 0° is R = A /A = 4.3. ij 1513 765 Substrate plane For the homeotropic LC cell, the intensity of the per- (a) (b) pendicular bands increases, at least, in comparison with the intensity of the parallel bands, owing to the differ- n ence in oscillator strength. In this case, the intensity ratio Rij = A1513/A765 = 1.1. For the SCE8 infiltrated into the porous matrix, this ratio of Rij = A1513/A765 = 0.4 is even smaller than that obtained for the homeotropic LC cell and indicates that the alignment of a ferroelectric liquid crystal in a confined porous silicon matrix is def- initely homeotropic with respect to the substrate’s Fig. 4. Side view of the orientation of (a) ferroelectric and plane (Fig. 4a). This result coincides with the align- (b) discotic liquid crystals in the macroporous silicon ment obtained for nematic LCs deposited on untreated matrix.

PHYSICS OF THE SOLID STATE Vol. 44 No. 6 2002 ORIENTATION OF DISCOTIC AND FERROELECTRIC LIQUID CRYSTALS 1201

0.025 (a) (b) 0.025 0.04 0.020 0.020 0.03 0.015 0.015 0.02 Absorbance 0.010 0.010

0.01 0.005 0.005

0 0 0 1540 1530 1520 1510 1500 1490 820 800 780 760 740 720 700 680 Wavenumber, cm–1

Fig. 5. FTIR spectra of the SCE8 ferroelectric liquid crystal infiltrated into porous silicon (short dotted line) and contained in ZnSe cells with planar (heavy solid line) and homeotropic (thin solid line) alignments. (Note the different scales for the heavy line in (b) and the fact that the absorption of the FLC cells decreases by a factor of 20 for both cells.) may give a homeotropic alignment of LCs formed from With this method, the alignments of two kinds of rodlike molecules. liquid crystals infiltrated into a macroporous silicon In the course of these investigations, we found matrix have been determined. The channel walls of strong intensity enhancement of the low-frequency macroporous silicon affect the orientation of LC mole- vibrational bands in the region 600Ð900 cmÐ1 for both cules such that the column axis of the discotic LC is types of liquid crystals in ma-PS. This enhancement perpendicular to the walls. The long molecular axis of becomes noticeable when comparing the intensity the rodlike molecules of the ferroelectric LC is aligned ratios for the parallel and perpendicular bands. As along the channel walls. A strong intensity enhance- already mentioned, this intensity ratio is much smaller ment of the low-frequency vibrational bands was than that obtained for both LC cells with homeotropic detected for both kinds of LCs infiltrated into the alignment. Moreover, this enhancement is observed for porous silicon matrix both parallel and perpendicular vibrational bands shown in the low-frequency region. A similar effect has ACKNOWLEDGMENTS been observed by Alieva et al. [20] for the species in the microcavity of a 1D photonic structure. It is worth not- The financial support of Enterprise Ireland through ing that the enhancement takes place as soon as the the International Collaboration Program IC/2001/042, light wavelength becomes close to the period of the the Russian State Program “Nanostructures in Physics,” artificial lattice (12 µm). Although the frequency range and the St. Petersburg Science Center Program “Low is, in our case, outside the main photonic band gap, Dimensional Structures” are gratefully acknowledged. which is expected in the lower frequency range, this observation seems to be related to the properties of REFERENCES holey wave guides [21]. 1. T. Bellini, N. A. Clark, C. D. Muzny, et al., Phys. Rev. Thus, we conclude that traditional IR methods can- Lett. 69 (5), 788 (1992). not establish the alignment of liquid crystals infiltrated 2. N. A. Clark, T. Bellini, R. M. Malzbender, et al., Phys. into a macroporous silicon matrix. We propose a new Rev. Lett. 71 (21), 3505 (1993). technique for determining the alignment which is based 3. G. S. Iannacchione, G. P. Crawford, S. Zumer, et al., on FTIR investigation of LCs in the mesophase and Phys. Rev. Lett. 71 (16), 2595 (1993). implies a comparison of the intensity of different vibra- 4. H. Xu, J. K. Vij, A. Rappaport, and N. Clark, Phys. Rev. tional bands, namely, those with the transition dipole Lett. 79 (2), 249 (1997). moment parallel and perpendicular to the core. This 5. P. Ziherl, A. Sarlah, and S. Zœ umer, Phys. Rev. E 58 (1), method is of particular interest for discotic liquid crys- 602 (1998). tals, since it allows the alignment to be determined 6. P. Ziherl and S. Zœ umer, Phys. Rev. Lett. 78 (4), 682 without heating a sample to the temperature of the iso- (1997). tropic phase or using oblique transmission IR spectros- 7. T. S. Perova, J. K. Vij, and A. Kocot, Adv. Chem. Phys. copy. 113, 341 (2000).

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