Annihilation of Positrons in Hydrogen-Saturated Titanium K

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 1Ð5. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 3Ð7. Original Russian Text Copyright © 2003 by Aref’ev, Boev, Imas, Lider, Surkov, Chernov.

METALS AND SUPERCONDUCTORS

Annihilation of Positrons in Hydrogen-Saturated Titanium K. P. Aref’ev*, O. V. Boev*, O. N. Imas*, A. M. Lider*, A. S. Surkov**, and I. P. Chernov* * Tomsk Polytechnical University, Tomsk, 634034 Russia ** Fraunhofer Institute of Nondestructive Control Methods, Saarbrücken, D-66123 Germany e-mail: [email protected] Received May 4, 2002

Abstract—The effect of atomic hydrogen on the electronic structure of α-titanium samples is studied using the electronÐpositron annihilation methods. It is shown that different states of hydrogen atoms are manifested in different ways in the positron lifetime distribution spectrum. The results of theoretical calculations of the ﬁrst component of the positron lifetime are in accord with the obtained experimental data. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION genated titanium and the experimental data. A unique instrument for solving this problem is the electronÐ Metals are the most important structural materials. positron annihilation (EPA) method, which directly However, hydrogen corrosion of metals can become a provides information on the electronic structure of hazard. In this respect, commercial operation of equip- defects in the material under investigation. In this work, ment used in the oil-and-gas, chemical, and nuclear- the EPA method was used in the diagnostics of the state power industries, where hydrogen and hydrogenous of titanium articles and structures in contact with media constitute a considerable part of the working hydrogenous media at the stage preceding brittle frac- atmosphere, can be highly dangerous. The change in ture. We measured the positron lifetime and Doppler the physical and mechanical properties of metals and broadening of the annihilation γ line (DBAL). The alloys under the action of hydrogen is a serious prob- DBAL parameters, the mean positron lifetime, and its lem. The materials used usually have to combine resis- different components associated with annihilation of tance to high stresses with an admissible high-temper- positrons in the region of positron-sensitive defects ature strain. However, the effect of hydrogen on their provide quantitative and qualitative information on the strength parameters is determined to a considerable type and concentration of such defects. extent by the chemical composition of the material. For Positron annihilation in metal hydrides was studied example, the limiting admissible concentration of earlier in [3Ð5] by analyzing the angular distribution of hydrogen is 0.5 at. % for commercially pure titanium annihilation photons (ADAP) and the positron lifetime. (BT1-0) and 1.0 at. % for the alloy TiV13Cr11Al13 [1]. The lifetime of quasi-free positrons was calculated the- This study is devoted to the effect of atomic hydro- oretically only for “complete” hydrides (TiH2) [6]. The gen on the mechanical properties of titanium. Upon the effect of hydrogen on the dynamics of formation of absorption of hydrogen under normal pressure, the defects (craters and cracks) in titanium BT1-0 was crystal lattice of α titanium expands and the ratio c/a of investigated visually in [2] with the help of scanning the hcp-lattice parameters decreases. Hydrogen disso- microscopy and by measuring the mean positron life- lution in a metal is characterized by a nonuniform dis- time. In this work, we investigated samples of commer- tribution of this gas from the surface to the bulk. It is cially pure titanium hydrogenated to the composition this nonuniform distribution that explains the occur- TiH0.01 and calculated the electronic structure, the rence of different extents of material degradation at the positron spectrum, and positron characteristics of α-Ti α surface and in the bulk. For example, after electrolytic and -TiH0.125, which were then compared with the saturation of titanium samples with hydrogen for experimental data obtained by the EPA method. 360 min, “traces of damage” can be ﬁxed on the surface in the form of an elevated concentration of dislocations [2]. However, we did not observe destruction of the 2. MATERIALS AND EXPERIMENTAL entire titanium sample in the atomic-hydrogen concen- TECHNIQUE tration range up to 11 × 10Ð5 at. %. We endeavored to The electronic band structures of pure α-Ti and α establish the mechanisms of variation of the properties -TiH0.125 were calculated using the self-consistent of titanium under the action of hydrogen by comparing method of linearized mufﬁn-tin orbitals in the atomic- the calculated electronic structures of ideal and hydro- sphere approximation (LMTO-ASA) with the Ceper-

2 AREF’EV et al.

Table 1. Parameters of the electronic structure of ideal α-Ti: abilities of positron location in various atomic spheres Γ Fermi energy EF relative to zero crystal energy, width EF Ð 1 (including the sphere occupied by a defect such as of the occupied part of the conduction band, total energy Etot, hydrogen), as well as the annihilation rates and the density of states N(EF) at the Fermi level, and hcp lattice positron lifetime. The method of calculating positron parameters a and c states is described in detail in [10, 11]. Simple Large Simple unit Annihilation parameters of positrons were mea- Parameter unit cell unit cell cell, APW sured on paired BT1-0 titanium samples (with contents (2 atoms/cell) (8 atoms/cell) method [8]* of Fe less than 0.18 at. %, Si less than 0.10 at. %, C less a, nm 0.2951 0.5902 0.2951 than 0.07 at. %, O less than 0.12 at. %, and N less than 0.04 at. %). A sample of BT1-0 was subjected to pre- c, nm 0.4684 0.4684 0.4684 liminary annealing at 750°C in vacuum, followed by ° EF, Ry 0.630 0.631 0.5953 slow cooling to 20 C. Paired samples were saturated Γ electrochemically with hydrogen in monomolar elec- EF Ð 1, Ry 0.475 0.474 0.466 trolyte LiOH (the electrolyte was thermostatically con- Etot, Ry Ð15.107 Ð15.101 trolled at 20°C) for 20, 60, 120, and 360 min. Immedi- N(E ), states/Ry 30.37 24.25 16.12 ately after the saturation of a sample with hydrogen, the F lifetime and Doppler spectra were measured simulta- * The augmented plane wave method. neously at T = 25°C. An Na22 isotope having an activity of the order of 106 Bq was placed between two identical τ parts of the sample under investigation and used as a Table 2. Theoretically calculated positron lifetime and the 22 positron probability distribution w over atomic spheres s, as source of positrons. The radioactive source was Na Cl well as the corresponding electron charge Q in atomic salt vapor-deposited on a 20-µm-thick aluminum foil. spheres and the probability ω of positron annihilation with The instant of position creation was detected by nuclear conduction electrons γ radiation with an energy of 1.28 MeV emitted almost simultaneously with the positron, while the instant of α-Ti α-TiH γ τ τ 0.125 annihilation was determined from the annihilation = 158.1 ps = 154.5 ps radiation with energy 0.511 MeV; the time resolution of s s s s the setup used was 240 ps. The positron lifetime distri- Ti Ti H E bution was processed using the standard Resolution w, % 100 32.25 4.90 62.85 program [12]. The DBAL was measured synchronously ω with the positron lifetime by using a Ge detector with a , % 100 42.19 5.35 52.46 resolution of 1.2 keV operating on the 0.5-MeV line. Q, el./atom 22 20.47 2.48 1.48 The DBAL spectrum was processed with the help of the LIFESPECFIT program [13, 14]. leyÐAlder exchange-correlation potential [7]. The crystal lattice was simulated by repetitive hexagonal large 3. DISCUSSION unit cells with eight titanium atoms (the lattice parameters for titanium are a = 0.2951 nm and c = 0.4684 nm In the low-hydrogen-concentration simulation of [8]). A hydrogen atom was placed in an octahedral pore the unit cell of the titanium crystal, an hcp unit cell with a doubled parameter a was used for calculating the with coordinates (1/4; 3 /12; c/4) in the units of the electronic structure. In order to unify the approach, the lattice parameter a. The remaining seven octahedral electronic structure of perfect α-Ti was calculated pores were filled with additional empty spheres (E) using the same algorithm. The correctness of this with zero charge density to include the crystal-potential approach is confirmed by the closeness of the results anisotropy in the model. The self-consistence proce- obtained with different schemes [calculations with a dure was carried out over 90 k points in the irreducible simple and a large unit cell (two and eight atoms per part of the Brillouin zone for an hcp unit cell and termi- unit cell, respectively)] in our work and in the literature nated when the change in the energy eigenvalues did [8] (Table 1). The electron energy spectrum and the not exceed 0.003 Ry and when the change in pressure lower positron band for TiH0.125 are shown in Fig. 1. calculated from the Pettifor formula [9] on each itera- The positron band has a parabolic form corresponding tion did not exceed 1 kbar. The quasi-free states of a to the quasi-free positron state [10]. Table 2 lists the positron in the crystal were described under the charge distribution, the positron probability distribution assumption that the effect of the positron on the elec- over atomic spheres, and the positron annihilation tron system is negligibly small; positron states were probability in the atomic spheres, as well as the calculated on the basis of the electron density, which positron lifetime, which will be henceforth associated was self-consistent in the absence of a positron. The with the first, short-lived time-distribution component τ α α positron potential and the positron wave function ( 1 in Table 3) for -Ti and -TiH0.125. For the given obtained in this way were used for calculating the prob- choice of the radii of atomic spheres (the radii of the Ti

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

ANNIHILATION OF POSITRONS IN HYDROGEN-SATURATED TITANIUM 3 and H atomic spheres and empty spheres E are identical), the calculations showed that valence electrons of 0.2 titanium are transferred in large numbers to empty interstitial spheres E and to the hydrogen sphere. A positron in a unit cell is distributed over the interstitial 0.1 region both in pure α-Ti and in hydrogenated Ti. It can be seen from Table 2 that, in the case of interstitial EF hydrogen, the positron location probability near hydro- 0 gen is half the corresponding value in the absence of hydrogen, which can be explained by the expulsion of the positron by the Coulomb field of the hydrogen nucleus to the region of lowest electron density. Calcu- –0.1 lations of the positron lifetime showed an insignificant decrease in the value of τ upon the introduction of hydrogen into the titanium structure. This fact can be , Ry –0.2 interpreted as follows: (i) in our calculations, we disre- E garded the lattice expansion due to hydrogenation of the titanium matrix, which occurs in the given case and –0.3 must cause a certain decrease in the electron density and a corresponding increase in the positron lifetime, and (ii) the electron concentration in the impurity –0.4 defect (hydrogen) and its electron density (one electron) are small against the background of the electrons in the titanium matrix, which is confirmed by the insig- –0.5 nificant variation of the first short-lived component (150Ð154 ps, Table 3). We also calculated the contribu- MGKHALM tions from the core electrons and valence electrons α α (90.32% in -Ti and 90.11% in -TiH0.125) to the EPA. Fig. 1. Electronic energy spectrum (dotted curves) and the α In spite of an insignificant discrepancy in the theoreti- lower positron band (solid curve) of -TiH0.125. cal, as well as experimental, data, our results confirm that hydrogen is a positron-sensitive defect. positron source and the samples). Since the third-com- Table 3 gives the parameters of experimentally mea- τ sured positron lifetime components. The first two rows ponent intensity is very low, a change in the value of 3 in the table correspond to samples cut from different even by 30% can be regarded as insignificant; we will regions of the same BT1-0 plate. Let us discuss the not return to the discussion of τ in our subsequent anal- τ 3 obtained results. The first component 1, associated ysis. More significant changes are observed in the val- with the quasi-free positron annihilation, displays sta- ues of τ (due to positron annihilation in the source and ble values in different samples and is in satisfactory 2 agreement with the calculated value of τ (Table 2). The in the region of structural vacancies in the sample under τ investigation) and, accordingly, in the mean positron changes in the long-lived component 3, associated with positron annihilation in air, may be due to changes lifetime τ. These changes are obviously due to differ- in the experimental geometry (different positions of the ences in the concentration and structure of the initial

Table 3. Experimental values of the three-component decomposition of the positron lifetime distribution and the parameters S and W of Doppler broadening of the annihilation γ line in α-titanium samples Mean DBAL parameters, Positron lifetime components, ps Component intensity, % Treatment lifetime, ps arb. units τ ± τ ± τ ± ± ± τ ± ± 1 1 2 10 3 I1 0.6 I2 0.6 I3 0.03 W 0.0003 S 0.0003 Without 150 390 1326 96.1 3.7 0.2 163 0.0757 0.4322 treatment 150 369 1419 94.3 5.3 0.3 167 0.0759 0.4332 Hydrogenation 152 356 1376 94.4 5.4 0.2 167 0.0748 0.4327 for 20 min 60 min 153 362 1344 94.0 5.7 0.3 170 0.0752 0.4328 120 min 154 321 1015 91.5 8.1 0.3 174 0.0766 0.4337 360 min 154 331 1298 91.7 8.0 0.3 174 0.0746 0.4347

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 4 AREF’EV et al. ing to annihilation of positrons with conduction elec- 1 1.5 trons and a Gaussian corresponding to the contribution 2 of core electrons (in analogy with the ADAP curves). This approach was used by us to determine the param- 3 eters S = SA/S0 and W = SB/S0, where S0 is the total area 1.0 under the curve, SA is the area under the parabolic component, and SB is the area under the wings of the Gaus- SA sian. It can be seen from Fig. 2 that hydrogenation of counts per channel 5 0.5 titanium does not lead to a considerable change in the shape of the DBAL line, which is in accord with the , 10

N S S general form of variation of the positron lifetime distri- B B bution. However, the parameters of the DBAL spectrum (S and W) increase upon the introduction of hydro- 508.16 511.00 513.84 E, keV gen into titanium samples (hydrogenation for 120 and 360 min); the integrated value of S in this case corre- γ 0 Fig. 2. Doppler broadening of the annihilation line for (1) × 6 × 6 × pure α-Ti and (2, 3) titanium hydrogenated for 120 and sponds to 5.6052 10 , 4.8682 10 , and 4.8358 360 min, respectively. 106 pulses for samples without and with hydrogenation for 120 and 360 min, respectively. This increase can be due to the following reasons (as in the case of the vacancies and indicate a nonuniform distribution of observed variations in the positron lifetime distribu- such defects in the BT1-0 plate. tion). The increase in the parameter S is associated with The saturation times 120 and 360 min of titanium the contribution to positron annihilation from electrons with hydrogen (during which the maximum hydrogen of hydrogen (both dissolved in the titanium matrix and concentration is attained) correspond to the chemical localized in the region of vacancies). The increase in the parameter W is associated with a distortion of the composition TiH0.01 in the surface layer and in a layer ~100 µm wide, which are in the sensitivity region of the positron wave function localized in the region of a EPA method. In this case, we also singled out three vacancy containing hydrogen atoms. Insignificant changes in these parameters indicate the absence of positron lifetime components which differed from the α initial components and were associated with the pres- defect clusters and the stability of the phase of tita- ence of hydrogen both in the matrix and in the region of nium in the bulk of the hydrogenated sample for a con- titanium vacancies. The small increase in the first com- centration of 11 × 10Ð5 at. % H. ponent τ1, along with the decrease in the corresponding values of I1, shows that no considerable changes in the 4. CONCLUSIONS free-electron density are observed during hydrogenation (nor in calculations). In hydrogenated samples, a Thus, the theoretical and experimental investiga- τ decrease in the second component 2 and, hence, an tions have revealed the sensitivity of the EPA method to increase in its intensity occur, as compared to the initial the presence and state of hydrogen atoms in titanium. titanium samples. This result can be explained only by The presence of hydrogen dissolved in the titanium an increase in the contribution from positrons annihilat- matrix is manifested in the first, short-lived, positron ing in the region of structural vacancies filled with lifetime component and in the parameter S of the hydrogen atoms. In this case, the electron density of a DBAL spectrum. Hydrogen localized in titanium vacancy increases, which must reduce the lifetime of vacancies makes a contribution to the second, long- positrons localized in such defects. The increase in the lived component of the positron lifetime distribution τ 2 component intensity shows that the positron capture and is manifested indirectly in the parameter W of the by these defects becomes more efficient when there are DBAL spectrum. hydrogen atoms present in them (obviously, due to an increase in their size), which is also manifested in a considerable increase in the mean positron lifetime. REFERENCES This point of view is logically confirmed by the results 1. N. D. Tomashov and R. M. Al’tovskiœ, Corrosion and obtained on the titanium samples with a lower hydro- Protection of Titanium (Moscow, 1963). gen concentration: the samples saturated for 60 and 2. I. P. Chernov, A. M. Lider, Yu. P. Cherdantsev, et al., Fiz. 20 min exhibit a smooth decrease in the mean lifetime Mezomekh. 3 (6), 97 (2000). and in the second-component intensity down to the ini- 3. V. I. Savin, R. A. Andrievskiœ, V. V. Gorbachev, and tial values for nonhydrogenated samples. A. D. Tsiganov, Fiz. Tverd. Tela (Leningrad) 14 (11), Figure 2 shows the DBAL spectrum for the initial 3320 (1972) [Sov. Phys. Solid State 14, 2815 (1973)]. and hydrogenated titanium samples. The DBAL spec- 4. A. D. Tsiganov, R. A. Andrievskiœ, V. V. Gorbachev, and trum for metals and alloys is usually approximated by a V. I. Savin, Izv. Akad. Nauk SSSR, Neorg. Mater. 10 (6), superposition of the parabolic component correspond- 1030 (1974).

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5. A. Gainotti, C. Ghezzi, M. Manfredi, and L. Zecchina, 11. O. V. Boev, M. J. Puska, and R. M. Nieminen, Phys. Rev. Nuovo Cimento B 56 (1), 47 (1968). B 36 (15), 7786 (1987). 6. S. E. Kul’kova, O. N. Muryzhnikova, and K. A. Beketov, 12. P. Kirkegaard, M. Eldrup, O. E. Mogensen, and Int. J. Hydrogen Energy 21 (11/12), 1041 (1996). N. J. Pedersen, Comput. Phys. Commun. 23, 307 7. D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45 (7), (1981). 566 (1980). 13. P. Kirkegaard, N. J. Pedersen, and M. Eldrup, Comput. Phys. Commun. 7, 401 (1974). 8. D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids (Plenum, New York, 14. R. Unger, POSIT: Programm zur Erfassung und Auswer- 1986). tung von Positronen-annihilationspektren. Benutzerhand- buch (Halle, Saale, 1994). 9. D. J. Pettifor, Commun. Phys. 1 (5), 151 (1976). 10. E. B. Boronski and R. M. Nieminen, Phys. Rev. B 34, 3820 (1986). Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 104Ð108. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 101Ð104. Original Russian Text Copyright © 2003 by Tereshina, Beskorovaœnaya, Pankratov, Zubenko, Telegina, Verbetsky, Salamova. MAGNETISM AND FERROELECTRICITY

Nitrogen-Containing Compounds RFe11TiNx (R = Gd or Lu) I. S. Tereshina1, 2, G. A. Beskorovaœnaya2, N. Yu. Pankratov2, 3, V. V. Zubenko2, I. V. Telegina2, V. N. Verbetsky2, and A. A. Salamova2 1 Baœkov Institute of Metallurgy and Materials Technology, Russian Academy of Sciences, Leninskiœ pr. 49, Moscow, 117119 Russia 2 Moscow State University, Vorob’evy gory, Moscow, 119899 Russia 3 International Laboratory of Strong Magnetic Fields and Low Temperatures, Wroclaw, 53-421 Poland Received January 9, 2002

Abstract—The structure and magnetic properties of RFe11TiN compounds (R = Gd or Lu) containing nitrogen are investigated. Magnetic measurements are performed on a magnetometer in magnetic ﬁelds up to 100 kOe in the temperature range from 4.2 to 750 K with the use of RFe11TiN single crystals, RFe11TiN powders placed in a ceramic cell, and samples oriented in an external magnetic ﬁeld. It is found that the nitridation leads to an increase in the Curie temperature and the saturation magnetization. The samples studied are uniaxial over the entire temperature range of magnetic ordering. The magnetic anisotropy decreases upon nitridation. It is demonstrated that, within the local anisotropy model, the decrease in the magnetic anisotropy constant K1 can be explained by the redistribution of the electron density in the vicinity of the crystallographic positions occupied by iron atoms. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION amounts leads to the formation of a compound with a stable structure of the formula RFe11T, where T is a sta- Nitrogen-containing iron compounds with rare- bilizing element (in our case, Ti). In this structure, iron earth metals (R) of the general formula RFe11TiNx have and titanium atoms occupy the 8f, 8i, and 8j positions, a ThMn12-type structure. Owing to their properties, whereas rare-earth metal atoms are located in the 2a these compounds are promising for use as permanent positions. Nitrogen atoms can be incorporated into the magnets [1]. The crystal structure of these materials 2b positions [3]. As a result, the nitrogen environment involves two sublattices, namely, a rare-earth metal involves two rare-earth metal atoms in the adjacent sublattice and a 3d transition metal sublattice. The mag- positions and four iron atoms in the 8j positions. Such netic properties of the iron sublattice, as a rule, are stud- an arrangement of nitrogen atoms should bring about ied using compounds with nonmagnetic rare-earth met- an increase in the unit cell volume, ordering of valence als, such as yttrium and lutecium. Earlier, Yang et al. [2] bonds between atoms of the elements entering into the performed a detailed investigation into the magnetic alloy composition, and a substantial change in the mag- properties of YFe11TiNx (x = 0.1) compounds. The pur- netic properties. In the present work, this effect of nitro- pose of the present work was to elucidate how nitrogen gen on the magnetic properties was investigated over affects the structure and magnetic properties of wide ranges of temperatures (from 4.2 K to TC) and LuFe11Ti and GdFe11Ti compounds. The study of gad- magnetic ﬁelds. olinium-based compounds is of special interest, because a Gd3+ ion has no orbital angular momentum; hence, the effect of a crystal ﬁeld on the rare-earth sub- 2. SAMPLE PREPARATION lattice can be ignored in investigating the magnetocrys- AND EXPERIMENTAL TECHNIQUE talline anisotropy of this material. Moreover, among the The experiments were performed with RFe Ti RFe Ti compounds, the GdFe Ti compound has the 11 11 11 alloys prepared by rf induction melting in a “Donets-1” highest temperature of transition to a magnetically apparatus in a special-purity argon atmosphere at the ordered state (the Curie temperature TC), whereas the pressure P = 1.1 atm. High-purity iron (99.9%), gado- LuFe11Ti compound is characterized by the lowest linium (99.5%), lutecium (99.5%), and titanium Curie temperature. (99.9%) served as the initial components. The weights of the metals in the batch did not exceed the calculated Compounds with a ThMn12-type structure have a body-centered tetragonal lattice (space group I4/mmm) values, because the saturation vapor pressure was rela- ° with a unit cell containing two formula units. As is tively small at a synthesis temperature of ~1600 C. known, no structures of the ThMn12 type are observed Nitrides of the RFe11Ti polycrystalline compounds in binary iron compounds of the general formula RFe12. were synthesized at temperatures ranging from 500 to However, the addition of a third component in small 600°C and pressures up to 50 atm. The samples were

NITROGEN-CONTAINING COMPOUNDS 105

Structure parameters and magnetic properties of RFe11TiNx compounds (R = Gd, Lu; x = 0.1) at T = 4.2 K K × 10Ð7, K × 10Ð7, Compound a, Å c, Å V, Å3 ∆V/V, % T , K σ , emu/g 1 2 C S erg/cm3 erg/cm3

LuFe11Ti 8.53 4.74 345 Ð 490 138 1.92 Ð LuFe11TiN 8.62 4.77 354 2.8 738 160 1.77 Ð GdFe11Ti 8.54 4.81 351 Ð 602 99 1.25 0.56 GdFe11TiN 8.67 4.80 361 2.8 >750 110 0.41 0.40 preliminarily ground, and the size of particles in pow- 3. RESULTS AND DISCUSSION dered samples did not exceed 5 µm. This made it possible to prepare samples with a sufficiently homogeneous It is found that the nitridation leads to a substantial nitrogen content. The amount of absorbed nitrogen was increase in the Curie temperature. For example, the calculated from the van der Waals equation p(V Ð nb) = Curie temperature increases from TC = 490 K for the nRT, where b = 38.6 cm3/mol is the van der Waals coef- LuFe11Ti initial compound to TC = 738 K for the ficient, p is the pressure (atm) of nitrogen in the system, LuFe11TiN nitride. We failed to measure the Curie tem- R = 82 × 106 cm3 atm./(mol K) is the universal gas con- perature TC for gadolinium nitrides precisely because of stant, V is the nitrogen volume (cm3), T is the tempe- the pronounced irreversible decrease in the magnetiza- rature (K), and n is the amount (mol) of nitrogen. The tion in the vicinity of 750 K. Upon cooling, the magne- nitrogen content in the nitrides prepared from the initial tization did not regain its original value completely. intermetallic compounds was equal to one nitrogen This suggests that, at the above temperature, either atom per formula unit. interstitial atoms leave the crystal lattice of the GdFe Ti compound or the samples undergo decompo- The structure investigation and quality control of the 11 initial and nitrided samples were performed using x-ray sition. The Curie temperature of the GdFe11Ti initial powder diffraction analysis on a DRON-2 diffractome- compound was determined to be TC = 602 K. As was ter (CuKα radiation) with powder samples. It was dem- noted above, this Curie temperature is highest for all the RFe11Ti compounds but is appreciably lower than the onstrated that RFe11TiN nitrides (R = Gd or Lu) retain a Curie point TC for pure iron (TC ~ 1050 K). Such a con- ThMn12-type tetragonal structure. The unit cell volume in the nitrided samples was approximately 2.8% larger siderable decrease in the temperature TC as compared to than that in the initial samples. The samples had a sin- pure iron can be associated with the antiferromagnetic gle-phase composition, and the x-ray powder diffrac- exchange interaction between individual atoms in the tion patterns contain no reflections of the α-Fe phase. iron sublattice in these compounds. It is known [4] that, in the case when the distance between Fe atoms is less The lattice parameters determined for the GdFe11TiN compound are in good agreement with the data avail- than a critical distance of 2.42 Å, the exchange interac- able in the literature. The presented data for the tion in FeÐFe pairs has negative sign. In RFe11Ti compounds, the distance between Fe atoms at the 8j posi- LuFe11TiN compound were obtained for the first time in this work (see table). tions is less than the critical distance. The incorporation of nitrogen brings about an expansion of the lattice to The magnetic properties were investigated using such an extent that the distance between Fe atoms single crystals of the initial alloys, nitride powders becomes larger than the critical value, the exchange placed in ceramic cells, and RFe11TiN powder particles interaction in FeÐFe pairs is enhanced, and, conse- oriented in a static magnetic field (~7 kOe). The orien- quently, the temperature T increases. tation of powder particles was fixed with the use of C epoxy resin. Samples were prepared in the form of We analyzed the temperature and field dependences spheres, which provided correct inclusion of the of the magnetization for RFe11Ti compounds and their demagnetizing factor for these samples. The Curie tem- nitrides. Figure 1 displays the temperature dependences peratures TC of the compounds under investigation of the magnetization for the GdFe11Ti and GdFe11TiN were determined using thermomagnetic analysis. The compounds. For comparison, this figure shows the temperature dependences of the magnetization were magnetization curve for the GdFe11TiH sample. It can measured on a pendulum magnetometer in magnetic be seen that the nitridation leads to a drastic increase in fields up to 12 kOe in the temperature range 78–750 K. the magnetization (see table). It can be seen from Fig. 1 The field dependences of the magnetization were mea- that, upon hydrogenation, the magnetization also sured using a capacitance magnetometer at the Interna- increases but to a smaller extent. As is known, the unit tional Laboratory of Strong Magnetic Fields and Low cell volume increases, on average, by approximately Temperatures (Wroclaw, Poland) in the temperature 1% upon hydrogenation and by 3% upon nitridation. range from 4.2 to 300 K in magnetic fields up to Thus, the magnetization changes in the following man- 100 kOe. ner: the larger the unit cell volume, the higher the mag-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 106 TERESHINA et al.

150 netic moments localized on iron atoms occupying the 8i, 8j, and 8f positions are equal to 1.99, 1.76, and µ 120 1.51 B, respectively. 3 According to Qi et al. [6], the nitridation is attended 90 2 1 by an increase in the magnetic moments of the atoms located at the 8i, 8j, and 8f positions owing to a narrow- , emu/g 60 σ ing of the 3d band upon volume expansion, which, in turn, leads to a total increase in the magnetization 30 observed in the experiments. Therefore, the lattice expansion due to the incorporation of nitrogen atoms 0 50 150 250 350 into octahedral interstices is the main factor responsible T, K for the increase in the magnetization. Figure 2 depicts the magnetization curves measured Fig. 1. Temperature dependences of the saturation magneti- for the GdFe11Ti initial compound along different crys- zation for (1) GdFe11Ti, (2) GdFe11TiH, and (3) GdFe11TiN tallographic directions in magnetic fields up to 100 kOe compounds. at temperatures of 4.2, 40, and 80 K. It can be seen from Fig. 2 that the [001] crystallographic direction coincides with the easy magnetization axis, whereas the [110] direction is the hard magnetization axis. The 100 effective magnetic anisotropy field is determined to be 80 [001] [001] [001] 60 [110] [110] [110] H0 = 90 kOe at T = 4.2 K, rapidly decreases with an 40 increase in the temperature, and reaches 60 kOe even at , emu/g 4.2 K 40 K 80 K σ 20 T = 80 K. Similar curves for an LuFe11Ti single crystal were obtained in our earlier work [7]. Unlike the R Fe 0 40 80 0 40 80 040 80 2 17 , kOe compounds [8], the above compounds do not exhibit an H anisotropy of saturation magnetization. The field dependences of the magnetization mea- Fig. 2. Magnetization curves for the GdFe11Ti single crystal along the [100] and [110] crystallographic directions at sured for the LuFe11Ti and GdFe11Ti compounds and temperatures of 4.2, 40, and 80 K. their nitrides along the easy and hard magnetization axes at T = 4.2 K are plotted in Fig. 3. As can be seen from this figure, the nitridation brings about an increase in the saturation magnetization and a decrease in the effective anisotropy field; however, all the crystals 160 3 4 under investigation remain uniaxial. It is worth noting 120 7 that, for yttrium compounds containing molybdenum 1 as a stabilizing element, the nitridation leads to a 2 5 80 8 change in the sign of the magnetic anisotropy constant , emu/g

σ 6 K1; in this case, the YFe11MoN compound is character- 40 (a) (b) ized by an in-plane anisotropy [9]. At the same time, yttrium compounds with a titanium stabilizing element 0 20 40 60 80 100 0 20406080100 retain axial anisotropy (the magnetic anisotropy con- H, kOe stant K1 for YFe11TiN decreases [2]). These ﬁndings indicate that the stabilizing dopant has a profound effect on the magnetic anisotropy constant K1. In order Fig. 3. Field dependences of the magnetization for (a) (1, 2) to determine the magnetocrystalline anisotropy con- the LuFe11Ti compound and (3, 4) the LuFe11TiN nitride stants in our case, we used the SucsmithÐThompson and (b) (5, 6) the GdFe11Ti compound and (7, 8) the GdFe11TiN nitride along (1, 3, 5, 7) the easy magnetization method [10], which involves a special mathematical axis and (2, 4, 6, 8) the hard magnetization axis at 4.2 K. processing of the magnetization curves measured along the easy and hard magnetization axes. The anisotropy constants for the initial compounds and their nitrides at netization of the studied compounds. The same is also T = 4.2 K are listed in the table. It follows from the table true for lutecium compounds [5]. that, upon nitridation, the magnetic anisotropy constants K1 decrease for both LuFe11TiN and GdFe11TiN Iron atoms in the ThMn12 structure are characterized compounds. by relatively small magnetic moments. For example, One of the known approaches to the interpretation the mean magnetic moment in the LuFe11Ti compound of the magnetocrystalline anisotropy is based on ﬁrst- µ µ µ is equal to 1.77 B (for pure iron, Fe = 2.2 B). The mag- principles band calculations. However, it should be

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 NITROGEN-CONTAINING COMPOUNDS 107 noted that rare-earth compounds are rather complex 8j objects for investigation using standard methods of the band theory. Despite recent advances made in the band Fe H Fe Fe N Fe theory [11], the anisotropy energy can be determined only to within an order of magnitude and exact calcula- c tions present considerable difficulties. For this reason, a simple qualitative treatment of the magnetocrystalline anisotropy in terms of the physically clear point-charge model can be used to advantage as before. According to 4f 4f this model, ions in a crystal are affected by a crystal electrostatic field whose potential V(r) is induced by charges of the nearest neighbor ions. Strong uniaxial H N magnetocrystalline anisotropy of the RFe11Ti initial Fe Fe Fe Fe compounds with nonmagnetic rare-earth elements and gadolinium (the orbital angular momentum of the 4f electron shell of gadolinium ions is equal to zero) can Fig. 4. A schematic diagram illustrating the interaction of be explained by a partial defreezing of the orbital angu- the 4f electron shell of a rare-earth ion and Fe ions located at the 8j positions with interstitial atoms (hydrogen and lar momentum L of iron ions in the anisotropic local nitrogen). crystal field, which significantly differs for crystallo- graphically nonequivalent positions occupied by iron atoms (8i, 8j, and 8f). In this case, a small defrozen orbital angular momentum (the component of the that the electric field gradient is also differently modi- orbital angular momentum L) is aligned with the easy fied at the iron ion sites 8j (Fig. 4). In Fig. 4, the rare- magnetization axis and, in turn, orients the total spin earth (Gd or Lu) ion with a spherical distribution of the angular momentum due to the spinÐorbit interaction electron density of the 4f shell is located at the center. [12]. The nearest environment of the rare-earth ion involves iron atoms located at the 8j positions. These iron atoms, The nitridation brings about a decrease in the mag- most probably, have a small, defrozen orbital angular netocrystalline anisotropy of the LuFe11TiN and momentum and are located in the immediate vicinity of GdFe11TiN compounds. This can be associated either the H and N interstitial atoms. It is known [14] that, in with the decrease in the orbital angular momentum L of metals, nitrogen atoms retain 2p orbitals that are iron ions or with the change in the local crystal field weakly distorted by the crystal lattice, because the p upon incorporation of nitrogen atoms into their nearest electron is tightly bound to the nitrogen anion. On the environment. The magnetic anisotropy constant K1 for other hand, upon hydrogenation, hydrogen atoms the RFe11Ti initial compound can be represented as the reside in electronic states with a low energy to which sum of three partial constants corresponding to the con- electrons can readily transfer from the 3d band. The tributions of the iron ions located at three crystallo- hydrogenation and nitridation bring about an increase graphically nonequivalent positions (8i, 8j, and 8f) [9]; in the magnetic moment at the 8j positions; however, that is, hydrogen and nitrogen have opposite effects on the K ()T = K ++K K . magnetocrystalline anisotropy. The introduction of 1 1i 1 j 1 f hydrogen atoms into the crystal lattice most likely leads Upon introduction of nitrogen atoms into the to an increase in the partial constant K1j and, conse- ThMn12 crystal structure, light interstitial atoms are quently, in the total constant K1 [5]. By contrast, the located in the closest proximity to the 8j positions and, incorporation of nitrogen atoms into octahedral holes hence, strongly affect the iron ions occupying these results in a decrease in the constant K1j and, hence, in positions. This results in the hybridization of the Fe d the total constant K1 for the LuFe11TiN and GdFe11TiN and N p states and the redistribution of the electron den- compounds. sity. As was noted earlier in [13], hydrogen and nitrogen atoms in the ThMn12 structure occupy the 2b positions along the c crystallographic axis. These atoms dif- 4. CONCLUSION ferently modify the electric field gradient at the rare- earth ion site and orient the quadrupole moment (and Thus, the hydrogenation and nitridation of RFe11Ti the related magnetic moment) of the rare-earth ions in compounds (R = Y, Gd, or Lu) have opposite effects on the direction parallel or perpendicular to the c axis, α the magnetocrystalline anisotropy. This can be depending on the Stevens factor J of the rare-earth ion. explained by the fact that the electric field gradients at In our previous work [13], we analyzed in detail the 8j positions occupied by iron ions significantly dif- how the hydrogenation and nitridation affect the mag- fer upon incorporation of hydrogen and nitrogen atoms netocrystalline anisotropy in RFe11Ti compounds in the due to the difference in the distributions of the electron case when R is a magnetic rare-earth ion. It seems likely density around these atoms.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 108 TERESHINA et al.

ACKNOWLEDGMENTS 6. Q. Nian Qi, Y. P. Li, and J. M. D. Coey, J. Phys.: Con- We would like to thank K.P. Skokov for preparing dens. Matter 4 (42), 8209 (1992). the samples used in our investigation, Yu.V. Skourski 7. S. A. Nikitin, I. S. Tereshina, Yu. V. Skourski, et al., Fiz. for his assistance in performing the experiments, and Tverd. Tela (St. Petersburg) 43 (2), 279 (2001) [Phys. S.A. Nikitin for his participation in discussions of the Solid State 43, 290 (2001)]. results and helpful remarks. 8. S. A. Nikitin, I. S. Tereshina, N. Yu. Pankratov, et al., Fiz. Tverd. Tela (St. Petersburg) 43 (9), 1651 (2001) [Phys. This work was supported by the State Program of Solid State 43, 1720 (2001)]. Support for Leading Scientiﬁc Schools of the Russian Federation (project no. 00-15-96695) and the Russian 9. E. Tomey, D. Fruchart, and J. L. Soubeyroux, IEEE Foundation for Basic Research (project nos. 99-02- Trans. Magn. 30 (2), 684 (1994). 17821 and 01-02-06490). 10. W. Sucsmith and J. E. Thompson, Proc. R. Soc. London, Ser. A 225, 362 (1954). 11. S. Buck and M. Fahnle, J. Magn. Magn. Mater. 166 (2), REFERENCES 297 (1997). 1. S. P. Eﬁmenko, Yu. K. Kovneristyœ, and I. M. Milyaev, 12. S. A. Nikitin, I. S. Tereshina, V. N. Verbetsky, and Fiz. Khim. Obrab. Mater., No. 3, 82 (1998). A. A. Salamova, Fiz. Tverd. Tela (St. Petersburg) 40 (2), 2. J. Yang, Sh. Dong, W. Mao, et al., Physica B (Amster- 285 (1998) [Phys. Solid State 40, 258 (1998)]. dam) 205, 341 (1995). 13. S. A. Nikitin, I. S. Tereshina, V. N. Verbetsky, and 3. Y. Yang, X. Zhang, L. Kong, and Q. Pan, Solid State A. A. Salamova, Metally, No. 1, 86 (2001). Commun. 78 (4), 313 (1991). 14. V. K. Grigorovich, Metallic Bonding and Structure of 4. I. B. Goodenough, Magnetism and the Chemical Bond Metals (Nauka, Moscow, 1988). (Interscience, New York, 1963). 5. I. S. Tereshina, S. A. Nikitin, N. Yu. Pankratov, et al., J. Magn. Magn. Mater. 231, 213 (2001). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 109Ð117. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 105Ð112. Original Russian Text Copyright © 2003 by Sukstanskiœ, Primak. MAGNETISM AND FERROELECTRICITY

Domain Structure in an Ultrathin Ferromagnetic Film: Three-Parameter Model A. L. Sukstanskiœ and K. I. Primak Donetsk Physicotechnical Institute, National Academy of Sciences of Ukraine, Donetsk, 83114 Ukraine e-mail: [email protected] Received January 22, 2002

Abstract—A stripe domain structure (DS) was studied in an ultrathin ferromagnetic film under an in-plane magnetic field. The basic structure parameters (the amplitude and period of the DS, the thickness of domain walls) were determined within a unified variational approach, and transitions from the DS state to a homogeneous canted and a homogeneous planar phase were studied. © 2003 MAIK “Nauka/Interperiodica”.

1. Recently, considerable attention has been given to infinitesimal, and the domain size D was taken to be a the properties of ultrathin magnetic films (UTMFs) a single variational parameter. However, such a trial few monolayers thick (≤10Ð7 cm) [1, 2]. This interest is function ignores the fact that the domain wall thickness stimulated by the development of novel UTMF technol- in ultrathin films is comparable to the thickness d of the ogies and experimental devices and, on the other hand, film itself and depends on the film parameters. More- by the great potential applications of UTMFs. over, as shown in [7Ð9], the DS that forms near a tran- Most of the studies of UTMFs are dedicated to their sition from a multidomain phase to a phase with a uni- domain structure (DS) [3Ð6]. The existence of DSs in form distribution of magnetization (the critical DS) is UTMFs was first predicted in [7, 8] by solving a com- sinusoidal in character. This structure cannot be plete set of micromagnetic (LandauÐLifshitz) equa- approximated by wide domains separated by narrow tions for the magnetization vector M and of the magne- domain walls; i.e., the concept of domain walls loses its meaning in this case. Therefore, the results of [11] are tostatics equations for the demagnetization field Hm (with allowance for the boundary conditions at the film inapplicable to the critical DS; neglect of the finite surface). In those works, the so-called critical DS thickness of a domain wall results in an incorrect (which arises in the vicinity of a second-order phase dependence of the domain size D on the film thickness transition to a homogeneous magnetic state with the even in a region far from the phase transition [12]. magnetization vector lying in the film plane) was stud- To describe the DS in an UTMF, a more intricate ied and the corresponding phase diagrams were con- trial function (with two variational parameters m and ∆) structed (the critical DS in thick films was considered was suggested in [12]: in [9]). The amplitude of the critical DS (i.e., the maximum value of the magnetization vector component nor- () 1/2 x Mz x = m M0sn --- m , (1) mal to the film surface) was assumed to be a small ∆ parameter, which made it possible to find an analytical | | solution to the micromagnetics equations. where M0 = M is the saturation magnetization, Mz is In general, analytical solution of these equations is the magnetization vector component along the normal impossible. Therefore, beginning with the pioneering to the film plane (axis z), and m is a parameter of the work of Kittel [10], the DS in magnets has been theo- elliptic function (this parameter is equal to the squared retically studied using the variational approach, within modulus of the elliptic integral [13]). The DS period which the character of the magnetization distribution is described by this function is 2D = 4∆K(m), where K(m) presupposed. A trial function describing the magnetiza- is a complete elliptic integral of the first kind. The ellip- tion distribution in a DS contains unknown parameters tic sine was chosen as a trial function, because the DS (typically, the domain size), whose equilibrium values problem in the absence of a magnetic field has an ana- are found by minimizing the energy of the magnet with lytical elliptic-sine-like solution if the demagnetizing respect to these parameters. field is ignored [14]. With such a trial function, the ∆ A typical example of this approach as applied to an thickness can be varied, which allows one to ade- analysis of the DS in an UTMF is the study made in quately describe both the critical and a finite-amplitude [11], where the simplest trial function corresponding to DS in an UTMF. a uniform magnetization distribution in domains was However, trial function (1) is inapplicable for the used, the domain wall thickness ∆ was assumed to be description of a DS in a film subject to an external in-

110 SUKSTANSKIŒ, PRIMAK

1/2 plane field, since the DS amplitude (m M0) and period placed in an external in-plane field. We write the film 2D are unambiguously related, which is not the case in energy as the presence of an in-plane field. This study is dedi- α β cated to a generalization of the approach advanced in ()∇ 2 2 1 WV= ∫d --- M Ð ---Mz Ð ---MHm Ð My Hy , (3) [12] to the case of the presence of an external in-plane 2 2 2 field. We use a trial function for which the DS amplitude and period are not related but are, rather, indepen- where M is the magnetization vector; α and β are the dent variational parameters. To describe the DS, a trial exchange and anisotropy constants, respectively; Hm is function with three variational parameters is chosen: the demagnetizing field, which can be determined from the magnetostatics equations; Hy is the field in the film  ≥ M ()x = aM sn x m . (2) plane (without loss of generality, we assume that Hy z 0 ∆--- 0); and the anisotropy axis is normal to the film surface (along the Z axis). These independent parameters are the DS amplitude Substituting Eq. (2) into Eq. (3), one arrives at the ∆ aM0, the domain wall thickness , and the elliptic-func- energy of the magnet (per unit length of the domain tion parameter m, which defines the DS period L = 2D = wall) as a function of parameters b, m, and u: 4∆K(m). It is noteworthy that the DS in a uniaxial fer- 2 WM= dσ()bmu,, , romagnet can be assumed to be one-dimensional only 0 in the case of sufficiently thin films, where the magne- α 2bK() m I ()bm, tization distribution over the film thickness (along axis σ()bmu,, = ------1 d2 u2 Z) can be considered uniform. Furthermore, this (4) description ignores the influence of morphological (), βb Em() I2 bm Ð ------1 Ð ------Ð h ------defects in an actual film (uncoated substrate areas, 2mKm() y Km() islands, etc.) and, hence, is inapplicable to the DS in a film that is not additionally processed to upgrade its 2n Ð 1 2n Ð π ∞ quality. For example, in annealed films, structural 2 q 1 Ð expÐ------4π bu u defects are significant only if the film is as thin as a + ------∑ ------, monolayer [6]; therefore, the DS in actual films can be 2() ()()2n Ð 1 2 mK m n = 1 2n Ð 1 1 Ð q adequately described [12]. An analysis of higher 2 ∆ dimensional DSs and DSs closely related to morpho- where b = a , u = D/d = 2K(m) /d, hy = Hy/M0, logical features of films is beyond the scope of this Km() study. cn2()ymdn2()ym I1 = ∫ ------dy, 1 Ð bcn2()ym The objective of this work is to determine the equi- 0 librium parameters a, ∆, and m as functions of the parameters characterizing the film (thickness, exchange Km() 2() and anisotropy constants) and of the external magnetic I2 = ∫ 1 Ð bsn ymdy, field. Special attention is paid to the problem of the 0 transition from the DS to a uniformly magnetized state. This transition can proceed in two ways: either through q = exp[]ÐπK()1 Ð m /Km(), the DS amplitude (i.e., a) tending to zero or through the and E(m) is a complete elliptic integral of the second DS period increasing without limit; i.e., ∆ ∞. In the kind. first case, the critical DS with a smooth (sinusoidal) The equilibrium values of the parameters b, m, and magnetization distribution emerges where, as indicated u are found from the variational equations above, the domain wall concept loses its meaning [7, 8]. The second transition occurs in “thick” films under an ∂σ ∂σ ∂σ ------===------0 . (5) external magnetic field perpendicular to the film plane ∂b ∂m ∂u [15]. However, as far as we know, these two types of transitions from the DS to a uniformly magnetized state As a rule, the domain size D in ultrathin films signifi- have not yet been considered within a unified approach. cantly exceeds the film thickness d. In this case, i.e., as the inequality We shall show that these transitions take place depending on the strength of the external in-plane field and on u ӷ 1 (6) the film parameters. is met, the exponential in the expression for the demag- σ 2. Thus, we consider a ferromagnetic film with netization energy m [the last term in Eq. (4)] can be uniaxial magnetic anisotropy of the easy-axis type expanded into a power series in the small parameter

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 DOMAIN STRUCTURE IN AN ULTRATHIN FERROMAGNETIC FILM 111 u−1 Ӷ 1. Restricting ourselves to the first two expansion where terms, we reduce σ to the form m 2 ∂ d h ()x f x˜ ==------, h˜ ------y , f =------i. (14) π3 π αβ β i ∂ σ 4 b Σ () Σ () ∗ ∗ x m = ------2 1 m +,------2 m (7) mK ()m 2u Thus, the DS of the film is determined by two where dimensionless physical parameters, x˜ and h˜ . We note β ∞ ∞ that these parameters can take on both positive (at ∗ > 2n Ð 1 ()2n Ð 1 β Σ q Σ 2n Ð 1 q 0) and negative (at ∗ < 0) values depending on the sign 1 ==∑ ------, 2 ∑ ------. ()2n Ð 1 2 ()2n Ð 1 2 ˜ n = 1 1 Ð q n = 1 1 Ð q of β∗; x˜ and h ∞ as β∗ 0. The first sum in Eq. (7) can be calculated exactly, 3. First of all, we consider the DS with a small and the energy takes on the form amplitude b Ӷ 1 (studied in [7, 8] within the micromagnetic approach). This DS is referred to as the critical DS α ()(), β () σ(),, 2bK m I1 bm ∗bEm (CDS). When studying the CDS, energy (4) can be bmu = ------Ð ------1 Ð ------d2 u2 2m Km() expanded into a power series in the small parameter b (8) (here, we do not restrict ourselves to the case of u ӷ 1): I ()bm, π4 2 2 b Σ () 2 Ð hy------Ð ------2 m , σ = gmu(), bg+ ()mu, b . (15) Km() muK2()m 1 β β π As shown in [7, 8], the CDS is described by a where ∗ = Ð 4 . smooth sinusoidal function which corresponds to the | From the equation ∂σ/∂u = 0, one finds an expres- limit m 0 [sn(x m) sin(x)] when the DS is sion for u, described by trial function (2). For this reason, we consider the parameters b and m to be of the same order of 3 α 2mK() m I ()bm, smallness and expand the functions g(m, u) and g1(m, u) u = ------1 -. (9) in Eq. (15) in a power series in parameter m: 2 π4Σ d 2 gmu(), = c ()u ()1 + m/8 + c ()u m2, (16) Upon substituting Eq. (9) into Eq. (8), the energy σ 0 1 becomes a function of only two parameters, b and m: απ2 () (), 6 + m hy g1 mu = ------+,------(17) 2 16d2u2 128 σ , d , β , , ()bm = ----- f 1()bm++∗ f 2()bm hy f 3(),bm (10) α where where απ2 β π () Ð hy  c0 u = ------Ð ------+ u1 Ð expÐ--- , π2 Σ2 4d2u2 4 u (), b 2 f 1 bm = Ð------, 2 ()5 (), 2 β 2m Km I1 bm 3π α Ð hy c ()u = ------Ð ------1 2 2 64 b Em() 128d u f ()bm, = Ð------1 Ð ------, (11) 2 2mKm() 3π 1 Ð expÐ------I ()bm, u π u f ()bm, = Ð------2 . + ------1 Ð expÐ--- + ------. 3 Km() 256u 3  Both parameters b and m can take on any value within the interval [0, 1]. Their equilibrium values m∗ From variational equations (5) and Eqs. (15)Ð(17), one finds equations for determining the equilibrium and b∗ are found from the variational equations values b and m: ∂σ ∂σ gmu(), ∂------== 0 , ------∂ 0. (12) b = Ð------(), , (18) m b 2g1 mu Equations (12) can be transformed as ∂ ∂g g1 ------+0.b------= (19) ()m ()m ˜ ()m ∂m ∂m x˜ f 1 ++f 2 h f 3 = 0, (13) Using the fact that b is small near the line of transi- ()b ()b ˜ ()b x˜ f 1 ++f 2 h f 3 = 0, tion to the uniformly magnetized phase, the equilibrium

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 112 SUKSTANSKIŒ, PRIMAK

β – hy Equations (22) and (23) define the relation between 14 the field and the parameters of the magnet for which the DS amplitude in the film vanishes. This relation can be written in the parametric form: 12 π β Ð23h = u Ð ()π + 3u exp Ð--- ≡ f β()u , y u β (24) – hy = fβ(u) π π Ð1/2 10 d ()π  ≡ () d ------= --- 2 u Ð + u exp Ð--- f d u . ------= f ()u α u u α d In fact, Eqs. (24) parametrically define the curve of 8 the phase transition from the state with DS to a uniformly magnetized phase (solid curve in Fig. 1). It is readily shown that condition (6) is equivalent to 6 condition d Ӷ α1/2 in the case of the CDS. As this inequality is met, the dependence of the critical value of β Ð hy on the film thickness d in the phase transition line 4 can be written in an explicit form: π2 2 π2d2 β π d β Ð4h = π Ð ------Ð4hy = Ð ------, (25) 2 y α α which is identical to the result obtained in [7] on the β basis of the micromagnetic approach (with Ð hy in place of β) and the result obtained in [12] within the 0 123456 variational method (the film thickness dependence d ------given by Eq. (25) is shown in Fig. 1). α It is significant that the domain size remains finite in the transition line, i.e., at b∗ = 0. Let us introduce the Fig. 1. Dependence of the critical anisotropy (field) on the film thickness. The solid line is the interface between the dimensionless film thickness d˜ and the dimensionless phase with DS and the homogeneous phase magnetized in ˜ the film plane, and the dashed line is the same interface in domain width D defined as the D/d ӷ 1 approximation. d β1/2 d˜ ==------, D˜ D------*-. (26) ()αβ∗ 1/2 α1/2 parameters m∗ and b∗ are found from Eqs. (16) and (19) as functions of parameter u: Then, if condition (6) is met, the relation between the CDS dimensionless period D˜ and the critical dimen- c ()u cr m∗ ==m∗()u Ð------0 -, (20) ˜ 16c ()u sionless thickness dcr (i.e., the film thickness at which 1 the transition to a homogeneous phase takes place) has () a simple form: ()c0 u c0 u 1 Ð ------()- 1 256c1 u D˜ = ------. (27) b∗ ==b∗()u Ð------. (21) cr ˜ (), () dcr 2g1 umu 4. Now, we consider the second possible type of the Along the transition line itself, we have b∗ = m∗ = 0, DS transition to the uniformly magnetized phase, i.e., from which we find an equation for the value u in this the transition through infinite growth of the structure line, period. This case corresponds to m 1 [we recall that K(m) ∞ and D ~ K(m) as m 1]. In the limiting () c0 u = 0. (22) case m 1, it is convenient to use the elliptic integral K(m) of the first kind rather than the quantity m as the The variational equation ∂σ/∂u = 0 in the transition variational parameter, i.e., to consider the structure line, i.e., at b∗ = m∗ = 0, takes on the form energy σ in Eq. (10) as a function of parameters b and K. ∂c ()u ≈ ------0 -0.= (23) Taking into account that E(m) ~ 1 and q(K) ∂u exp(−π2/2K), (1 Ð m)K ≈ 0 for m 1, as well as the

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 DOMAIN STRUCTURE IN AN ULTRATHIN FERROMAGNETIC FILM 113 Σ ∞ fact that the sum 2 in the case under consideration can K can be met only for x˜ 0. Hence, the tran- be approximated by the expression (see [12]) sition to a homogeneous phase through unlimited growth of the domain size can occur only for x˜ 0. 2K2 Σ ()K ≈ ------()lnKc+ , From Eq. (28), we find an expression for the equilib- 2 π4 rium value K = K∗ as a function of variables b, x˜ , and h˜ : ≈ where c 0.7, we find the functions f1, 2, 3(b, K) in Eq. (10) to be ()1 Ð b 1/2 2x˜ r2 ∂lnI K∗ = ------1 Ð ------1 Ð ------1 2 1/2 ˜ I ()b, 1 ∂ bKc()ln + 11 Ð K ()1 Ð b Ð h 1 lnb f ()bK, ==Ð------, f ()bK, ------, (31) 1 (), 2  1/2 KI1 b 1 2 K ˜ ln()1 Ð b arcsinb Ð h------Ð ------, (), ()1/2 1/2 f 3 bK 41Ð b 2b where r is defined by Eq. (30). It is evident that in the ()1 Ð b 1/2()K + ln()1 Ð b 1/2 + b1/2arcsinb1/2 = Ð------, limit of x˜ 0 and K ∞, the equilibrium ampli- K tude b remains finite and is given by (), (), I1 b 1 = lim I1 bm 2 m → 1 b∗ = 1 Ð h˜ . (32) 1 1 Ð b 1 + b1/2 Thus, as in the case of zero field [12], the DS period = --- 1 Ð.------ln ------b 1/2 ()1/2 becomes infinitely large in the limit of infinitesimal b 1 Ð b film thickness. In this case, in fact, the DS degenerates The equations for the equilibrium values of the into a single domain wall, whose equilibrium amplitude parameters b and K, b∗ is defined by Eq. (32). ∂σ()bK, ∂σ()bK, 5. In Sections 3 and 4, we considered the DS near ------0,==------0, ∂b ∂K the boundaries of the domain of its existence. Now, we will study Eqs. (13) for the DS parameters for arbitrary take on the form values of parameters m and b [condition (6) is consid- ()2 ∂()(), ered to be met; therefore, the parameter u is given by x˜ lnKc+ lnI1 b 1 1 Ð K Eq. (9)]. ------(), -1 Ð ------∂()- + ------I1 b 1 lnb 2 In view of the rather complex structure of the func- (28) 1/2 1/2 tions f , it is difficult to solve Eqs. (13) analytically h˜ K + ln()1 Ð b arcsinb 1, 2, 3 --- + ------1/2 Ð0,------1/2 = with respect to m and b. For this reason, in order to ana- 2 ()1 Ð b 2b lyze Eqs. (13), we solve these equations with respect to the parameters x˜ and h˜ : x˜ b []()2 ()b ------(), lnKc+ Ð 2 lnKc+ Ð --- I1 b 1 2 (29) ∆ ()bm, ˜ (), x 1/2 1/2 1/2 x bm = ------, (33a) + h˜ []1/2() 1 Ð b ln()1 Ð b + b arcsinb = 0. ∆∞()bm, The solution to Eq. (29) meeting the condition ∆ (), ˜ h bm lnK ӷ 1 is h()bm, = ------, (33b) ∆∞()bm,  I ()b, 1 where lnKc+ = rbx(),,˜ h˜ ≡ 11+  + ------1 2x˜ ()b ()m ()m ()b  ∆ ()bm, = f f Ð f f , (30) x 2 3 2 3 ()1/2 () 1/2  ∆ (), ()b ()m ()m ()b × ˜ 1 Ð b ln 1 Ð b arcsinb h bm = f 1 f 2 Ð f 1 f 2 , 1 Ð h------2+ ------. b b1/2  ∆ (), ()b ()m ()m ()b ∞ bm = f 3 f 1 Ð f 3 f 1 , In the absence of an external field (h˜ = 0), Eq. (30) the functions f1, 2, 3 are defined by formulas (11), and reduces to the corresponding expression for the param- the superscripts (b) and (m) indicate differentiation of eter K derived in [12]: these functions with respect to corresponding parameters. The problem to be solved is to determine the equi- 1 1/2 lnK = 1 Ð1c + + ------. ˜ ˜ 4x˜ librium parameters m∗ = m∗(x˜ , h ) and b∗ = b∗(,x˜ h ), i.e., to determine the functions inverse to x˜ (b, m) and Analysis shows that the radicand in Eq. (30) is bounded for any value of b and, therefore, the condition h˜ (b, m) given by Eqs. (33).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 114 SUKSTANSKIŒ, PRIMAK

m The subscript 0 indicates that the parameter h˜ is 1.0 0.9 µ zero in the 0(b) curve [according to (33)], and the 0.99 N superscript ∞ indicates that the parameter x˜ diverges in –1 µ 0.8 0(b) ˜ the µ∞(b) curve (as does h , except in the intersection 0.999 µ 1.000 ∞(b) 0.3 0.2 µ points with the 0(b) curve). These two curves cross 0.4 0.6 0.998 each other at the origin of coordinates (0, 0), at point (1, 1.5 0.5 1), and at a point N(bN, mN) within the domain of defi- N 0.996 ≤ ≤ 0.4 µ∞(b) nition of the parameters b and m (we recall that 0 b µ0(b) 1, 0 ≤ m ≤ 1), thus partitioning this domain of definition 0.994 into four regions (see below). 0.2 1.5 –1 0.992 According to Eq. (34), the region in which x˜ (b, m) 0.92 0.94 0.96 0.98 1.00 and h˜ (b, m) have physical meaning is defined by the 0 0.2 0.4 0.6 0.8 1.0 b ≥ µ µ condition m 0(b). The ∞(b) curve, in turn, divides µ this region into two parts, (m > µ∞(b)) and (m < µ∞(b)), Fig. 2. Equilibrium DS parameters m∗ = h(b∗) at fixed values of the field. Dashed lines correspond to the cases with corresponding to domain structures existing at β∗ > 0 rather high anisotropy (in comparison with the external and and β∗ < 0, respectively. demagnetization fields), and dash-dotted lines represent the typical dependences for films with low anisotropy. The equilibrium parameters m∗ and b∗ are further analyzed for a fixed value of one of the physical param- ∆ ˜ An analysis showed that x(m, b) > 0 for arbitrary eters h and x˜ . values of m and b belonging to the domain of definition ˜ ≤ ≤ ≤ ≤ ∆ (i) First, we consider the case of fixed h . Equa- (0 b 1, 0 m 1), while the signs of ∞(m, b) and µ ∆ tion (33b) describes the curve m∗ = h(b∗), where the h(m, b) can be arbitrary. Hence, the sign of function x˜ (b, m) is dictated only by the sign of function subscript h indicates that the parameter h˜ is constant µ ˜ along this curve. Figure 2 shows m∗ = h(b) curves con- ∆∞(m, b), while the sign of h (b, m) is determined by ∆ ∆ structed using a numerical solution to Eq. (33b) for var- both the signs of ∞(m, b) and h(m, b). However, as follows from definition (14), the signs of the parameters ious fixed values of h˜ . Each point in this curve corre- x˜ and h˜ are controlled by the sign of β∗; these values sponds to equilibrium values m∗ and b∗ at a value of x˜ are positive for β∗ > 0 and negative for β∗ < 0. Hence, which can be calculated by substituting m∗ and b∗ into only those values of m and b for which the signs of the Eq. (33a). µ functions x˜ (b, m) and h˜ (b, m) coincide will have phys- The h(b∗) curves all originate at the point N(bN, µ µ ical meaning in solutions (33): mN) where the 0(b) and ∞(b) curves cross each other; as the µ∞(b) curve is approached, the parameter x˜ sgn()x˜ ()bm, = sgn()h˜ ()bm, . (34) ∞, since ∆∞ 0. Thus, as the point N is approached ˜ If the value of x is fixed, then Eq. (33a) will para- along a curve corresponding to a fixed value of h˜ , the metrically describe the family of curves m∗ = µ (b∗) in x thickness x˜ increases. Therefore, condition (6), defin- the (b, m) plane for equilibrium values m∗ and b∗, while ing the applicability of the approximation used, is vio- Eq. (33b) will give the dependence of the parameter h˜ lated in a certain vicinity of the point N. µ on the equilibrium parameters m∗ and b∗. If, con- The h(b∗) curves describe the dependence of the DS equilibrium parameters m∗ and b∗ [as well as u∗, versely, the value of h˜ is fixed, then Eq. (33b) will para- µ ˜ metrically describe the family of curves m∗ = h(b∗) in according to Eq. (9)] on x˜ at a fixed value of h . The µ the (b, m) plane for equilibrium values m∗ and b∗, while shape of the h(b∗) curves depends on the value and ˜ µ Eq. (33a) will give the dependence of the parameter x˜ sign of the parameter h . There exist two typical h(b∗) on the equilibrium parameters m∗ and b∗. dependences (see below). First, we consider two characteristic curves, m = In the case of rather high anisotropy µ (b) and m = µ∞(b), defined by 0 β > h + 4π, (37) ∆ (), y h mb = 0 (35) ˜ µ and which corresponds to 0 < h < 1, the h(b∗) curves end ∆ (), 2 ∞ mb = 0, (36) at the point m = 1, b = 1 Ð h˜ . As follows from the anal- respectively. ysis of this extreme case performed above, these equi-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 DOMAIN STRUCTURE IN AN ULTRATHIN FERROMAGNETIC FILM 115 librium values of the DS parameters are (formally) D˜ achieved at a zero film thickness. This result is in line 5 with the conclusion [7] that in thin films with high 0.999 anisotropy, the DS exists no matter how small the film 0.5 0.3 thickness. As indicated in Section 4, in the limit of zero 1.5 film thickness, the DS represents a single domain wall 4 and the extreme value of amplitude (32) is equal to the 1.7 amplitude of a homogeneous canted phase (see, e.g., 1.9 [7]). Obviously, the transition into a homogeneous 3 –0 phase in an actual film of finite size takes place at a finite film thickness, when the domain width becomes –0.5 equal to the film dimensions. 2 –1 β π In a film with low anisotropy, < hy + 4 (which ˜ ˜ corresponds to two ranges of the parameter h : h > 1 1 and h˜ < 0), the µ (b∗) curves end at the point m = 0, b = 0.2 0.3 0.4 0.5 0.6 0.7 0.8 h ˜ 0. This means that the CDS arises in such films (this d extreme case is considered in Section 3 in detail). In this case, the DS transforms into a homogeneous phase Fig. 3. Dependence of the domain size on the film thickness with its magnetization vector lying in the film plane at fixed values of the field. The line labels indicate the val- (b = 0) as the film thickness decreases to a certain crit- ues of the parameter h˜ . The dashed line represents the rela- ical value d ; the DS period remains finite (D ~ tion between the critical domain size and the critical film cr thickness. K(m) π/2), and the DS itself becomes sinusoidal. µ Using the h(b∗) dependences, Eq. (9) for u, and Eq. (33) for x˜ (b∗, m∗), we find the parametric depen- m 1.0 0.01 µ dence of the equilibrium dimensionless period D˜ on ∞(b) 0.1 N µ ( ) ˜ –1 0 b the dimensionless film thickness d , which is defined, 0.8 1.000 1 0.1 ˜ 0.998 according to formula (27), at a fixed value of field h : –0.15 0.6 µ∞(b) N 0.996 2µ ()b K3 I ()b, µ ()b D˜ = ------h 1 h , 0.994 π4Σ ()µ () (), µ () 0.4 2 h b x˜ b h b (38) µ 0.992 1 0(b) ˜ ˜ (), µ () 0.990 d = x b h b . 0.2 –1 0.988 The corresponding curves are shown in Fig. 3 for 0.92 0.96 1.00 ˜ various values of the parameter h . One can see that, in 0 0.2 0.4 0.6 0.8 1.0 b the case of high anisotropy, the period D˜ increases infi- µ nitely as the thickness d˜ decreases to zero. In the case Fig. 4. Equilibrium values of the DS parameters m∗ = x(b∗) of low anisotropy, there exists a certain critical thick- at fixed values of the film thickness (the values of the parameter x˜ are indicated near the lines). ness dcr at which the DS transforms into a homogeneous phase: the magnetization distribution becomes uniform at thicknesses smaller than d . The critical val- cr remember that the film morphology can have a signifi- ˜ ˜ ues correspond to the Dcr (dcr ) curve, which is cant effect on DS formation in monolayer films). As the described by simple analytical expression (27) in the field strengthens to the value at which inequality (37) is case of u ӷ 1. We note that in the case of β > 4π (β∗ > violated, the magnetization normal component disap- 0), the dependence of the DS period on the film thick- pears at a certain critical thickness depending on the ness can be switched (by varying the field h˜ ) between field according to formula (25). the modes corresponding to the cases of high and low (ii) Now, we consider solutions to Eqs. (33) at vari- anisotropy. This switching should experimentally man- ous fixed values of the parameter x˜ . Figure 4 displays ifest itself as follows. In a film with high anisotropy, the m = µ (b ) curves constructed by using a nume- i.e., when inequality (37) is met, the normal magnetiza- ∗ x ∗ tion component should exist in films with arbitrarily rical solution to Eq. (33a), which parametrically small (down to monolayers) thicknesses (one should describes these curves at a fixed value of x˜ (the sub-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 116 SUKSTANSKIŒ, PRIMAK

D˜ which, as was already indicated, corresponds to the π 0.01 equilibrium values of the DS parameters in a zero field 3.2 10 and at various values of the film thickness. The increase 0.05 3.0 in the field from zero to the value at which the CDS 2.8 0.3 arises (and the transition to a homogeneous phase takes µ 8 ˜ 2.6 place) corresponds to motion along the x(b∗) curve D> 0.2 µ 2.4 –0.3 from the point 0(b∗) to point (0, 0). D˜ 0.1 2.2 < 6 –0.5 It is noteworthy that the difference between the 2.0 µ µ x(b∗) curves and the ∞(b∗) curve decreases as the 1.8 ||˜ 4 0.3 0 0.5 1.0 1.5 2.0 parameter x increases. However, one should remember that inequality (6), on which the theory presented ˜ µ D< above is based, is violated in the vicinity of the ∞(b∗) 2 –0.5 curve. β 0123 4h˜ In the case of low anisotropy ( ∗ < 0), the DS can exist in films for which the relation ||x˜ > 1/π2 is valid. Fig. 5. Dependences of the domain size on the field at fixed Films with smaller values of ||x˜ are in the state with ˜ ˜ values of the film thickness. D> and D< are the lines of the uniform magnetization in the film plane even in a zero transition between the homogeneous phase and the DS β β field (the formation of a DS in such films can be condi- under the field in the cases ∗ > 0 and ∗ < 0, respectively. tioned by the morphological properties of the film). µ Using the x(b∗) dependences, Eq. (9) for u, and ˜ script x indicates that the parameter x is fixed along the Eq. (33b) for h˜ (b , m ), we find the parametric depen- µ ∗ ∗ curve). Substituting x(b∗) into Eq. (33b), we obtain the dence of the DS dimensionless period D˜ on the nor- dependence of the DS parameters on the field h˜ at a malized field h˜ : fixed value of the thickness x˜ . The motion along the µ (b∗) curve corresponds to the variation in the equilib- x µ () 3 (), µ () ˜ 2 x b K I1 b x b rium values of the DS parameters caused by an h˜ vari- D =,------π4Σ ()µ ()b x˜ 1/2 (39) ation at a fixed value of the parameter x˜ . The initial 2 x µ µ ˜ ˜ (), µ () points of all the x(b∗) curves lie in the 0(b∗) curve, h = h b x b .

∆˜ ∆˜ 1.50 0.01 0.9 1.5 3

1.25 0.5 1.7

2 1.00 –0 0.3 –0.5 0.05 0.75 –1 1 –0.2 –0.3 0.3 0.50 –0.5

0.25 012345 0 0.25 0.50 0.75 h˜ d˜ Fig. 7. Dependence of the domain wall thickness on the Fig. 6. Dependence of the domain wall thickness on the film field at fixed values of the film thickness. Dashed lines are thickness at fixed values of the field. The dashed line is the the dependences of the domain wall thickness on the critical dependence of the domain wall thickness on the film thick- field for the critical DS in the cases β∗ > 0 (upper line) and ness for the critical DS. β∗ < 0 (lower line).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 DOMAIN STRUCTURE IN AN ULTRATHIN FERROMAGNETIC FILM 117

Figure 5 shows the D˜ (h˜ ) curves (the subscript x Thus, by using a trial function with three variational x parameters, we have described two probable scenarios indicates that the parameter x˜ is fixed). As indicated of the DS transition to a homogeneous magnetized ˜ ˜ above, these curves originate at h = 0 and end at hcr (x), phase within a unified approach and taken into account at which point the magnetization vector becomes paral- the influence of the field on the equilibrium DS param- lel to the film plane (and the CDS arises). The critical- eters. ˜ ˜ field dependence of the critical period Dx (hcr ) can be derived from Eq. (24) (taking into account the condi- REFERENCES tion u ӷ 1) to be 1. T. Duden, R. Zdyb, M. Altman, and E. Bauer, Surf. Sci. 480 (3), 145 (2001). β Ð1/2 ˜ π˜ ∗ 2. W. Kuch, J. Gilles, F. Offi, et al., Surf. Sci. 480 (3), 153 Dcr = hcr Ð ------. (40) β∗ (2001). 3. E. Allenspach, M. Stampanoni, and A. Bishof, Phys. ˜ These curves are labeled in Fig. 5 by D> for β∗ > 0 and Rev. Lett. 65 (26), 3344 (1990). ˜ β 4. M. Stampanoni and E. Allenspach, J. Magn. Magn. by D< for ∗ < 0. Mater. 104Ð107 (3), 1805 (1992). It is noteworthy that, when studying the DS in a 5. V. Grolier, J. Ferre, A. Maziewski, et al., J. Appl. Phys. uniaxial film, the domain wall thickness is often 73, 10 (1993). assumed to be sufficiently small in comparison with the 6. M. Speckmann, H. P. Oepen, and H. Ibach, Phys. Rev. DS period and is neglected. However, the analysis we Lett. 75 (10), 2035 (1995). carried out shows this assumption to be valid only in the 7. V. G. Bar’yakhtar and B. A. Ivanov, Zh. Éksp. Teor. Fiz. case of high anisotropy (37) and small film thickness, 72 (4), 1504 (1977) [Sov. Phys. JETP 45, 789 (1977)]. i.e., when the condition 8. B. A. Ivanov, V. P. Krasnov, and A. D. Sukstanskiœ, Fiz. Nizk. Temp. 4 (2), 204 (1978) [Sov. J. Low Temp. Phys. D ----= 2 Km() ӷ 1 (41) 4, 101 (1978)]. ∆ 9. V. V. Tarasenko, I. E. Dikshteœn, and E. V. Chenskiœ, Zh. Éksp. Teor. Fiz. 70 (6), 2178 (1976) [Sov. Phys. JETP 43, is met. 1136 (1976)]. As Fig. 6 shows, the domain wall thickness essen- 10. C. Kittel, Phys. Rev. 70 (11Ð12), 965 (1946). tially depends on the film thickness and, even in the 11. B. Kaplan and G. A. Gehring, J. Magn. Magn. Mater. case of high anisotropy, increases by a few times with 128 (1), 111 (1993). small changes in the film thickness, with condition (41) 12. A. L. Sukstanskii and K. I. Primak, J. Magn. Magn. remaining valid. This means that the domain wall Mater. 169 (1), 31 (1997). energy does not remain unchanged as the film thickness 13. Handbook of Mathematical Functions, Ed. by changes, which should be taken into account in consid- M. Abramowitz and I. A. Stegun (National Bureau of ering the DS. Standards, Washington, 1964; Nauka, Moscow, 1979). The field dependence of the domain wall thickness 14. J. Kaczer, M. Zeleny, and P. Suda, Czech. J. Phys. 13 (4), is shown in Fig. 7. One can see that the domain wall 579 (1963). thickness is maximum for the CDS. For films with a 15. K. Kooy and U. Enz, Philips Res. Rep. 15 (1), 7 (1960). small thickness, the domain wall thickness increases by ˜ a few times as the field changes from 0 to hcr . Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

METALS AND SUPERCONDUCTORS

Electron Density Variation at Copper Sites at the Superconducting Transition in Cuprates N. P. Seregin Institute of Analytical Instrumentation, Russian Academy of Sciences, Rizhskiœ pr. 26, St. Petersburg, 198103 Russia Received March 15, 2002

Abstract—The temperature dependence of the center of gravity S of the Mössbauer spectrum produced by 67 2+ Zn impurity ions occupying copper and yttrium sites in YBa2Cu3O6.6, YBa2Cu3O6.9, YBa2Cu4O8, Bi2Sr2CaCu2O8, HgBa2CuO4, and HgBa2CaCu2O6 at temperatures above the superconducting transition point Tc was shown to be dominated by the second-order Doppler shift. For T < Tc, the quantity S is affected by the band mechanism associated with the formation of Cooper pairs and their Bose condensation. The variation of electron density at a metal site of the crystal was found to be related to the superconducting transition temperature. The variation of electron density created by the Cooper pair Bose condensate in compounds with two structurally inequivalent copper positions was shown to be different for these copper sites. The experimental dependence of the fraction of superconducting electrons on temperature agrees with the analogous dependence following from BCS theory for all the sites studied. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. EXPERIMENTAL TECHNIQUE The electron density distribution over lattice sites is The Mössbauer sources were prepared by diffusing radioactive 67Cu and 67Ga into polycrystalline samples different for the superconducting and normal (metallic) ° phases, and this difference can be measured using in evacuated quartz ampules at 450 C for 2 h. The starting samples were synthesized using the techniques Mössbauer spectroscopy. It was shown in [1] that the described in [4Ð8]. As reference samples that do not conditions for detecting Cooper pairs in cuprates using undergo the superconducting transition, we used mate- Mössbauer spectroscopy are most favorable for the rials obtained by annealing the starting (superconduct- 67Zn isotope if one uses the emission variant of spec- ing) samples in air at 600°C for 2 h. 67 troscopy with the parent nuclei Cu (with the daughter The 67Zn Mössbauer spectra were measured using probe 67Zn2+ sitting at copper sites) or 67Ga (with the an MS-2201 commercial spectrometer with a modiﬁed daughter probe 67Zn2+ occupying yttrium or rare-earth energy-modulating system. The modulator chosen was metal sites). a piezoelectric transducer based on PZT ceramics. The maximum scan velocity was ±150 µm/s. The spectrom- We report here on a study of electron density distri- eter was calibrated against the 67Zn spectrum with a bution at copper lattice sites of the high-temperature 67Cu source. The gamma quanta were detected with a superconductors YBa2Cu3O6.9 (Tc = 90 K), YBa2Cu3O6.6 Ge(Li) semiconductor detector sensitized in the (Tc = 50 K), and YBa2Cu4O8 (Tc = 80 K) carried out 100-keV region. The Mössbauer spectra were taken using Mössbauer emission spectroscopy on the with a 67ZnS absorber. The sample and the absorber 67Cu(67Zn) and 67Ga(67Zn) isotopes. In these com- were cooled in a cold helium ﬂow, and the source heat- pounds, the copper atoms occupy two structurally ing was achieved with an electric furnace. The temper- inequivalent positions, Cu(1) and Cu(2), with the corre- ature was monitored with a semiconductor pickup. The ± sponding population ratio 1 : 2 (for YBa Cu O and temperature of the absorber was 10 1 K, and the tem- 2 3 6.9 ± YBa Cu O ) [2] or 1 : 1 (for YBa Cu O ) [3]. It was perature of the source could be varied from 10 2 to 2 3 6.6 2 4 8 90 ± 2 K. assumed that the variation of the electron density at these sites at the transition of the compound to the superconducting state would be different. Also studied 3. EXPERIMENTAL RESULTS for comparison were the compounds HgBa2CuO4 (Tc = AND DISCUSSION 79 K), HgBa2CaCu2O6 (Tc = 93 K), and Bi2Sr2CaCu2O8 The 67Cu(67Zn) Mössbauer spectra of all the com- (Tc = 80 K), in which copper occupies the only position pounds studied were either quadrupole triplets (for [4Ð6]. compounds with only one copper position) or a super-

12 SEREGIN

The ﬁrst term in Eq. (1) describes the dependence of the isomer shift on volume V and becomes manifest in 4 structural phase transitions. The second term is due to the temperature dependence of the second-order Dop- 2 pler shift D and, in the Debye approximation, can be cast as [15]

m/s 0 2

µ ()δ δ ()()Θ D/ T P = Ð 3k0E0/2Mc FT/ , (2) , 1 –2S 2 where k0 is the Boltzmann constant, E0 is the isomer 3 transition energy, M is the probe nucleus mass, c is the –4 4 speed of light in free space, Θ is the Debye temperature, 5 Θ 6 and F(T/ ) is the Debye function. The Debye model –6 satisfactorily describes the vibrational properties of primitive lattices only; nevertheless, Eq. (2) fits quite 0 20 40 60 80 100 well to the experimental S(T) relations found for crys- T, K tallochemically complex compounds [1]. This is accounted for by the fact that the D(T) relation is dom- Fig. 1. Temperature dependences of the center of gravity S inated by the short-wavelength region of the phonon of the Mössbauer spectra of 67Zn2+ at the Cu(1) [1, 4], Cu(2) [2, 5], and Y [3, 6] sites as measured relative to its spectrum, which is described sufficiently well by the value at 50 K for (1Ð3) YBa2Cu3O6.6 and (4Ð6) Debye approximation. Finally, the third term in Eq. (1) YBa2Cu3O6.5. Solid line is the theoretical temperature takes into account the temperature dependence of the dependence of S for the case of a second-order Doppler shift isomer shift and is induced by the electron density vari- for Θ = 420 K. ation at the Mössbauer nucleus sites, which is expected to occur in the host matrix undergoing the superconducting transition. position of two quadrupole triplets (for compounds having two structurally inequivalent copper positions). Figure 1 presents typical S(T) relations for the The fine structure of the spectra and assignment of the Cu(1), Cu(2), and Y sites in the lattices of YBa2Cu3O6.5 triplets to the corresponding copper site were described and YBa2Cu3O6.6. It was found that the temperature in [9Ð12]. The isomer shift (IS) of all spectra can be dependence of the spectrum center of gravity S mea- identified with the 67Zn2+ ions (IS Ӎ 67Ð77 µm/s rela- sured relative to its value at Tc is fitted well by Eq. (2) tive to the spectrum of the 67Ga source in copper). It in the temperature interval 10Ð90 K for all reference samples if one uses the Debye temperatures 420 ± 10 K was assumed that, in the course of diffusion doping, the ± ± 67 (for YBa2Cu3O6.5), 400 10 K (for YBa2Cu4Ox), 260 parent Cu atoms enter the copper sites (this assump- ± tion is borne out by the data from [6, 7]) and, hence, that 10 K (for Bi2Sr2CaCu2Ox), and 360 10 K (for the 67Zn2+ probe formed in the 67Cu disintegration occu- HgBa2CuOx and HgBa2CaCu2Ox). (These Debye tem- pies a copper site. peratures are in accord with the literature values 67 67 derived from heat capacity measurements [16Ð19].) In The Ga( Zn) Mössbauer spectra of YBa2Cu3O6.9, other words, isomer shift variations, both due to a YBa2Cu3O6.6, and YBa2Cu4O8 were quadrupole triplets change in the volume and originating from temperature with the isomer shift corresponding to the 67Zn2+ ions variations, virtually do not affect the S(T) dependence (IS Ӎ 100Ð107 µm/s). The fine structure and assign- for the nonsuperconducting samples. Because the com- ment of the triplets to 67Zn2+ centers at yttrium sites pounds studied do not undergo structural phase transi- were described in [10, 13]. tions in the interval 10Ð90 K [2, 3], this behavior of The quadrupole interaction constants C for the S(T) was to be expected. 67Zn2+ centers at both the copper and yttrium sites are For all superconducting samples, the S(T) depen- practically temperature-independent. This can be dence for T > Tc is likewise described by the second- attributed to the fact that the electric field gradient at the order Doppler shift (2) and the Debye temperatures do 67Zn nuclei detected by the 67Zn2+ probe is generated not differ from those for the reference samples. In the primarily by the lattice ions and the lattice constants of T < Tc region, S depends on temperature more strongly the compounds studied vary very little in the tempera- than would follow from Eq. (2); as a result, one must ture interval 4.2Ð100 K [2, 3]. now take into account, in Eq. (1), the third term, which The temperature dependence of the center of gravity describes the temperature dependence of the isomer S of the 67Zn Mössbauer spectrum at constant pressure shift. P can be determined using [14] To describe the observed phenomenon qualitatively, ()δ δ []δ()δ ()δ δ we introduce the following parameters: the isomer shift S/ T P = IS / lnV T lnV/ T P (1) [IS]T at the given temperature T, found as the difference ()δ δ []δ()δ + D/ T P + IS / T V . [IS]T = ST Ð DT (where ST and DT are the center of grav-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTRON DENSITY VARIATION AT COPPER SITES 13 ity and Doppler shift of the spectrum at temperature T, 10 0.25 respectively), and the limiting isomer shift IS for T 0, deﬁned as the difference [IS]0 = S0 Ð D0 (here, S0 and 8 0.20 D0 are the center of gravity and Doppler shift of the spectrum for T 0, respectively). As seen from 3 Fig. 2, [IS] is larger, the higher the superconducting 0 6 9 0.15 m/s transition temperature. The value of [IS]0 also depends 8 , a.u. µ 2 |

, 6 on the actual site occupied by the Mössbauer probe. 0 7 (0) [IS]0 is the largest for the Cu(2) sites, substantially 4 5 0.10 [IS] smaller for the Cu(1) sites, and the smallest for the Y 4 ∆|ψ sites, if one compares sites in the same lattice 2 1 0.05 (YBa2Cu3O6.6, YBa2Cu3O6.9, or YBa2Cu4O8; see table). 2 According to BCS theory, as the temperature is lowered and passes through Tc, Cooper pairs appear in the 0 superconductor (the coupling electrons have opposite 0 0.02 0.04 0.06 0.08 1.00 –1 –1 momenta and opposite spins, so that the total momen- T c , K tum, orbital angular momentum, and spin of a Cooper pair are zero) to form the Bose condensate [20]. This ∆|Ψ |2 Ð1 should change the electron density at lattice sites: the Fig. 2. Dependence of [IS]0 and (0) on Tc . Symbols conduction electrons of the metallic phase are are data for (1) Cu in Nd1.85Ce0.15CuO4, (2) Cu in described by Bloch wave functions for temperatures La1.85Sr0.15CuO4, (3) Cu(2) in YBa2Cu3O6.9, (4) Cu(2) in YBa2Cu3O6.6, (5) Cu(2) in YBa2Cu4O8, (6) Cu in T > Tc and by a common coherent wave function for T < Bi2Sr2CaCu2O8, (7) Cu in Tl2Ba2CaCu2O8, (8) Cu in Tc. The isomer shift found as HgBa2CuO4, and (9) Cu in HgBa2CaCu2O6. The data for α∆ Ψ()2 Nd1.85Ce0.15CuO4, La1.85Sr0.15CuO4, and Tl2Ba2CaCu2O8 IS = 0 (3) were taken from [1]. (here, ∆|Ψ(0)|2 is the difference between the relativistic electron densities at the nuclei of interest in two sam- µ |Ψ |2 ples and α is a constant dependent on the nuclear where IS is measured in m/s, (0) in atomic units, parameters of the isotope used) is directly connected and Tc in kelvins. with the change in the electron density at the 67Zn We readily see that the experimental value of the 67 nucleus sites, with the value of [IS]0 characterizing the change in the electron density at a Zn nucleus site electron density generated by the Bose condensate occurring at the superconducting transition does not under the conditions where all conduction electrons are exceed 0.2 a.u. and corresponds to the minimum possi- ∆|Ψ |2 bound in Cooper pairs. In converting [IS]0 to (0) , ble size of a Cooper pair. The existence of such a mini- we made use of the value of α taken from [21]. Figure 2 mum size is apparently associated with the physical ∆|Ψ |2 Ð1 impossibility of existence of Cooper pairs with a dis- plots the dependence of (0) on T c ; we see that tance between the components of less than a certain ∆|Ψ |2 |Ψ |2 |Ψ |2 the quantity (0) = c(0) Ð 0(0) increases with critical length. increasing Tc, which reflects the growth of electron den- BCS theory permits one to find the temperature sity at the 67Zn nuclei in going over from the nonsuper- dependence of the effective density of superfluid elec- |Ψ |2 67 conducting [electron density 0(0) at the Zn nuclei] to superconducting phase [with electron density |Ψ |2 67 The values of [IS] for various lattice sites of YBa Cu O , c(0) at the Zn nuclei]. 0 2 3 6.6 YBa2Cu3O6.9, and YBa2Cu4O8 The dependence of ∆|Ψ(0)|2 on T becomes clear if c µ we take into account that the standard correlation Compound Site [IS]0, m/s ξ length 0 (the “size” of a Cooper pair for T 0) for YBa Cu O , T = 50 K Cu(1) 1.8(3) anisotropic superconductors scales as ξ ~ TÐ1 and, 2 3 6.6 c 0 Cu(2) 3.7(3) thus, that Fig. 2 reflects the dependence of [IS] and 0 Y 1.0(3) ∆|Ψ |2 Ð1 (0) on T c and on the standard correlation length YBa Cu O , T = 90 K Cu(1) 2.9(3) ξ 2 3 6.9 c 0. This dependence has an exponential character: Cu(2) 6.6(3) [] [] IS 0 = 7.9exp Ð 31.4/T c Y 1.9(3) YBa Cu O , T = 80 K Cu(1) 2.2(3) or 2 4 8 c Cu(2) 4.8(3) ∆Ψ()2 [] 0 = 0.2exp Ð 31.4/T c , Y 1.0(3)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 14 SEREGIN

sity of superfluid electrons exhibit satisfactory agree- 1 6 ment for both the Cu(1) and Cu(2) (as well as for the Y) 1.0 2 7 positions. 3 8 4 9 0.8 5 10 0 4. CONCLUSION Thus, it has been established that the temperature /[IS] 0.6 T dependence of the center of gravity S of the Möss- 67 2+ [IS] bauer spectrum of the Zn crystal probe is domi- 0.4 nated for the YBa2Cu3O6.6, YBa2Cu3O6.9, YBa2Cu4O8, Bi2Sr2CaCu2O8, HgBa2CuO4, and HgBa2CaCu2O6 0.2 compounds above the superconducting transition point Tc by the second-order Doppler shift. In the T < Tc region, the quantity S is affected by the band mecha- 012 x nism associated with the formation of Cooper pairs and their Bose condensation. The change in the electron density at a metal site of a crystal is related to the super- Fig. 3. Dependence of [IS]T/[IS]0 on the parameter x = ∆ 1.76(k0T/ ). Solid line shows a theoretical dependence of conducting transition temperature. As shown for the the effective density of superfluid electrons on the parame- YBa2Cu3O6.6, YBa2Cu3O6.9, and YBa2Cu4O8 com- ter x. Symbols refer to data obtained for the following sites: pounds having two structurally inequivalent positions (1) Cu(1) in YBa2Cu3O6.6, (2) Cu(1) in YBa2Cu3O6.9, (3) for the copper atoms, the variation of electron density Cu(1) in YBa2Cu4O8, (4) Cu(2) in YBa2Cu3O6.6, (5) Cu(2) produced by the Bose condensate of Cooper pairs is dif- in YBa2Cu3O6.9, (6) Cu(2) in YBa2Cu4O8, (7) Cu in ferent for these cooper sites, as well as for the yttrium Bi2Sr2CaCu2O8, (8) Cu in HgBa2CuO4, (9) Cu in sites. The change in the electron density is the largest Hg2Ba2CaCu2O6, and (10) Y in YBa2Cu3O6.9. for the Cu(2) sites, considerably smaller for the Cu(1) sites, and the smallest for the Y sites. The experimentally established temperature dependence of the frac- trons ρ(T) [20]; on the other hand, one may expect that ρ tion of superconducting electrons for all the sites stud- (T) ~ [IS]T/[IS]0. Figure 3 presents a theoretical ρ ∆ ∆ ied [Cu(1), Cu(2), Y] is in satisfactory agreement with dependence of on the parameter x = 1.76(k0T/ ) ( is the analogous dependence following from BCS theory. the energy gap in the elementary excitation spectrum of a superconductor taken from [20]) together with our data on the dependence of [IS]T/[IS]0 on the parameter ACKNOWLEDGMENTS x for various cuprates. The calculated and experimental This study was supported by the Ministry of Educa- temperature dependences of the effective density of tion of the Russian Federation, project no. E 00-3.4-42. superfluid electrons are seen to be in satisfactory agreement. This comes as a surprise; indeed, the problem of the inapplicability of BCS theory to the description of REFERENCES the properties of high-temperature superconductors has 1. N. P. Seregin, F. S. Nasredinov, and P. P. Seregin, Fiz. been discussed in the literature in considerable detail Tverd. Tela (St. Petersburg) 43, 587 (2001) [Phys. Solid [22]; therefore, one should consider with caution our State 43, 609 (2001)]. observation of agreement between the theoretical and 2. J. Capponi, C. Chaillout, A. Hewat, et al., Europhys. experimental dependences of the effective density of Lett. 3, 1301 (1987). superfluid electrons on the parameter x for the group of 3. E. Kaldis, P. Fischer, A. W. Hewat, et al., Physica C superconductors under study and the possibility of (Amsterdam) 159, 668 (1989). applying BCS theory in its unmodified form to the description of high-temperature superconductivity. 4. O. Chmaissem, Q. Huang, S. N. Putilin, et al., Physica C (Amsterdam) 212, 259 (1993). This agreement should apparently be treated as an indication that the formation of Cooper pairs and their Bose 5. E. V. Antipov, J. J. Capponi, C. Chaillout, et al., Physica condensation takes place in any theory of high-temper- C (Amsterdam) 218, 348 (1993). ature superconductivity. 6. J. M. Tarascon, W. R. McKinnon, P. Barboux, et al., Phys. Rev. B 38, 8885 (1988). A specific feature of the compounds illustrated in 7. J. D. Jorgensen, B. W. Veal, A. P. Paulikas, et al., Phys. Fig. 3 is the difference in the value of [IS]0 observed to Rev. B 41, 1863 (1990). exist between the Cu(1) and Cu(2) sites. This is obvi- 8. P. Gaptasarma, V. R. Palhar, and M. S. Multane, Solid ously a consequence of a spatial inhomogeneity in the State Commun. 77, 769 (1991). electron density produced by the Bose condensate of 9. P. P. Seregin, V. F. Masterov, F. S. Nasredinov, et al., Fiz. Cooper pairs. Nevertheless, the calculated and experi- Tverd. Tela (St. Petersburg) 36, 769 (1994) [Phys. Solid mental temperature dependences of the effective den- State 36, 422 (1994)].

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTRON DENSITY VARIATION AT COPPER SITES 15

10. F. S. Nasredinov, V. F. Masterov, N. P. Seregin, and 17. H. M. Ledbetter, S. A. Kim, and R. B. Goldfarb, Phys. P. P. Seregin, J. Phys.: Condens. Matter 11, 8291 (2000). Rev. B 39, 9689 (1989). 11. F. S. Nasredinov, V. F. Masterov, N. P. Seregin, and 18. A. Junod, T. Craf, D. Sanchez, et al., Physica C (Amster- P. P. Seregin, J. Phys.: Condens. Matter 12, 7771 (2000). dam) 165/166, 1335 (1990). 12. V. F. Masterov, F. S. Nasredinov, N. P. Seregin, and 19. S. J. Collocott, R. Driver, C. Audrikidis, and F. Pavese, P. P. Seregin, Zh. Éksp. Teor. Fiz. 114, 1079 (1998) Physica C (Amsterdam) 156, 292 (1989). [JETP 87, 588 (1998)]. 13. V. F. Masterov, F. S. Nasredinov, N. P. Seregin, and 20. J. R. Schrieffer, Theory of Superconductivity (Benjamin, P. P. Seregin, Fiz. Tverd. Tela (St. Petersburg) 38, 1986 New York, 1964; Nauka, Moscow, 1970). (1996) [Phys. Solid State 38, 1094 (1996)]. 21. A. Svane and E. Antoncik, Phys. Rev. B 34, 1944 (1986). 14. J. S. Shier and R. D. Taylor, Phys. Rev. 174, 346 (1968). 22. Physical Properties of High Temperature Superconduc- 15. D. L. Nagy, in Mössbauer Spectroscopy of Frozen Solu- tors, Ed. by D. M. Ginsberg (World Scientiﬁc, Sin- tions, Ed. by A. Vértes and D. L. Nagy (Akadémiai gapore, 1989; Mir, Moscow, 1990). Kiadó, Budapest, 1990; Mir, Moscow, 1998). 16. T. Sasaki, N. Kobayashi, O. Nakatsu, et al., Physica C (Amsterdam) 153Ð155, 1012 (1988). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 118Ð123. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 113Ð118. Original Russian Text Copyright © 2003 by Dubinin, Arkhipov, Teploukhov, Mukovskiœ. MAGNETISM AND FERROELECTRICITY

Ferromagnetic Superstructure of an La0.85Sr0.15MnO3 Manganite Single Crystal S. F. Dubinin*, V. E. Arkhipov*, S. G. Teploukhov*, and Ya. M. Mukovskiœ** * Institute of Metal Physics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoœ 18, Yekaterinburg, 620219 Russia ** Moscow Institute of Steel and Alloys, Leninskiœ pr. 4, Moscow, 117935 Russia Received February 14, 2002

Abstract—The elastic thermal-neutron scattering patterns of a La0.85Sr0.15MnO3 manganite orthorhombic single crystal are investigated in the temperature range 4.2Ð300 K. It is found that, in addition to the known ferro- π magnetic ordering (TC = 240 K), this compound exhibits a ferromagnetic superstructure with the (010)2 /b 16 wave vector (in the Pnma setting of the space group D2h ). The ferromagnetic superstructure is observed in the studied crystal at temperatures ranging from 4.2 to 200 K. It is shown that the formation of the ferromagnetic superstructure in this compound is directly associated with a 1/8-type charge ordering of Mn3+ and Mn4+ ions. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION These authors revealed two series of superstructure reflections with the fractional indices The considerable interest expressed in neutron diffraction investigation of doped manganites of the for- ()h ± 1/2,, 0 k ± 1/2 (2) mula and 3+ 2+ ()h ± 1/4,, 0 k ± 1/4 . (3) La0.85Sr0.15MnO3, (1) In [1], series (3) of nuclear superstructure reflections in which a number of Mn3+ ions acquire a valence of 4+ was treated as an indication of the existence of charge in order to satisfy the electroneutrality condition, is (polaron) ordering in the crystal, i.e., as an indication of associated with the giant magnetoresistance effect the formation of chains composed of identical ions observed in ferromagnetic compounds of this class. In alternating in the order recent years, the giant magnetoresistance effect has assumed great practical importance. Mn3+ÐMn4+ÐMn3+ (4) Neutron diffraction investigations of doped manga- in the initial lattice. Periods of type (4) appear in the nites are also of particular scientific interest, because neutron diffraction pattern of the crystal, because Mn4+ the question as to the nature of charge ordering in the ions do not affect the surrounding oxygen octahedra 2+ La3+ Sr MnO compound remains open. In this (since these ions have no orbital angular momentum), 0.85 0.15 3 whereas the JahnÐTeller Mn3+ ions strongly distort their respect, special mentioned should be made of two environments due to interaction of the ionic orbital recent works [1, 2] devoted to neutron diffraction anal- d(z2) with the crystal lattice. ysis of the magnetic and structural states of La Sr MnO manganite single crystals. More recently, Vasiliu-Doloc et al. [2] also thor- 0.85 0.15 3 oughly studied the magnetic and nuclear structures of Yamada et al. [1] investigated the structure and 3+ 2+ an La0.85Sr0.15MnO3 single crystal in the temperature magnetic properties of La0.85Sr0.15MnO3 crystals in the temperature range 10Ð300 K in the a*c* plane of the range 10Ð380 K. It is interesting to note that, as in [1], reciprocal orthorhombic lattice (unlike the notation these researchers observed superstructure reflections of used in [1], the designations of the reciprocal lattice type (2) in the diffraction pattern of the crystal. How- axes are given here and below in the Pnma setting of the ever, no superstructure reflections of type (3) were 16 revealed in the diffraction pattern measured in [2]. This space group D2h for convenience of comparison with can be explained, for example, by the poor quality of our results). According to the results obtained in [1], the the crystal or the deviation of the chemical composition studied crystal, in its magnetic properties, is a collinear of the manganite studied in [1] from the composition ferromagnet with the Curie temperature TC = 240 K. specified by formula (1). Most likely, the absence of

FERROMAGNETIC SUPERSTRUCTURE 119 superstructure reflections of type (3) in the diffraction (a) pattern was the reason why charge ordering in the stud- (004) (404) ied compound was not considered in [2]. Vasiliu-Doloc et al. [2] analyzed in detail the origin of the weak magnetic superstructure reflection (030), which appears in the neutron diffraction pattern of the crystal at temperatures below TCA = 205 K. According to the authors’ opinion [2], this reflection can be associated with the formation of a weak antiferromagnetic component or a canted magnetic structure in the studied ferromagnet (000) (400) (TC = 235 K) at temperatures below TCA. From the foregoing, it is seen that the neutron dif- (b) fraction investigations performed in [1, 2] led to contra- (040) (440) dictory results regarding the occurrence of charge 3+ 4+ 3+ 2+ ordering of Mn and Mn ions in La0.85Sr0.15MnO3 single crystals. In this respect, we undertook a detailed study of the structural and magnetic states in this manganite over a wide range of temperatures. In our previ- (000) (400) ous experimental work [3], we measured the scattering patterns of thermal neutrons of crystals at relatively Fig. 1. A schematic representation of the neutron scattering patterns of the La0.85Sr0.15MnO3 manganite orthorhombic high temperatures (in the range 190Ð380 K). It should single crystal at 4.2 K in (a) the a*c* plane and (b) the a*b* be noted that the diffraction patterns obtained in [3] are plane. similar to those measured by Vasiliu-Doloc et al. [2]. The results obtained in [3] indicate that charge ordering 3+ 2+ the neutron diffraction pattern of the single crystal. For in the La0.85Sr0.15MnO3 manganite is characterized by example, the relative intensity of Bragg reflections cor- a long-range order. Moreover, in [3], we determined the responding to the wavelength λ/2 in the diffraction pat- correlation lengths of the long-range order at tempera- tern of the reference crystal reached only 0.02% of the tures above the Curie point for the compound under intensity of the main Bragg reflections. (In other words, investigation. spurious peaks in the neutron diffraction pattern of a However, in order to solve the problem of charge single crystal will arise only in the case when the inten- ordering in the LaÐSr manganite conclusively, it is nec- sity of the main Bragg reflections is greater than essary to perform systematic neutron diffraction exper- 105 pulses.) This appreciably increased the sensitivity iments at liquid-helium temperature and to compare the of our neutron diffraction technique. experimental results with the data obtained in [1, 2]. 3. RESULTS AND DISCUSSION 2. SAMPLES AND EXPERIMENTAL TECHNIQUE The elastic neutron scattering patterns of the 3+ 2+ In this work, we examined a manganite single crystal La0.85Sr0.15MnO3 manganite orthorhombic single with the chemical composition specified by formula (1). crystal were measured at a temperature of 4.2 K in the The manganite single crystal was grown by zone melt- two principal planes a*c* and a*b* of the reciprocal ing. The misorientation of mosaic blocks in the crystal lattice. The results obtained are schematically repre- did not exceed 25′. A sample was prepared in the form sented in Fig. 1. In this figure, the main Bragg reflec- of a cylinder with linear sizes d = 3 mm and l = 10 mm. tions of the orthorhombic crystal are depicted by large The [110] crystallographic direction in a perovskite closed circles. The superstructure reflections are indi- cubic reference sample deviated from the cylinder axis cated by other symbols. For example, the superstruc- ° of the sample by approximately 15 . ture reflections associated with the rotational and Q2 Experiments on elastic scattering of thermal neu- JahnÐTeller modes, which are well known in com- trons were performed using a special multichannel dif- pounds of this class [4], are shown by squares and open fractometer designed for single-crystal investigations. circles, respectively. The wavelength λ of neutrons incident on the sam- It is worth noting that superstructure reflections of ple was equal to 1.567 Å. This was provided by a dou- type (3), which should be observed midway between ble crystal monochromator prepared from pyrolytic the main Bragg reflections in the reciprocal lattice and carbon and germanium. The efficient monochromatiza- the superstructure reflections indicated by closed tion of the primary beam and the appropriate choice of rhombs, are absent in Fig. 1. As an example, Fig. 2 the wavelength of monochromatic neutrons made it shows two neutron diffraction patterns measured at possible to suppress multiple diffraction harmonics in 4.2 K along the paths depicted by dashed lines in

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

120 DUBININ et al.

(2 0 2) (3 0 1) (0 1 0) 200 500 (2.5 0 1.5) (a) (0 3 0) (0 5 0) 400 150 300 100 (a) 200 50 100 200 0 150 2.00 2.25 2.50 2.75 3.00 100 (b)

ξ ξ Counts [ 0 ] 50 (1 0 1) (2 0 2) 200 Counts 150 500 (b) 100 (c) 400 50 300 (1.5 0 1.5) 200 0.8 1.0 1.2 2.9 3.1 4.8 5.0 5.2 100 [0 ξ 0] 0 1.00 1.25 1.50 1.75 2.00 [ξ 0 ξ] Fig. 3. (a) Experimental intensities of the (010), (030), and (050) superstructure reflections in the neutron diffraction Fig. 2. Neutron diffraction patterns of the pattern of the La0.85Sr0.15MnO3 single crystal at 4.2 K; (b) La0.85Sr0.15MnO3 single crystal at 4.2 K in the a*c* plane calculated intensities of the (010), (030), and (050) reflec- of the reciprocal lattice: (a) scanning between the reflections within the model of their magnetic nature; and (c) cal- tions (202) and (301) and (b) scanning between the reflec- culated intensities of the (010), (030), and (050) reflections tions (101) and (202). within the model of structural distortions.

Fig. 1a. These scanning paths cover the reciprocal lat- and (050) reciprocal lattice points in addition to the tice points at which Yamada et al. [1] measured the tem- reflections observed in [2]. perature dependences of nuclear superstructure reflections of type (3). As can be seen from Fig. 2, reflections The experimental neutron diffraction data obtained of series (3) are absent in our neutron diffraction pat- at a temperature of 4.2 K are shown by open circles in tern, which is in agreement with the results obtained in Fig. 3a. Figure 4 depicts the temperature dependence of [2]. the peak intensity of the (010) superstructure reflection. Consequently, there are good grounds to believe As can be seen from Fig. 4 and the results reported in that, if charge ordering actually occurs in the manganite our earlier work [3], the (010) weak reflection disap- crystal under investigation, the superstructure reflec- pears at a temperature of 200 K. This critical tempera- tions depicted by closed rhombs in Fig. 1 should pro- ture agrees well with the temperature Tcrit = 205 K vide information on this ordering. The original reflec- determined in [2] from the experimental temperature tions of this series are as follows: dependence of the intensity of the (030) reflection. Therefore, we can make the inference that the series of ()∗ q1 = 0 b 0 (5) () q2 = a*/2 0 c*/2 . (6) 200 According to the experimental results obtained in our earlier work [3], the intensities of superstructure reflections of type (6) involve both nuclear and magnetic components. Note that the nuclear component of 150 these reflections is rather complicated, because it is determined by the distortion of oxygen octahedra and their spatial arrangement in the crystal. In the present Counts 100 work, as in [2], primary attention was concentrated on T the origin of superstructure reflections of type (5), CO which are located along the b* direction in the reciprocal lattice. The choice of this direction was motivated 50 by the fact that the nuclear superstructure reflections 0 50 100 150 200 (0 k 0) (associated with oxygen ion displacements of T, K the Q2 type [4]) with odd indices k are forbidden by Pnma symmetry in the scattering pattern of the studied Fig. 4. Temperature dependence of the peak intensity of the crystal. It should be noted that, in this work, we mea- (010) superstructure reflection in the neutron diffraction sured the intensities of neutron scattering at the (010) pattern of the La0.85Sr0.15MnO3 single crystal.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 FERROMAGNETIC SUPERSTRUCTURE 121 weak reflections shown in Fig. 3a is associated with the 10000 3+ 2+ same superstructure observed in the La0.85Sr0.15MnO3 manganite orthorhombic single crystal. 8000 In order to elucidate the nature of the superstructure observed, we compared the experimental data pre- 6000 sented in Fig. 3a and the results of simple model calculations. Counts 4000 If we assume, for example, that the diffraction pattern shown in Fig. 3a is completely magnetic in nature, 2000 the intensity of superstructure (SS) reflections at different transferred momenta (κ = 4πsinθ/λ) can be represented by the standard relationship [5] 0 0 50 100 150 200 250 ∼ 2()κ ()κ T, K If SS P SS , (7) κ where f ( SS) is the magnetic form factor of the manga- Fig. 5. Temperature dependence of the intensity of the fer- κ romagnetic component of the (200) reflection in the neutron nese ion and P( SS) is the integral factor. In Fig. 3b, the diffraction pattern of the La0.85Sr0.15MnO3 manganite solid lines correspond to the integrated intensities of orthorhombic single crystal. Points are the experimental reflections of the series under consideration. These data taken from [2], and the dashed line corresponds to the intensities were calculated within the model of their results of calculations. magnetic nature. As can be seen from Figs. 3a and 3b, the results of calculations are in poor agreement with the experimental data. For example, the calculated magnetic and structural components. As is clearly seen intensity of the (030) reflection is almost half as high as from this figure, the magnetic contribution to the total the experimental intensity and the (050) reflection is intensity of the (010) superstructure reflection is greater virtually absent in Fig. 3b. than 90%. This circumstance is of fundamental impor- With the aim of fitting the experimental data to the tance, because it provides a means for constructing a calculated values, we assume that a magnetic density model of the magnetic structure of the LaÐSr man- wave superposes on a nuclear density wave with the ganite. same period. Under this assumption, the nuclear super- For this purpose, it is reasonable to compare the structure reflections can be associated only with longi- intensities of the structure (ST) ferromagnetic reflection tudinal ion displacements [with respect to the wave (020) and the superstructure reflection (010); that is, vector (5)] in the crystal. In this case, the dependence of F 2 2 2()κ ()κ the intensity of the superstructure reflections on the I()020 F()020 M()020 f ST P ST ------= ------2 2 2 , (9) transferred momentum can be represented by the I()010 F()M()f ()κ P()κ expression [5] 010 010 SS SS where F and F are the superstructure and struc- ∼ ()κ 2 ()κκ ∼ 2 ()κ (010) (020) I SSu P SS SSP SS , (8) ture scattering amplitudes, respectively, and M(010) and M are the vectors dependent on the mutual orienta- where u is the vector of ion displacements along the κ (020) SS tion of the scattering vector and the direction of the direction. In Fig. 3c, the dashed lines show the intensi- magnetic moments in the crystal [5]. ties of the (010), (030), and (050) reflections, which were calculated within the model of longitudinal struc- The magnitudes of the vectors M(hkl) for particular tural distortions of the crystal (according to our experi- crystallographic planes substantially depend on the mental data, the calculations were performed under the domain structure of a magnetic material. For example, assumption that the intensity of the (050) superstruc- in crystals with a random orientation of magnetic ture reflection is governed only by the nuclear compo- domains with respect to the crystallographic axes, the nent). effective value M2 is constant and equal to 2/3 for all By combining the curves depicted in Figs. 3b and (h k l) indices. For the manganite orthorhombic crystal 2 3c, it can be easily shown that the results of simple cal- studied in this work, the factor M()hkl can significantly culations carried out in our work are in good agreement differ from 2/3. As an example, we analyze the temper- with the experimental data presented in Fig. 3a. This ature dependence of the intensity of the ferromagnetic fact counts in favor of the magnetic and nuclear origin component of the (200) reflection (in this case, the scat- of the superstructure under consideration and is the first tering vector k coincides with the a* direction). The new result of the present work. experimental data taken from [2] are presented in Now, we discuss one of the most interesting results Fig. 5. The dashed line in Fig. 5 shows the dependence of analyzing the diffraction patterns displayed in Fig. 3. of the quantity proportional to the square of the magne- Let us consider the experimental (010) peak and its tization of the manganite orthorhombic single crystal.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 122 DUBININ et al. magnetic superstructure. The superstructure with ferro- 6 7 4+ magnetic order is directly associated with a 1/8-type Mn 3+ 4+ Mn3+ charge ordering of Mn and Mn ions in the studied compound. The magnetic moments of trivalent and tet- µ µ 2c0 5 ravalent manganese ions are equal to 4 B and 3 B, 2 respectively. The superstructure unit cell in the a0c0 plane is shown in Fig. 6. The closed and open circles in 4 8 this figure indicate the positions of the Mn4+ and Mn3+ ions, respectively. According to formula (6) for the 1 3 wave vector, the superstructure parameters are equal to 2a0 twice the initial orthorhombic unit cell parameters. The size of the superstructure unit-cell along the axis b0 = Fig. 6. A view of the unit cell of the ferromagnetic super- 2bC (where bC is the parameter of the perovskite cubic structure of the La0.85Sr0.15MnO3 manganite in the a0c0 unit cell) coincides with that of the initial unit cell. In plane. this case, only the Mn3+ ions are located at a height 1/2b0 and the arrangement of manganese ions in the unit cell at a height b is identical to that represented in This curve is calculated within the mean-field approxi- 0 mation for spin S = 2. As can be seen from Fig. 5, the Fig. 6. Thus, the unit cell of the ferromagnetic super- calculated and experimental data obtained in the low- structure consists of 16 manganese ions (their number- temperature range (10Ð120 K) are in close agreement. ing in the a0c0 plane is shown in Fig. 6), of which 2 ions However, at temperatures above 200 K, the experimen- are tetravalent and 14 ions have a valence of 3+. Within tal intensities substantially exceed the calculated val- this model of the magnetic cell, the structure factors can ues. This can be explained only by a considerable be represented in the form change in the domain structure of the ferromagnetic 2 2 () []()µ µ 2 F 010 = 24 B Ð 3 B , crystal. In other words, the factor M() for the (200) (10) hkl 2 []×µ()µ 2 reflection plane depends on the temperature and, conse- F()020 = 27 4 B + 3 B . quently, differs from 2/3. It is evident that the simplest By substituting expressions (10) into relationship (9) 2 2 ratios of the factors M()hkl in relationship (9) should be and taking into account the obvious equality M()010 = obtained for reflections corresponding to parallel crys- 2 2 2 M()020 , we obtain the following intensity ratio: tallographic planes, for example, M()020 /M()010 . ()F F 2 I()020 /I()010 exp = 373. (11) It is well known that the structure factor F()010 depends on the magnetic structure of the material. For Now, we compare the calculated intensity ratio (11) the manganite under investigation, there exist only two with the experimental data. The intensities of the (020) variants of magnetic ordering. One variant is the canted Bragg reflection in the neutron diffraction pattern of the spin system proposed in [2]. For a canted magnetic 3+ 2+ La0.85Sr0.15MnO3 single crystal are presented in Fig. 7. structure of the crystal, the structure factor contains the The closed and open circles indicate the results of mea- antiferromagnetic component as a fitting parameter. surements at temperatures of 4.2 and 300 K, respec- The antiferromagnetic component obtained by fitting to tively. (Recall that the Curie temperature of the studied the experimental data takes on a small fractional value. manganite is 240 K.) The integrated intensity of the In our opinion, the formation of a canted magnetic (020) structure ferromagnetic reflection is equal to the 3+ 2+ structure in the La0.85Sr0.15MnO3 compound at rela- difference between the integrated intensities of the reflections shown in Fig. 7. Since the times of measur- tively high temperatures (TCA = 200 K) is physically improbable. Indeed, it is common knowledge that the ing the reflections displayed in Figs. 3 and 7 are identi- Néel temperature of antiferromagnetic ordering of solid cal, we can easily obtain the experimental intensity 2+ ratio solutions in the La3+ Sr MnO system (T = 140 K 1 Ð x x 3 N ()F F at x = 0) decreases with an increase in the dopant con- I()020 /I()010 exp = 358. (12) centration [1]. This decrease in the temperature TN can The experimental error in the determination of the be explained by the increase in the concentration of intensity ratio (12) is equal to only 4.2%. The main Mn4+ ions, which encourage the formation of ferromag- advantage of our model of magnetic ordering resides in netic ordering. the fact that the close agreement between the experi- In the case of ferromagnetic ordering, the mental and calculated data is achieved (without resort- 3+ 2+ ing to fitting parameters) only with allowance made for 4+ 3+ La0.85Sr0.15MnO3 manganite can exhibit only a ferro- the 1/8-type stoichiometric ratio of Mn and Mn ions

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 FERROMAGNETIC SUPERSTRUCTURE 123

(0 2 0) transformation into the phase Q*, which was experi- 60000 mentally observed by Vasiliu-Doloc et al. [2]. 4.2 K 300 K 50000 4. CONCLUSION Thus, it was revealed that, in addition to the known 40000 ferromagnetic ordering at the Curie temperature TC = 3+ 2+ 30000 240 K, the La0.85Sr0.15MnO3 manganite orthorhombic

Counts single crystal exhibits a ferromagnetic superstructure π 20000 with the (010)2 /b wave vector. The ferromagnetic superstructure is observed in the studied crystal at temperatures ranging from 4.2 to 200 K. It was demon- 10000 strated that the formation of the ferromagnetic superstructure in this compound is directly associated with a 1/8-type charge ordering of Mn3+ and Mn4+ ions. 1.9 2.0 2.1 [0 ξ 0] ACKNOWLEDGMENTS Fig. 7. Intensities of the (020) Bragg reflection in the neu- This work was supported by the State Scientific and tron diffraction pattern of the La0.85Sr0.15MnO3 single crys- Technical Program “Topical Directions in the Physics tal at 4.2 and 300 K. of Condensed Matter: Neutron Investigations of Con- densed Matter” (project no. 107-19(00)-P-D01), the State Program of Support for Leading Scientific in relationships (10) and their ordered arrangement in Schools of the Russian Federation (project no. 00-15- the manganite crystal. It is this circumstance that 96581), and the Russian Foundation for Basic Research should be considered main evidence of the occurrence (project no. 99-02-16280). 3+ 2+ of charge ordering in the La0.85Sr0.15MnO3 manganite at temperatures below the transition point TCO = 200 K. REFERENCES This is the principal result of the present work. 1. Y. Yamada, O. Hino, S. Nohdo, et al., Phys. Rev. Lett. 77 (5), 904 (1996). In our opinion, charge ordering is responsible for the 2. L. Vasiliu-Doloc, J. W. Lynn, A. H. Moudden, et al., aforementioned fact that longitudinal waves of static Phys. Rev. B 58 (22), 14913 (1998). displacements of octahedral complexes (Fig. 3c) appear 3. S. F. Dubinin, V. E. Arkhipov, Ya. M. Mukovskiœ, et al., in the manganite compound at temperatures below TCO. Fiz. Met. Metallogr. 93 (3), 60 (2002). In turn, these waves directly affect the lattice parame- 4. J. L. Garcia-Munos, M. Suaaidi, J. Fontcuberta, and ters of the orthorhombic crystal. Actually, the energy of J. Rodriguez-Carvajal, Phys. Rev. B 55, 34 (1997). the crystal with ordered charges reaches a minimum 5. Yu. A. Izyumov, V. E. Naœsh, and R. P. Ozerov, Neutron when the charges are equally spaced in the lattice. This Diffractometry of Magnetic Materials (Atomizdat, Mos- situation can occur only in compounds with cubic sym- cow, 1981). metry. To put it differently, the formation of charge ordering in the orthorhombic phase Q' can cause its Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

MAGNETISM AND FERROELECTRICITY

The Dynamics of Domain Walls in an Easy-Plane Magnet in the Field of an Acoustic Wave V. S. Gerasimchuk and A. A. Shitov Donbass State Academy of Civil Engineering, Makeevka, Donetsk oblast, 86123 Ukraine e-mail: [email protected] Received February 20, 2002

Abstract—The drift of a 180° domain wall is studied in an easy-plane weak two-sublattice ferromagnet subject to an elastic-stress ﬁeld generated by an acoustic wave. The dependences of the drift velocity on the amplitude and polarization of the acoustic wave are found. The conditions of the drift of a stripe domain structure are determined. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tic subsystem on the magnetic subsystem, neglecting In this paper, we study the interaction between a the backward action of the magnetic system on the elas- domain wall (DW) and an elastic wave in a weak easy- tic system. We consider the wavelength of the acoustic plane ferromagnet. One of the most interesting magnets wave to be much larger than the DW width, which makes it possible not to consider the internal structure of this type is iron borate FeBO3 [1]. This compound is of particular interest, because FeBO is transparent in of the DW. It has been shown that there is a lattice 3 potential relief for the DW motion in a magnet [10]. If the visible spectral region and has a Néel temperature the DW velocity is sufficient to overcome the lattice TN = 348 K, which makes it possible to study its prop- barrier, the DW can move in fields which are much erties at room temperature. In particular, the magne- smaller than the coercive force associated with the toelastic properties of FeBO3 were studied in [2]. given potential barrier. In what follows, an external Elastic strains produced by an acoustic wave change field is necessary only to compensate for dynamic the energy of a DW and cause the DW to move [3]. losses. In this paper, we consider the steady motion of Steady-state motion of the DW is possible at velocities a DW. As indicated below, the DW drift velocity Vdr is smaller than the critical velocity V˜ , determining the much smaller than the critical velocity V˜ . magnetoelastic gap in the DW velocity spectrum [4]. The nonlinear macroscopic dynamics of a two-sub- The dependence of the velocity V˜ on the compression lattice, weak easy-plane ferromagnet in an acoustic stress in the plane of a sample was measured experi- wave can be described on the basis of the Lagrangian mentally in [5]. As the DW velocity reaches the critical density L{l} expressed in terms of the antiferromag- value V˜ , the 180° DW may decay into two 90° DWs netism unit vector l [11]. If the vector l is defined by two yielding a new 90° domain. angular variables θ and ϕ as The direct influence of an elastic wave on an iso- l + il ==sinθexp()iϕ , l cosθ, (1) lated 180° DW was experimentally studied in [6]. It x y z was established that the directional motion of the DW the Lagrangian density can be written in the form is possible under the action of a longitudinal acoustic  α wave. Such a directional motion of the DW was theo- {}θϕ, 2 ()θú 2 ϕ2 2 θ L = M0------+ ú sin retically investigated in [7Ð9]. In particular, the drift of 2c2 the DW in a uniaxial ferromagnet was analyzed on the basis of the averaged Slonczevski equations [7]. By α β 2 []()∇θ 2 ()∇ϕ 2 2 θ λ d 2 θ using the Lagrangian formalism and directly solving Ð --- + sin Ð --- Ð + ------cos the equations of motion of the magnetization vector, the 2 2 2δ drift of the DW in a weak ferromagnet, such as rare- γ [ θ()ϕ ϕ 2 θ earth orthoferrite, was studied [8, 9]. Ð sin2 uzxcos + uyzsin + uzzcos (2) 2 θ()]2 ϕ ϕ 2 ϕ 2. MODEL AND EQUATION OF MOTION + sin uxxcos + uxysin2 + uyysin Let us consider an arbitrary polarized acoustic  wave. We suppose that the wave is defined as an exter- ---+ 2λθsin2 cos2 ϕ  , nal field and take into account the influence of the elas- 

THE DYNAMICS OF DOMAIN WALLS 125 α λ ± where dots indicate differentiation with respect to time, where z0 = /4 is the DW thickness, R = 1 is the M0 is the magnitude of the sublattice-magnetization topological charge of the DW, and ρ = ±1 is a parame- αδ 1/2 vector, c = gM0( ) /2 is the minimum spin-wave ter defining the direction of rotation of the vector l in phase velocity (with δ and α being the homogeneous- the DW. and inhomogeneous-exchange coupling constants, The 180° DWs separating domains with opposite respectively), g is the gyromagnetic ratio, d is the Dzya- β magnetization directions in a stripe domain structure loshinskiœ interaction constant, is the uniaxial-anisot- have opposite topological charges R. The vector l can ropy constant, λ is the effective orthorhombic-anisot- γ rotate in the domain wall from ÐR to +R (or in the oppo- ropy constant, uik is the elastic-strain tensor, and is the site direction) either through the positive or the negative magnetoelastic constant. direction of the y axis; the direction of the rotation is Let us introduce a dissipative function defined by the parameter ρ. For this reason, to neigh- ˜ ˜ boring DWs in the stripe domain structure with the vec- λM . λM 2 F ==------0I2 ------0()θú + ϕú 2 sin2 θ , (3) tor l rotating in the xy plane correspond the values ±∞ − 2g 2g lx(z = ) = +R and one of the two values ly(z = 0) = ±ρ ˜ . For the isolated DW studied in [6], we have R = +1 where λ is the Gilbert damping constant. and ρ = +1. With due regard for the dynamic losses, the equations of motion are expressed in terms of the angular variables as 3. AN ACOUSTIC WAVE PROPAGATING IN THE PLANE OF A DOMAIN WALL α∆θ1 úúθ θθα1 ϕ2 ()∇ϕ 2 Ð ---- + sin cos ---- ú Ð c2 c2 In order to solve equations of motion (4) and (5), we consider a monochromatic acoustic wave propagating 2 ω λϕ2 β λ d in the DW plane: u(r⊥, t) = u0exp(ik⊥r⊥ Ð i t), where -+4 cos ++Ð 2 ----δ- (4) k⊥r⊥ = kxx + kyy. Let us introduce a collective variable Z(r⊥, t) as the coordinate of the DW center. The DW γ [ θ()2 ϕ ϕ 2 ϕ Ð sin2 uxxcos + uxysin2 + uyysin Ð uzz drift velocity is determined as the instantaneous velocity V(r⊥, t) = Zú (r⊥, t) averaged over a period of oscilla- λ˜ + 2cos 2θ()]u cosϕ + u sinϕ = ------θú , (), xz yz tions, Vdr = V r⊥ t (the bar means averaging over a gM0 period of oscillations). Supposing the amplitude of the 2 α d 2 2 α∇[] ∇ϕsin θ Ð ------()ϕú sin θ Ð 4λθsin sinϕϕcos acoustic wave to be sufficiently small, we seek a solu- c2 dt tion to the system of Eqs. (4) and (5) in terms of power- series expansions: Ð γθ[][sin 2 {}()u Ð u sin2ϕ + 2u cos2ϕ (5) yy xx xy θ(), π θ ()θξ,, ()…ξ,, r t = /2 + 1 r⊥ t + 2 r⊥ t + , λ˜ θ()]ϕ ϕ ϕ 2 θ + sin2 uzycos Ð uzxsin = ------ú sin . ϕ(), ϕ()ξ ϕ()ϕξ,, ()…ξ,, gM0 r t = 0 + 1 r⊥ t ++2 r⊥ t , The equations of motion have two types of solutions … describing planar 180° DWs in the ground state. Since VV= 1 ++V 2 , (8) the condition (β Ð 2λ + d2/δ) > 4λ > 0 is fulfilled, the where ξ = z Ð Z(r⊥, t) and indices n = 1, 2, … indicate vector l is collinear to the x axis in the absence of exter- the order of smallness with respect to the acoustic-wave nal fields and a domain wall in which l rotates in the xy ϕ ξ plane is stable far away from the spin-reorientation amplitude. The function 0( ) describes the motion of θ θ π the nondisturbed DW and has a structure similar to that region [12]. For this DW, we have = 0 = /2 and the ϕ of the static solution to Eq. (7). The higher order func- equation for the angular variable 0(z) takes the form θ ϕ tions n and n (n = 1, 2 …) describe motion of the dis- αϕ λϕ ϕ 0'' Ð0,4 sin 0 cos 0 = (6) turbed DW and the excitation of spin waves. where the prime indicates differentiation with respect ϕ ∞ to z. Using the boundary conditions 0(Ð ) = 0, ϕ ∞ π 3.1. Linear Approximation 0(+ ) = , we find the magnetization distribution in the static Bloch-type 180° DW: The equations of the first-order perturbation theory have the form 1 1 Ð1 z ϕ' ==----Rsinϕ ()z ----Rρcosh ----, 0 z 0 z z 0 0 0 ∂2 ω ∂ ˆ 1 r ϕ ()ξ,, z (7) L + ------+ ------1 r⊥ t ϕ () ---- ω2 ∂ 2 ω2 ∂t cos 0 z = ÐRtanh , 0 t 0 z0

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 126 GERASIMCHUK, SHITOV

∞ ϕ ()ξ ∂2 ∂ ()ξ ()ξ ()ξ Rsin 0 Z1 Z1 2 2 tanh /z0 sin p Ð pz0 cos p = ------------+ ω ------Ð ω z ∆⊥Z (9) D ()ξ = ------d()pz , 2 2 r ∂ 0 0 1 2 ∫ π 0 z ω ∂t t pz0 0 0 Ð∞ ()λ Ð q sinh------p 2 γ Ð ------[]()u Ð u sin2ϕ ()ξ + 2u cos2ϕ ()ξ , λ yy xx 0 xy 0 ω 2 ωω 2 qq===+ iq , q ------, q ------r . 1 2 1 ω 2 2 0 ω ∂2 ω ∂ 0 ˜ σ 1 r θ ()ξ,, L ++------+ ------1 r⊥ t ω2 ∂t2 ω2 ∂t Supposing the amplitude of the Goldstone mode to 0 0 (10) be zero [8, 9], the following equation for the determination of the DW velocity can be derived from Eq. (9): γ = ------()u cosϕ + u sinϕ , λ xx 0 yz 0 ∂2 ∂ 2 Z1 Z1 2 2 ------+0.ω ------Ð ω z ∆⊥Z = (12) 2 2 2 2 2 Ð1 2 r ∂ 1 0 1 where ∆⊥ = ∂ /∂x + ∂ /∂y , σ = (β Ð 2λ + d /δ)(4λ) , ∂t t ω 0 = c/z0 is the activation frequency of the lowest This equation indicates that, in the linear approxi- ω λ˜ δ branch of the bulk-spin-wave spectrum, r = gM0/4 mation with respect to the acoustic-wave amplitude, the is the characteristic relaxation frequency, k⊥ is the wave sound does not cause the DW to move but results in the vector of the sound wave, and k⊥ = |k⊥| = ω/s, with s excitation of spin waves described by Eqs. (11). These being the acoustic-wave velocity. excitations are caused by both longitudinal and transverse acoustic waves. ˆ 2 2 ξ2 2 ()ξ The operator L = Ðz0 d /d + 1 Ð 2/cosh /z0 has the well-known eigenfunctions 3.2. Second-Order Approximation 1 ξ f ()ξ = ------cosh Ð1---- , 0 z The second-order-approximation equation for the 2z0 0 ϕ ξ function 2( , r⊥, t) has the form ξ ()ξ 1 ()ξ f p = ------tanh---- Ð ipz0 exp ip ∂2 ω ∂ z0 1 r b p L Lˆ + ------+ ------ϕ ()ξ,,r⊥ t ω2 ∂ 2 ω2 ∂t 2 λ λ 0 t 0 (L is the crystal length) and eigenvalues 0 = 0 and p = b2 = 1 + p2 z2 . We seek a solution to the first-order- p 0 ∂2 ∂ ϕ ∂ approximation equations (9) and (10) in the form of Z2 Z2 2 2 Rsin 0 2 Z1 = ------+ ω ------Ð ω z ∆⊥Z ------+ ------power-series expansion in terms of the complete set of ∂ 2 r ∂t 0 0 2 ω2 ω2∂t t 0z0 0 eigenfunctions of the operator Lˆ . The result is × ϕú ' ()ξ,, θ ()θξ,, ' ()ξ,, ϕ  iγ 1 r⊥ t Ð 2z0 1 r⊥ t 1 r⊥ t Rsin 0 ϕ ()ξ, ρ  [()()ξ 1 t = R Re ------λ kyu0y Ð kxu0x D1 8 ∂ 2 ϕ 1 Z1 sin2 0 2 + ∆⊥Z Ð ------+ ϕ ()ξ,,r⊥ t sin2ϕ 1 c ∂t 2 1 0  ()()]ξ ()ω -Ð kxu0y + kyu0x D2 exp ik⊥r⊥ Ð i t ,  (13) ϕ' ()ξ,,∂2 ∂ 1 r⊥ t Z1 ω Z1 ω2 2∆ + ------------+ r------Ð 0z0 ⊥Z1 (11) ω2 ∂t2 ∂t γR  ik u 0 θ ()ξ,  y 0z ϕ ()ξ 1 t = ------λRe ------σ - sin 0 4  Ð q γ + ------[()θu cosϕ Ð u sinϕ ()ξ,,r⊥ t 2λ zy 0 xz 0 1 ik u  x 0z ϕ ()ξ ()ω  + ------cos 0 exp ik⊥r⊥ Ð i t , + ()ϕ()u Ð u cos2ϕ + 2u sin2ϕ ()]ξ,,r⊥ t . 1 + σ Ð q  xx yy 0 xy 0 1 We do not present the second-order-approximation where θ ξ equation for the function 2( , r⊥, t), because it does not ∞ tanh()ξ/z cos()pξ + pz sin()pξ define the DW motion. Since we are interested in the D ()ξ = ------0 0 -d()pz , induced DW motion only, it will suffice, when deter- 1 ∫ πpz 0 ()λ 0 mining V2(t), to find the coefficient corresponding to Ð∞ p Ð q cosh------ϕ ξ 2 the Goldstone mode in the expansion of 2( , r⊥, t) in

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 THE DYNAMICS OF DOMAIN WALLS 127

y y

u k k u ϕ ϕ s z s z

x x

Fig. 1. Transverse wave. The displacement vector is perpen- Fig. 2. Transverse wave. The displacement vector lies in the dicular to the DW plane. DW plane.

ˆ µ ω terms of the eigenfunctions of the operator L and to set where i( ) are the nonlinear DW mobilities, this coefﬁcient equal to zero [8, 9]. As a result, we have q2 µ ()ω = Ð µ------2 -, ∂2Z ∂Z 1 0[]σ()σ 2 2 []()2 2 ρ 2 ω 2 ω2 2∆ 1 + Ð q1 + q2 Ð q1 + q2 R ------+ r------Ð 0z0 ⊥Z2 ∂t2 ∂t (14) τ (17) µ ()ω ≈ 2 ------. ()ω ()ω ()2 2 = NN+ 1 exp 2i t + N2 exp Ð2i t . 1 Ð q1 + q2

µ π γ2 λλ˜ τ ηµ ≈ × 5 Here, Here, 0 = z0g M0/(16 ), = 0 1.2 10 cm/s, and η is a numerical parameter. In estimating the drift ∞ θ θ velocity, we used the following parameters of iron 2 1* 1' Ð4 σ ≈ × 4 × N = ω z 2z Re dξ f ()ξ ------sinϕ borate [11, 13]: z0 ~ 10 cm, 2 10 , g = 2.94 0 0 0 ∫ 0 2 0  6 Ð1 γ 2 7 3 µ ≈ × 19 Ð∞ 10 (s Oe) , M0 ~ 10 erg/cm , 0 2.35 10 cm/s, ω ≈ × 10 Ð1 ω ≈ × 6 Ð1 0 1.4 10 s , and r 9.2 10 s . ϕ ()ϕ ∂2 ∂ sin2 0 1* ' Z1 Z1 2 2 Since the wavelength of the acoustic wave is much Ð ϕ*ϕ ------Ð ------------+ ω ------Ð ω z ∆⊥Z 1 1 4z ω2 ∂ 2 r ∂t 0 0 1 larger than the DW width (the relationship ω Ӷ 1010 sÐ1 0 4 0z0 t Ӷ corresponds to this condition), we have q1, q2 1. γ (15) Hence, µ (ω) depends on the frequency only weakly. [()θϕ ϕ *()ξ,, 2 Ð ------λ uzycos 0 Ð uxzsin 0 1 r⊥ t 2 At the sound velocity s = 4.71 × 105 cm/s [11] and Ð5 the limiting value of the strain tensor ku0 ~ 10 , the DW  µ ω Ð5 ()ϕ()ϕ ϕ ()]ξ,, drift velocity due to 2( ) can be as high as 10 cm/s. + uxx Ð uyy cos2 0 + 2uxysin2 0 1* r⊥ t ,  The presence of the factor Rρ in Eq. (16) for the DW drift velocity indicates that the stripe domain structure where asterisks imply complex conjugation. We do not can drift as a whole. For this to happen, the vector l present explicit expressions for the coefﬁcients N1 and should rotate from one domain to another in the same N2, because they vanish after subsequent averaging of direction. the solution to Eq. (14). Integrating Eq. (14) and aver- Now, let us elucidate the dependence of the DW aging the obtained solution over a period of the acoustic drift velocity on the acoustic-wave polarization. Since wave yields the following expression for the DW drift the wave propagates in the plane of the boundary (xy velocity V = V = ∂Z /∂t: plane), the wave vector can be represented as k = (kx, ky, dr 2 2 ϕ ϕ 0) = k(cos s, sin s, 0). The following polarizations of ρµ ()ω ()2 ρµ ()ω the wave are possible (Figs. 1Ð3). V dr = R 1 kxky u0z + R 2 (16) (1) A transverse wave with the displacement vector × []()2 2 ()2 2 kxky u0y Ð u0x + u0yu0x ky Ð kx , perpendicular to the DW plane: u = u0(0, 0, 1). In this

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 128 GERASIMCHUK, SHITOV

above can be generalized to this case. In the ﬁrst-order y perturbation theory, the sound also does not generate the DW drift but excites spin waves described by the relations u ϕ ()ξ, 1 t = 0, ϕ k s πγ 2 (21) θ ()ξ, k z0 {}()ξ []()ω 1 t = ------λ Re B exp ikZÐ t , ϕ 8 s z where

()ξ z0[]ρ ()ξ ()ξ B = Ð---π- u0y D3 + iRu0x D4 x ()ξ ()ξ + b1 f k + b2 f 0 , Fig. 3. Longitudinal wave. ∞ ()Ωξ (), ()ξ f p pq D3 = L ∫ ------π ()dp, z0 pkÐ case, the drift velocity given by Eq. (16) can be repre- Ð∞ cosh------sented as 2 ρ ∞ R µ ()ω ()2 ϕ f ()Ωξ ()pq, V dr = ------1 ktu0 sin2 s, (18) ()ξ p 2 D4 = L ∫ ------π ()dp, z0 pkÐ ω Ð∞ sinh------where kt = /st, with st being the velocity of transverse 2 sound. Ω , (2) A transverse wave with the displacement vector 2iR Lu0x ()kq ϕ ϕ b1 = Ð------π -, in the DW plane: u = u0(Ðsin s, cos s, 0). It follows kz0 from Eq. (16) that the drift velocity in this case is  Rρ 2 2z Ru iρu V = ------µ ()ω ()k u sin4ϕ . (19) b = ------0 ------0x - + ------0y - , dr 2 2 t 0 s 2 σ Ð qπkz πkz cosh------0 sinh------0 (3) A longitudinal wave with the displacement vec- 2 2 tor u = u (cosϕ , sinϕ , 0). The drift velocity of this 0 s s Ω λ λ σ Ð1 λ 2 2 wave is (n, q) = {n ( n Ð q + )} , k = kz, and n = 1 + n z0 ρ (n = p, k). R µ ()ω ()2 ϕ V dr = Ð------2 klu0 sin4 s. (20) It is seen from these expressions that, in contrast to 2 the case of the sound propagating in the DW plane, spin ω waves are excited only by transverse components of the Here, kl = /sl, where sl is the speed of the longitudinal sound. field. It follows from Eqs. (18)Ð(20) that in the case of a In the second-order perturbation theory, the DW sound wave propagating along a direction parallel to drifts with velocity the DW plane, the DW can drift in both a longitudinal µ ()ω ()2 V dr = xx ku0x and a transverse acoustic wave. Equations (19) and (20) (22) indicate that the drift of the domain structure defined by ρµ ()ω ()()µ()ω ()2 µ ω + R xy ku0x ku0y + yy ku0y , 2( ) occurs in opposite directions depending on the acoustic-wave type (transverse or longitudinal). where, at ω Ӷ 1010 sÐ1, we have ()2 kz0 kz0q2 4. AN ACOUSTIC WAVE PROPAGATING µ ==η µ ------, µ Ðη µ ------, xy 1 0 σ2 xx 2 0 σ2 PERPENDICULAR TO THE PLANE (23) OF A DOMAIN WALL kz q µ = Ðη µ ------0 -2. Let us find solutions to the equations of motion (4) yy 3 0 σ2 and (5) for the case when the sound wave propagates η η η perpendicularly to the DW plane, u = u0exp[i(kzz Ð The numerical coefficients 1 ~ 1, 2 ~ 0.05, and 3 ~ ωt)]. We introduce the collective coordinate of the DW 0.4 are obtained as a result of estimating the expres- µ ω center Z(t), which, in contrast to the case considered sions for ij( ) in the long-wave limiting case. The non- µ above, does not depend on r⊥. The theory developed linear mobility xy gives the main contribution the drift

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 THE DYNAMICS OF DOMAIN WALLS 129 velocity. The drift velocity due to this mobility is Vdr ~ 3. V. G. Bar’yakhtar and B. A. Ivanov, Fiz. Met. Metall- 0.1 cm/s. oved. 39 (4), 478 (1975). 4. A. K. Zvezdin, V. V. Kostyuchenko, and A. A. Mukhin, Preprint No. 209, FIAN SSSR (Lebedev Institute of 5. CONCLUSION Physics, Academy of Sciences of USSR, Moscow, 1983). On the basis of the analysis carried out, it has been 5. M. V. Chetkin, V. V. Lykov, and V. D. Tereshchenko, Fiz. established that in a weak easy-plane ferromagnet, an Tverd. Tela (Leningrad) 32 (3), 939 (1990) [Sov. Phys. isolated DW and a stripe domain structure can drift in Solid State 32, 555 (1990)]. an acoustic wave propagating along a direction either 6. M. V. Chetkin, V. V. Lykov, A. A. Makovozova, and perpendicular or parallel to the DW plane. In the former A. G. Belonogov, Fiz. Tverd. Tela (Leningrad) 33 (1), case, the directional motion of DWs is caused by the 307 (1991) [Sov. Phys. Solid State 33, 177 (1991)]. transverse components of the acoustic wave. The drift 7. S. I. Denisov, Fiz. Tverd. Tela (Leningrad) 31 (11), 270 velocity is maximum in an acoustic wave with two non- (1989) [Sov. Phys. Solid State 31, 1992 (1989)]. zero transverse components of the amplitude, u0x and 8. V. S. Gerasimchuk and A. L. Sukstanskiœ, Zh. Éksp. Teor. u0y. When the acoustic wave propagates in the DW Fiz. 106 (4), 1146 (1994) [JETP 79, 622 (1994)]. plane, the DW drift is due to both transverse and longi- 9. V. S. Gerasimchuk and A. L. Sukstanskiœ, Zh. Éksp. Teor. tudinal components of the ﬁeld. In this case, however, Fiz. 118 (6), 1384 (2000) [JETP 91, 1198 (2000)]. the DW drift velocity is much smaller than in the case 10. T. Egami, Phys. Status Solidi B 57, 211 (1973). of the acoustic wave propagating perpendicular to the 11. B. A. Ivanov, A. L. Sukstanskiœ, and A. V. Vershinin, Fiz. DW plane. Tverd. Tela (Leningrad) 33 (7), 1978 (1991) [Sov. Phys. Solid State 33, 1114 (1991)]. 12. A. L. Sukstanskiœ, Fiz. Tverd. Tela (Leningrad) 27 (11), REFERENCES 3509 (1985) [Sov. Phys. Solid State 27, 2119 (1985)]. 1. I. Bernal, C. W. Struck, and J. G. White, Acta Crystal- 13. M. P. Petrov, G. A. Smolenskiœ, A. P. Paugurt, et al., Fiz. logr. 16, 849 (1963). Tverd. Tela (Leningrad) 14 (1), 109 (1972) [Sov. Phys. Solid State 14, 87 (1972)]. 2. A. M. Kadomtseva, R. Z. Levitin, Yu. F. Popov, et al., Fiz. Tverd. Tela (Leningrad) 14 (1), 214 (1972) [Sov. Phys. Solid State 14, 172 (1972)]. Translated by A. Poushnov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 130Ð137. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 124Ð130. Original Russian Text Copyright © 2003 by Aliev, Abdulvagidov, Batdalov, Kamilov, Gorbenko, Amelichev, Kaul’, Kurbakov, Trunov.

MAGNETISM AND FERROELECTRICITY

Effect of a Magnetic Field on the Thermal and Kinetic Properties of the Sm0.55Sr0.45MnO3.02 Manganite A. M. Aliev*, Sh. B. Abdulvagidov*, A. B. Batdalov*, I. K. Kamilov*, O. Yu. Gorbenko**, V. A. Amelichev**, A. R. Kaul’**, A. I. Kurbakov***, and V. A. Trunov*** * Institute of Physics, Dagestan Scientiﬁc Center, Russian Academy of Sciences, ul. 26 Bakinskikh Komissarov 94, Makhachkala, 367003 Dagestan, Russia ** Moscow State University, Vorob’evy gory, Moscow, 119899 Russia *** St. Petersburg Institute of Nuclear Physics, Russian Academy of Sciences, Gatchina, Leningrad oblast, 188300 Russia e-mail: [email protected] Received November 27, 2001; in ﬁnal form, April 1, 2002

Abstract—The temperature and magnetic-field dependences of the heat capacity, thermal conductivity, thermopower, and electrical resistivity of the Sm0.55Sr0.45MnO3.02 ceramic material are studied in the temperature range 77–300 K and in magnetic fields up to 26 kOe. It is revealed that the quantities under investigation exhibit anomalous behavior due to a magnetic phase transition at the Curie temperature TC. An increase in the magnetic ∆ field strength H leads to an increase in the Curie temperature TC and a jump in the heat capacity Cp at TC. The temperature dependences of the measured quantities are characterized by hystereses that are considerably suppressed in a magnetic field of 26 kOe and depend neither on the thermocycling range nor on the rate of change in the temperature. The thermal conductivity K at temperatures above TC shows unusual behavior for crystalline solids (dK/dT > 0) and, upon the transition to a ferromagnetic state, drastically increases as a result of a decrease in the phonon scattering by JahnÐTeller distortions. It is demonstrated that the hystereses of the studied properties of the Sm0.55Sr0.45MnO3.02 manganite are caused by a jumpwise change in the critical temperature due to variations in the lattice parameters upon the magnetic phase transition. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION respect, many researchers are inclined to believe that the great variety of properties of manganite materials Rare-earth substituted manganites with a perovskite cannot be interpreted in the framework of one particular structure are characterized by a unique combination of model. Our interest in the Sm1 Ð xSrxMnO3.02 (x = 0.45) properties inherent in metals and dielectrics (and also manganite studied in the present work is associated properties exhibited by ionic and covalent crystals), the with the following two factors: (i) owing to the large coexistence of ferromagnetic and antiferromagnetic difference between the ionic radii of samarium and ordering, phase separation, and giant magnetoresis- strontium (rSm = 1.132 Å and rSr = 1.310 Å), the inti- tance effects. This explains the considerable interest mate interrelation between the electronic, magnetic, expressed by researchers in these physical objects. and phonon subsystems (with the leading role played However, up to now, the nature of the anomalous prop- by the magnetic interactions) most clearly manifests erties observed in manganites has remained unclear. In itself in this compound, and (ii) the ferromagnetic particular, the giant magnetoresistance effect in con- metallic and antiferromagnetic dielectric phases can ventional magnetic semiconductors is usually attrib- coexist in this system [8, 9]. The aim of the present uted to the strong indirect exchange interaction and the work was to reveal these specific features. For this exchange-induced two-phase magnetic state. For man- purpose, we experimentally investigated the tempera- ganites, this pattern is complicated by manifestations of ture and magnetic-field dependences of the heat capac- the JahnÐTeller effect and a relative softness of the ity, thermal conductivity, thermopower, and electrical crystal lattice. As a consequence, the crystal structure resistivity of the Sm Sr MnO ceramic material. of these compounds can undergo transformations under 0.55 0.45 3.02 the action of temperature, magnetic fields, and pressure. A large number of models have been proposed for 2. SAMPLE PREPARATION interpreting the unusual physical properties of mangan- AND EXPERIMENTAL TECHNIQUE ites (see, for example, reviews [1Ð3]). In particular, the model of phase separation offers a satisfactory explana- The sample to be studied was prepared according to tion of many experimental findings. However, this the procedure described earlier in [10]. Here, it should model also runs into certain problems [4Ð7]. In this only be mentioned that the oxygen stoichiometric index

EFFECT OF A MAGNETIC FIELD ON THE THERMAL AND KINETIC PROPERTIES 131

100

80 K

2 , J/mol p C

100 150 200 250 T, K

Fig. 1. Temperature dependences of the heat capacity of the Sm0.55Sr0.45MnO3.02 manganite in magnetic ﬁelds H = (1) 0 and (2) 26 kOe.

was determined by iodometric titration [11] and proved Sm0.55Sr0.45MnO3.02 sample both in the presence and in to be 3.02(1). The content of Mn4+ ions was determined the absence of a magnetic field exhibit anomalies with from data on the oxygen stoichiometry (with due regard a hysteresis in the vicinity of the Curie temperature TC. for both the accuracy in the determination of the oxy- The main specific features in the behavior of the heat gen stoichiometry and the electroneutrality condition) capacity are as follows: the critical temperature and amounted to 47Ð49% of the total content of manga- depends on the magnetic field strength and the direction nese ions. The heat capacity was measured using ac cal- of the change in the temperature; in this case, an orimetry [12]. The thermal conductivity was deter- increase in the magnetic field strength is accompanied mined either by the method of stationary heat flux or as by an increase in the critical temperature and a decrease the product of the heat capacity Cp and the thermal dif- in the hysteresis width. Since the hysteresis in magnetic fusion coefficient η according to the technique materials could be caused by long-term relaxation, the described in [12]. This technique provides a means of measurements were performed at different rates of performing measurements with a small temperature change in the temperature and within different ther- ∆ gradient across the sample ( T < 0.05 K), which is mocycling ranges; however, the temperatures TC and, especially important in studies in the vicinity of phase ∆ hence, the values of TC remained unchanged. Note transitions. The electrical resistivity was measured also that experimental points in the heat capacity curve using the standard four-point probe method. The mag- for the Sm0.55Sr0.45MnO3.02 manganite at temperatures netic field in the course of measurements was always below and above the transition cannot be approximated directed perpendicularly to the heat flux. by a curve described by the Debye interpolation formula, as is the case with manganites of other composi- 3. RESULTS AND DISCUSSION tions [13, 14]. This suggests that the phonon spectrum of the Sm0.55Sr0.45MnO3.02 manganite substantially Figure 1 displays the temperature dependences of changes at the Curie temperature TC and that the the heat capacity of the Sm0.55Sr0.45MnO3.02 manganite observed hystereses are associated with these changes. in magnetic fields H = 0 and 26 kOe in the temperature range 80Ð280 K upon cooling and heating. The mea- The temperature dependences of the electrical resis- surements of the heat capacity in a magnetic field were tivity of the Sm0.55Sr0.45MnO3.02 manganite in magnetic carried out after cooling of the sample to 77 K without fields H = 0 and 26 kOe upon cooling and heating are a magnetic field. As can be seen from Fig. 1, the tem- plotted in Fig. 2. As can be seen from this figure, the perature dependences of the heat capacity of the dependence of the electrical resistivity also exhibits a

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

132 ALIEV et al.

8 very sharply, whereas the reverse crossover (upon heating) proceeds in a gradual manner (Fig. 2). 6 H ρ

1 / In our earlier work [10], the experimental results 10 4 were interpreted in the framework of the electron ∆ρ – 2 model of phase separation. Within this model, the

cm Tc Sm Sr MnO manganite at T < T can be repre- 0 0.55 0.45 3.02 C Ω 1 sented as a dielectric matrix with embedded ferromag- , 100 120 140 160 ρ 100 T, K netic metallic clusters. However, detailed analysis of more recent experimental data demonstrated that the 2 results of measurements have defied unambiguous interpretation only in terms of the phase separation model. According to neutron diffraction investigations 10–1 performed recently (the results of these investigations 100 150 200 250 will be published in a separate work), no phase separa- T, K tion occurs in the Sm0.55Sr0.45MnO3 manganite. At temperatures below TC, the Sm0.55Sr0.45MnO3 compound Fig. 2. Temperature dependences of the electrical resistivity undergoes a phase transition to a homogeneous ferro- of the Sm0.55Sr0.45MnO3.02 manganite in magnetic fields magnetic state. This state is characterized by the mag- H = (1) 0 and (2) 26 kOe. The inset shows the temperature netic moment M = 3.36(5)µ /mol (where µ is the Bohr ρ ρ ρ B B dependence of the relative magnetoresistance ( 0 Ð H)/ H magneton) upon saturation at T = 4 K and corresponds upon heating in a field of 26 kOe. to a nearly complete magnetic ordering without indications of the antiferromagnetic phase below TC. Figure 3 displays the neutron diffraction patterns measured for * 152 152 12000 * the Sm0.55Sr0.45MnO3.02(1) manganite with a Sm isotope at different temperatures and a neutron wave- * * * * length of 2.343 Å. The measurements were carried out 9000 2 K on a G4.2 high-resolution neutron powder diffractome- 50 ter situated in the neutron guide room of an ORPHEUS 100 105 reactor (LLB, Saclay, France). The crystal structure for 6000 110 all temperatures is described by the space group Pnma. 115 120 No additional peaks attributed to antiferromagnetic 125 phases are observed at temperatures below T ≈ 130 K. 130

Intensity, arb. units 3000 150 The result obtained turned out to be somewhat unex- 200 pected, because the investigations into the magnetic 250 300 structure and transport properties of a sample with a close composition, namely, 154Sm Sr MnO , 0 20 40 60 80 100 120 140 0.55 0.45 3 2θ, deg showed that, in this manganite, the low-temperature phase is magnetically inhomogeneous and the magnetization is substantially less than the theoretical value Fig. 3. Experimental neutron diffraction patterns of the [15–17]. In order to confirm the reliability of the results 152 Sm0.55Sr0.45MnO3.02(1) manganite at different tempera- obtained, we performed neutron diffraction analysis of tures. Asterisks mark the first (most intense) reflections a sample with a slightly different composition, namely, attributed to the ferromagnetic phase. Arrows indicate positions at which the first reflections associated with the A-type Sm0.525Sr0.475MnO3. For this compound, the results antiferromagnetic phase (determined from the neutron dif- were identical to those obtained earlier, which sug- fraction pattern of the manganite with x = 0.60) should be gested the formation of a collinear ferromagnetic struc- observed. ture. Therefore, in SmÐSr manganites, the homogeneous ferromagnetic state is observed in a narrow concentration range (in the vicinity of x = 0.45). hysteresis. This hysteresis is suppressed by the mag- Furthermore, in [8, 18], high magnetization (~3.3µ ) netic field and shifted to the high-temperature range, as B was observed for a sample with a very close composi- is the case with the other quantities measured in our tion (x = 0.44). It seems likely that the high sensitivity work. It can also be seen from Fig. 2 that the external of the physical properties to doping level is a specific magnetic field strongly suppresses the electrical resis- feature of compounds in the Sm Sr MnO system. tivity, thus giving rise to a giant magnetoresistance 1 Ð x x 3 effect, which is maximum in the vicinity of the Curie Moreover, the experimental neutron diffraction data temperature TC (see the inset in Fig. 2). Moreover, the obtained at temperatures close to TC indicate a drastic crossover from semiconductor conductivity to metallic decrease in the molar volume (lattice contraction) of conductivity in the absence of a magnetic field occurs the Sm0.55Sr0.45MnO3.02 manganite upon transition to

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EFFECT OF A MAGNETIC FIELD ON THE THERMAL AND KINETIC PROPERTIES 133 the ferromagnetic phase. In our opinion, this is associ- 7 68 ated with the giant spontaneous magnetostriction. The T = 147 K space group (Pnma) remains the same over the entire temperature range; i.e., no structural phase transition ρ 66 occurs in accordance with the Ehrenfest and Landau 6 K

cm classiﬁcations of phase transitions. The lattice parame- Ω ters change in a speciﬁc manner: the rhombic base of , 64 ρ J/mol

the unit cell contracts sharply (the temperature depen- 5 ,

Hc p dences of the parameters a and c exhibit jumps), C whereas the parameter b changes only slightly. The 62 C transition to the ferromagnetic phase is attended by 4 p ordering of MnO6 octahedra. As a consequence, JahnÐ Teller distortions of the lattice jumpwise decrease. 60 0 5 10 15 20 25 A change in the interatomic distances and angles in H, kOe the plane of the unit cell base leads to an increase in the energy of the indirect exchange interaction between Fig. 4. Field dependences of the heat capacity and the elec- manganese ions and, hence, in the Curie temperature TC trical resistivity of the Sm Sr MnO manganite. corresponding to the ferromagnetic phase. The reverse 0.55 0.45 3.02 transition from the ferromagnetic phase to the paramagnetic phase occurs in the crystal lattice with a different (higher, as follows from our data) temperature 2.2 2 T C' . This results in a hysteresis in the behavior of the studied properties of the Sm Sr MnO mangan- 2.0 0.55 0.45 3.02 1 K ∆ ' ite. The decrease in the difference TC = T C Ð TC in a 1.8 magnetic field can be explained by the following circumstance. Apart from the increase in the temperatures , W/m 1.6 ' K TC and T C in a magnetic field, the lattice parameters of 1.4 the Sm0.55Sr0.45MnO3.02 manganite at temperatures below and above the phase transition point become 1.2 closer to each other in magnitude owing to the giant 100 150 200 250 magnetostriction, as is the case with a typical ferromag- T, K net. Consequently, as follows from the above reason- Fig. 5. Temperature dependences of the thermal conductiv- ing, the temperatures TC and T C' also become closer to ∆ ity of the Sm0.55Sr0.45MnO3.02 manganite in magnetic each other and the difference TC decreases. In this fields H = (1) 0 and (2) 26 kOe. respect, it should be noted that the notion of the Curie temperature TC used in analyzing the magnetic phase transitions becomes conventional for systems with magnetic phase and that the charge carriers in this large hystereses. One further feature in the behavior of phase possess a high mobility. the heat capacity of the Sm0.55Sr0.45MnO3.02 manganite ∆ is an increase in the jump Cp(T) in a magnetic field. In Although it is universally accepted that the giant our opinion, this can be caused by a drastic increase in magnetoresistance effect and other unusual properties the contribution of magnetic fluctuations to the heat of manganites should be interpreted in terms of the dou- capacity as the critical temperatures of the ferromag- ble exchange mechanism with due regard for the elec- netic and paramagnetic phases approach each other. tronÐphonon interaction and that investigation of the thermal conduction is a proven way of revealing the The field dependences of the heat capacity and the specific features of this interaction, the thermal conduc- electrical resistivity (Fig. 4) are also worthy of notice. tion in manganites remains poorly understood [19Ð22]. In a zero field, the Sm0.55Sr0.45MnO3.02 sample at T = The temperature dependences of the thermal conduc- 147 K occurs in the paramagnetic state. An increase in tivity of the Sm0.55Sr0.45MnO3.02 manganite upon heat- the magnetic field to a certain critical field Hc(T) ing and cooling in a zero magnetic field and a field of (dependent on the temperature) leads to a magnetic 26 kOe are shown in Fig. 5. The following specific fea- phase transition, as judged from the increase in the heat tures in the temperature dependence of the thermal con- capacity at Hc. The electrical resistivity begins to ductivity are noteworthy: a sharp change in the phase decrease drastically almost simultaneously with an transition range, anomalously small magnitudes increase in the heat capacity. This suggests that the (≤2 W/m K), unusual behavior of the dependence K(T) delocalization of charge carriers occurs in the ferro- (dK/dT > 0) for crystalline solids at temperatures above

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 134 ALIEV et al.

2.4 thermal conductivity of amorphous solids (for example, T = 113 K fused silica [23]). In these materials, the mean free path 147 K of phonons is limited by the size of structural units and 147 K a decrease in the heat transfer with a decrease in the temperature is due to a decrease in the thermal conduc- 2.0 tivity. A similar behavior is typical of ceramic samples

K in the case when grain-boundary scattering is the dominant mechanism of phonon scattering. However, taking into account the ratio of the phonon mean free path , W/m × 4 K (~5.6 Å, see below) to the mean grain size (~2 10 Å) 1.6 and the same behavior of the thermal conductivity observed for manganite single crystals [19, 20], the aforementioned circumstance cannot be used to explain the dependence K(T) observed for manganites.

Hc The mean free path of phonons lph can be estimated 1.2 0 5 10 15 20 25 from the Debye relationship for phonon thermal con- H, kOe 1 ductivity K = --- C ν l (where C is the heat capacity ph 3 v s ph v Fig. 6. Field dependences of the thermal conductivity of the ν per unit volume and s is the velocity of sound), our Sm0.55Sr0.45MnO3.02 manganite at different temperatures. experimental data on Kph and Cv, and the available data ν on s [13, 24, 25]. In particular, from the data obtained at × 6 3 TC, and the temperature hysteresis correlating with the T = 200 K (Cv = 1.86 10 J/m K, Kph = 1.75 W/m K, ρ ν × 3 ≈ hystereses of Cp(T) and (T). and s = 5 10 m/s), we found lph 5.6 Å. Hence, we For magnetic materials, the total thermal conductiv- can assume that the magnitudes of the structural distor- ity can be represented as the sum of the electron (K ), tions limiting the mean free path of phonons in manga- e nites are of the order of the lattice parameter. The role phonon (Kph), and magnon (Km) components: K = Ke + of structural distortions can be played by JahnÐTeller Kph + Km. These components each can contribute to the local distortions of MnO6 oxygen octahedra, which can anomalies observed in the dependence K(T). The esti- significantly change upon phase transitions and in mation of the electron component Ke of the thermal response to a magnetic field [24]. These distortions can conductivity with the use of the WiedemannÐFranz drastically affect phonon heat transfer in these materi- ρ relationship Ke = LT/ (where L is the Lorentz number) als and bring about a sharp decrease in the phonon com- gives K /K < 0.1%. Consequently, the electron compo- e ponent Kph upon transition to a paramagnetic (dielec- nent Ke does not make a noticeable contribution to the tric) phase. According to the neutron diffraction data, dependence K(T). The magnon component can be esti- the transition to a ferromagnetic state is accompanied mated from the expression of the kinetic theory for by symmetrization in the arrangement of oxygen octa- thermal conductivity of magnons, which relates the hedra due to a decrease in the JahnÐTeller distortions, ∆ heat capacity jump C at TC to the magnetic contribu- which, in turn, results in a substantial increase in the 1 thermal conductivity at temperatures below TC. Visser tion: K = --- ∆Cν τ , where ν and τ are the velocity m 3 m m m m et al. [19] and Cohn [20] analyzed the temperature and of longitudinal magnons and their relaxation time, magnetic-field dependences of the thermal conductivity respectively. By using the experimental heat capacity for different manganites (LaÐSrÐMnÐO, LaÐCaÐMnÐ jump ∆C = 2.5 J/m3 K and the characteristic values of O, and other systems) and also reached the conclusion ν τ × Ð12 that the main mechanism limiting the phonon heat m = 600 m/s and m = 2 10 s taken from [20], we ≈ transfer in the materials studied is phonon scattering by obtain Km 0.014 W/m K. This value is negligible as JahnÐTeller static distortions. Upon transition to the compared to the anomalously large change in the ther- ferromagnetic phase, these distortions disappear and, mal conductivity (∆K ≈ 0.8 W/m K) at temperatures hence, the thermal conductivity increases. close to T . Furthermore, the measurements of the C As was noted above, the hysteresis in the behavior dependence K(T) for an La Ca MnO polycrystal 0.9 0.1 3 of K(T) is associated with the difference in the critical revealed no noticeable anomalies at the Curie tempera- temperatures corresponding to the ferromagnetic and ture TC [19]. Therefore, we can draw the inference that paramagnetic phase. the temperature dependence of the thermal conductiv- The most important results were obtained in the ity of the Sm0.55Sr0.45MnO3.02 manganite is determined by the specific features of phonon scattering. study of the effect of a magnetic field on the thermal conductivity (Fig. 6). It can be seen from Fig. 6 that, in The low values of the thermal conductivity and its the ferromagnetic state (T = 113 K), the magnetic field behavior at T > TC (kD/dT > 0) recall the behavior of only slightly affects the thermal conductivity K. This is

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EFFECT OF A MAGNETIC FIELD ON THE THERMAL AND KINETIC PROPERTIES 135 indirect evidence that the concentration of the ferro- should bring about the formation of holes and that, magnetic phase in the sample is close to saturation. At therefore, holes are charge carriers in manganites. The temperatures above TC, the field dependence of the sign of the charge carriers is usually determined from thermal conductivity K is similar to the field depen- the experimental data on the Hall effect or from the sign dence of the heat capacity Cp(H). An increase in the of the thermopower. In magnetic materials, the mea- magnetic field to a certain critical value does not affect surements of the Hall effect involve certain problems the thermal conductivity K. When the magnetic field associated with the contribution of the so-called anom- strength H becomes high enough to restore the struc- alous Hall effect, which is difficult to separate from the tural and magnetic orderings destroyed by the tempera- conventional Hall effect. For this reason, the sign of the ture, i.e., reaches the critical value Hc, the thermal con- charge carriers is often judged from the results of ther- ductivity increases drastically. The last circumstance is mopower measurements. However, the experimental of particular importance: the observed effect of the data on the thermopower of manganites are inconsistent magnetic field on the lattice thermal conductivity with the concept of hole conduction. As a rule, the hole directly indicates that the lattice dynamics should be conduction manifests itself in lightly doped manganites taken into account when constructing the theory explaining the mechanism of a giant magnetoresistance (x < 0.2) and a complex dependence of the ther- effect in perovskite manganites. mopower on the temperature and magnetic field is characteristic of manganites with an intermediate doping The sensitivity of the Sm1 Ð xSrxMnO3 system to level (0.2 < x < 0.5) [26Ð28]. In this case, the ther- doping level can also be judged from the comparison of mopower S can reverse sign depending on the magnetic our results with the data obtained by Kasper et al. [26]. field H and the temperature T. Hundley and Neumeier These authors measured the dependences Cp(T), K(T), [28] interpreted their results on the thermopower of and S(T) for a sample of similar composition, namely, La1 Ð xCaxMnO3 manganites in the framework of a Sm0.56Sr0.44MnO2.98 (in terms of the manganese model according to which an increase in the tempera- valence, this sample efficiently corresponds to ture is accompanied by a redistribution of charge states Sm0.6Sr0.4MnO3), and did not observe specific features of manganese ions in such a way that, in addition to in the behavior of the studied quantities at temperatures Mn4+ and Mn3+ ions, there appear Mn2+ ions corre- close to TC. It seems likely that, in [26], the authors did sponding to hole carriers. Bivalent manganese ions pos- not perform measurements under different temperature sess a high mobility and provide the observed sign of conditions (upon cooling and heating). This can the carriers. Bebenin et al. [29, 30] measured the Hall account for the absence of hystereses but cannot effect and the thermopower in manganites (both these explain the absence of pronounced anomalies in the quantities change sign with an increase in the tempera- behavior of C and K. Possibly, the ratio of trivalent to p ture) and made the inference that there is a crossover tetravalent manganese ions in the sample plays a certain from hole conduction to electron conduction. Accord- role. It is known that, in the case when these ions are contained in equal amounts, there occurs charge ordering to the authors’ opinion, the change in the conduc- ing accompanied by orbital ordering and the formation tion type is caused by a shift of the mobility edge upon of a magnetic zigzag structure, which is referred to as transition to the paramagnetic phase. Most likely, in the the CE structure. The number of Mn4+ ions in our sam- general case, the dependence S(T) for manganites ple amounts to 47Ð49% of the total number of manga- should be interpreted in terms of the two-band model, nese ions; hence, charge ordering and related phenom- according to which carriers of different types predomi- ena can play an important role (for comparison, this nate in different temperature ranges. Since charge car- number in the sample studied in [26] amounts to 40% riers of both signs are involved in the conduction and of the total number). However, we believe that such a the thermopower is determined as the difference large difference in the physical properties of between the corresponding contributions, the positive Sm0.55Sr0.45MnO3 and Sm0.6Sr0.4MnO3 samples is asso- or negative sign of the thermopower implies the domi- ciated with the qualitative difference in the magnetic nation of the electron or hole contribution to the depen- structures of these manganites. According to the neu- dence S(T). Our experimental dependences of the ther- tron diffraction data obtained in [15, 17], the manganite mopower of the Sm0.55Sr0.45MnO3.02 sample on the tem- sample with x = 0.40 is characterized by three magnet- perature T and the magnetic field H are plotted in ically ordered phases, namely, ferromagnetic (below Fig. 7. It can be seen that the thermopower is negative 130 K), A-type antiferromagnetic (below 120 K), and (S < 0) over the entire temperature range studied, is CE-type antiferromagnetic (below 160 K) phases, not very high (characteristic of metals) at tempera- whereas the sample with x = 0.45 exists in a homoge- tures below T , and drastically increases in magnitude neous ferromagnetic state at temperatures below the C Curie point T . upon transition to the paramagnetic phase. Moreover, C the dependence of the thermopower exhibits a hyster- It is generally believed that doping of lanthanum esis similar to that observed in the dependences ρ(T), 3+ manganites with bivalent elements replacing La Cp(T), and K(T).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 136 ALIEV et al.

0 magnetic phase transition. (iii) The anomalously small ρ 2 6 magnitudes of the thermal conductivity K and the fea- –10 tures revealed in its temperature dependence are associ- T = 147 K

cm –15 ated with local distortions of the crystal lattice, speciﬁ-

–10 5 V/K Ω µ cally with distortions of MnO6 octahedra due to mani-

, –20 , 3+ ρ S S festations of the JahnÐTeller effect on Mn ions. V/K –25 µ 4 (iv) The dependence of the thermopower S(H, T) , –20 S should be interpreted in the framework of the two-band 1 0 5 15 25 H, kOe model. –30 ACKNOWLEDGMENTS This work was supported by the Russian Foundation –40 80 120 160 200 240 280 for Basic Research, project nos. 00-07-90241, 00-15- T, K 96662, 02-02-17895, and 02-07-06048.

Fig. 7. Temperature dependences of the thermopower of the REFERENCES Sm0.55Sr0.45MnO3.02 manganite in magnetic fields H = (1) 0 and (2) 26 kOe. The inset shows the field dependences of 1. E. L. Nagaev, Phys. Rep. 346, 387 (2001). the thermopower and the electrical resistivity. 2. Yu. A. Izyumov and Yu. N. Skryabin, Usp. Fiz. Nauk 171, 121 (2001) [Phys. Usp. 44, 109 (2001)]. The dependences S(T, H) can be explained using the 3. V. M. Loktev and Yu. G. Pogorelov, Fiz. Nizk. Temp. 26, Mott formula relating the quantity S and ρ; that is, 231 (2000) [Low Temp. Phys. 26, 171 (2000)]. 4. N. N. Loshkareva, Yu. P. Sukhorukov, S. V. Naumov, 2 σ ()ε π kBT ' F et al., Pis’ma Zh. Éksp. Teor. Fiz. 68, 89 (1998) [JETP S = Ð------, Lett. 68, 97 (1998)]. 3 e σε() F 5. A. I. Abramovich, R. V. Demin, L. I. Koroleva, et al., where kB is the Boltzmann constant, e is the elementary Pis’ma Zh. Éksp. Teor. Fiz. 69, 375 (1999) [JETP Lett. σ ε 69, 404 (1999)]. charge, ( F) is the conductivity at the Fermi level, and ∂ 6. I. F. Voloshin, A. V. Kalinov, S. E. Savel’ev, et al., Pis’ma σ ε σ ε '( F) = ----- ( ). Zh. Éksp. Teor. Fiz. 71 (3), 157 (2000) [JETP Lett. 71, ∂ε 106 (2000)]. Following Asamitsu and Moritomo [27], we can 7. A. Machida, Y. Moritomo, E. Nishibori, et al., Phys. Rev. σ ε σ ε assume that '( F) = const and the conductivity ( F) = B 62, 3883 (2000). σ (as in the isotropic case) is inversely proportional to 8. C. Martin, A. Maignan, M. Hervieu, and B. Raveau, the electrical resistivity; i.e., σÐ1 = ρ. Therefore, the Phys. Rev. B 60, 12191 (1999). dependence of the thermopower S on H is completely 9. A. I. Abramovich, L. I. Koroleva, A. V. Michurin, et al., determined by the dependence of the resistivity ρ(H). Fiz. Tverd. Tela (St. Petersburg) 42, 1451 (2000) [Phys. Indeed, as can be seen from the inset in Fig. 7, the Solid State 42, 1494 (2000)]. dependence S(H) in a certain temperature range is sim- 10. A. M. Aliev, Sh. B. Abdulvagidov, A. B. Batdalov, et al., ilar to the dependence ρ(H). Furthermore, the inequal- Pis’ma Zh. Éksp. Teor. Fiz. 72, 668 (2000) [JETP Lett. ∆ ∆ρ 72, 464 (2000)]. ity S > is satisfied. Therefore, the giant mag------ρ- 11. A. A. Bosak, O. Yu. Gorbenko, A. R. Kaul, et al., S0 J. Magn. Magn. Mater. 211, 61 (2000). netothermopower effect can be considered in addition 12. Sh. B. Abdulvagidov, G. M. Shakhshaev, and to the giant magnetoresistance effect. I. K. Kamilov, Prib. Tekh. Éksp., No. 5, 134 (1996). 13. A. P. Ramirez, P. Schiffer, S.-W. Cheong, et al., Phys. 4. CONCLUSIONS Rev. Lett. 76, 3188 (1996). 14. M. N. Khlopkin, G. Kh. Panova, A. A. Shikov, et al., Fiz. Thus, the above analysis of the experimental data on Tverd. Tela (St. Petersburg) 42, 111 (2000) [Phys. Solid the heat capacity, thermal conductivity, thermopower, State 42, 114 (2000)]. and electrical resistivity of the Sm0.55Sr0.45MnO3.02 15. V. V. Runov, V. Yu. Chernyshev, A. I. Kurbakov, et al., manganite sample allowed us to draw the following Zh. Éksp. Teor. Fiz. 118, 1174 (2000) [JETP 91, 1017 conclusions. (i) The temperature dependences of the (2000)]. studied quantities exhibit hystereses that correlate with 16. S. M. Dunaevskiœ, A. I. Kurbakov, V. A. Trunov, et al., each other and are suppressed by the magnetic field. Fiz. Tverd. Tela (St. Petersburg) 40, 1271 (1998) [Phys. (ii) The observed hystereses are caused by the change Solid State 40, 1158 (1998)]. in the temperature TC with variations in the lattice 17. I. D. Luzyanin, V. A. Ryzhov, D. Yu. Chernyshov, et al., parameters (spontaneous magnetostriction) upon the Phys. Rev. B 64, 094432 (2001).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EFFECT OF A MAGNETIC FIELD ON THE THERMAL AND KINETIC PROPERTIES 137

18. F. Damay, N. Nguen, A. Maignan, et al., Solid State 26. N. V. Kasper, A. Kattwinkel, N. Hamad, et al., Physica B Commun. 98, 997 (1996). (Amsterdam) 292, 54 (2000). 19. D. W. Visser, A. P. Ramirez, and M. A. Subraminian, 27. A. Asamitsu and Y. Moritomo, Phys. Rev. B 53, R2952 Phys. Rev. Lett. 78, 3947 (1997). (1996). 20. J. L. Cohn, J. Supercond. 13, 291 (2000). 28. M. F. Hundley and J. J. Neumeier, Phys. Rev. B 55, 21. K. H. Kim, M. Uehara, and S.-W. Cheong, Phys. Rev. B 11511 (1997). 62, R11945 (2000). 22. S. Ulenbruck, B. Buchner, R. Gross, et al., Phys. Rev. B 29. N. G. Bebenin, R. I. Zaœnullina, V. V. Mashkautsan, et al., 57, R5571 (1998). Fiz. Tverd. Tela (St. Petersburg) 43, 482 (2001) [Phys. 23. R. Berman, Thermal Conduction in Solids (Clarendon, Solid State 43, 501 (2001)]. Oxford, 1976; Mir, Moscow, 1979). 30. N. G. Bebenin, R. I. Zaœnullina, V. V. Mashkautsan, et al., 24. Yu. P. Gaœdukov, N. P. Danilova, A. A. Mukhin, and Zh. Éksp. Teor. Fiz. 113, 981 (1998) [JETP 86, 534 A. M. Balbashov, Pis’ma Zh. Éksp. Teor. Fiz. 68, 141 (1998)]. (1998) [JETP Lett. 68, 153 (1998)]. 25. R. G. Radaelli, M. Marezio, H. Y. Hwang, et al., Phys. Rev. B 54, 8992 (1996). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 138Ð140. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 131Ð133. Original Russian Text Copyright © 2003 by Myagkov, Polyakova, Bondarenko, Polyakov.

MAGNETISM AND FERROELECTRICITY

Self-Propagating High-Temperature Synthesis and Magnetic Features of FeÐAl2O3 Granulated Films V. G. Myagkov*, K. P. Polyakova*, G. N. Bondarenko**, and V. V. Polyakov* * Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia ** Institute of Chemistry and Chemical Technology, Siberian Division, Russian Academy of Sciences, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received April 18, 2002

Abstract—The classical solid-phase reaction between Fe2O3 and Al layers in thin films is initiated. It is shown that, in the reaction products, Fe granulated films are formed in the Al2O3 nonconducting matrix. Analysis of the reaction equation demonstrates that the volume fraction of iron in the granulated films is less than the percolation threshold. This determines the magnetic properties of iron clusters in a superparamagnetic state. It is assumed that the nanocrystalline microstructure exists in thin films after solid-phase reactions proceeding under conditions of self-propagating high-temperature synthesis. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Earlier [8, 9], it was demonstrated that solid-phase Granulated films consisting of ferromagnetic nano- reactions in bilayer thin films at high heating rates pro- clusters contained in a nonconducting matrix have been ceed under conditions of self-propagating high-temper- intensively studied in recent years, because they exhibit ature synthesis. The self-propagating high-temperature very interesting properties, such as giant magnetoresis- synthesis in thin films proceeds in the form of a surface tance [1], giant anomalous Hall effects [2], specific combustion wave. This process is characterized by the optical properties [3], and quantum-confinement initiation temperature T0. In our case, the reagents for effects [4]. From the practical point of view, these films reaction (1) were bilayer and multilayer Al/Fe2O3 film can be used as media for magnetic and magneto-optical samples, which were prepared by dc ion-plasma sput- recording [5, 6]. Granulated magnetic films that contain tering through successive deposition of Fe2O3 and alu- nanoclusters of Fe, Ni, Co, and their alloys in SiO2 and minum layers without vacuum deterioration. The tar- Al2O3 matrices have been the main subject of recent gets used in producing Al and Fe2O3 films were pre- investigation in this field. A conventional method of pared in the form of wafers cut respectively from producing metallic nanoclusters inside a nonconduct- monolithic aluminum and a conducting ceramic coat- ing matrix in thin films is codeposition of a metal and ing of iron oxide, which was obtained by plasma depo- an insulator onto a substrate. In this case, the nanoclus- sition [10]. ters are randomly distributed over the matrix and their size depends on the conditions of thermal annealing The films were evaporated on glass substrates at a temperature of ~320 K in an argon atmosphere at a and deposition. However, the successive deposition of × Ð2 a metal and an insulator can result in self-organization pressure of 5 10 Pa. The evaporation rate was in the formation of metallic nanoclusters [7]. The devel- 0.3 nm/s for aluminum and 0.2 nm/s for iron oxide. The opment of new techniques for preparing granulated total thicknesses of the aluminum and iron oxide films samples is very important, because this makes it possi- lay within the ranges 15Ð50 and 40Ð120 nm, respec- ble to extend the range of structural characteristics tively. The samples obtained were heated at a rate no determining the physical properties of nanocomposites. less than 20 K/s in order to initiate the wave of self- propagating high-temperature synthesis. For bilayer The purpose of this work was to investigate Fe nan- film samples, the solid-phase reaction proceeded at ogranulated films formed inside the Al2O3 matrix. temperatures above the initiation temperature T0 ~ 800 K under conditions of self-propagating high-tem- 2. SAMPLE PREPARATION perature synthesis. The front of the propagation of AND EXPERIMENTAL TECHNIQUE high-temperature synthesis, which was observed visually, is typical of self-propagating high-temperature The preparation procedure is based on initiating the synthesis in thin films [8, 9]. For multilayer film sam- classical solid-phase reaction in thin films: ples, the observation of the front of self-propagating Fe2O3 + 2Al = Al2O3 + 2Fe. (1) high-temperature synthesis was hampered. For this rea-

SELF-PROPAGATING HIGH-TEMPERATURE SYNTHESIS AND MAGNETIC FEATURES 139

1000 α-Fe O 2 3 800 Fe O (a) 3 4 600

3 400 200 0

α , emu/cm -Al2O3 –200 M , arb. units (b) I Fe –400 –600 –800 –1000 –3 –2 –1 0 1 2 3 H, kOe 30 35 40 45 50 55 60 2θ, deg

Fig. 1. X-ray diffraction patterns of the [Al(20 Fig. 2. Magnetization curve for the [Al(20 nm)/Fe O × 2 3 nm)/Fe2O3(40 nm)] 4 multilayer film: (a) initial sample (40 nm)] × 4 multilayer film after thermal annealing at a and (b) sample after thermal annealing at a temperature of temperature of 800 K for 10 min. The magnetization is 800 K for 10 min. given per unit volume of iron. son, these samples were subjected to thermal annealing in a nonconducting matrix below the percolation through heating to a temperature 10Ð20 K higher than threshold. the initiation temperature T0 and heat treatment at this temperature for 50 min (the time required for complet- The size of iron grains in the direction perpendicular ing the reaction and subsequent annealing). to the sample surface was determined from the broadening of the diffraction peak of Fe(110) according to the Scherrer formula. The mean grain size was found to 3. RESULTS AND DISCUSSION be 28 nm. As was shown by Chen et al. [12], particles with a smaller size exist in a one-dimensional state. Figure 1a shows a typical x-ray diffraction pattern Under the assumption that the iron cluster in other × for the [Al(20 nm)/Fe2O3(40 nm)] 4 multilayer film directions has the same size, the cluster should exhibit sample prior to the solid-phase reaction. This pattern a magnetic single-domain structure. In this case, the α contains reflections attributed to -Fe2O3. The forma- sample can be treated as an ensemble of single-domain tion of Fe3O4 in small amounts is also possible in the particles and its behavior can be described in terms of initial samples. The absence of reflections attributed to the StonerÐWohlfahrt theory. The magnetization curve aluminum implies that aluminum is in an amorphous or for an ensemble of single-domain particles has the form finely crystalline state. After the solid-phase reaction of a hysteresis loop with the following parameters: α (Fig. 1b), reflections from -Fe2O3 and Fe3O4 disap- Mr = 0.5MS and Hc = 0.958 K/MS (see, for example, pear and the reflections assigned to iron and amorphous [13]). Figure 2 depicts a typical magnetization curve for α -Al2O3 appear in the x-ray diffraction pattern. The FeÐAl2O3 films obtained as a result of reaction (1). The presence of the reflections associated with Fe(110) in absence of a hysteresis loop permits us to assume that the x-ray diffraction patterns, along with other weak the iron clusters predominantly occur in the superpara- reflections or their complete absence, indicates that iron magnetic state. A similar hysteresis curve, which can be nanocrystallites are characterized by a preferred orien- considered as a superposition of magnetization curves tation of the (110) plane aligned parallel to the substrate for iron nanoclusters in superparamagnetic and mag- surface. The size distribution of nanoclusters and the netic states, was observed earlier for CoÐAg granulated relative volume of the metallic fraction x are important films [14]. The coexistence of the superparamagnetic factors responsible for the magnetic properties of gran- and magnetic states can also be observed in ferromag- ulated materials. It follows from the reaction equation netic nanoclusters inside a nonconducting matrix [15]. that xr = 0.365. This value is less than the percolation The critical size below which the iron nanoclusters in threshold xp. For many granulated systems that contain the Al2O3 matrix are in the superparamagnetic state is metallic nanoclusters in a dielectric matrix, the perco- approximately equal to 10 nm [12]. This suggests that lation threshold is estimated as xp = 0.5Ð0.6 [11]. This the size distribution of iron nanoclusters in the samples suggests that the metallic nanoclusters are separated under investigation is rather broad and covers sizes from each other. The separation of iron nanoclusters is from several to tens of nanometers. It should be noted confirmed by a high resistivity of the samples, namely, that no hysteresis loop is observed even at a tempera- Ð6 Ð4 ρ = 10 Ð10 Ω m, which is typical of metallic clusters ture of 77 K. It follows that the blocking temperature TB

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

140 MYAGKOV et al. for iron nanoparticles in the superparamagnetic state is between Al and Fe2O3 layers. Analysis of the reaction below this temperature. equation proved that the volume fraction of iron is less The reaction products are formed solely at the front than the percolation threshold. In the studied samples, of self-propagating high-temperature synthesis. The the (110)-textured iron nanoclusters are in superpara- characteristic reaction time is determined from the for- magnetic and magnetic states. τ 2 τ mula t = /V f where is the thermal diffusivity of the bilayer film sample and Vf is the front velocity at the ACKNOWLEDGMENTS initiation temperature of self-propagating high-temper- This work was supported by the Krasnoyarsk τ ature synthesis. Let us assume that is determined by Regional Foundation for Research, project no. 11F001C. the thermal diffusivity of the metal oxide; i.e., τ ~ (1Ð5) × Ð7 2 × Ð2 10 m /s and Vf ~ 1 10 m/s [8, 9]. As a result, the characteristic reaction time is estimated as t = (1Ð5) × REFERENCES 10Ð3 s. Under the assumption that the reaction proceeds 1. S. Mitani, S. Takahashi, K. Takahashi, et al., Phys. Rev. in the solid phase at a temperature close to the melting Lett. 81 (13), 2799 (1998). point of the metal and that the diffusion coefficient is 2. A. B. Pakhomov, X. Yan, and Y. Xu, J. Appl. Phys. 79 (8), determined to be D ~ 10Ð12Ð10Ð14 m2/s, the size of 6140 (1996). metallic clusters is estimated as r ~ Dt = (3Ð70) nm. 3. A. N. Drachenko, A. N. Yurasov, I. V. Bykov, et al., Fiz. Tverd. Tela (St. Petersburg) 43 (5), 897 (2001) [Phys. This estimate is in good agreement with the aforemen- Solid State 43, 932 (2001)]. tioned sizes of iron nanoclusters formed as a result of reaction (1). The dependence of the front velocity of 4. B. Raquet, M. Goiran, N. Negre, et al., Phys. Rev. B 62 (24), 17144 (2000). self-propagating high-temperature synthesis Vf on the substrate temperature T (T > T ) follows a law close to 5. S. H. Liou and C. L. Chien, Appl. Phys. Lett. 52 (6), 512 S S 0 (1988). the Arrhenius law. Consequently, this velocity should 6. Z. S. Jiang, G. J. Jin, J. T. Ji, et al., J. Appl. Phys. 78 (1), determine the size distribution function and the mean 439 (1995). size of iron nanoparticles. Therefore, the microstructure can be controlled both by varying the substrate 7. D. Babonneau, F. Petrov, J.-L. Maurice, et al., Appl. Phys. Lett. 76 (20), 2892 (2000). temperature TS upon initiating the self-propagating high-temperature synthesis in Al/Fe O bilayer films 8. V. G. Myagkov and L. E. Bykova, Dokl. Akad. Nauk 354 2 3 (6), 777 (1997). and by choosing the conditions of subsequent heat treatment. In turn, the microstructure determines the 9. V. G. Myagkov, V. S. Zhigalov, L. E. Bykova, and V K. Mal’tsev, Zh. Tekh. Fiz. 68 (10), 58 (1998) [Tech. magnetic properties of FeÐAl2O3 granulated films. Phys. 43, 1189 (1998)]. Note that the nanocrystalline microstructure should 10. A. A. Lepeshev, V. N. Saunin, S. V. Telegin, et al., Zh. be predominant after passing the wave of self-propagat- Tekh. Fiz. 70 (5), 130 (2000) [Tech. Phys. 45, 653 ing high-temperature synthesis in all the samples con- (2000)]. taining more than two phases in the reaction products. 11. C. L. Chien, J. Appl. Phys. 69 (8), 5267 (1991). In particular, similar microstructure and magnetic prop- 12. C. Chen, O. Kitakami, and Y. Shimada, J. Appl. Phys. 84 erties are observed in FeÐTiO2 films after the solid- (4), 2184 (1998). phase reaction between Ti and Fe2O3 layers. 13. M. Prutton, Thin Ferromagnetic Films (Butterworths, Washington, 1964; Sudostroenie, Leningrad, 1967). 4. CONCLUSIONS 14. S. Honda, M. Nawate, M. Tanaka, and T. Okada, J. Appl. Phys. 82 (2), 764 (1997). It was demonstrated that the solid-phase reactions 15. Y. Xu, B. Zhao, and X. Yan, J. Appl. Phys. 79 (8), 6137 proceeding under conditions of self-propagating high- (1996). temperature synthesis can be used for preparing granulated media. Specifically, the FeÐAl2O3 granulated films were obtained after the solid-phase reaction Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 141Ð145. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 134Ð138. Original Russian Text Copyright © 2003 by Zalesskiœ, Frolov, Khimich, Bush. MAGNETISM AND FERROELECTRICITY

Composition-Induced Transition of Spin-Modulated Structure into a Uniform Antiferromagnetic State in a Bi1 Ð xLaxFeO3 System Studied Using 57Fe NMR A. V. Zalesskiœ*, A. A. Frolov*, T. A. Khimich*, and A. A. Bush** * Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 119333 Russia e-mail: [email protected] ** Moscow State Institute of Radio Engineering, Electronics, and Automation (Technical University), pr. Vernadskogo 78, Moscow, 117454 Russia Received March 20, 2002; in ﬁnal form, April 26, 2002

Abstract—By analyzing the NMR line shape, the transformation of a spatially spin-modulated magnetic structure in BiFeO3 into an ordinary spatially uniform structure of the LaFeO3 orthoferrite in Bi1 Ð xLaxFeO3 solid solutions is studied. The measurements are made using a spin-echo technique at temperatures of 77 and 4.2 K on ceramics with compositions x = 0, 0.1, 0.2, 0.61, 0.9, and 1.0 enriched by the 57Fe isotope. It is shown that the spin-modulated structure disappears near the concentration x = 0.2, which corresponds, according to the published data, to the phase transition with a change in the unit-cell symmetry R3c C222. A formula is obtained describing the NMR absorption line shape for the spin-modulated structure with account of local linewidth. Theoretical spectra adequately describe the evolution of the experimental spectrum in the concentration range 0 ≤ x ≤ 0.2. Highly nonuniform local magnetic ﬁelds in the intermediate compositions make it impossible to detect NMR signals in a sample with x = 0.61. A uniform magnetic structure characterized by a single narrow line arises in the range of existence of a phase with the symmetry Pnma typical of the pure orthoferrite LaFeO3. © 2003 MAIK “Nauka/Interperiodica”.

The structure and properties of BiFeO3 and its solid Indirect evidence of the existence of a spatially spin- solutions were widely studied in the 1960s in connec- modulated structure (SSMS) in BiFeO3 follows from tion with a great interest expressed at that time in mag- studies of the mangnetoelectric effect, which is still netoelectrics, i.e., compounds possessing both mag- given much attention [5Ð7]. netic (spin) and electric-dipole long-range ordering (see, e.g., monograph [1]). Bismuth ferrite BiFeO3 is The existence of an SSMS in BiFeO3 has recently the most typical and extensively investigated represen- been confirmed in 57Fe NMR studies [8, 9]. The rota- tative of this class of compounds. In an early stage of tion of spins in the plane of a cycloid results in the studies, it was not known that BiFeO3 also had a unique appearance of an NMR frequency band due to the magnetic structure characterized by a spatial spin mod- anisotropic contribution to the local magnetic field Hn ulation of the cycloidal type, which was established at 57Fe nucleus sites. The anisotropic NMR frequencies using neutron diffraction in 1982 [2]. Antiferromagnet- in BiFeO are described by the expression [8] ically ordered spins in such a structure rotate onto the 3 plane containing the threefold axis (the c axis) of the νθ()≈ν ν ()θν ν 2 δν2 θ rhombohedral unit cell of BiFeO3, with the wave vector || Ð || Ð ⊥ sin = || Ð sin , (2) directed perpendicularly to this axis and the period λ ≈ 620 Å being incommensurate with the lattice spacing. where ν|| and ν⊥ are the frequencies corresponding to A cycloid can be described by the equation [3, 4] the parts of the cycloid where the spins are oriented parallel and perpendicular to the c axis, respectively. Due () θ() ±4Km , to the specific density distribution of H (or of the NMR cosx = sn------xm, (1) n λ frequencies) in the coordinate x or in the angle θ for the θ cycloid, two edge absorption peaks arise in an NMR where is the angle of spin rotation relative to the c ν ν axis, x is the coordinate along the cycloid propagation frequency band at the frequencies || and ⊥ with a char- direction, sn(x, m) is the Jacobian elliptic function, m is acteristic dip between them. The experimental NMR its parameter, and K(m) is a complete elliptic integral of spectrum for BiFeO3 at 77 K is shown in Fig. 1a (curve 1). the first kind. The profile of such an NMR line P(ν) for the SSMS in

142 ZALESSKIŒ et al.

(a) (b)

Intensity, arb. units 1

2 3

74 75 76 77.45 77.5077.55 77.60 ν, MHz

57 Fig. 1. Experimental Fe NMR spectra for Bi1 Ð xLaxFeO3 solid solutions with x = (1) 0, (2) 0.1, (3) 0.2, (4) 0.9, and (5) 1.0 at 77 K. the case where the local line shape is represented by a NMR technique is not presently available. That is why δ function was calculated in [8] to be we chose a simpler scenario, wherein the transition to a uniform magnetic structure is caused by a change in P()ν (3) the composition of a binary solid solution in which one 1/2 1/2 1/2 Ð1 of the components is BiFeO and the other component ∼ν{}()|| Ð ν ()ννÐ ⊥ []1/m Ð ()ννÐ ⊥ /δν , 3 is an antiferromagnet with a uniform magnetic struc- where ν is the running frequency. ture. As the latter component, the LaFeO3 orthoferrite The important feature of such a spectrum is the pos- was chosen. The structure and properties of the sibility of estimating the degree of anharmonicity of the Bi1 − xLaxFeO3 system were studied in [10Ð14]. cycloid by analyzing the shape of the spectrum. From Ceramic samples of this system were studied in [10Ð Eqs. (1) and (3) it follows that if m 0, the cycloid 12]. Four almost equal compositional phase ranges is sinusoidal and the spectrum has a symmetrical shape were found: 0 < x < 0.19 (I), 0.2 < x < 0.5 (II), 0.55 < with edge peaks of equal heights and a minimum x < 0.73 (III), and 0.75 < x < 1.0 (IV). In range I, there between them. If m 1, the cycloid becomes more exists a phase with a rhombohedral unit cell (space anharmonic, the ratio of the height of the higher fre- group R3c) inherent in the pure BiFeO compound, the ν 3 quency peak at the frequency || to that of the lower fre- compounds in range IV have the orthorhombic struc- quency peak at ν⊥ increases, and the dip minimum ture (space group Pnma) inherent in orthoferrites, and shifts towards ν⊥. Spectrum 1 in Fig. 1a gives evidence the intermediate phases, II and III, are characterized, as of an essential anharmonism of the cycloid in BiFeO3. in [12], as orthorhombic. From above, it follows that NMR allows one, com- Within the whole concentration range 0 ≤ x ≤ 1.0, paratively easily, (in contrast with neutronography) not there exists a magnetic ordered state; the Néel temper- only to identify SSMS but also to obtain additional ature varies almost linearly with increasing x: from information about its properties. 643 K for BiFeO3 to 738 K for LaFeO3. Ferroelectric In this connection, it would be advisable to use properties disappear at x = 0.5 according to [10, 11] or NMR for studying the transitions associated with the at x = 0.75 (in the vicinity of a boundary between appearance or disappearance of an SSMS, for example, phases III and IV) according to [12]. the transition to an ordinary spatially uniform magnetic Publications [13, 14] were devoted to studies of the structure. According to magnetoelectric-effect data [5Ð Bi1 Ð xLaxFeO3 system performed on single crystals 7], such a transition in BiFeO3 takes place in strong grown using spontaneous crystallization from the melt. magnetic ﬁelds (≈200 kOe). Unfortunately, a high-ﬁeld Instead of the four phases IÐIV, the authors of [13, 14]

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 COMPOSITION-INDUCED TRANSITION OF SPIN-MODULATED STRUCTURE 143 found six phases, among them a low-symmetry triclinic Table 1. Symmetry; lattice parameters a, b, c, α; and unit- phase (space group P1) in the range 0.06 < x < 0.24 into cell volume V/z of Bi1 Ð xLaxFeO3 solid solutions (z is the number of formula units in the unit cell) which the BiFeO3 structure transforms at x > 0.06. Ground crystals of one of the compositions (Bi0.93La0.07FeO3) Symmetry, a, b, c, Å; α, x V/z, Å3 were studied using neutron diffraction [15]. However, phase deg the question of the existence of an SSMS in this composition remained open. 0 Rhombohedral, a = 3.963(2); V/z = 62.2(2), We studied the NMR spectra for samples with com- R3c, I α = 89.43(3) z = 1 positions x = 0, 0.1, 0.2, 0.61, 0.9, and 1.0. The sample 0.1 Rhombohedral, a = 3.959(2); V/z = 62.1(2), with x = 0 was the same polycrystalline BiFeO3 that R3c, I α = 89.50(3) z = 1 was used in our previous NMR studies [8, 9]. The sam- 0.2 Orthorhombic, a = 5.598(5), V/z = 61.4(2), ples were prepared using the standard ceramic technol- C222, II b = 5.617(5), z = 2 ogy from the starting reagents Bi2O3, La2O3, and Fe2O3. 57 c = 3.904(2) In all cases, Fe2O3 enriched by the Fe isotope up to 95.43 wt % was used. Solid-state synthesis and the mix- 0.61 Orthorhombic, a = 5.544(5), V/z = 60.8(2), ture sintering were performed by increasing the tem- C2221, III b = 5.576(5), z = 4 perature of calcination stepwise followed by cooling of c = 7.863(3) the sample in a turned-off furnace. The sintering temperature increased with increasing La content from 0.9 Orthorhombic, a = 5.543(5), V/z = 60.2(2), 800°C (x = 0) to 1250°C (x = 1). Pnma, IV b = 5.564(5), z = 4 The phase composition and the crystallographic c = 7.856(3) characteristics of samples were determined with the 1.0 Orthorhombic, a = 5.549(5), V/z = 60.7(2), help of an x-ray diffractometer DRON-4 (CuKα radia- Pnma, IV b = 5.564(5), z = 4 tion) by using the powder method with Ge powder as an internal reference. The lattice parameter of cubic Ge c = 7.859(3) crystals was taken to be a = 5.653 Å. Within an accuracy of 1Ð3%, the samples were found to be perovskite-like single-phase Bi La FeO rf power) for the samples with x = 0.9 and 1.0 (possess- 1 Ð x x 3 ing weak ferromagnetism) corresponded to the NMR solid solutions. Their structural characteristics are lines for the nuclei in domains. Substitution of Bi for La given in Table 1. (sample with x = 0.9, curve 4 in Fig. 1b) results only in NMR was observed using a spin-echo method in a zero an asymmetric broadening of the NMR line and a slight magnetic field with the help of a semicommercial fre- shift of the center of gravity of the line towards lower quency-scanning pulsed radiospectrometer ISSh-1-13M frequencies due to the appearance of a nonequivalent (manufactured at the Institute of Radio Engineering surrounding of oxygen anions involved in spin density and Electronics, Russian Academy of Sciences). The transfer via the FeÐOÐFe superexchange coupling. measurements were carried out at 77 and 4.2 K. The NMR spectra at 4.2 K have the same shape as The spectra obtained at 77 K for compositions x = those at 77 K, but all resonance frequencies are shifted ν ν ν 0.1 and 0.2 are shown in Fig. 1a (curves 2, 3, respec- towards higher values. The frequencies ||, ⊥, and max tively). For the Bi0.9La0.1FeO3 sample, the double- for the samples at 4.2 and 77 K are given in Table 2. peaked NMR spectrum typical of the SSMS is In order to theoretically describe the changes occur- observed. For the Bi0.8La0.2FeO3 sample, whose com- ring in spectra upon the substitution of La for Bi, one position corresponds to the transitional range between should pass from simple equation (3), in which the δ phases I and II, the spectrum consists of a single broad function is taken as a local line shape function f[ν Ð ν ≈ maximum at a frequency of max 75 MHz and has no ν(θ)], to a model which takes into account a finite local edge peaks. The attempt to detect an NMR signal from linewidth. For this purpose, we apply the method of cal- the sample with x = 0.61 was unsuccessful despite the culation of the NMR absorption line shape used by But- close-to-100% abundance of the 57Fe isotope, which ler [16] in studying NMR in domain walls whose mag- was most likely associated with the highly nonuniform netic structure possesses a similar 180° periodicity. In local fields Hn in the intermediate orthorhombic phases. contrast to domain walls, there is no need to consider As the composition approached that of lanthanum the enhancement factor, since, as was shown in [8], the orthoferrite in the range of phase IV, a single narrow enhancement mechanism is ineffective in BiFeO3. line characteristic of a uniform magnetic structure According to [16], the profile P(ν) of an NMR absorp- appeared. For pure LaFeO3, a narrow symmetric line tion line for a 180°-periodic structure is given by with a halfwidth of only 500Ð600 kHz was observed π (curve 5, Fig. 1b; we note that the frequency scales in ()ν ()νθ []θ νθ() Figs. 1a, 1b differ by a factor of 10). The echo excita- P = ∫I Ð d , (4) tion conditions (pulses were 15 and 30 µs long at a high 0

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 144 ZALESSKIŒ et al. ν ν ν Table 2. Frequencies ||, ⊥, and max for samples of various compositions of the Bi1 Ð xLaxFeO3 system at 77 and 4.2 K ν ν ν ||, MHz ⊥, MHz max, MHz x 77 K 4.2 K 77 K 4.2 K 77 K 4.2 K 0 75.35 75.58 74.60 75.00 0.1 75.25 75.50 74.60 74.90 0.2 74.9Ð75.0 75.3Ð75.4 0.9 77.55 77.80 1.0 77.56 77.81 where I(θ) is the intensity of the signal in the part of the From Eq. (8), it follows that ξ = Ð1 corresponds to the cycloid where spins make an angle θ with the c axis. frequency ν⊥ (θ = π/2) and ξ = 0, to the frequency ν|| The function I(θ) for the cycloidal magnetic structure in (θ = 0). The integral in Eq. (7) reduces to BiFeO was calculated in [8] to be 3 π Ð1/2 Ð1 2 Ð1/2 I()θ ∝ ()mÐ1 Ð 1 + sin2 θ . (5) P()ν ∝ ∫()m Ð 1 + sin θ (9) In order to take into account the finite width of a 0 2 Ð1 local line, we will describe the line using a Lorentzian × []1 + ()δ/ν/∆ 2()sin2 θξÐ dθ. with width ∆ independent of the angle θ: Calculated curves 1Ð3 in Fig. 2 correspond to exper- ∆ 1 f []ννθÐ () = ------. (6) imental spectra 1Ð3 in Fig 1a, respectively (the values π∆2 + ()ννθÐ ()2 of parameters m and δν/∆ used in calculating the integral in Eq. (9) are given in the caption to Fig. 2). Theo- Substituting Eqs. (2) and (5) into Eq. (4), we obtain retical spectra well reproduce the line shape evolution π caused by the substitution of La for Bi atoms within the Ð1 2 Ð1/2 concentration range 0 < x < 0.2. The main cause of P()ν ∝ ∫()m Ð 1 + sin θ changes in the NMR spectra is the increased nonunifor- 0 (7) mity of local fields Hn, which results in an increase in the linewidth ∆. Indeed, for BiFeO , the value of ∆ in ×∆[]2 ()νν δν2 θ 2 Ð1 θ 3 + Ð || + sin d . frequency units is 75 kHz, whereas for the composition x = 0.1, ∆ = 125 kHz. Furthermore, the fact that the The integration is conveniently performed by introduc- peaks tend to be equal in height and that m decreases ing dimensionless parameters: from 0.83 to 0.5 indicates that the cycloid becomes ∆ ξνν= ()Ð || /δν; δν/∆. (8) more harmonic. When the local line shape becomes comparable in magnitude with the frequency anisotropy δν = 0.75 MHz, the spectrum smears into a single broad line (curves 3 in Figs. 1a, 2) and looses the basic feature (two edge peaks) inherent in an SSMS. How- ever, the fact that the NMR frequencies remain distributed over the range between ν|| and ν⊥ suggests that the sample with composition x = 0.2 contains regions with 1 spins distributed in the angle θ, although according to x-ray analysis, the crystal structure of the sample is orthorhombic (Table 1). Obviously, a further increase 2 in the nonuniformity of Hn with increasing La concentration should result in such a large spread of NMR fre- Intensity, arb. units quencies that the conditions for the formation of a pro- 3 nounced NMR line will no longer be able to be fulfilled. This is the reason why NMR was not observed in the sample with x = 0.61. The conditions for NMR obser- –3 –2 –1 0 1 2 3 vation are restored only for compositions close to that ξ of pure lanthanum orthoferrite. From the results of the present NMR studies it fol- Fig. 2. Theoretical profile of the NMR lines obtained by calculating the integral in Eq. (9) with parameters δν/∆ and m lows that the SSMS is destroyed in a concentration equal to (1) 10, 0.83; (2) 6, 0.5; and (3) 1, 0.5, respectively. range close to x = 0.2, which, according to the data pre-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 COMPOSITION-INDUCED TRANSITION OF SPIN-MODULATED STRUCTURE 145 sented in [10Ð12], corresponds to the transformation of 5. Yu. F. Popov, A. K. Zvezdin, G. P. Vorob’ev, et al., rhombohedral phase I into orthorhombic phase II. Our Pis’ma Zh. Éksp. Teor. Fiz. 57 (1), 65 (1993) [JETP Lett. results do no confirm the existence at 0 < x < 0.2 of the 57, 69 (1993)]. triclinic phase identified in single crystals in [13, 14]. 6. Yu. F. Popov, A. M. Kadomtseva, G. P. Vorob’ev, and For the theoretical analysis of the NMR spectra, the A. K. Zvezdin, Ferroelectrics 162, 135 (1994). local linewidth and its dependence on x had to be taken 7. A. M. Kadomtseva, Yu. F. Popov, G. P. Vorob’ev, and into account. With an increase in La concentration A. K. Zvezdin, Physica B (Amsterdam) 211, 327 (1995). above x = 0.2 (in regions II and III), a large nonunifor- 8. A. V. Zalessky, A. A. Frolov, T. A. Khimich, et al., Euro- phys. Lett. 50 (4), 547 (2000). mity of Hn arises, which hinders the NMR observation. A uniform antiferromagnetic structure forms only in 9. A. V. Zalesskiœ, A. K. Zvezdin, A. A. Frolov, and region IV, where a narrow single-peaked NMR line is A. A. Bush, Pis’ma Zh. Éksp. Teor. Fiz. 71 (11), 682 observed. (2000) [JETP Lett. 71, 465 (2000)]. 10. Yu. E. Roginskaya, Yu. N. Venevtsev, S. A. Fedulov, and G. S. Zhdanov, Kristallografiya 8 (4), 610 (1963) [Sov. ACKNOWLEDGMENTS Phys. Crystallogr. 8, 490 (1963)]. 11. Yu. E. Roginskaya, Yu. N. Venevtsev, and G. S. Zhdanov, This study was supported by the Russian Founda- Zh. Éksp. Teor. Fiz. 44 (4), 1418 (1963) [Sov. Phys. tion for Basic Research, project no. 02-02-16369. JETP 17, 955 (1963)]. 12. M. Polomska, W. Kaczmarek, and Z. Pajak, Phys. Status Solidi A 23, 567 (1974). REFERENCES 13. V. A. Murashov, D. N. Rakov, I. S. Dubenko, et al., Kri- stallografiya 35 (4), 912 (1990) [Sov. Phys. Crystallogr. 1. Yu. N. Venevtsev, V. V. Gagulin, and V. R. Lyubimov, 35, 538 (1990)]. Magnetoelectrics (Nauka, Moscow, 1982). 14. Z. V. Gabbasova, M. D. Kuz’min, A. K. Zvezdin, et al., 2. I. Sosnowska, T. Peterlin-Neumaier, and E. Steichele, Phys. Lett. A 158, 491 (1991). J. Phys. C 15, 4835 (1982). 15. I. Sosnovska, R. Przenioslo, F. Fisher, and V. A. Mura- shov, J. Magn. Magn. Mater. 160, 384 (1996). 3. I. Sosnowska and A. K. Zvezdin, J. Magn. Magn. Mater. 140Ð144, 167 (1995). 16. M. A. Butler, Int. J. Magn. 4, 131 (1973). 4. M.-M. Tehranchi, N. F. Kubrakov, and A. K. Zvezdin, Ferroelectrics 204, 181 (1997). Translated by A. Zalesskiœ

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 146Ð153. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 139Ð146. Original Russian Text Copyright © 2003 by Barilo, Gatal’skaya, Shiryaev, Bychkov, Kurochkin, Ustinovich, Szymczak, Baran, Krzyma«nska.

MAGNETISM AND FERROELECTRICITY

A Study of Magnetic Ordering in LaMnO3 + d Single Crystals S. N. Barilo*, V. I. Gatal’skaya*, S. V. Shiryaev*, G. L. Bychkov*, L. A. Kurochkin*, S. N. Ustinovich*, R. Szymczak**, M. Baran**, and B. Krzyma«nska** * Institute of Solid-State and Semiconductor Physics, National Academy of Sciences, Minsk, 220072 Belarus e-mail: [email protected] ** Institute of Physics, Polish Academy of Sciences, Warsaw, 02-668 Poland Received May 10, 2002

Abstract—The magnetic state of LaMnO3 + δ single crystals with different contents of nonstoichiometric oxygen grown by electric deposition is studied by measuring the temperature and ﬁeld dependences of magnetization for different orientations of a magnetic ﬁeld. The results obtained are analyzed in terms of a two-phase magnetic-state model. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION in the structural, magnetic, and transport properties of LaMnO3. The continuous interest in substituted lanthanum Nonstoichiometric LaMnO δ typically contains manganites La A MnO (A stands for the doping diva- 3 + 1Ðx x 3 manganese ions in the +3, +4, and +2 valence states, as lent or monovalent ion) stems from the fact that these well as vacancies on the cation sublattice [6, 7]. Its for- compounds exhibit a remarkable variety of properties mula can be written more exactly as (LaMn) δO , or important both from the standpoint of science and 1 Ð 3 La3/(3 + δ)Mn3/(3 + δ)O3 for the case of La : Mn = 1 : 1. As application. In particular, these compounds exhibit δ colossal magnetoresistance (CMR). The CMR is inter- is increased, the strongly distorted orthorhombic O' preted usually in terms of the double-exchange (DE) lattice transfers to orthorhombic (Pbmn, a ≤ c/2 < b), model [1, 2] or noncollinear antiferromagnetism (AF) pseudocubic (Pbmn, a ≈ b ≈ c/2 ), and, subsequently, [3]. The models involving a change from the polaron- 4+ mediated to hopping conduction near the paramagnetic to rhombohedral [7Ð10]. As the Mn concentration (PM)-to-ferromagnetic (FM) phase transition at tem- decreases and the magnetization of LaMnO3 + δ decreases, AF ordering sets in; for δ = 0.05Ð0.07, the perature Tc and melting of a charge-ordered state (see review [4] and references therein) likewise fail in satis- magnetic moment was found to contain AF and FM factorily explaining the nonstandard behavior of man- components, which were assigned to the magnetic ganites. A model of a crystal state with two magnetic structure being noncollinear [11] or to electronic phase separation [12]. Note that the few available studies on phases originating from strong sÐd exchange [5] has the magnetic transport properties of LaMnO δ pro- frequently been invoked in recent years to account for 3 + the anomalous properties of the manganites. vided controversial results. Most of the investigations were performed on ceramic LaMnO3 + δ samples, and LaMnO3 + δ is a model system for studying the mag- information on measurements of lanthanum manganite netic interactions and disorder in mixed-valence man- single crystals with nonstoichiometric oxygen are prac- ganites. Mn3+ can transfer to Mn4+, in particular, in the tically lacking. To understand more clearly the pro- presence of nonstoichiometric oxygen in undoped cesses occurring in the manganites and depending on 4+ the interaction between the magnetic, charge, and local LaMnO3 + δ with a nominal Mn content roughly equal δ structures, one has to carry out measurements on high- to 2 . Stoichiometric LaMnO3 contains only ferromag- quality single crystals. netically ordered Mn3+ ions in the (ab) MnÐO planes, which form, in turn, an AF lattice along the c axis, thus There are very few papers dealing with LaMnO3 + δ producing an A-type AF coupling with T ~ 135 K [4]. single crystals. Weak FM [13] and electron spin reso- N nances [14, 15] have been studied in single crystals The FM ordering of Mn3+ ions is initiated by the Mn3+Ð 3+ grown using the ﬂoating-zone technique with radiation OÐ Mn anisotropic superexchange interaction, which heating. Among the numerous disadvantages involved is stabilized at room temperature by JahnÐTeller (JT) in this method is that a crystal thus grown has twins of distortions in the O' structure (Pbmn, c/2 < a < b). An various types. increase in temperature or an excess of oxygen favors McCarroll et al. [16] proposed electrochemical dep- suppression of the JT effect, which results in a change osition on a high-temperature melt solution serving as

A STUDY OF MAGNETIC ORDERING 147

Table 1. Structural data for LaMnO3 + δ single crystals Crystal 1 2 3 δ 0.01 0.02 0.085 Mn4+%2417 3+ 4+ 3+ 4+ 3+ 4+ Formula unit La0.997 Mn0.975Mn0.022 O3 La0.993 Mn0.954Mn0.049 O3 La0.972 Mn0.807Mn0.165 O3 Space group 300 K Pbnm Pbnm 300 K a, Å 5.529 5.493 3.894α = 90°56′ b, Å 5.572 5.536 c, Å 7.756 7.821 V/Z, Å3 59.73 59.45 59.06 Structure O' OR

300 K m(MnÐOI), Å 1.95 1.95 s(MnÐOII), Å 1.93 1.956 l(MnÐOII), Å 2.02 1.958 Rt = l/s 1.047 ~1 an anode as a method for growing lanthanum mangan- ples to be single-phase orthorhombic or rhombohedral ite single crystals. Among the merits of this method is single crystals [17]. The measurements were performed the absence of twins in the grown crystals. We report on three LaMnO3 + δ single crystals with different oxy- here on an investigation of LaMnO3 + δ single crystals gen contents: crystal 1 (δ = 0.01), crystal 2 (δ = 0.02), with various oxygen contents prepared using a refined and crystal 3 (δ = 0.085). Crystal 1 had orthorhombic electrochemical deposition method described in detail O' structure, crystal 2 had orthorhombic O structure, in [17] to study the effect of oxygen nonstoichiometry and crystal 3 had rhombohedral structure. Table 1 lists on the magnetic properties and magnetic states of these the crystallographic characteristics of the three single crystals. Particular attention was focused on the LaMnO3 + δ single crystals, including the distance m magnetic anisotropy of LaMnO3 + δ single crystals. between the Mn ion and the apical oxygen ion (MnÐ OI), as well as the short (s) and the long (l) distances between the Mn and OII ions in the MnO4 plane. The 2. SAMPLES AND EXPERIMENTAL values of m, s, and l were calculated using the technique TECHNIQUE proposed in [19]. We chose Rt = 1/s as a parameter characterizing the JT distortions, as in [7]. The LaMnO3 + δ single crystals were grown by electrochemical deposition from a melt solution using plat- As follows from Table 1, the distortions of the oxy- inum electrodes and a platinum crucible. The technique gen octahedra around the manganese ions decrease employed to obtain single crystals and the determina- with increasing content of nonstoichiometric oxygen; tion of the cation and anion composition are described the O' structure (crystal 1, δ = 0.01) transfers to the O in considerable detail in [17]. The single crystals thus structure (crystal 2, δ = 0.02), and then, at the maxi- grown were of a lustrous black and their shape was mum value δ = 0.085, we have rhombohedral crystal 3. close to cubic. The cation composition was determined The magnetization of the single crystals was deter- using x-ray fluorescence. The average manganese mined with a SQUID magnetometer (Quantum Design, valence was derived in two steps; first, the oxidizing MPMS-5) in magnetic fields of up to 5 T and at temper- capacity was found using iodometric titration, after atures ranging from 4.2 to 300 K for various magnetic- which the total content of manganese was measured field orientations with respect to the principal crystallo- using the photocolorimetry of a solution of permanga- graphic axes. nic acid upon complete oxidation of the manganese [18]. Assuming vacancies on the La and Mn sublattices to be present in equal numbers, the oxygen content was 3. EXPERIMENTAL RESULTS found based on the average manganese valence, with 3.1. Temperature Dependences of the Magnetization the concentration Mn4+% = 2δ × 100%. of Single Crystals 1, 2, and 3 for Various The phase composition, lattice parameters, and Magnetic Field Orientations crystal orientation were determined using the x-ray The temperature dependences of the magnetization technique. The x-ray structural measurements made on M(T) of the three single crystals were measured in powder samples of crystalline material showed all sam- weak fields under field cooling (FC) and zero-field

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 148 BARILO et al.

0.5 5 No. 1 4 H || (ab) 0.4 3 No. 1 , emu/g M 2 No. 2 1 0.3 H || c 0 80 100 150 200 250 300 , emu/g T, K M 0.2

× 0.1 4

0 50 100 150 200 250 300 T, K

Fig. 1. Temperature dependences of the magnetizations MFC (open symbols) and MZFC (filled symbols) for LaMnO3 + δ single crystals No. 1 at 200 Oe and No. 2 at 10 Oe (H || c); the inset shows crystal No. 1 for H || (ab). cooling (ZFC). As seen from Fig. 1, lowering the tem- momagnetic irreversibility below 150 K. Note that at perature entails a fairly sharp PM-to-FM transition at ~200 K, all the measured M (T) and M (T) curves || FC ZFC Tc= 125 K (crystal 1) and 170 K (crystal 2) for H c. display a certain bend (Fig. 2a). This feature, as well as For T < Tc, the MZFC(T) and MFC(T) curves exhibit a the difference between MFC and MZFC, becomes less clearly different behavior. In particular, MFC grows with pronounced in higher fields (Fig. 2b), until in such high decreasing T < Tc practically down to the lowest tem- fields as 6 and 50 kOe, all the curves practically coin- peratures reached. At the same time, the MZFC curve cide throughout the temperature interval covered, − reveals a broad maximum and for T < Tf = 155 K (crys- 5 300 K. tal 1) or 160 K (crystal 2), the M (T) and M (T) ZFC FC Figure 3a plots the temperature dependence of vol- cease to coincide. Below Tf , MZFC decreases to reach ume magnetic susceptibility χ (T) = M /H measured very low values at 5 K. In the case of the magnetic field FC FC in fields of 6 and 50 kOe (H is directed along the c axis). H lying in the (ab) plane in crystal 1, the behavior of the The inset shows the temperature dependence of inverse MFC(T) and MZFC(T) relations at T < Tf = 120 K is quite χ susceptibility 1/ FC(T), whose extrapolation to zero different (see inset to Fig. 1). First, there is practically Θ Θ yields the PM Curie temperature p. The values of p no difference between the MFC(T) and MZFC(T) curves. ≈ equal to ~145 K (H = 6 kOe) and ~135 K (H = 50 kOe) Both curves pass a sharp maximum at Tf 120 K. Sec- ond, both M and M decrease by more than four are positive and higher than Tc, which implies an FM ZFC FC character of exchange coupling between the spins. For times with a further decrease in temperature. Θ crystals 2 and 3, p = 177.3 and 203.4 K, respectively. Figure 2a displays the temperature dependence of Table 2 lists the Curie constant C and the effective num- the magnetization of crystal 3 measured in a weak mag- ber of Bohr magnetons calculated for all crystals in the netic field of 10 Oe in the temperature interval 5Ð 300 K PM state. It is instructive to follow the effect of the for three magnetic field orientations. Single crystal 3 magnetic field on the magnetization of crystal 1 in the has rhombohedral structure (Table 1). During the mea- case H lying in the (ab) plane (Fig. 3b). Note that the surements, the magnetic field was oriented perpendicu- field of 6 kOe shifts the maximum in MFC(T) by 10 K lar to the (200) growth planes. The crystal was observed toward lower temperatures, while the magnetization to transfer to the FM state at Tc = 190 K in all three mag- decreases by a factor of ~1.4 as the sample is cooled to netic field configurations. Below ~240 K, both M and FC 5 K; therefore, the decrease in MFC(T) at 6 kOe is three MZFC increase strongly in magnitude for all magnetic times smaller than that in a field of 200 Oe (inset to field orientations; below ~160 K, MFC has a constant Fig. 1). A further increase of the magnetic field in mag- value different from those of MZFC. For the first orienta- nitude brings about the disappearance of the maximum tion of the magnetic field in Fig. 2a, the MFC(T) and in the MFC(T) curve for H = 50 kOe in the (ab) plane, MZFC(T) curves follow a similar behavior, whereas in while a weak maximum in MZFC(T) at ~50 K is still the other two orientations, MFC and MZFC exhibit ther- seen (inset to Fig. 3b).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 A STUDY OF MAGNETIC ORDERING 149

(a) No. 3 (b) H = 100 Oe 0.6 H = 10 Oe 5

0.5 4

0.4 3

, emu/g 0.3 M 2 ZFC || 0.2 FC H [200] ZFC || ZFC FC H [200] || FC H [020] 1 ZFC 0.1 || ZFC FC H [020] || FC H [002]

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

Fig. 2. Temperature dependences of MFC and MZFC magnetizations obtained for LaMnO3 + δ single crystal No. 3 in a field of (a) 10 Oe for three different magnetic-field directions and (b) 100 Oe for two field directions.

0.010 35 80 (a)100 (b) H = 50 kOe 30 60 H || c FC H || ( ) FC No. 1 FC ab || χ 40 H || (ab)ZFC 1/ Oe H c 25 50 20 20

0.005 , emu/g , emu/g M 15 0 50 100 150 200 250 300 FC H = 6 kOe 0 50 100 150 200 250300 No. 1 χ T, K H = 50 kOe 10 H || (ab), H = 6 kOe 5 FC

0 50 100 150 200 250 300 0 50 100 150 200 250 300 T, K T, K χ χ Fig. 3. Temperature dependences (a) of the susceptibility FC and 1/ FC (inset) obtained on LaMnO3 + δ single crystal No. 1 in ﬁelds || || of 6 and 50 kOe (H c) and (b) of the magnetization obtained on LaMnO3 + δ single crystal No. 1 for H (ab) at 6 kOe and (inset) 50 kOe.

3.2. Isotherms of the Field Dependences of Extrapolation of M(H) from the saturation region to Magnetization of Single Crystals 1, 2, and 3 H = 0 yielded an estimate for the spontaneous magneti- || || zation Ms = 70 emu/g (H c) and 67 emu/g (H (ab)). The hysteresis loops measured by us (not shown in Note that at 5 K, the magnetization saturates in ﬁelds Fig. 4) are symmetric with respect to the origin; there- || || Hs ~ 5Ð6 kOe (H c) and ~10 kOe (H (ab)). The the- fore, Fig. 4 shows only part of the M(H) curves t obtained in the measurements. For sample 1 (H || c), the oretical value of saturation magnetization Ms at T = 0 3+ coercive ﬁeld Hc is ~1.5 kOe at 5 K and decreases to as calculated for total FM spin ordering of Mn (S = 2) ~100 Oe at 50 K. Measured in the H || (ab) orientation, and Mn4+ (S = 3/2) in single crystal 1 is 91.6 emu/g, the value of Hc at 5 K is substantially smaller, about which is ~25% higher than the observed Ms at 5 K. 200 Oe. In the other crystals, Hc is low and, at 5 K, is Such a low value of Ms may be due to a sizable part of ~30 Oe for sample 2 and less than 10 Oe for crystal 3. the sample being in the AF state. The values of Ms(FC)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 150 BARILO et al.

Table 2. Magnetic characteristics of LaMnO3 + δ single crystals Crystal/δ 1/0.01 2/0.02 3/0.085 H orientation ||c ||(ab) ||c ||[200] PM region C, (emu K)/mole 4.13 4.73 4.77 4.49 Θ p, K 144.5 131.7 177.3 203.4 µ Peff, B 5.5 6.15 6.18 5.99 FM region

Tc, K 125 ~125 170 190 Ms(FC) 70 67 88 89 Ms(ZFC) 63.6 63.6 86 87 µ µ , B 3.03 2.9 3.75 3.77 Hc (5 K), Oe 1500 200 ~30 <10 Hs, kOe 5 10 7 8 χ 3 × Ð4 × Ð4 × Ð4 × Ð5 HF(5 K), cm /g 2 10 9.6 10 2.2 10 (6Ð8) 10 JF/kB 8.4 10.9 JA/kB Ð1.22 ~Ð0.5

|| χ and Ms(ZFC) and the corresponding effective magnetic and then falls off rapidly. For H (ab), the HF(T) rela- moment are presented in Table 2. Note that the low- tion follows a more complicated course for T < Tc, with temperature magnetization is higher than the magneti- an additional maximum appearing at ~50 K. In crystal 2, ≈ χ || zation measured at T 100–120 K only in fields above the HF(T) curve measured for H c passes through a ~1000 Oe (Fig. 4b), whereas in lower fields, the mag- maximum at Tmax = 200 K (above Tc) and remains prac- netization at T ≈ 100Ð120 K is noticeably higher than tically constant below 100 K. The high-field suscepti- that at lower temperatures. bility of crystal 3 in the single-domain state behaves in µ a similar way (Fig. 5). For crystal 2, Ms(5 K) is ~88 emu/g or 3.75 B/Mn. The values of the saturation magnetization at 5 K derived from the field dependences M(H) for all three magnetic-field orientations in crystal 3 (Fig. 4c) vary 4. DISCUSSION OF RESULTS within the interval 87Ð89 emu/g, which corresponds to The observed behavior of magnetization in a magnetic moment of 3.68Ð3.77µ at 5 K. B LaMnO3 + δ single crystals can be understood by invok- ing the model of the two-phase magnetic state, which assumes separation of a system, at low enough temper- 3.3. Temperature Dependence of the High-Field atures, into an AF matrix and FM clusters embedded in Susceptibility of Single Crystals 1, 2, and 3 it. The excess oxygen in LaMnO3 + δ gives rise to It is well known that the differential magnetization vacancy formation on the cation (La and Mn) sublat- χ = [M(H ) Ð M(H )]/(H Ð H ) for a fixed value of ∆H = tices [6]. Each vacancy has a charge of Ð3e and acts as 1 2 1 2 an acceptor for the holes doped into the system, with H1 Ð H2 is a characteristic of the magnetic state of a Ӷ χ the result that hole-enriched FM clusters form near the sample. In weak fields (H Hs), the susceptibility of vacancies. The AF matrix is free of holes and is cou- a crystal with Ms > 0 contains contributions due to pled to the FM clusters by superexchange interaction domain wall displacement and domain magnetization at the interface separating the two phases [10]. These vector rotation toward the magnetic-field direction. In assumptions, which permit one to explain the magnetic high fields, χ = χ characterizes the sample suscepti- HF properties of LaMnO3 + δ, draw from measurements, bility in the single-domain magnetic state. Figure 5 dis- for instance, of inelastic neutron scattering in plays temperature dependences of the high-field sus- La Ca MnO [21], which revealed the existence of χ 0.95 0.05 3 ceptibility HF(T) of crystal 1 measured for two mag- an anisotropic AF phase and an isotropic FM phase netic-field orientations. The fields H1 and H2 are chosen with an FM correlation length of 8–10 Å in all direc- Ӷ χ to be 6 and 50 kOe, respectively. For T Tc, HF for tions. Neutron diffraction measurements performed on || H c is nearly constant; close to Tc, the susceptibility polycrystalline LaMnO3 + δ samples [8] likewise sup- grows to a maximum value at Tmax = 140 K (this is in port the existence of phase separation, with the latter agreement with the behavior of an FM material [20]) depending strongly on the actual conditions of crystal

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 A STUDY OF MAGNETIC ORDERING 151

80 No. 1 (a)No. 1 (b) H || c FC H || (ab) FC

40 , emu/g M

T = 5 K 20 T = 50 K T = 100 K T = 50 K T = 125 K T = 115 K T = 100 K T = 130 K T = 120 K T = 115 K T = 150 K T = 125 K T = 120 K 0 20 40 04020 H, kOe 100 (c)

80 100 H || [020] 50 K No. 3 80 100 K 60 T = 5 K 150 K 60 175 K , emu/g

M 200 K 40 40 H || [200] 230 K H || [020] H || [002] 20 270 K 20

0 10 20 30 40 50

0 1020304050 H, kOe

|| || Fig. 4. Field dependences of the magnetization of LaMnO3 + δ single crystals (a, b) No. 1 for (a) H c and (b) H (ab) and (c) No. 3 for three magnetic-field orientations (T = 5 K); inset: M(H) plotted for the region 50Ð270 K. growth and heat treatment. This accounts for the scatter cluster (for instance, for a rhombohedral structure) of literature data on the crystal and magnetic properties appears. However, in the latter case, AF regions still of LaMnO3 + δ. It should be stressed that the crystal persist, as found in neutron diffraction studies made on phases revealed in [8] coincide in structure with the La1 Ð xSrxMnO3 for x > 0.175 [22] despite the fact that ones identified in the present study (except the P1121/a the sample with rhombohedral structure was in the phase). In what follows, we use the model proposed in metallic phase. [10] to explain our data. The distribution of holes and the spin configuration For small values of δ and low enough tempera- of the AFÐFM system depend on temperature; for T ӷ tures, the FM clusters are separated from one another; Tc, the spins are in the random state and the holes are as δ (and, hence, the cation vacancy concentration) distributed uniformly. When the sample is cooled increases, however, the distance between the FM clus- (T Tc), the holes are concentrated near the cation ters decreases, making it possible for some of them to vacancies to form clusters with FM-ordered spins merge into larger clusters until, eventually, an infinite (through strong DE interaction). The magnetic moments

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 152 BARILO et al. ibility between M and M will be large (Fig. 1, H || 0.0010 H1 = 50 kOe, H2 = 6 kOe H || (ab) FC ZFC H || c c; Table 2). The difference between MFC and MZFC || 0.0008 H [200] decreases with increasing magnetic ﬁeld (Fig. 2; inset H || [020]

/g to Fig. 3b), a feature characteristic of the presence of 3 0.0006 FM clusters in the matrix [24]. No. 3 , cm no. 2 Note that for H || (ab) (crystal 1), FM ordering at

HF 0.0004

χ ≈ Tc 125 K is accompanied by the appearance of an 0.0002 No. 1 additional maximum at T = 120 K in the MFC(T) relation measured at 200 Oe (inset to Fig. 1). The magni- 0 50 100 150 200 250 300 tude of Tmax depends on the magnetic field and shifts from 120 to 110 K in a field of 6 kOe (Fig. 3b). In addi- T, K tion, MFC(T) and MZFC(T) are not thermomagnetically Fig. 5. Temperature dependences of the high-field suscepti- irreversible in this field configuration (inset to Fig. 3b). χ Thus, the FM cluster contribution is the lowest. Such an bility HF(T) measured on LaMnO3 + δ single crystals Nos. 1, 2, and 3. anomaly in the behavior of MFC(T) has been observed || in single-crystal La0.9Sr0.1MnO3 for H (ab) [25], where FM ordering at Tc = 150 K was accompanied by of the host ions are ordered antiferromagnetically. The the appearance of a maximum in the MFC(T) curve at exchange interactions in an AFÐFM system may be 110 K and the field dependences M(H) in the interval considered the sum of a strong positive JFC interaction 105Ð118 K had a stepwise character. Metamagnetic inside the FM clusters and a negative interaction JA and transitions in high fields and the additional maximum in χ a positive interaction JF between spins along and per- the temperature dependence of HF(T) for T < Tc were pendicular to the c axis, respectively, in the AF matrix assigned [26] to a structural transition in single-crystal [23], as well as of the exchange interaction JB > 0 at the La0.9Sr0.1MnO3 from the pseudocubic O" to orthorhom- boundary separating the two phases [10]. We estimated bic O' structure occurring in the interval 100Ð110 K. the exchange constants JF and JA using the values of Tc The field dependences of M(H) of single crystal 1 Θ δ and p measured for = 0.01 and 0.02 and came to the exhibit neither a stepwise pattern nor metamagnetic following relations [13]: transitions in high fields. Nevertheless, an anomaly in || ()() the behavior of MFC(T) does exist for T < Tc (H (ab)) T c = 2/3SS+ 1 4 JF + 2 J A /kB, and the value of Tmax depends on H (Fig. 3b); also, the Θ ()() χ p = 2/3SS+ 1 4J F + 2J A /kB. HF(T) relation follows a complicated pattern in the low-temperature region (Fig. 5). The values thus obtained (Table 2) correlate well with As δ (i.e., the cation vacancy concentration) the results for LaMnO3 + δ [7] found for the corresponding values of R = l/s (Table 1). The spins of the AF increases, (i) the FM cluster density increases and, t accordingly, the number of AF-matrix spins per FM matrix near the FM clusters are canted relative to the cluster decreases; (ii) the spin canting angle in the AF spins in the FM clusters because of the exchange inter- matrix increases; and (iii) the crystal orthorhombicity action JB, and the canting angle grows with increasing decreases and, as a consequence, the crystallographic FM cluster density and applied magnetic field [10]. magnetic anisotropy decreases. All these processes δ The actual character of the temperature depen- bring about a decrease in Hc with increasing (Table 2). dences of M and M should depend on the relative FC ZFC The growth of the saturation magnetization Ms with magnitude of the contributions of the AF matrix and the increasing δ (Table 2) is associated not only with a FM clusters for a given magnetic-field orientation. In growth in the number of completely field-aligned spins the FC case, the cluster spins align with the field with in the FM clusters but also with an increase in the cant- decreasing temperature T < Tc, so that the contribution ing angle in the AF matrix. The feature observed for || δ to MFC will be the largest for H c. In the ZFC case, T > Tc in the single crystal with = 0.085 (Fig. 2) can where the sample is cooled in a zero field, there is no also be assigned to the presence of clusters. As the preferential orientation for spins in the FM clusters and temperature increases, the long-range order in the the contribution that the FM phase provides to MZFC matrix breaks down faster than the short-range order should be the lowest for H || c (down to the lowest pos- in the clusters; therefore, the M(T) curve exhibits two sible temperatures). If a weak field is applied to the magnetic-ordering temperatures. In polycrystalline sample, the magnetization will depend strongly on the LaMnO3 + δ [12], such an anomaly was observed in χ system anisotropy. In the case of strong anisotropy, a ac(T) curves. The size of the clusters can be estimated weak magnetic field will not be able to rotate the spins using the technique [27] relating the cluster magnetic and MZFC will be small. Because the magnitude of the moment to the concentration of cation vacancies. For coercive field Hc is due to the magnetic anisotropy of δ = 0.085, this concentration is ~0.025 and, hence, the the AF phase, for H < Hc, the thermomagnetic irrevers- clusters are about 15–20 Å in size. The nonlinearity of

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 A STUDY OF MAGNETIC ORDERING 153 ӷ the M(H) curves for T Tc is likewise associated with 10. M. Muroi and R. Street, Aust. J. Phys. 52 (2), 208 (1999). the presence of FM clusters (Fig. 4). 11. J. P. Mitchel, D. N. Argyriou, C. D. Potter, et al., Phys. Rev. B 54 (9), 6172 (1996). 12. A. N. Pirogov, A. E. Teplykh, V. I. Voronin, et al., Fiz. 5. CONCLUSION Tverd. Tela (St. Petersburg) 41 (1), 103 (1999) [Phys. Thus, we have reported on a study of magnetic Solid State 41, 91 (1999)]. ordering in LaMnO3 + δ single crystals grown by elec- 13. V. Skumryev, F. Ott, J. M. D. Coey, et al., Eur. Phys. J. B trochemical deposition. Single crystals with the struc- 11 (3), 401 (1999). tures O' (δ = 0.01), O (δ = 0.02), and R (δ = 0.085) 14. S. Mitsudo, K. Hirano, H. Nojiri, et al., J. Magn. Magn. exhibit FM ordering at Tc = 125, 170, and 190 K, Mater. 177Ð181 (1), 877 (1998). respectively. A strong thermomagnetic irreversibility is 15. A. M. Balbashov, M. K. Gubkin, V. V. Kireev, et al., Zh. δ observed between MFC(T) and MZFC(T) for = 0.01 and Éksp. Teor. Fiz. 117 (3), 542 (2000) [JETP 90, 474 || δ || 0.02 for H c. For = 0.01 and H (ab), MFC(T) was (2000)]. observed to behave anomalously for T < Tc. The tem- 16. W. H. McCarroll, K. V. Ramanujachary, I. D. Fawcett, perature and field dependences of magnetization and M. Greenblatt, J. Solid State Chem. 145 (1), 88 (1999). obtained for LaMnO3 + δ single crystals are accounted for within the model of a two-phase magnetic system 17. D. D. Khalyavin, S. V. Shiryaev, G. L. Bychkov, et al., in containing an insulating AF matrix with embedded Proceedings of Fourth International Conference “Single hole-enriched FM clusters. Crystal Growth and Heat & Mass Transfer,” Obninsk, 2001, Vol. 2, p. 372. 18. A. G. Soldatov, S. N. Barilo, and S. V. Shiryaev, Zh. ACKNOWLEDGMENTS Anal. Khim. 56 (12), 1245 (2001). This study was partially supported by the Scientific 19. A. K. Bogush, V. I. Pavlov, and L. V. Balyko, Cryst. Res. Research Committee, project no. 5P03B01620. Technol. 18 (3), 589 (1983). 20. K. P. Belov, Magnetic Transitions (Fizmatgiz, Moscow, 1959; Consultants Bureau, New York, 1961). REFERENCES 21. M. Hennion, F. Moussa, J. Rodriguez-Carvajal, et al., 1. C. Zener, Phys. Rev. 82 (3), 403 (1951). Phys. Rev. B 56 (1), R497 (1997). 2. P. W. Anderson and H. Hasegawa, Phys. Rev. 100 (1), 22. D. Louca, T. Egami, E. L. Brosha, et al., Phys. Rev. B 56 675 (1955). (14), R8475 (1977). 3. P. G. De Gennes, Phys. Rev. 118 (1), 141 (1960). 23. F. Moussa, M. Hennion, J. Rodriguez-Carvajal, et al., 4. J. M. D. Coey, M. Viret, and S. von Molnár, Adv. Phys. Phys. Rev. B 54 (22), 15149 (1996). 48 (2), 167 (1999). 24. L. Ghivelder, L. A. Castillo, M. A. Gusmâo, et al., Phys. 5. E. L. Nagaev, Usp. Fiz. Nauk 166 (8), 833 (1996) [Phys. Rev. B 60 (17), 12184 (1999). Usp. 39, 781 (1996)]. 25. K. Ghosh, R. L. Greene, S. E. Lofland, et al., Phys. Rev. 6. J. A. M. van Roosmalen and E. H. P. Cordfunke, J. Solid B 58 (13), 8206 (1998). State Chem. 110 (1), 109 (1994). 26. A. V. Korolyov, V. Ye. Arkhipov, V. S. Gaviko, et al., 7. C. Ritter, M. R. Ibarra, J. M. De Teresa, et al., Phys. Rev. J. Magn. Magn. Mater. 213 (1), 63 (2000). B 56 (12), 8902 (1997). 27. J. Z. Sun, L. Krusin-Elbaum, A. Gupta, et al., Appl. 8. Q. Huang, A. Santoro, J. W. Lynn, et al., Phys. Rev. B 55 Phys. Lett. 69 (7), 1002 (1996). (22), 14987 (1997). 9. F. Prado, R. D. Sánchez, A. Caneira, et al., J. Solid State Chem. 146 (2), 418 (1999). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 154Ð157. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 147Ð150. Original Russian Text Copyright © 2003 by Kamzina, Kraœnik. MAGNETISM AND FERROELECTRICITY

Mechanism of the Polarization Response in the Relaxor State of Lead Scandotantalate Single Crystals with Different Levels of Ion Ordering L. S. Kamzina and N. N. Kraœnik Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received November 13, 2001; in ﬁnal form, March 20, 2002

Abstract—The dielectric properties and optical transmission of stoichiometric lead scandotantalate (PST) single crystals in strong electric fields was studied above the temperature of the spontaneous ferroelectric phase transition (Tsp). It is shown that the mechanism of polarization response directly above Tsp is related to induced polarization effects and macrohysteretic behavior only in ac fields above 5 kV/cm. An analysis of reciprocal dielectric permittivity carried out over a broad temperature range far above the temperature at which the dielectric permittivity passes through a maximum revealed that specific features of the relaxor behavior manifest themselves up to 400°C even in highly ordered PST crystals. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The mechanism of polarization response in ac fields and the temperature dependence of ε in the relaxor state The nature of the dielectric response in disordered in compounds of the type PbB' B'' O , which occupy media, to which ferroelectric relaxors belong, has 1/2 1/2 3 remained a topical problem in the physics of ferroelec- an intermediate position between normal ferroelectrics tricity for many years. However, obtaining an unambig- and typical relaxors, are not fully understood. Among uous answer to the question of which “relaxing” ele- these substances is lead scandotantalate, PbSc1/2Ta1/2O3 ments are responsible for the dielectric response within (PST). There is only one communication that reports on a broad temperature range is a very difficult problem. the effect of the ac field amplitude 0 < E < 4 kV/cm on This is accounted for by the fact that interactions the dielectric response of PST crystals near the sponta- among the relaxing elements play a major role at low neous phase transition temperature (Tsp) [5]. In that temperatures and as the temperature varies, the nature study, it was shown that in fields of up to 4 kV/cm and itself of these elements may change from single dipoles at temperatures slightly above Tsp, the observed nonlin- at high temperatures to domains and domain walls at earity is most likely associated with the dynamics of low temperatures. At the same time, in the region of domain walls and phase boundaries. In addition to the comparatively weak interactions in the ergodic phase of typical relaxor behavior, these substances also exhibit a the relaxors (at high temperatures), one can obtain an spontaneous phase transition from the relaxor state to explicit answer to this question; this could prove an the ferroelectric phase at a temperature below the max- important step in understanding the nature of relaxors. imum in ε. By properly changing the degree of ordering s of the Sc3+ and Ta5+ ions, one can vary not only the Such studies have been carried out on lead mag- phase transition temperature but also the character of noniobate (PMN), which is a classical relaxor [1, 2]. It the transition. As s decreases, the specific features of has been shown experimentally that the nonlinear the relaxor behavior characteristic of the diffuse ferro- dielectric properties in the ergodic phase revealed in ac electric transition are seen more clearly. An increase in electric fields 0 < E < 2 kV/cm are due only to the s (s 1) weakens the relaxor properties. In partially motion of phase boundaries of polar regions rather than ordered PST compounds, the relaxor state extends over to thermally activated orientation of the local spontane- a fairly broad temperature interval above Tsp [6, 7]. ous polarization vector in polar regions. This study deals with the mechanism of the polar- Among the specific features of relaxors (PMN and ization response of PST crystals as a function of ac field similar compounds) is that the temperature dependence amplitude and with the effect of the degree of ion order- of reciprocal dielectric permittivity ε in the ergodic ing s on the behavior of dielectric permittivity at tem- ε peratures above T . phase immediately above the maximum in (Tmax ε) is sp quadratic [3, 4] and that only at high temperatures in the We employed not only the classical dielectric meth- paraelectric phase does this dependence obey the ods but also an optical technique, namely, measurement CurieÐWeiss law, as is the case with conventional ferro- of the optical transmission (OT); the latter method is electrics. more sensitive to changes in the size of the inhomoge-

MECHANISM OF THE POLARIZATION RESPONSE 155 neities. If the size of the scatterers changes at the phase transition, this will entail changes not only in light scat- 1 tering but also in the optical transmission. Scattering 1' depends on the ratio of the dimensions of the scattering 2 particle to the wavelength of light. If this ratio is small, the scattered intensity will be weak and the sample will be virtually transparent. In the absence of an electric field, such a state is characteristic of the relaxor phase. 3 As the particle size increases, the scattering intensity grows strongly, particularly in directions making small angles with the propagation vector of the incident light. This results in a decrease in the OT. We showed in previous communications [8, 9] that the spontaneous phase transition in stoichiometric PST crystals is of percolation type. In phase transitions of the percolation Optical transmission, arb. units type, the average size of new-phase clusters at percola- Tsp tion threshold approaches the size of the sample and a large-scale nonuniform structure (“infinite cluster”) forms. In this case, the phase transition should be 20 30 40 50 accompanied by the appearance of anomalously narrow T, °C peaks in small-angle light scattering intensity and, hence, by a minimum in the OT. Fig. 1. Temperature dependences of optical transmission in PST crystals (s = 0.9) measured at different ac electric field amplitudes E: (1) 0, (1') 4, (2) 6, and (3) 8 kV/cm. 2. EXPERIMENTAL TECHNIQUE AND THE SAMPLES USED spontaneous phase transition Tsp. The width of the min- We used stoichiometric PST crystals with different imum is ~1Ð2°C. In this region, a large-scale structure degrees of ion ordering (0.3 < s < 0.98). Dielectric and forms, small-angle scattering increases sharply, and the optical measurements were conducted on PST single × × crystal becomes practically opaque. Immediately above crystals 1 2 2 mm in size which were not subjected T (percolation threshold), the scatterer size decreases to any mechanical processing. The electric field was sp and the OT increases. An increase in the field amplitude applied along the 〈100〉 crystallographic directions, and to 4 kV/cm (curves 1, 1') virtually does not affect the the light was propagated along 〈001〉. The dielectric magnitude of OT at temperatures above T . The change permittivity was measured at frequencies of 1 kHz and sp 1.3 MHz on a sample heated at a rate of 2Ð4°C/min in in the OT curve shape observed to occur below Tsp in a the temperature range from 0 to 700°C. field of 4 kV/cm is associated with an increase in domain size. The behavior of the OT at temperatures The optical studies were conducted with a HeÐNe slightly below T was discussed by us in more detail laser. A low-power laser beam was passed through a sp elsewhere [5]. In fields above 5 kV/cm, the OT falls off sample mounted in a temperature-controlled cryostat considerably (curves 2, 3) directly above T . The with an aperture of ~1.5°. This aperture permitted us to sp effectively exclude the contribution from Rayleigh increase in domain and polar region size, which entails scattering on small particles to optical transmission. On the formation of a macrodomain cluster, may account passing through the crystal, the light was detected by a for the above-mentioned decrease in OT. This conjec- photodiode, whose output was fed to a lock-in detector ture is also corroborated by the double dielectric hyster- with a time constant of 1 s. The laser beam modulation esis loops observed immediately above Tsp in polycrys- frequency was 1 kHz. A 50-Hz electric field with an talline PST samples [10]. It follows from the data pre- amplitude of up to 8 kV/cm was applied to the sample. sented in [10] that strong nonlinearity and hysteresis The time-averaged signal was measured. All the optical effects are induced only in fields above 5 kV/cm. Dou- measurements were performed in integrated mode. ble hysteresis loops appear, the inhomogeneities grow After each application of the electric field, the samples in size, and a large-scale macrodomain structure forms. were depolarized by heating to 100°C prior to subse- The increase in inhomogeneity size brings about an quent measurement. increase in small-angle scattering and a decrease in the OT; this exactly is observed experimentally. This type of nonlinearity is associated with induced polarization 3. EXPERIMENTAL RESULTS switching and domain growth. AND DISCUSSION As follows from the hysteresis loops, in weak fields Figure 1 presents OT temperature dependences (below 4 kV/cm), the P vs. E relation is linear and the obtained at different ac electric field amplitudes. The size of the inhomogeneities does not change; therefore minima in the OT curves lie at the temperature of the at temperatures above Tsp, no change in the OT is

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

156 KAMZINA, KRAŒNIK α relaxor state exists but also a spontaneous phase transi- 1.6 tion occurs, no quadratic dependence of εÐ1 on temper- 1.5 α = 1 ature has been observed [12]. Figure 2 shows the recip- 40 1.4 rocal dielectric permittivity plotted vs. temperature on –4 1.3 the logÐlog scale for PST crystals with different values 10 30 0 0.5 1.0

× of s. It is seen that the slope of the straight lines (equal ) s α 1 20 to ) varies depending on s; more speciﬁcally, the Ð max ε α = 2 larger the degree of ordering s at temperatures above

– α α Tmax ε, the closer the parameter to unity ( = 1.3 for –1

ε α ( 10 s = 0.9, curve 1). As s decreases, the parameter 123 approaches 2 (α = 1.58 for s = 0.3, curve 3). This is seen more clearly in the inset to Fig. 2. The reciprocal dielectric permittivity in crystals with different s is observed to depend on temperature with α = 1 only at tempera- 100 1000 tures above 400°C, which implies a transition to a uni- α ° (T Ð Tmax ε) , C form paraelectric phase. Thus, the PST crystals studied in this work retain specific features of the relaxor behavior up to 400°C even in highly ordered com- εÐ1 εÐ1 Fig. 2. Reciprocal dielectric permittivity ( Ð max ) plot- pounds. α ted vs. (T Ð Tmax ε) on a logÐlog scale for PST single crystals with the degree of ordering s: (1) 0.9, (2) 0.7, and (3) We used dielectric permittivity measurements per- 0.3. Dashed lines are calculated for α = 1 and 2. Inset: plot formed with different ac electric fields to calculate the of the parameter α vs. s. α coefficient for temperatures above Tsp. It was found that changing the field strength from 1 to 600 V/cm vir- α observed within the experimental accuracy (curve 1' in tually does not affect the magnitude of . This may Fig. 1). At the same time, in [5], slight variations were mean that ac fields of up to 600 V/cm do not induce the noticed in the dielectric permittivity of this crystal in formation of new polar clusters and only slightly fields below 600 V/cm; these variations are most likely increase the size of those that already exist. This is also associated with a change in the dynamics of domain or confirmed by our optical measurements carried out in interphase states. The polarization induced at low tem- fields of up to 4 kV/cm (curve 1' in Fig. 1). Otherwise, peratures in the ferroelectric state deviates from its pre- α would have grown with increasing ac field to ferred direction already in fields as weak as ~4 kV/cm approach 2, as was observed in [13] to occur in a PZT- for T > Tsp and undergoes reorientation. based ferroelectric ceramic. The upper temperature boundary of existence of an intermediate relaxor state in PST, above which the crys- 4. CONCLUSION tal transfers to a paraelectric state, is a debatable topic. In [6], indications of a relaxor behavior could be traced Thus, our dielectric and optical studies of stoichio- ° up to 170 C, whereas in [7], the paraelectric phase metric disordered PST single crystals with different ° formed only above 400 C. An answer to this problem values of s clearly indicate that the mechanism of polar- can be found by studying the behavior of dielectric per- ization response in the relaxor state immediately above mittivity over a broad temperature range, including the T is related to induced polarization effects and macro- relaxor and paraelectric states, and by investigating the sp effect of ion ordering s on the character of this behavior. hysteretic behavior only in ac fields above 5 kV/cm. It is known that in compounds with a diffuse phase tran- Growth of domains and polar regions in size, which sition the temperature dependence of the reciprocal gives rise to the formation of macrodomain clusters above T , may account for the observed decrease in dielectric permittivity above Tmax ε is described by the sp OT. εÐ1 εÐ1 α ε expression [11] ( Ð max ) = A(T Ð Tmax ε) , where max ε Analysis of the behavior of the dielectric permittiv- is the value of at the maximum, Tmax ε is the temperature of the maximum in ε, α is a critical parameter vary- ity over a broad temperature interval considerably ing in the interval of 1 to 2, and A is a constant. In typ- above Tmax ε showed the relaxor-behavior features in ical relaxors, such as PMN, the dielectric permittivity PST crystals to persist up to 400°C even in highly follows a quadratic relation with α = 2 at temperatures ordered compounds. This behavior of ε at high temper- ° approximately 200 C above Tmax ε. Only at higher tem- atures in crystals with s 1 appears to be of interest ° peratures (360Ð400 C above Tmax ε) is the CurieÐWeiss and requires further study to gain insight into the nature law met and does the parameter α approach unity. In of relaxor phenomena in inhomogeneous ferroelectric compounds of the type of PST in which not only the media.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 MECHANISM OF THE POLARIZATION RESPONSE 157

ACKNOWLEDGMENTS 7. L. S. Kamzina, N. N. Kraœnik, L. M. Sapozhnikova, et al., Pis’ma Zh. Tekh. Fiz. 14 (19), 1760 (1988) [Sov. This study was supported by the Russian Founda- Tech. Phys. Lett. 14, 764 (1988)]. tion for Basic Research, project no. 01-02-17801. 8. L. S. Kamzina and A. L. Korzhenevskiœ, Pis’ma Zh. Éksp. Teor. Fiz. 50 (3), 146 (1989) [JETP Lett. 50, 163 (1989)]. REFERENCES 9. L. S. Kamzina and N. N. Kraœnik, Fiz. Tverd. Tela 1. A. K. Tagantsev and A. E. Glazounov, Phys. Rev. B 57 (St. Petersburg) 42 (1), 136 (2000) [Phys. Solid State 42, (1), 18 (1998). 142 (2000)]. 2. A. E. Glazounov, A. K. Tagantsev, and A. J. Bell, Phys. 10. F. Chu, N. Setter, and A. K. Tagantsev, J. Appl. Phys. 74 Rev. B 53 (17), 11281 (1996). (8), 5129 (1993). 3. G. A. Smolensky, J. Phys. Soc. Jpn., Suppl. 28, 26 11. R. L. Moreira and R. P. S. M. Lobo, J. Phys. Soc. Jpn. 61 (1970). (6), 1992 (1992). 4. V. V. Kirillov and V. A. Isupov, Ferroelectrics 5, 3 (1973). 12. A. Sternberg, L. Shebanovs, E. Birks, et al., Ferroelec- trics 217, 307 (1998). 5. L. S. Kamzina and N. N. Kraœnik, Fiz. Tverd. Tela 13. A. V. Shilnikov, I. V. Otsarev, A. I. Burkhanov, et al., Fer- (St. Petersburg) 43 (10), 1880 (2001) [Phys. Solid State roelectrics 235, 125 (1999). 43, 1958 (2001)]. 6. F. Chu, C. R. Fox, and N. Setter, J. Am. Ceram. Soc. 81 (6), 1577 (1998). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 158Ð162. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 151Ð155. Original Russian Text Copyright © 2003 by Dul’kin, Raevskiœ, Emel’yanov. MAGNETISM AND FERROELECTRICITY

Acoustic Emission and Thermal Expansion of Pb(Mg1/3Nb2/3)O3 and Pb(Mg1/3Nb2/3)O3ÐPbTiO3 Crystals E. Dul’kin*, I. P. Raevskiœ**, and S. M. Emel’yanov** * Advanced School of Applied Science, The Hebrew University, Jerusalem, 91904 Israel e-mail: [email protected] ** Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344090 Russia e-mail: [email protected] Received April 3, 2002

Abstract—The temperature dependence of the elongation per unit length for Pb(Mg1/3Nb2/3)O3 crystals unan- nealed after growth and mechanical treatment is investigated in the course of thermocycling. It is revealed that this dependence deviates from linear behavior at temperatures below 350°C. The observed deviation is characteristic of relaxors, is very small in the ﬁrst cycle, increases with increasing number n of thermocycles, and reaches saturation at n ≥ 3. In the ﬁrst cycle, a narrow maximum of the acoustic emission activity is observed in the vicinity of 350°C. In the course of thermocycling, the intensity of this maximum decreases and becomes zero at n > 3. For (1 Ð x)Pb(Mg1/3Nb2/3)O3ÐxPbTiO3 crystals, the dependence of the temperature of this acoustic emission maximum on x exhibits a minimum. It is assumed that the phenomena observed are associated with the phase strain hardening due to local phase transitions occurring in compositionally ordered and polar nanoregions. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ordering in nanoregions is characterized by a ratio of 1 : 1 and occurs in accordance with the random-site Lead magnoniobate Pb(Mg1/3Nb2/3)O3 (PMN) is a model [3, 10, 11]. Within this model, the crystal lattice typical representative of relaxor ferroelectrics. These of PMN in an ordered state consists of two sublattices, materials have been extensively investigated in recent namely, the sublattice B' containing Nb5+ cations alone years [1Ð3]. The permittivity ε of relaxor ferroelectrics and the sublattice B'' whose sites are randomly occu- shows a smeared maximum whose temperature Tm and pied by Nb5+ and Mg2+ cations in the ratio 1 : 2; as a ε height m depend on the frequency of the measuring result, the composition of lead magnoniobate is field. Unlike ferroelectrics, which exhibit an abrupt described by the formula Pb(Nb)1/2(Nb1/3Mg2/3)1/2O3. phase transition, relaxors do not undergo macroscopic structural transformations at temperatures close to T . At temperatures below Td, the temperature depen- m dences of the birefringence and the elongation per unit The specific features in the properties of relaxors are ∆ associated with their inhomogeneity on the mesoscopic length L/L for PMN deviate from linear behavior level, particularly with the presence of polar and com- (characteristic of higher temperatures), which is associ- positionally ordered nanometer-sized regions in the ated with the formation of polar nanoregions and an paraelectric matrix [2Ð11]. The polar regions are increase in their total volume with a decrease in the ≈ temperature [2Ð4, 12Ð18]. The maximum deviation of formed in the course of cooling at a temperature Td ∆ ° the dependence L/L(T) from linearity at temperatures 350 C, which is considerably higher than the tempera- below T agrees with the electrostrictive constants of ture T (T ≈ Ð10°C for PMN). The number and the d m m PMN in order of magnitude [2, 4, 18]. It should be mean size of these regions gradually increase with noted that, at temperatures above Td, the linear thermal decreasing temperature [2Ð4, 8, 9]. By contrast, the α mean size (2Ð5 nm) and volume content (20Ð30% expansion coefficients determined both with the according to different estimates [5, 8]) of composition- dilatometric technique [2, 12, 14, 17] and from the tem- ally ordered regions remain nearly constant with a perature dependences of the lattice parameter [15, 16, change in the temperature from 200 to 800 K [8]. For 18] are in close agreement. At the same time, the dilato- these materials, the ordered regions were long thought metric data obtained in different works [2, 12, 14, 17] to be nonstoichiometric in composition with the ratio for temperatures below Td differ significantly. Mg : Nb = 1 : 1 and the net uncompensated charge was Earlier, it was repeatedly noted that the properties of considered to be one of the main factors limiting their relaxors substantially depend on the prehistory of the size [2, 3, 5]. In recent years, there has appeared evi- studied samples, specifically on the conditions of pre- dence that, although PMN ferroelectrics have a compo- liminary annealing and mechanical treatment [2]. sition with the ratio Mg : Nb = 2 : 1, compositional Although the available data on this dependence corre-

ACOUSTIC EMISSION AND THERMAL EXPANSION 159 spond to temperatures below Tm, we can assume that a The measurements were performed according to a similar effect also manifests itself at higher tempera- combined technique described earlier in [24]. A crystal tures. Note that, in specific cases, considerable differ- was mounted on the polished surface of a cylindrical ences between the dependences ∆L/L(T) measured for quartz acoustic waveguide, which was vertically different crystals (or even for the same crystal in differ- inserted into the bottom of the furnace. An acoustic ent coolingÐheating cycles) are observed in ferroelec- emission detector made from a TsTS-19 piezoelectric tric materials characterized by an abrupt phase transi- ceramic material was cemented to the lower cold sur- tion. One of the reasons for these differences is the face of the waveguide. Two rods of a differential phase strain hardening, which consists in generating dilatometer with a sensitivity higher than 10Ð7 were misfit dislocations due to disturbance of coherence at inserted into the top of the furnace. The temperature the interfaces of ferroelectric nuclei or between an anti- was measured using a chromelÐalumel thermocouple ferroelectric phase and a paraelectric matrix upon a whose junction was in close contact with the crystal. first-order phase transition [19–21]. As a rule, crystals The thermal elongation per unit length ∆L/L and the contain grown-in dislocations. In the course of ther- acoustic emission activity were measured simulta- mocycling, these dislocations interact (in particular, neously during cooling of the crystals in the course of annihilate) with misfit dislocations, which is attended thermocycling in the temperature range 200Ð500°C at a by acoustic emission [19Ð22]. Acoustic emission max- rate of 1Ð2 K/min. ima are frequently observed even in the case when anomalies in the elongation per unit length ∆L/L are either absent or weakly pronounced [22]. The acoustic 3. RESULTS emission activity Nú (the number of pulses per unit The results of dilatometric measurements of PMN time) is proportional to the number of annihilating dis- crystals in the first three thermocycles are presented in locations. Consequently, the maximum of the acoustic Fig. 1. In the first thermocycle, the temperature depen- emission activity, as a rule, is well pronounced only in dence ∆L/L(T) only slightly deviates from linear behav- the first heating–cooling cycles and decreases in the ior. In the second and third thermocycles, the depen- course of phase strain hardening. The acoustic emission dence ∆L/L(T) appreciably deviates from linearity; in technique, as applied to relaxor ferroelectrics, makes it this case, the deviation increases with an increase in the possible to determine the phase transition temperature number of thermocycles. After the third thermocycle, and to investigate manifestations of phase strain hard- the dependence ∆L/L(T) ceases to change. The devia- ening. tion of the temperature dependence ∆L/L(T) for PMN crystals from linear behavior after the third thermocy- The aim of this work was to investigate the thermal cle is in reasonable agreement in sign and order of mag- expansion and acoustic emission of Pb(Mg Nb )O 1/3 2/3 3 nitude with the results obtained in [2, 12Ð14]. The devi- and (1 Ð x)Pb(Mg1/3Nb2/3)O3ÐxPbTiO3 (PMNÐPT) crys- ation of the dilatometric curves from linearity is tals in the course of thermocycling in the vicinity of the observed beginning at 350°C and is attended by a dras- temperature Td under nonequilibrium conditions of tic increase in the acoustic emission activity. The relaxation of growth defects and defects produced by acoustic emission activity decreases with an increase in mechanical treatment. the number of thermocycles. After the third thermocycle, the acoustic emission is virtually absent. 2. SAMPLE PREPARATION Figure 2 shows the dependences ∆L/L(T) and Nú (T) AND EXPERIMENTAL TECHNIQUE for (1 Ð x)Pb(Mg1/3Nb2/3)O3ÐxPbTiO3 (x = 0.25 and 0.40) crystals in the course of cooling in the first ther- Transparent yellow crystals of Pb(Mg1/3Nb2/3)O3 mocycle. As in the case of the PMN crystal, the depen- and (1 Ð x)Pb(Mg1/3Nb2/3)O3ÐxPbTiO3 (x = 0.25 and dence ∆L/L(T) measured for a crystal with x = 0.25 0.40) in the form of cubes with edges up to 6 mm in (corresponding to relaxor compositions [3]) in the first length and faceting along the (001) planes of the per- thermocycle exhibits a nearly linear behavior. A radi- ovskite basis were grown through spontaneous crystal- cally different curve ∆L/L(T) is obtained for a crystal lization from a solution in melt with the use of a PbOÐ with x = 0.40, which is similar in structure and proper- B2O3 mixture as a solvent [23]. The composition of ties to conventional ferroelectrics [3, 7]. A jump corre- PMNÐPT crystals was determined using a “Camebax- sponding to the ferroelectric phase transition in the Micro” scanning microscope–microanalyzer. Samples vicinity of 220°C is observed for the elongation per unit in the form of plates 4 × 4 × 1 mm in size were cut par- length ∆L/L even upon the first cooling. Moreover, the allel to the (001) natural faces of the crystal and were dependence ∆L/L(T) deviates from linearity in the then ground. It should be noted that, following the pur- vicinity of 330°C, i.e., at a lower temperature than in pose in hand, the studied crystals after growth and the case of the PMN crystal. Unlike the PMN crystal, mechanical treatment were not annealed prior to mea- the crystals with x = 0.25 and 0.40 are characterized by surements, unlike in the universally accepted proce- two maxima of the acoustic emission activity. For the dure. crystal with x = 0.40, the first maximum in the depen-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 160 DUL’KIN et al.

(a) (b) 2.5 1 30 2.0

3 20 1.5 2 –1 10 ×

3 . , s L N / 1.0 L 2

∆ 10 0.5 1 3 0 0 200 250 300 350 400 450 200 250 300 350 400 450 T, °C T, °C

Fig. 1. Temperature dependences of (a) the elongation per unit length and (b) the acoustic emission activity Nú for the Pb(Mg1/3Nb2/3)O3 crystal during cooling in different heatingÐcooling cycles. Numerals near the curves indicate the cycle number.

6 (a) (b)

3 5 40 10

× 4

–1 L / . , s 20

L 3 N ∆ 2 1 0 150 250 350 450 150 250 350 450 T, °C T, °C

400 (c) 300 200 C , °C

T 100 Rh T 0 –100 0 0.1 0.2 0.3 0.4 x Fig. 2. Temperature dependences of (a) the elongation per unit length and (b) the acoustic emission activity Nú for (1 Ð x)Pb(Mg1/3Nb2/3)O3ÐxPbTiO3 crystals with x = 0.25 (dashed lines) and x = 0.40 (solid lines) during cooling in the ﬁrst heating– cooling cycle. (c) Temperatures of deviation of the dependences a*(T) from linear behavior according to the data taken from [16] (closed circles) and the temperature of the acoustic emission activity maximum (open triangles) as a function of the PbTiO3 content in (1 Ð x)Pb(Mg1/3Nb2/3)O3ÐxPbTiO3 crystals. The dashed line shows the xÐT phase diagram of the (1 Ð x)Pb(Mg1/3Nb2/3)O3Ð xPbTiO3 system according to the data taken from [16, 25, 26]. Designations: C is the cubic phase, Rh is the rhombohedral phase, and T is the tetragonal phase. dence Nú (T) corresponds to a ferroelectric phase transi- Thus, the dependence of the temperature of the maxi- tion at 220°C and the second maximum is observed at mum in Nú (T) on the titanium content in PMNÐPT crys- ° 330 C, i.e., at a temperature lower than that for the tals exhibits a minimum (Fig. 2c). PMN crystal. The main maximum in the dependence Nú (T) for the crystal with x = 0.25 occurs at 250°C, 4. DISCUSSION which is considerably lower than that for PMN. A weak maximum of the acoustic emission activity at a temper- A comparison of the temperatures of the maxima ature of approximately 330°C, most likely, can be revealed in the dependence Nú (T) and the xÐT phase caused by concentration inhomogeneities in the crystal. diagram of solid solutions in the (1 Ð x)PMNÐxPT sys-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ACOUSTIC EMISSION AND THERMAL EXPANSION 161 tem [16, 25, 26] demonstrates that only the low-temper- explanation of the nonmonotonic dependence Td(x) for ature acoustic emission maximum and its attendant the PMNÐPT system. It is known that the concentration jump in the dependence ∆L/L(T) for crystals with x = dependence of the phase transition temperature for 0.40 correspond in temperature to the known phase solid solutions of an antiferroelectric and a ferroelectric transition from the tetragonal phase to the cubic phase. passes through a minimum (the most known examples All the other anomalies in the acoustic emission are are provided by the NaNbO3ÐKNbO3 and PbZrO3Ð observed at temperatures close to Td. At these tempera- PbTiO3 systems [1, 27]). Upon addition of PbTiO3 to tures, as follows from the high-temperature x-ray dif- compositionally ordered antiferroelectrics, such as fraction data obtained in [16], the dependence of the PbMg1/2W1/2O3 and PbYb1/2Nb1/2O3, this minimum in mean unit-cell parameter a* = V1/3 (where V is the unit the concentration dependences becomes more pro- cell volume) for the studied crystals deviates from lin- nounced [3, 27]. ear behavior (Fig. 2c). It should be noted that the formation of polar regions A decrease in the acoustic emission activity due to in the paraelectric matrix can also be considered a local phase strain hardening was previously observed upon ferroelectric phase transition; moreover, compared to thermocycling in ferroelectric crystals [19] and ferro- the antiferroelectric phase transitions, the local ferroelectric ceramic materials [20] undergoing abrupt electric phase transitions should make a substantially phase transitions. In particular, for (Na,Li)NbO3 larger contribution to the acoustic emission and phase ceramic samples, a maximum in the acoustic emission strain hardening [19Ð21]. This can be judged, in partic- activity was revealed in the first thermocycle, whereas ular, from the more pronounced strain hardening a minimum manifested itself after the third thermocy- observed in the 0.6PMNÐ0.4PT crystal. For this crystal, cle, as is the case with our crystals. If the change ∆ ∆ the dependence L/L(T) strongly deviates from linear observed in the dependences L/L(T) for PMN crystals behavior even in the first cooling cycle. This can be is caused by phase strain hardening, we should assume explained by the occurrence of the normal ferroelectric that a phase transition occurs near Td. This assumption transition, which is accompanied by a strong spontane- is consistent with data available in the literature on the ous strain and, consequently, by a more appreciable anomalies observed in the temperature dependences of change in the dependence ∆L/L(T). On the other hand, the permittivity ε(T) [12], the mean unit-cell parameter since the temperatures of the local ferroelectric transi- a*(T) [16], and the heat capacity [15] for PMN crystals tions lie in a very wide range [2], their related acoustic at temperatures close to Td. However, the absence of emission manifests itself only as a background weakly macroscopic structural transformations in PMN crys- dependent on the temperature. A narrow maximum in tals at these temperatures suggests a local character of the acoustic emission activity and anomalies in the the phase transition. As was noted above, the PMN dependence a*(T) in the temperature range of phase crystals involve nanoregions of two types, namely, transitions occurring in compositionally ordered polar and compositionally ordered nanoregions. Since regions can be observed because the total volume of the volume content of polar nanoregions in the vicinity these regions is sufficiently large (up to 30% of the of Td is close to zero [2Ð4, 9], it seems likely that the crystal volume [5]) and the temperatures of the phase narrow maximum at 350°C in the dependence Nú (T) transitions in all the regions are close to each other. corresponds to a phase transition in compositionally ordered nanoregions whose volume content is relatively large and does not depend on the temperature. 5. CONCLUSIONS Apparently, this phase transition is antiferroelectric in Thus, we investigated the thermal expansion and nature, because, otherwise, the temperature depen- acoustic emission of PMN and PMNÐPT crystals under dence of the birefringence proportional to the mean nonequilibrium conditions of relaxation of growth square of the polarization should exhibit a jump at a defects and defects produced by mechanical treatment. temperature close to Td; however, no jumpwise increase It was found that the shape of the curve ∆L/L(T) in the birefringence occurs [2, 4]. Furthermore, a num- depends on the number of thermocycles and that the ber of authors [3, 6Ð8, 11] have observed ion displace- acoustic emission activity exhibits a maximum at T ≈ ments of the antiferroelectric type in compositionally Td. For (1 Ð x)PMNÐxPT crystals, the dependence of the ordered nanoregions of the PMN crystals. Precision x- temperature of this maximum in the acoustic emission ray diffraction investigations [16] also demonstrated activity on x shows a minimum. Analysis of the results that the deviation of the temperature dependence a*(T) obtained and the data available in the literature on the from linearity upon cooling of PMN crystals is pre- structural, dielectric, and thermal properties of PMN ceded by a jumpwise decrease in the parameter a*, materials [12, 13, 16] permit the assumption that the which is characteristic of antiferroelectric phase transi- acoustic emission maxima observed in the vicinity of tions [1]. Td are associated with the local antiferroelectric phase The assumption regarding the antiferroelectric transitions occurring in compositionally ordered nan- nature of the phase transition in compositionally oregions. The change in the dependence ∆L/L(T) upon ≈ ordered nanoregions at T Td also offers a satisfactory thermocycling is caused by the phase strain hardening

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 162 DUL’KIN et al. induced by local phase transitions in both the composi- 10. Y. Yan, S. J. Pennycook, Z. Xu, and D. Viehland, Appl. tionally ordered and polar nanoregions of the crystal. Phys. Lett. 72, 3145 (1998). The considerable scatter in the data on the thermal 11. S. Miao, J. Zhu, X. Zhang, and Z.-Y. Cheng, Phys. Rev. expansion of PMN crystals at temperatures below Td is B 65, 052101 (2001). explained by the fact that the deviation of the curve 12. N. N. Krainik, L. A. Markova, V. V. Zhdanova, et al., Fer- ∆ roelectrics 90, 119 (1989). L/L(T) from linear behavior at temperatures below Td strongly depends on the degree of relaxation of the 13. A. Fouskova, V. Kohl, N. N. Krainik, and I. E. Mylnik- defect structure (in our case, on the conditions of ova, Ferroelectrics 34, 119 (1981). annealing of the crystals after the growth, mechanical 14. H. Arndt and F. Schmidt, Ferroelectrics 79, 149 (1988). treatment, and other actions). 15. P. Bonnneau, P. Garnier, C. Calvarin, et al., J. Solid State Chem. 91, 350 (1991). 16. O. Bunina, I. Zakharchenko, S. Yemelyanov, et al., Fer- ACKNOWLEDGMENTS roelectrics 157, 299 (1994). This work was supported in part by the Russian 17. M. Damdekalne, K. Bormanis, L. Chakare, and A. Stern- Foundation for Basic Research (project no. 01-03- berg, Ferroelectrics 186, 293 (1996). 33119) and the Ministry of Education of the Russian 18. A. E. Glazounov, J. Zhao, and Q. M. Zhang, in Proceed- Federation (project no. E00-3.4-287) ings of 5th Williamsburg Workshop on First-Principles Calculations for Ferroelectrics, Ed. by R. E. Cohen (American Inst. of Physics, Woodbury, 1998); AIP Conf. REFERENCES Proc. 436, 118 (1998). 19. V. G. Gavrilyachenko, E. A. Dul’kin, and A. F. Semen- 1. G. A. Smolenskiœ, V. A. Belov, V. A. Isupov, N. N. Kraœ- chev, Fiz. Tverd. Tela (St. Petersburg) 37, 1229 (1995) nik, R. E. Pasynkov, A. I. Sokolov, and N. K. Yushin, [Phys. Solid State 37, 668 (1995)]. Physics of Ferroelectric Phenomena (Nauka, Leningrad, 20. E. A. Dul’kin, L. V. Grebenkina, I. V. Pozdnyakova, 1985). et al., Pis’ma Zh. Tekh. Fiz. 25 (2), 68 (1999) [Tech. 2. L. E. Cross, Ferroelectrics 76, 241 (1987). Phys. Lett. 25, 70 (1999)]. 3. I. W. Chen, J. Phys. Chem. Solids 61, 197 (2000). 21. E. A. Dul’kin, V. G. Gavrilyachenko, and O. E. Fesenko, 4. G. Burns and F. H. Dacol, Solid State Commun. 48, 853 Fiz. Tverd. Tela (St. Petersburg) 39, 740 (1997) [Phys. (1983). Solid State 39, 654 (1997)]. 22. E. A. Dul’kin, I. P. Raevskiœ, and S. M. Emel’yanov, Fiz. 5. E. Prouzet, E. Husson, N. de Mathan, and A. Morell, Tverd. Tela (St. Petersburg) 39, 363 (1997) [Phys. Solid J. Phys.: Condens. Matter 5, 4889 (1993). State 39, 316 (1997)]. 6. E. Husson, M. Chubb, and A. Morell, Mater. Res. Bull. 23. S. M. Emel’yanov, N. P. Protsenko, V. A. Zagoruœko, 23, 357 (1988). et al., Izv. Akad. Nauk SSSR, Neorg. Mater. 27, 431 7. S. B. Vakhrushev, Author’s Abstract of Doctoral Disser- (1991). tation (St. Petersburg, 1998). 24. E. A. Dul’kin, Mater. Res. Innovations 2, 338 (1999). 8. A. Tkachuk, H. Chen, P. Zschack, and E. Colla, in Fun- 25. S. W. Choi, T. R. Shrout, S. J. Jang, and A. S. Bhalla, Fer- damental Physics of Ferroelectrics 2000: Aspen Center roelectrics 100, 29 (1989). for Physics Winter Workshop, Ed. by R. E. Cohen 26. E. V. Colla, N. K. Yushin, and D. Viehland, J. Appl. Phys. (American Inst. of Physics, Melville, 2000); AIP Conf. 83, 3298 (1998). Proc. 535, 136 (2000). 27. V. A. Isupov, Phys. Status Solidi A 181, 211 (2000). 9. G. K. Smolenskiœ, N. K. Yushin, S. I. Smirnov, and S. N. Dorogovtsev, Dokl. Akad. Nauk SSSR 294, 1366 (1987) [Sov. Phys. Dokl. 32, 501 (1987)]. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 16Ð20. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 17Ð21. Original Russian Text Copyright © 2003 by Samoœlov, Sukhov, Rakhmanov. METALS AND SUPERCONDUCTORS

Metastable Phases in YBCO Films Created by Short-Term Annealings M. I. Samoœlov*, V. A. Sukhov**, and A. L. Rakhmanov* * Institute of Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 127412 Russia ** Moscow Power Engineering Institute, Moscow, 111250 Russia Received December 28, 2001; in ﬁnal form, May 21, 2002

Abstract—The effect of short-term low-temperature annealing in air and in vacuum on the properties of HTSC films of YBCO is studied. It is shown that, under certain conditions of preparation of initial samples, a transition from the HTSC phase with the superconducting transition temperature Tc = 90 K to a phase with Tc = 60 K occurs without a noticeable change in the oxygen content. It is found that, as a result of short-term annealings, a transition from the HTSC phase with Tc = 60 K to the phase with Tc = 90 K can occur only through the vacuum annealing stage, which converts the sample into the superconducting state. Short-term annealings lead to multiple reversible “switching” of the films from one phase to another. The obtained results are of practical interest, since the proposed method can be used to quickly obtain superconducting YBCO films in various phase states. It is shown, in addition, that the annealing procedure makes it possible not only to increase the oxygen concentration but also to produce a structural rearrangement of a YBCO film. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION We studied a large number of superconducting films of the YBCO system which were subjected to short- It has been generally accepted that the supercon- term (5Ð15 min), low-temperature (400Ð700°C) ducting transition temperature Tc of bulk oxide-based annealing in air and in vacuum. The superconducting high-temperature superconductors (HTSCs) is practi- transition temperature Tc was determined using the cally in one-to-one correspondence with the oxygen standard four-probe method. The lattice parameter c concentration x (or the crystal lattice parameter c) [1]. was determined using x-ray diffractometry. It was In bulk YBa Cu O (YBCO) single crystals, for exam- 2 3 x shown that the T (c) dependence for high-quality, sin- ple, the superconducting phase with T ≈ 90 K exists for c c gle-phase films displays a hysteresis; i.e., supercon- 6.8 < x < 7.0 (1.168 nm < c < 1.170 nm) and the super- ducting phases with T ≈ 90 and ≈60 K may exist at the conducting phase with T ≈ 60 K corresponds to 6.6 < c c same value of c. Moreover, we established a certain x < 6.8 (1.172 nm < c < 1.176 nm), while for concentra- exclusion principle. For example, short-term anneal- tions x below 6.4Ð6.5 (c > 1.178 nm), a semiconducting ings make it possible to obtain the 60-K phase and a (nonsuperconducting) phase is formed. However, this regularity is not universal and the one-to-one corre- semiconducting nonsuperconducting phase from the spondence between the superconducting transition 90-K phase; the 60-K phase can be transformed only to temperature and the lattice parameter c (and, appar- a semiconducting phase, and the semiconducting phase ently, the oxygen concentration x) is violated under cer- can be used to obtain the 90- and 60-K HTSC phases. tain conditions. For example, this is the case with bulk Thus, a direct transition from the 60-K phase to the samples from which oxygen is removed according to 90-K phase with the help of shot-term annealings is for- the so-called soft method, i.e., when a sample is sub- bidden, while the transition from the semiconducting jected to short-term low-temperature annealing under phase to the 90-K HTSC phase is allowed. special conditions or when oxygen is removed by sub- The technological process worked out by us makes stitution of hydrogen or azobenzene [1Ð5]. it possible to obtain films with high values of the super- Another object in which the Tc(c) dependence dif- conducting transition temperature Tc and a large lattice fers from that for single crystals is a superconducting parameter c directly in the course of sputtering (without film. The superconducting phase with Tc = 80Ð90 K subsequent processing) [7Ð10]. The possibility of mul- exists in films up to values of c = 1.180Ð1.182 nm [6]. tiple irreversible switching of the samples from the Unfortunately, it is quite difficult to determine the oxy- 90-K HTSC phase to the 60-K and semiconducting gen concentration in films using direct methods in view phases and back indicates that no damage or contami- of their small mass. For this reason, the relation nation of the films occurs as a result of the annealing between the lattice parameter c and the oxygen content regimes used by us. The obtained results combined x in films has not been established reliably. with the available data from the literature (see above)

METASTABLE PHASES IN YBCO FILMS 17

Table 1. Comparison of properties of ﬁlms obtained with and without post-annealing

Sample Tc, K c, nm x* Regime YBCO (A) 90 1.174 6.6 Post-annealing in regime II YBCO (B) 75 1.171 6.8 Without post-annealing; 40 vol % oxygen in the working mixture * Oxygen content is recalculated from the x(c) relation for bulk samples [1]. suggest that metastable superconducting phases may (100 mA) was triggered. In the second case (regime II), exist in the YBCO system. the working-mixture pressure was raised to 1000 Pa due to oxygen puffing. After post-annealing, the films were cooled to room temperature over 30 min. 2. SAMPLE PREPARATION AND METHODS Repeated annealings were carried out in the working OF MEASUREMENT chamber (vacuum annealing) or in a special furnace Films were obtained using dc magnetron sputtering (annealing in air). on lithium niobate substrates (monocrystalline LiNbO6 Resistance measurements were made using the stan- coated with a ZrO2 layer oriented in the (100) direction) dard four-probe technique. X-ray measurements were and on monocrystalline MgO (100) substrates. The made on a Dron-4 diffractometer (CuKα radiation) and sputtering system consisted of a vacuum chamber were used to monitor the single-phase form of a sample 300 mm in diameter with a bracket for clamping a and the orientation of the crystallographic axis c and to cylindrical magnetron to the chamber and a heater determine the crystal lattice parameter c. The standard mounted above it, whose temperature was registered methods were also used for x-ray measurements. with a thermocouple. The system also included blocks of evacuation, gas supply, and pressure control. A thermal molecular pump with an evacuation rate of 20 sÐ1 3. RESULTS OF MEASUREMENTS Ð1 and a rotary pump with an evacuation rate of 100 s We systematically studied the effect of technologi- × Ð5 ensured a residual pressure down to 2 10 Pa in the cal parameters of the process on the value of Tc and on working chamber. A target with a stoichiometric com- the results of x-ray structural analysis. In our experi- position was soldered to a water-cooled magnetron ments, we varied the separation between the target and with the help of indium solder. Substrates were fixed the substrate and their relative position, the current and directly to a quartz thermal ballast supplied with a voltage of the magnetron discharge, the composition of heater with a nichrome strip as the resistive element. the working mixture, the evacuation rate, and the tem- The temperature of the substrate was controlled with a perature of the substrate and determined the optimal chromelÐalumel thermocouple. The target diameter values of these parameters (for the given setup). The was 100 mm, the thickness was 7 mm, and the substrate experiments proved that the properties of the films × size was 20 10 mm. The substrate was mounted above depend on the above parameters insignificantly. This the center of the target. Two gas flow rate regulators fact considerably facilitates the obtaining of samples monitored the gas mixture composition in the chamber. with a good reproducibility of their characteristics. It The films were grown in situ using the following should be noted, however, that overly strong deviations procedure [7Ð10]. A substrate was fixed to the heater, of parameters from their optimal values considerably after which the working chamber was evacuated to a affect the properties of the samples. For example, fol- pressure of 2 × 10Ð5 Pa. Then, the substrate was heated lowing [6], we varied the oxygen content in the films by to the sputtering temperature 700°C and a working changing its concentration in the gas mixture, which mixture containing 70 vol % Ar and 30 vol % O was proved to be ineffective, since the change in the work- introduced into the chamber. The working mixture ing-mixture composition affected the parameters of the pressure was 20 Pa. The magnetron discharge current magnetron discharge and sharply deteriorated the was 0.8 A, and the voltage was 80 V. Prior to the next reproducibility of the results. On the contrary, the post- series of sputtering, the target was presputtered in pure annealing parameters significantly affect only the char- argon for 5 h. The film deposition time was 3Ð5 h, and acteristics of the films. By varying the conditions of the thickness of the prepared films was 100 nm. After post-annealing, we can purposefully modify the prop- deposition, the films were cooled for 10 min under the erties of the samples over wide limits. It should be pressure at which the sputtering was carried out; then, noted that optimal post-annealing is a more effective the temperature of the films remained unchanged for method for increasing the superconducting transition 15 min. For the sake of brevity, we will call this process temperature than variation of the oxygen pressure in the post-annealing. We used two regimes for such pro- working mixture. Table 1 gives an example of such a cesses. In the first case (regime I), the working-mixture comparison for films obtained with post-annealing in pressure remained unchanged but a collector discharge regime II and for similar films obtained at an elevated

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

18 SAMOŒLOV et al.

Table 2. Effect of annealing on the properties of a sample obtained with post-annealing in regimes I and II. It can (successive annealings in air, each annealing lasting 15 min; be stated that such annealings did not change the cation the sample was obtained in regime I) composition and caused no contamination or mechani- Annealing cal failure of the samples, since by using this method, a Tc, K c, nm ° sample can be transformed repeatedly from the super- temperature, C ≈ conducting state with Tc 90 K to the superconducting ≈ 89.6 1.172 <400 state with Tc 60 K and to a semiconducting (nonsuper- 88.5 1.174 450 conducting) state and back. The absence of phases 71.3 1.175 530 other than the stoichiometric phase 123 was confirmed by the x-ray structural data. 62.4 1.175 580 It was found that annealing of a sample in air at 400Ð 580°C for less than 15 min makes it possible to change Table 3. Effect of annealing on the properties of a sample the value of Tc considerably. Table 2 gives an example (single annealing in air for 15 min; the sample was obtained of such a study for a sample obtained with post-anneal- in regime I) ing in regime I. It can be seen that, when the annealing temperature exceeds 400°C, the superconducting tran- Annealing sition temperature decreases monotonically. This effect Tc, K c, nm ° temperature, C was observed for a series of samples both with repeated 90.0 1.172 Without annealing and single annealings (Table 3). The regular dependence of T on the annealing temperature indicates the 60.8 1.175 580 c formation of a stable superconducting phase under the action of only short-term heating in air under normal Table 4. Effect of annealing on the properties of a sample pressure. The decrease in Tc was accompanied by an (single annealing in air for 15 min; the sample was obtained insignificant decrease in the resistivity ratio γ = ρ ρ γ ≈ ≈ γ ≈ ≈ in regime II) 300 K/ 100 K from 2 at Tc 90 K to 1.6 for Tc 60 K. It is worth noting that a conventional correlation Annealing between the superconducting transition temperature Tc, K c, nm ° temperature, C and the lattice parameter c is observed for the samples 89.0 1.174 Without annealing obtained with post-annealing in regime I: the value of T decreases upon an increase in c (c = 1.72 nm for the ≈60 1.174 500 c 90-K phase and c = 1.75 nm for the 60-K phase; see Tables 2, 3). oxygen concentration in the gas mixture (40 vol % For the films obtained with post-annealing in instead of 30 vol %) but without post-annealing. Tech- regime II, the superconducting transition temperature nological investigations are described in greater detail also decreases monotonically with increasing anneal- in [7Ð10]. ing temperature (Table 4). An important distinguishing feature of the films prepared in regime II is that the Thus, using the two regimes with short-term (not value of Tc for such films decreases at a constant value exceeding 15 min) post-annealings at a relatively low of the crystal lattice parameter c. Thus, the crystal oxygen pressure, we prepared series of homogeneous structure and its evolution in the course of short-term YBCO films with a stable, high superconducting tran- annealings differ noticeably for samples obtained in sition temperature Tc (88Ð90 K). It should be noted that regimes I and II, while the electrical parameters are no systematic difference in the currentÐvoltage charac- similar. teristics and in the values of Tc was observed for films Using short-term annealings in air and in vacuum, obtained in regimes I and II. X-ray measurements superconducting films can be reversibly switched from proved, however, that all the films obtained in regime II one state to another. Table 5 and the figure show an have larger values of the crystal lattice parameter c than example of this type of complicated sequence of the samples prepared in regime I: c = 1.172 nm (regime I) switchings for one of the samples. We denote the state and c = 1.174 nm (regime II). It should be noted that our of the initial sample by A. The initial film was prepared films are distinguished by a relatively large value of in regime I; the parameters of the film are given in parameter c [which would correspond to an oxygen ≈ ≈ Table 5, and the temperature dependence of resistance content x 6.8 (regime I) or x 6.6 (regime II) for bulk R(T) is shown in the figure (curve A). Then, the film samples]. Henceforth, repeated heating of a film cooled was annealed at 580°C in air and was transformed into to room temperature after its preparation will be the 60-K phase (state B in Table 5 and curve B in the fig- referred to as annealing. ° ure). Annealing at 400 C in air led to an increase in Tc Then, we studied the effect of short-term (5Ð15 min) but did not return the sample to the initial 90-K phase annealing (quenching) at temperatures of 400Ð700°C (state C). It was found that, in order to return the film to in air and in vacuum on the properties of the films the initial state, it must first be transformed into the

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 METASTABLE PHASES IN YBCO FILMS 19

Table 5. Transitions of a film from one state to another as a It should be noted that a superconducting sample ≈ result of short-term annealings with Tc 90 K can be transformed to a semiconducting State tetragonal phase through short-term annealing in vac- T , K c, nm x Annealing regime of sample c uum and returned to the initial 90-K superconducting phase through short-term annealing in air. ≈ A 89 1.172 6.8 Without annealing The switchings of a film from one state to another B 60 1.175 ≈6.6 15-min annealing described above were carried out by us two, three, and in air at 580°C even four times on the same sample. It is noteworthy C 77 1.175 ≈6.6 15-min annealing that the reversibility of transformations is quite high in air at 400°C (see Table 5 and figure). For the same sample in the D Ð 1.189 ≈6.0 5-min annealing 90-K phase, the values of parameter c and the R(T) in vacuum at 630°C dependences for the initial and annealed states coincide E 89 1.172 ≈6.8 15-min annealing within the accuracy of our measurements. in air at 400°C 4. CONCLUSIONS semiconducting phase with the help of short-term Thus, the results of our experiments indicate the annealing in vacuum at a sufficiently high temperature richness of the phase diagram of superconducting (state D) and then annealed in air to transform the film YBa2Cu3Ox films for a constant or slightly varying oxy- into the 90-K phase (state E). gen content x. The observed effects are probably associated with the phase separation characteristic of The following interesting fact is worth noting. strongly correlated electron systems, e.g., with the Short-term annealing does not return a 60-K film to the ordering of copper ions with different valences in our 90-K phase, but exactly the same annealing converts a deficiently doped samples or with the formation of the semiconducting film into the superconducting phase so-called stripe structures [1]. It is also probable that a transition from the 60- to the 90-K phase without a with Tc = 89 K. This sort of exclusion principle is change in the oxygen content can be explained in the observed for all samples (prepared in regimes I and II) framework of the model proposed in [1, 11Ð13] for for different sequences of switchings. Moreover, even describing neutron and Raman scattering spectra. In 1-h annealing in oxygen of a sample transformed to this model, it is assumed that the so-called apical oxy- state B (60-K phase) did not return it to the initial 90-K gen [O (4) ion in the BaÐO plane] may be in two states superconducting state. The superconducting transition separated by a potential barrier of the order of 0.1 eV. temperature remained at a level of 70Ð77 K. More definite conclusions as to the reasons for the observed effects require further investigations. The obtained results are also of purely practical 2000 importance, since, first, the technique proposed by us makes it possible to obtain superconducting YBCO films with satisfactory characteristics quite rapidly and, second, these results can be used for optimization of the D film preparation process. Indeed, the short-term annealing procedure used in this work enabled us to clearly 500 trace the second stage of the two-stage (sputteringÐ 6 B annealing) process of preparing HTSC films with a

Ω C , high Tc. It is shown that annealing increases the oxygen R A content x and is also required for a certain structural 4 E rearrangement of the YBCO crystal lattice.

2 ACKNOWLEDGMENTS The authors are grateful to D.A. Kozlov (Moscow Institute of Steel and Alloys) for performing the x-ray 0 structural measurements. 50 100 150 200 250 300 T, K REFERENCES Temperature dependence of the ﬁlm resistance for different annealing conditions of the initial sample. The notation on 1. N. M. Plakida, High-Temperature Superconductivity the curves corresponds to the states of the sample and the (Mezhdunarodnaya Programma Obrazovaniya, Mos- annealing regimes listed in Table 5. cow, 1996; Springer, Berlin, 1995).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 20 SAMOŒLOV et al.

2. Yu. M. Baœkov, S. K. Filatov, V. V. Semin, et al., Pis’ma 9. M. I. Samoilov, in Abstracts of 5th International Work- Zh. Tekh. Fiz. 16 (14), 56 (1990) [Sov. Tech. Phys. Lett. shop on High-Temperature Superconductors and Novel 16, 544 (1990)]. Inorganic Materials, Moscow, 1998, W. 51. 3. A. A. Konovalov, A. A. Sidel’nikov, and R. T. Pav- 10. M. I. Samoœlov, S. P. Kobeleva, M. P. Beketov, et al., in lyukhin, Sverkhprovodimost: Fiz., Khim., Tekh. 7, 517 Proceedings of International Conference “Physicotech- (1994). nical Problems of Electromechanical Materials and 4. V. N. Kuznetsov, Pis’ma Zh. Tekh. Fiz. 27 (10), 1 (2001) Components,” Moscow, 1999, p. 180. [Tech. Phys. Lett. 27, 394 (2001)]. 11. Yu. M. Baœkov, Sverkhprovodimost: Fiz., Khim., Tekh. 5. Yu. N. Drozdov, S. A. Pavlov, and A. E. Paraﬁn, Pis’ma 7, 1208 (1994). Zh. Tekh. Fiz. 24 (1), 55 (1998) [Tech. Phys. Lett. 24, 24 (1998)]. 12. A. P. Saœko and V. E. Gusakov, Fiz. Nizk. Temp. 22, 748 6. D. K. Aswal, S. K. Gupta, S. N. Naranga, et al., Thin (1996) [Low Temp. Phys. 22, 575 (1996)]. Solid Films 292, 277 (1997). 13. M. A. Obolenskiœ, D. D. Balla, A. V. Bondarenko, et al., 7. M. I. Samoilov, V. A. Sukhov, and V. A. Ryzhkov, Super- Fiz. Nizk. Temp. 25, 1259 (1999) [Low Temp. Phys. 25, lattices Microstruct., Suppl. A 21, 307 (1997). 943 (1999)]. 8. M. I. Samoilov and V. A. Sukhov, in Proceedings of 3rd International Summer School on High-Temperature Superconductivity, Eger, 1997, p. 51. Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 163Ð166. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 156Ð159. Original Russian Text Copyright © 2003 by Sharkov, Galkina, Klokov, Klevkov.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Nonequilibrium Phonon Propagation in High-Purity CdTe A. I. Sharkov, T. I. Galkina, A. Yu. Klokov, and Yu. V. Klevkov Lebedev Physical Institute, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia e-mail: [email protected] Received January 25, 2002

Abstract—Propagation of nonequilibrium acoustic phonons in samples of high-purity CdTe (impurity content ~1016 cmÐ3) was studied using the heat pulse technique under pulsed photoexcitation. An analysis of nonequilibrium phonon propagation made by comparing the experimental response with Monte Carlo calculations assuming samples to be without twins provided an estimate for the spontaneous anharmonic phonon decay con- × Ð52 Ð1 Ð5 stant AL = 2 10 s Hz . The probability of free phonon transit through a twin boundary in a sample with twin structure was estimated as AC = 0.96. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION improve the technology of development of such struc- Wide-band-gap IIÐVI compounds have been wit- tures. nessing renewed interest in recent years. This stems The present study is aimed at investigating the spe- from the need to develop both high-efficiency injection cific features in the propagation of nonequilibrium lasers (based on ZnSe, ZnSeS) that operate in the blue phonons in CdTe crystals, both having a twin structure and violet spectral regions and (CdTe-, CdZnTe-based) and without it, which is needed for further research in detectors of x-ray and nuclear radiation. Application of the field of nanostructures, for instance, of ZnTe/CdTe the IIÐVI compounds meets with difficulties, because [2]. samples obtained by using traditional high-temperature methods are heavily defected and possess, as a rule, a very high electrical resistivity. The latter is due to self- 2. EXPERIMENT compensation, an effect common to all wide-band-gap Propagation of nonequilibrium phonons in CdTe compounds. was studied using the heat pulse technique. This tech- The purest IIÐVI polycrystalline compounds are nique and the main processes involving nonequilibrium presently produced using low-temperature methods, for phonons are illustrated schematically in Fig. 1. Non- instance, repeated resublimation [1]. In order to equilibrium phonons are generated in sample S by improve the growth technology of the IIÐVI com- pulsed excitation P. In the series of experiments pounds, the complex processes of interaction responsi- reported here, phonons were excited by light with pho- ble for the significant part of the electronic spectrum of ton energy in excess of the band gap of the material these materials, e.g., interactions between various impurities with one another and with defects, have to be understood. Investigation of pulsed heat transport, i.e., of nonequilibrium phonon propagation modes, by S 1 3 using the heat pulse technique may provide additional information on the content and nature of defects in the IIÐVI compounds. P 2 D Moreover, devices based on these materials are, as a rule, heterostructures, which makes characterization of 4 the interfaces an extremely important problem. Investi- 5 gation of phonon propagation through interfaces provides additional information on the interface characteristics. At the same time, studies of phonon transit across interfaces of different nature, such as grain boundaries, 5' twinning planes, and interfaces in heterostructures and superlattices, are hindered by the fact that each sample is characterized by a unique combination of the defect Fig. 1. Main processes taken into account in the modeling: content, interface imperfections, etc. Therefore, under- (1) spontaneous decay, (2) elastic scattering, (3) effects at sample boundaries, (4) free transit through block bound- standing the phonon behavior near interfaces even on a aries, and (5, 5') backward and forward scattering from qualitative level may yield information required to block boundaries, respectively.

164 SHARKOV et al.

1.0 ated with crystal growth at the interfaces separating two coexisting phases. The formation of twins in single- 4 crystal grains of high-purity CdTe polycrystals with textured structure in the [111] growth direction was frequently observed by us to occur (see [10]) and, as we believe, was related to nonequilibrium conditions of 0.5 vapor transport to the site of crystallization [the vapor 3 flux density was ~1017 atoms/s cm2 at a recrystallization 2 temperature of (0.5Ð0.6)Tmelt]. The twin axis usually coincided with [111], and the twinning plane was par- 1 allel to the (111) plane. The twin boundaries on the Norm. response, arb. units twinning plane were visualized by etching the samples 0 2000 4000 6000 8000 in K2Cr2O7 : H2O : H2SO4 + AgNO3 (EAg-1). t, ns We chose for the study samples both with and without a twin structure. Fig. 2. Comparison of the experimental response for a twin- free sample (circles) with Monte Carlo calculations made The arrival of nonequilibrium acoustic phonons was with A = 3 × 10Ð53, 2 × 10Ð53, 1 × 10Ð53, and 3 × 10Ð54 sÐ1 detected with a meander-shaped, thin-film, supercon- L ducting bolometer, 0.35 × 0.50 mm in size, made of HzÐ5 for curves 1Ð4, respectively. Crosses plot the experimental response [10] for a twinned sample. ~300 Å-thick granular aluminum, which was thermally evaporated on one of the sample faces. The sample surface was excited by an LGI-21 nitrogen laser (λ = τ studied; in these conditions, phonons are generated 337 nm, P ~ 7.5 ns). The response was measured with directly in the sample in the process of cooling of hot a V9-5 computer-controlled stroboscopic voltage con- carriers or nonradiative recombination. The phonons verter. The measurements were carried out at 1.7 K. thus produced propagate through the sample and undergo (1) spontaneous anharmonic decay, (2) elastic scattering, etc. (Fig. 1). By absorbing in detector D, 3. EXPERIMENTAL RESULTS they induce a signal. The shape of the time-resolved AND DISCUSSION signal carries information on the specific features of the The experimental bolometer response to pulsed processes occurring with nonequilibrium phonons and excitation of a twin-free sample is shown in Fig. 2 (cir- on their propagation mode. cles). We readily see that while the signal is fairly long, The signals can be analyzed by comparing the the pulse of the arriving LA phonons (shown with an experimental detector response with Monte Carlo cal- arrow) is resolved. The beginning of the pulses corre- culations [3]. This method permits one to estimate the sponds to the phonon ballistic transit time (310 ns for scattering intensity of nonequilibrium acoustic LA and 550 ns for TA phonons). phonons from point defects in the sample under study The response signals were analyzed in terms of a [4, 5], to determine the characteristic dimensions of the model [3] taking into account spontaneous anharmonic grains making up the sample [6], etc. decay and elastic scattering of phonons from point The polycrystalline CdTe samples of stoichiometric defects. The mean free times with respect to spontane- τ composition used in our experiment were prepared ous decay ( L) and to elastic scattering from point τ through repeated low-temperature sublimationÐcrystal- defects ( S) are known to depend strongly on phonon lization processes [7]. The total content of 65 residual Ð1 Ð1 frequency (τ = A ν 5, τ = A ν4). impurities was 10Ð3Ð10Ð4 wt %. The polycrystalline L L S S ingots prepared at T ~ 600°C had a textured structure In our earlier modeling of the propagation of non- with the single-crystal grains growing in the [111] equilibrium acoustic phonons in silicon and diamond direction. [4, 5], we used the known values of the constants AL and The polycrystalline ingots were sliced perpendicu- varied AS, which permitted us to determine the constant lar to the single-crystal growth direction into wafers AS for specific samples with an unknown impurity con- 1 mm thick, of which samples for measurements with tent. For CdTe, we did not succeed in finding calculated × dimensions 8 8 mm were cut. Following grinding and values of AL in the literature. We also did not locate the polishing, the samples were etched in a BrÐmethanol values of the third-order elastic constants necessary for solution to remove the damaged surface layer. X-ray calculating AL, as was done, for instance, in [8]. At the analysis showed the single-crystal grains to be ~1.5Ð same time, as already mentioned, the studied samples 2.0 mm in diameter. Some grains were twinned. of stoichiometric CdTe had an extremely low concen- Twinning is a characteristic feature of the IIÐVI tration of impurities (~1016 cmÐ3). It should be pointed compounds, including CdTe. The formation of twins is out that while in diamond, for instance, there are only usually attributed to nonequilibrium processes associ- three isotopes, with the content of the dominant isotope

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

NONEQUILIBRIUM PHONON PROPAGATION 165

1.0 1.0

3 3 0.5 2 0.5 2 1 1 Norm. response, arb. units Norm. response, arb. units

0 2000 4000 6000 8000 0 2000 4000 6000 8000 t, ns t, ns

Fig. 3. Comparison of the experimental response for a sam- Fig. 4. Comparison of the experimental response [10] for a ple with twin structure (circles) with Monte Carlo calcula- sample with twin structure (circles) with Monte Carlo cal- tions made for different values of the phonon mean free path culations made for different probabilities of free phonon λ: 100, 70, 50, and 30 µm for curves 1Ð4, respectively (see transit through the twin boundary AC: 0.97, 0.96, and 0.95 [10, Fig. 5]). for curves 1Ð3, respectively.

6 C12 being about 99% (whereas the content of nitrogen form the leading edge of the response for times close to is not less than 1017 cmÐ3), both cadmium and tellurium that of the phonon ballistic transit through the sample. in CdTe have eight stable isotopes each, with the con- Attempts [10] to fit the calculated response to the tent of the most abundant ones being about 30%. All experimental data by inclusion of spontaneous decay this gives one grounds, in calculations of the constant and elastic scattering alone on point defects (processes characterized by a strong dependence on phonon fre- AS, to take into account phonon scattering from the isotopes only and neglect the impurity scattering. Calcula- quency) did not meet with success. Therefore, nonequi- tions made in accordance with [9] yield for the constant librium phonon propagation was simulated in [10] in terms of another model assuming phonon propagation of elastic phonon scattering from the isotopes, AS = 2.5 × 10Ð40 sÐ1 HzÐ4. to be diffusive, the only variable parameter being the phonon mean free path λ independent of phonon fre- We used the calculated value of AS and varied AL in quency. the modeling; this was carried out for the actual experimental geometry. It was assumed (due to a lack of lit- Figure 3 (cf. [10, Fig. 5]) compares the experimental erature data for CdTe) that the relative probabilities of response (circles) with calculations made for several phonon decay are the same as in silicon. Figure 2 compares the calculated response (solid lines) with the experimental data (circles). The best fit is seen to be 1.0 × Ð53 Ð1 Ð5 achieved for AL = 2 10 s Hz , so that, for instance, for LA phonons of frequency 1 THz, the mean 3 free path limited by phonon decay is ~160 µm. 2 Also shown in Fig. 2 is the experimental response (crosses) obtained on a sample with a twin structure. 0.5 One readily sees the following significant differences. 1 (1) Although the thickness of the sample with twin structure (800 µm) is less than that of the twin-free one (1000 µm), the response is markedly longer (7500 ns vs. 4500 ns). Norm. response, arb. units (2) The beginning of the response is delayed sub- 0 2000 4000 6000 8000 stantially (by more than 500 ns). t, ns (3) The pulse corresponding to the arrival of LA phonons is not resolved. Fig. 5. Comparison of the experimental response [10] for a sample with twin structure (circles) with Monte Carlo cal- The smooth rise delayed by a time longer than the culations made for the same probability of free phonon tran- ballistic transit time implies the existence of one more sit through the twin boundary AC = 0.96 but different values × Ð52 × Ð53 × Ð54 Ð1 Ð5 scattering mechanism limiting the mean free path of of AL: 2 10 , 2 10 , and 2 10 s Hz for curves phonons, including low-frequency phonons, which 1Ð3, respectively.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 166 SHARKOV et al. λ values of the mean free path . It is seen that the response curves calculated with the same AC = 0.96 but response obtained with such modeling reproduces quite for different values of AL. Considered from the physical well both the shift of the leading edge and the general standpoint, this means that phonon propagation in pattern of the experimental response, the best ﬁt value CdTe is essentially affected by scattering not only from being λ = 50 µm. twin boundaries but also from point defects (isotopes) The process which only weakly depends on phonon in the bulk of the sample. frequency may be associated with scattering from plane interfaces. These cannot, however, be grain boundaries, because, as already mentioned, their dimensions 4. CONCLUSION exceed the obtained value of λ by more than an order of Thus, we have found the anharmonic decay con- × Ð52 Ð1 Ð5 magnitude. At the same time, this scattering can take stants of acoustic phonons AL = 2 10 s Hz for place on twinning planes present in the sample. pure cadmium telluride (N ~ 1016 cmÐ3). Comparison of Because the distance between the twinning planes is 2Ð µ λ the experimental data obtained on CdTe samples, both 8 m, which is markedly smaller than the value of with twins and twin-free, permits the conclusion that obtained, it can be assumed that the probability of twinning planes provide a contribution to phonon scat- phonon scattering from twinning planes is much less tering in addition to the dominant scattering from iso- than unity. topes. The above estimate of the spontaneous phonon decay constant AL in CdTe permits one to simulate the propagation of nonequilibrium phonons in this com- ACKNOWLEDGMENTS pound with allowance for the spontaneous anharmonic The authors are indebted to N.N. Sentyurina for decay, elastic scattering, and the existence of plane chemical processing of the samples and to V.P. Marto- interfaces in the sample (such a model has been pro- vitskiœ for x-ray diffraction measurements. posed for analysis of nonequilibrium phonon propaga- This study was supported by the Russian Founda- tion in CVD diamond and is described in considerable tion for Basic Research, project nos. 99-02-17183 and detail in [6]). 01-02-16500. This model assumes that the sample is made up of blocks, shown schematically by dotted lines in Fig. 1, and that phonons can pass freely from one block to REFERENCES another (example 4) and scatter elastically at the block 1. S. A. Medvedev, Yu. V. Klevkov, V. S. Bagaev, and boundaries (5, 5'). We note that there is no consistent A. F. Plotnikov, Nauka Proizvod. 6, 16 (2000). microscopic description of high-frequency phonon 2. V. S. Bagaev, V. V. Zaitsev, E. E. Onishchenko, and scattering from grain boundaries. Yu. G. Sadofyev, J. Cryst. Growth 214/215, 250 (2000). The modeling was performed under the assumption 3. M. M. Bonch-Osmolovskiœ, T. I. Galkina, A. Yu. Klokov, that the sample is a plate, the twinning planes are paral- et al., Fiz. Tverd. Tela (St. Petersburg) 38 (4), 1051 lel to the sample surface with a separation of 5 µm, and (1996) [Phys. Solid State 38, 582 (1996)]. that the grain dimensions in other directions are 4. M. M. Bonch-Osmolovskii, T. I. Galkina, A. Yu. Klokov, µ 500 m. The probability AC of free phonon transit et al., Cryogenics 34, 855 (1994). through the plane interfaces was a variable parameter. 5. T. I. Galkina, A. Yu. Klokov, R. A. Khmelnitskii, et al., The value of AL used in the modeling was taken to be Proc. SPIE 3484, 222 (1998). × that obtained on the sample free of twins, AL = 2 6. A. I. Sharkov, T. I. Galkina, A. Yu. Klokov, et al., Dia- 10−52 sÐ1 HzÐ5. mond Relat. Mater. 9 (3Ð6), 1100 (2000). Figure 4 compares the responses calculated in this 7. A. V. Kvit, Yu. V. Klevkov, S. A. Medvedev, et al., Fiz. model for A = 0.97, 0.96, and 0.95 with the experimen- Tekh. Poluprovodn. (St. Petersburg) 34 (1), 19 (2000) C [Semiconductors 34, 17 (2000)]. tal data. We readily see that the calculations reproduce both the shift of the beginning of the response and the 8. S. Tamura, Phys. Rev. B 31 (4), 2574 (1985). 9. S. Tamura, Phys. Rev. B 30 (2), 849 (1984). response duration. The best-ﬁt value is AC = 0.96. It is essential that in such a modeling the value of A 10. A. I. Sharkov, A. Yu. Klokov, T. I. Galkina, and L Yu. V. Klevkov, J. Russ. Laser Res. 21, 478 (2000). should be estimated preliminarily, because the shape of the response depends strongly on the magnitude of AL. This conclusion follows from Fig. 5, which displays Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 167Ð170. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 160Ð162. Original Russian Text Copyright © 2003 by Flerov, Burriel, Gorev, Isla, Voronov.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Low-Temperature Specific Heat of the Rb2KScF6 Elpasolite I. N. Flerov*, R. Burriel**, M. V. Gorev*, P. Isla**, and V. N. Voronov* * Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: ﬂ[email protected] ** Instituto de Sciencia de Materiales de Aragon, CSIC-Universidad de Zaragoza, Zaragoza, 50009 Spain Received February 27, 2002

Abstract—The speciﬁc heat of single-crystal Rb2KScF6 is measured using ac calorimetry in the range 4Ð280 K. The results are discussed in the context of a group-theoretical analysis of possible distortions of the elpasolite structure and of experimental data obtained earlier with an adiabatic calorimeter in a narrower temperature region. © 2003 MAIK “Nauka/Interperiodica”.

5 Space group Fm3 m (Oh ) is characteristic of several space groups were analyzed altogether. The lowest families of perovskite-like crystals, for instance, of symmetry of a distorted elpasolite structure was found those with the structure of elpasolite, cryolite, ordered to be triclinic (P1 ). However, the limiting case of ReO3, and antiﬂuorite. The rich variety of possible dis- superposition of rotations corresponds to trigonal sym- tortions in these structures is one of the factors accounting for the existence of sequences of phase transitions metry (space group R3 ) when rotations of the type which may result from changes in temperature, pres- (∆∆∆) are involved. sure, and/or atomic substitutions [1]. As the tempera- Structural studies of a large number of halogen-con- ture is lowered, the original G0 structure undergoes taining elpasolites with atomic cations, in which consecutive distortions, which are usually accompa- sequential or single displacive-type phase transitions nied by a gradual lowering of the symmetry. can occur, showed their lowest symmetry to be mono- ϕϕψ clinic (P21/n, rotations) [1]. However, since most A comprehensive group-theoretical and symmetry crystals have not been studied below 70Ð100 K, one analysis of vibration representations in crystals with cannot exclude the possibility that their structure under- 5 space group Fm3 m (Oh ) was reported in [2Ð4]. The G0 goes a more complex distortion as the temperature is structure, which is characteristic, in particular, of the lowered further. elpasolites with a general formula A2BB'X6, was treated One such crystal is the Rb2KScF6 elpasolite. Opti- as being made up of rigid octahedral ionic groups B'X6 cal, thermophysical, and structural studies carried out + + and A and B ions. The B'X6 octahedra are more rigid in the temperature region 100 < T < 310 K revealed that this crystal undergoes two sequential phase transitions: structural elements than BX6 because of the large difference in charge between the B and B' ions. It was estab- Fm3 m (G0) I4/m (G1) P21/n (G2) [5]. In lished that the structures of distorted elpasolite phases accordance with the entropy changes occurring at the may be considered as resulting from either simple rota- temperatures T1 = 252.4 K and T2 = 222.8 K, both struc- tions of the octahedra by a small angle about one of the tural transformations were assigned to displacive phase fourfold axes of the unit cell or from a superposition of transitions. We note that Raman investigations showed these rotations about several axes. One may conceive of the octahedra to remain regular (rigid) structural ele- situations for which rotations of the octahedra are ments in both the cubic and distorted phases of + accompanied by displacements of the A ion. Rb2KScF6 [1, 6]. This experimental conclusion permits Possible combinations of two basic octahedron rota- one to consider that structural distortions in the tetrag- tions, ϕ and ψ, were considered in [2–4]. In the ﬁrst onal and monoclinic phases are connected only with octahedron rotations of the type (000) (00ϕ) case, the octahedra in adjacent layers are rotated in ϕϕψ opposite directions, and in the second case, in the same ( ). sense. Translational symmetry is changed only in the In accordance with [2Ð4], the phase alteration case of rotations of the ψ type. Complex combinations sequence observed in the Rb2KScF6 crystal [5] could be of simultaneous rotations of the same octahedra, ∆ = ϕ continued below 100 K along two paths (Fig. 1); in ± ψ, were also found to be possible. Nineteen combina- other words, one could conceive, besides the known tions of rotations corresponding to thirteen types of phase transitions, of two or three additional phase trans-

168 FLEROV et al.

–– few millikelvins. The temperature oscillations were P1(Z = 2) P1(Z = 4) ϕϕ∆ ϕ∆∆ detected with a differential chromelÐalumel thermocouple made of wires 0.025 mm in diameter, whose – junctions were fixed to the sample and the surrounding P21/n (Z = 2) R3(Z = 4) ϕϕψ ∆∆∆ copper block with GE7031 varnish. Gaseous helium filling the measurement cell at a low pressure (10 mbar) – provided heat removal from the sample to the copper P1(Z = 4) ψ∆∆ block. The absolute value of the temperature of the block was determined with a platinum and a germanium resistance thermometer. The temperature of the sample was slightly higher than that of the copper block Fig. 1. Possible paths of further distortion of the monoclinic ∆ phase of crystals with elpasolite structure [2Ð4]. and was found as TB + TDC, where TB is the temperature of the copper block measured with the resistance ∆ thermometer and TDC is the constant component of the formations. The low-temperature phase was found to be temperature difference measured by the thermocouple. 3+ triclinic in related ammonium-based (NH4)2NH4M F6 There is a frequency range with characteristic time crystals (M3+ = Ga, Sc) [7, 8] as a result of phase-tran- short as compared with the relaxation time of the sam- sition-induced ordering of the octahedral and tetrahe- ple temperature to the copper block temperature, but dral ionic groups. Note that all the phase transforma- long as compared with the internal relaxation time. tions depicted schematically in Fig. 1 are clearly pro- With appropriate excitation frequencies, the specific nounced first-order transitions, because the number of heat is inversely proportional to the amplitude of tem- components of the distortion that was already present in ∆ perature oscillations TAC measured with the differen- the Gi Ð 1 phase changes in the Gi phase. tial thermocouple. Measurements at different tempera- Calorimetry is the most convenient method of tures were made at frequencies from 2 to 20 Hz, with searching for new phase transitions, because it reliably the larger frequency values lying in the low-tempera- detects heat capacity anomalies caused by transforma- ture region. The temperature variation rates chosen in tions of any nature. This stimulated our present mea- the heating and cooling runs were 10 to 30 K/h. The surements of the specific heat of Rb KScF in the tem- sample temperature was corrected taking the corre- 2 6 ∆ perature interval 4Ð280 K by using a calorimetric sponding value of TDC into account. technique based on alternating heat current (ac calorim- These measurements yielded relative values of the eter). The goals of the study were as follows: (i) to specific heat, which were reduced to absolute values by search for heat capacity anomalies at temperatures comparing the specific heat obtained at 200 K with that below 100 K, which could be associated with a possible measured by us earlier using adiabatic calorimetry [5]. further symmetry lowering, and (ii) to analyze the Above and below this temperature, the specific heat results, taking account of the earlier data obtained by measured with the latter method was found to be using adiabatic calorimetry, in order to find whether slightly larger. This difference reaches its largest value, inclusion of the low-temperature specific heat affects 1%, at 110 and 275 K. the earlier determined entropy associated with phase transitions. The temperature dependence of the specific heat of elpasolite Rb2KScF6 measured using the dynamic The sample was prepared in two stages. First, method is presented graphically in Fig. 2. In the region Rb2KScF6 was synthesized from a melt of correspond- 4Ð100 K, which has not been covered previously in adi- ing amounts of the anhydrous starting components abatic-calorimetry measurements, no specific-heat RbF, KF, and ScF3 in a graphite crucible in an argon anomalies were detected beyond the experimental scat- environment. Next, a transparent, colorless Rb2KScF6 ter. This implies that no structural transformations with single crystal was grown using the Bridgman method in an enthalpy greater than 0.5 J/mol occur in this temper- an evacuated and sealed platinum ampule. X-ray stud- ature region. Thus, the monoclinic phase of the ies carried out at room temperature (in the cubic phase) Rb2KScF6 elpasolite remains stable down to very low showed the single crystal thus prepared to contain no temperatures, at least to 4 K. foreign phases. It appears of interest to compare our observations Dynamic measurements of the Rb2KScF6 specific with the recent nonempirical calculations of the static heat were performed in the range 4Ð280 K on a modi- and dynamic properties of Rb2KScF6 in the three fied Sinku Riko ACC-1VL calorimeter. The sample for phases [9]. The results of the first-principles calcula- study, cut from a bulk single crystal, was a plane-paral- tions made within a microscopic ionic-crystal model lel platelet measuring 2 × 2 × 0.2 mm. The light beam which takes into account the deformability and polariz- emitted by a stabilized halogen lamp was chopped and, ability of ions are in qualitative agreement with experi- on passing through an optical waveguide, produced a mental data related to the determination of the unit-cell periodic variation of the sample temperature within a parameters, angles of rotation of the octahedra, and

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 LOW-TEMPERATURE SPECIFIC HEAT 169 phase transition temperatures. Calculations of the total 300 lattice vibration spectrum revealed the existence of unstable vibrational modes in the cubic and tetragonal 250 phases of Rb2KScF6; these modes condense at the center and edge of the Brillouin zone, respectively. As for Κ the monoclinic phase, no unstable modes were found in 200 its vibration spectrum, because, in accordance with the above calculations, transitions to lower symmetry 0.8 phases in Rb KScF are impossible because the P2 /n 150 2 6 1 0.6 phase is stable down to 0 K. R / 0.4 As expected from the study of Rb KScF using adi- 100 S 2 6 ∆

Specific heat, J/mol 0.2 abatic calorimetry [5], our calorimetric measurements Τ Τ 2 1 made above 100 K revealed two anomalies in the spe- 50 0 cific heat, which are in agreement with the known phase 150 200 250 transitions between the cubic, tetragonal, and mono- Τ, Κ clinic phases. The temperatures of the maxima in the ± ± 0 50 100 150 200 250 specific heat, T1 = 253.4 0.2 K and T2 = 224.9 0.2 K, ± T, K differ from the values T1 = 252.4 0.1 K and T2 = 222.8 ± 0.1 K found using adiabatic calorimetry [5]. Note that the samples used in the present measurements Fig. 2. Temperature dependence of the specific heat of a and in [5] were cut from different single crystals. It is Rb2KScF6 crystal. Dashed line is the lattice specific heat. known [1] that the phase transition temperatures deter- Inset: temperature dependence of the excess entropy. mined in different samples of fluorine-containing perovskite-like crystals, including the elpasolites, can dif- ∆ ∆ fer by up to 5Ð7 K. gration of the Cp(T) and ( Cp/T)(T) functions yielded values of the enthalpy Σ∆H = ∆H + ∆H = 1250 ± 70 The change in enthalpy at a first-order phase transi- i 1 2 Ð1 Σ∆ ∆ ∆ ± tion (latent heat) cannot be measured using the above J mol and of the entropy Si = S1 + S2 = 6.0 0.3 method of ac calorimetry; as a result, the shape of the J (mol K)Ð1. The temperature dependence of the dimen- specific-heat peak derived in this experiment may differ sionless excess entropy ∆S/R is shown graphically in from that found from adiabatic measurements. Indeed, the inset to Fig. 2; its value at saturation (0.73) is in as seen from Fig. 2, the maximum value of the specific good agreement with that derived in the adiabatic calo- heat at T2 (which corresponds to a first-order phase rimeter experiments (0.71) [5]. transition to the monoclinic phase [5]) is 260 J/mol K; this value is substantially below the value (≥310 J/mol Thus, our present studies and the analysis of the K) found using adiabatic calorimetry [5]. This differ- low-temperature specific heat of the Rb2KScF6 elpaso- ence becomes particularly noticeable when comparing lite, considered together with the data from [5, 9], show the excess specific heats, whose determination requires that the monoclinic phase of this crystal remains stable evaluation of the lattice contribution. down to 0 K and that allowance for the data on the Cp(T) relation below 100 K does not influence the accuracy The lattice specific heat was determined by fitting the Debye and Einstein functions, C (T) = A D(Θ /T) + with which the thermodynamic functions of the L 1 D sequential phase transitions in this crystal are deter- A E(Θ /T), to the experimental values of the specific 2 E mined. heat measured at a sufficient distance from T1 and T2. The CL(T) expression chosen describes the temperature dependence of the experimental specific heat satisfac- ACKNOWLEDGMENTS torily. Within the temperature region 4Ð110 K, the average deviation does not exceed 1%. Varying the temper- This study was supported by the Russian Founda- ature interval that corresponds to the phase transition tion for Basic Research (project no. 00-15-96790), region and is excluded from the fitting procedure from INTAS (grant no. 97-10177), and the Ministry of Sci- 110Ð270 to 140Ð270 K did not bring about marked Θ Θ ence and Technologies of Spain (grant no. MAT2001- changes in the parameters D and E; these parameters 3607-C02-02). were found to be 148.5 and 335.5 K, respectively. Because the excess specific heats associated with the high- and low-temperature phase transitions overlap in REFERENCES the region between T1 and T2 (Fig. 2), we determined the excess thermodynamic functions corresponding to 1. I. N. Flerov, M. V. Gorev, K. S. Aleksandrov, et al., the phase transition sequence G0 G1 G2. Inte- Mater. Sci. Eng. R 24 (3), 81 (1998).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 170 FLEROV et al.

2. S. V. Misyul’, Kristallograﬁya 29, 941 (1984) [Sov. 7. S. V. Mel’nikova, S. V. Misyul’, A. F. Bovina, and Phys. Crystallogr. 29, 554 (1984)]. M. L. Afanas’eva, Fiz. Tverd. Tela (St. Petersburg) 42 3. K. S. Aleksandrov and S. V. Misyul’, Kristallograﬁya 26, (2), 336 (2000) [Phys. Solid State 42, 345 (2000)]. 1074 (1981) [Sov. Phys. Crystallogr. 26, 612 (1981)]. 8. M. V. Gorev, I. N. Flerov, S. V. Mel’nikova, et al., Izv. 4. M. Couzi, S. Khaïroun, and T. Tressaud, Phys. Status Ross. Akad. Nauk, Ser. Fiz. 64 (6), 1104 (2000). Solidi A 98 (2), 423 (1986). 5. I. N. Flerov, M. V. Gorev, S. V. Mel’ nikova, et al., Fiz. 9. V. I. Zinenko and N. G. Zamkova, Fiz. Tverd. Tela Tverd. Tela (St. Petersburg) 34 (7), 2185 (1992) [Sov. (St. Petersburg) 41 (7), 1297 (1999) [Phys. Solid State Phys. Solid State 34, 1168 (1992)]. 41, 1185 (1999)]. 6. A. N. Vtyurin, A. Bulou, A. S. Krylov, and V. N. Voro- nov, Fiz. Tverd. Tela (St. Petersburg) 43 (11), 2066 (2001) [Phys. Solid State 43, 2154 (2001)]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 171Ð175. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 163Ð167. Original Russian Text Copyright © 2003 by Kashirina, Lakhno, Sychev.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Electron Correlations and Instability of a Two-Center Bipolaron N. I. Kashirina*, V. D. Lakhno**, and V. V. Sychev** * Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Kiev, 03028 Ukraine e-mail: [email protected] ** Institute of Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow oblast, 142290 Russia Received March 14, 2002

Abstract—The energy of a large bipolaron is calculated for various spacings between the centers of the polarization potential wells of the two polarons with allowance made for electron correlations (i.e., the explicit dependence of the wave function of the system on the distance between the electrons) and for permutation symmetry of the two-electron wave function. The lowest singlet and triplet 23S states of the bipolaron are consid- η ≤ η ≈ ered. The singlet polaron is shown to be stable over the range of ionic-bond parameter values m 0.143 η ε ε ε ε ( = ∞/ 0, where ∞ and 0 are the high-frequency and static dielectric constants, respectively). There is a single energy minimum, corresponding to the single-center bipolaron conﬁguration (similar to a helium atom). The η 4 ប2 ε2 binding energy of the bipolaron for 0 is Jbp = Ð0.136512e m*/ ∞ (e and m* are the charge and effective mass of a band electron), or 25.8% of the double polaron energy. The triplet bipolaron state (similar to an orthohelium atom) is energetically unfavorable in the system at hand. The single-center conﬁguration of the triplet bipolaron corresponds to a sharp maximum in the distance dependence of the total energy Jbp(R); therefore, a transition of the bipolaron to the orthostate (e.g., due to exchange scattering) will lead to decay of the bound two-particle state. The exchange interaction between polarons is antiferromagnetic (AFM) in character. If the conditions for the Wigner crystallization of a polaron gas are met, the AFM exchange interaction between polarons can lead to AFM ordering in the system of polarons. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION polarization potential wells of two polarons has not yet been investigated with allowance for electron correla- Interest in polarons was rekindled after the discov- tions (here and henceforth, electron correlations are ery of high-temperature superconductivity (HTSC). taken to mean an explicit dependence of the wave func- Some vexatious computational errors made in early tion (WF) of the two-electron system on the distance papers on the subject of bipolarons (see review [1]) between the electrons). were corrected in [2]. The minimum found in [2] for the energy of the single-center bipolaron (or the Pekar When the dependence of the energy of a system of bipolaron [3]) is significantly lower than the energy two polarons on the distance between the centers of the minimum of the two-center bipolaron first considered two polarization wells is calculated using the varia- in [4]. Later, the results of [2] were reproduced in [5] tional method, preference should be given to the deep- using both Pekar and Gaussian functions. In spatial est minimum. In solving this problem, success will be configuration, the single-center bipolaron is similar to a achieved if, by taking into account electron correla- helium atom and the two-center bipolaron, to a hydro- tions, we reproduce (or improve) the results of varia- gen molecule. tional calculations performed to date for any distance between the centers of the polarization wells. Only in Since the energy of the ground state of the single- that case can one answer the question of whether the center bipolaron was found to be significantly lower atomic or molecular bipolaron configuration is favor- than that of the two-center bipolaron, investigation of able. the molecular configuration of the bipolaron has virtually ceased. We may only cite the papers by Mukho- Recently, interest in the subject of polarons and morov (see, e.g., [6, 7] and references therein), in which bipolarons has also been inspired by studies on the the study into the two-center configuration was contin- properties of these particles in anisotropic crystals, ued despite the fact that the energy minimum found in low-dimensional structures, and systems with quantum [2] was considerably lower than that obtained in [6, 7]. wells [8Ð13]. The proper choice of the bipolaron con- However, the energy of the two-electron system as a figuration and the electron correlation effect are also of function of the distance between the centers of the importance in such systems.

172 KASHIRINA et al.

–0.105 ប2 1 T = Ð------〈〉Ψ()∆r , r + ∆ Ψ()r , r , 2m* 1 2 1 2 1 2 –0.110 (2) 2 –0.115 V = Ψ()r , r e Ψ()r , r , ee 1 2 ε------1 2 ∞r12 )

R –0.120 ( 2 2 2 bp Ψ(), Ψ(),

J 2e r1 r2 r3 r4 –0.125 V ef = Ð------∫------dt12dt34. ε˜ r13 –0.130 2 In what follows, atomic units with an energy unit 4 ប2 ε2 ប2ε 2 –0.135 e m*/ ∞ and an effective Bohr radius a0* = ∞/m*e 01 3 5 7 as a unit of length are used. R As a trial WF, we take the following linear combination of Gaussian functions: Fig. 1. Dependence of the bipolaron energy on the distance Ψ ()Φ, (), ()SΦ(), between the centers of the polarization wells (1) without S r1 r2 = r1 r2 + Ð1 r2 r1 , (3) and (2) with regard for electron correlations calculated for Φ(), n = 5 in Eq. (4). r1 r2 n (4) ()2 () 2 2. BASIC EQUATIONS = ∑ Ci exp Ða1ira1 Ð 2a2i r1r2 Ð a3irb2 , The Hamiltonian of the system consisting of two i = 1 electrons and a phonon field is taken in the form where S = 0 for the singlet state (symmetric with respect to the permutation of the coordinates of the H = បω∑a+a + ∑V ()a Ð a+ [exp()ikr electrons), S = 1 for the triplet (antisymmetric) state of k k k k Ðk 1 the bipolaron, and r (r ) is the position vector of the k k 1 2 first (second) electron, with the origin taken at a point ប2 ប2 2 midway between the points a and b (the centers of the () ∆ ∆ e + exp ikr2 ] Ð ------1 Ð ------2 + ------, (1) polarization wells). The z axis passes from the point a 2m* 2m* ε∞ r Ð r 1 2 to the point b. The distance between these points is R. The quantities C , a , a , and a are variational param- e 2πបω 1 1 1 i 1i 2i 3i V k ==Ði------, ------Ð ---- . eters. Electron correlations are taken into account by k ε˜ ε˜ ε∞ ε V 0 the term exp[Ð2a2i(r1r2)] in Eq. (4). The polaron WF is Here, V is the volume of the crystal; ω is the frequency taken in the form of an optical phonon; k is the wave vector of a phonon; n + Ψ ()r = ∑ c exp()Ðα r2 , (5) ak and ak are the creation and annihilation operators, p i i ε ε i = 1 respectively, for a phonon with wave vector k; ∞ and 0 α are the high-frequency and static dielectric constants, where ci and i are variational parameters. respectively; r1 and r2 are the position vectors of the electrons; and m* is the effective mass of an electron. 3. RESULTS OF COMPUTATIONS In Eq. (1), the first term is the Hamiltonian of optical phonons, the second term is the Frölich electron– 3.1. Singlet Bipolaron phonon interaction Hamiltonian for the two-electron Figure 1 shows the dependence of the energy of the system, the third and fourth terms are the kinetic energy ground (singlet) state of the bipolaron on the distance of the electrons, and the last term describes the Cou- between the centers of the polarization wells calculated η ε ε lomb repulsion between the electrons. for = ∞/ 0 = 0 using wave function (3) with n = 5 in We perform a canonical transformation of the Eq. (4) without regard for electron correlations (a1i = Hamiltonian (1) exp(Sa)Hexp(ÐSa) with Sa = a3i, a2i = 0, i = 1, …, n) and with allowance for them. It + can be seen from Fig. 1 that as the distance between the C (a Ð a ), vary the Hamiltonian with respect to ∑k k k k polarons increases, the effect of electron correlations the parameters Ck (characterizing the shift transforma- decreases; the energy functional of the bipolaron tion), and take the average over the phonon variables. approaches the product of the two functionals corre- As a result, we obtain the following functional for the sponding to two noninteracting polarons and the bipo- ground state of the bipolaron: laron energy tends to twice the polaron energy calculated within this approximation (Jp = Ð0.0542564). We Jbp = TV++,ee V ef note that this value of the polaron energy is calculated

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTRON CORRELATIONS AND INSTABILITY 173 by us using a polaron wave function approximation (5) 0.010 with ﬁve exponentials and exactly reproduces the result η obtained in [14]. 0.15 0 Thus, the minimum corresponding to the two-center 0.12 bipolaron state in the distance dependence of the 0.09 E

∆ –0.010 energy calculated without regard for electron correla- 0.06 tions (curve 1 in Fig. 1) is due to a poor choice of the trial electron wave function. Such minima were consid- 0.03 ered in [4, 6, 7]. We note that, to our knowledge, –0.020 Vinetskiœ and Gitterman [4] were the first to demon- 0.00 strate (on the basis of microscopic calculations) the –0.030 possible existence of a bipolaron, which sets off their 01357 paper from the later publications devoted to bipolarons, R in spite of the fact that the choice of the trial WF in [4] was not the best. Fig. 2. Dependence of the bipolaron binding energy on the distance between the centers of the polarization wells calcu- When comparing the results of variational calcula- lated for various values of the ionic-bond parameter η and tions based on different trial WFs, one should compare n = 5 in Eq. (4). the absolute values of the minima of the energy functional under study. However, the binding energy is presented, as a rule, in units of the polaron energy calcu- 3.2. Triplet Bipolaron lated within the same approximation. The results Ψ When studying photoconductivity in YBCO, Dev- obtained in [2] using the Pekar trial WF (r1, r2) = γ α α α ing and Salje [16] observed a wide absorption band in N(1 + r12)(1 + r1)(1 + r2)exp[Ð (r1 + r2)] were reproduced in [5]. With this trial function, it was found the infrared region with a peak near 5.5 × 103 cmÐ1 and ∆ ≈ η assigned this peak to transitions of bosons from the that E/2Jp 0.22 for = 0. The binding energy of the ∆ ground (singlet) state to the excited metastable triplet bipolaron is defined as E = Jbp Ð 2Jp, where Jp = −0.0542564 is the exact value of the polaron energy state. It was also assumed in [16] that, in addition to sin- calculated in [14] using a numerical method within the glet bipolarons, there exist triplet bipolarons in a certain strong-coupling approximation. The region of exist- temperature range and that it is the triplet bipolarons ence of the bipolaron with the Pekar trial WF is η ≤ that are responsible for the broadening of the NMR η m = 0.125. In [2], the absolute values of the ground- lines of Cu and O in YBCO. The population of the trip- state energy of the bipolaron were not presented and the let levels was assumed to increase with temperature, so binding energy and the range of existence of the bipo- that at T ≈ 200 K, the conductivity was due predomi- ∆ ≈ η ≈ laron ( E/2Jp 0.25, m 0.14) were somewhat over- nantly to the triplet bipolarons. Later, the change in the estimated, because the binding energy of the bipolaron shape of the conductivity versus temperature curve pre- was calculated using a somewhat overestimated value dicted in [16] was indeed observed in the vicinity of the of the polaron energy, which was found for the trial WF temperature indicated above (see review [17] and refer- in the form (1 + αr)exp(Ðαr). The bipolaron binding ences therein). energy for η 0 presented in [15] is also overestimated (∆E/2J ≈ 0.22, J = 1/6π), because the calcu- We calculated the energy of the triplet bipolaron pol pol (similarly to an orthohelium atom). The lowest numer- lations were performed with the polaron WF taken in η the form of a single Gaussian. If we replace J by the ical value of the energy obtained using WF (3) for = pol 0 for the single-center (R = 0) configuration of the trip- exact strong-coupling approximation value of Jp for the let bipolaron was Jor = Ð0.076082. ϕ γ 2 µ2 2 two-electron WF (r1, r2) = N(1 + r12 )exp(Ð (r1 + As the distance between the centers of the polariza- 2 ∆ ≈ r2 )) used in [15], we will obtain E/2Jp 0.193. tion wells increases, the energy corresponding to the triplet bipolaron term decreases monotonically (in per- The absolute value of the ground-state energy found 3Σ fect analogy with the u term of the hydrogen mole- by us using WF (3) with n = 11 in Eq. (4) is Jbp = −0.136512 or, in dimensionless units, ∆E/2J ≈ 0.258, cule). At R = 0, a fairly sharp peak is observed on the p energy versus distance curve. This peak is indicative of and η ≈ 0.143. m the instability of the triplet 23S state, which can occur, Figure 2 shows the dependence of the bipolaron for example, under nonequilibrium conditions where binding energy on the distance between the centers of exchange scattering of band electrons by bipolarons the polarization wells for various values of η and n = 5 takes place. In this case, bipolarons break down into in Eq. (4). single polarons.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 174 KASHIRINA et al. 3.3. Interaction between Polarons sibility of the insulatorÐsuperconductor transition in the at Large Distances process of destruction of a polaron lattice was substan- Now, we will show that there is antiferromagnetic tiated theoretically. Without going into details, we will (AFM) exchange interaction between polarons. Indeed, simply note that when the conditions for the Wigner at sufficiently large distances between polarons, the crystallization are met, the AFM exchange interaction correlation effects become insignificant and the bipo- between polarons can cause AFM ordering to occur in laron WF can be written as a symmetrized or antisym- a system of polarons. Antiferromagnetism in the elec- metrized product of the polaron WFs for the singlet and tronic Wigner crystal was discussed in [22, 23]. triplet states, respectively. In this case, to within the The AFM interaction between polarons can also terms quadratic in the overlap integral K, the interaction lead to a decrease in the paramagnetic component of the energy of the two polarons has the form magnetic susceptibility of a polaron gas with increasing polaron concentration, even if bound bipolaron states Eint = E1 Ð JexS1S2, (6) do not arise. 3 where S1 and S2 are the spins of the first and second The instability of the 2 S term of the bipolaron with electrons, respect to its decay into single polarons, shown by us in 2 2 terms of the Fröhlich Hamiltonian describing the inter- 1 a()1 b()2 action of electrons with optical phonons, does not rule E1 = ----ε ∫------dt12, (7) 0 r12 out the possible formation of bound triplet states of the two-electron system [13] or the existence of triplet 1 4 bipolarons resulting from the interaction of electrons Jex = ----- K1 Ð ---K2K, ε∞ ε˜ with elementary excitations of other types, e.g., with spin waves. The formation of self-localized electronic a()1 b()1 a()2 b()2 states in antiferromagnets with a low Néel temperature K1 = ∫------dt12, (8) r12 (a spin polaron in an AFM crystal) was considered in [24, 25]. In [17], it was supposed that high-temperature a()1 b()1 b()2 2 superconductivity is due to a triplet bound state of spin K ==------dt , K a()1 b()1 dt . 2 ∫ r 12 ∫ 1 polarons, which forms in much the same way as the 12 bipolaron. Here, we introduced the notation common for two-cen- Based on the Fröhlich Hamiltonian, we can consider ter systems: a(1) and b(1) are the polaron WFs centered ≡ Ψ only large polarons and bipolarons. The continuum at the points a and b, respectively; i.e., a(1) p(ra1) approximation is applicable if the effective polarization ≡ Ψ and b(1) p(rb1). potential in which self-trapped electrons move varies In Eq. (8), the first term corresponds to ferromag- smoothly over distances of the order of the lattice con- netic Coulomb exchange and the second term describes stant b [3]. In [3], the effective polarization radius rp of AFM interaction between the polarons via phonons. an electron was defined as the length over which the Thus, at large distances, the polarons repel each self-consistent polarization potential decreases by a other and the spin-dependent part of the interaction factor of 2. For the simplest, hydrogenic, polaron WF, ≈ ប2 ε 2 (total exchange) is antiferromagnetic in nature (at we have rp 10 ˜ /m*e . η ≈ ε ≈ 2 0, we have E1 1/ 0R, Jex Ð3K /R). We also The continuum approximation can be applied if rp > note that there is a potential barrier to the formation of ≤ b. In the opposite case of rb b, the spatial dispersion the bipolaron state. ε ε of the dielectric constants 0 and ∞ becomes signifi- ε ε cant; the difference in value between 0 and ∞ 4. DISCUSSION decreases with decreasing size of the region of polaron ε ≈ ε ≈ Thus, polarons repel one another at large distances localization, and we have 0 ∞ 1. In most ionic and, therefore, a system of polarons may behave as an crystals, SrTiO3, and layered cuprates (along an “easy- electron gas with Coulomb repulsion between particles. plane” direction), the value of rp is approximately 3Ð If the concentration of polarons is sufficiently low, a 5 lattice constants. transition may occur (as in an electron gas [18]) to the The spatial dispersion of the dielectric constants can 2 ε Wigner crystal state, provided that kBT < e / 0a (a is the be included in a qualitative way by using the phenome- distance between particles). There are a number of nological interpolation model by Inkson [26]. In this papers devoted to the Wigner crystallization of a case, an increase in the electronÐphonon coupling is polaron gas. For example, it was shown in [19, 20] that compensated by the dielectric-constant dispersion; as a a system of polarons (considered within a continuum result, if we take into account only the interaction of ≤ approximation) can crystallize into a hexagonal lattice electrons with optical phonons, the case of rp b is not with a period which depends on the polaron concentra- realized in nearly all ionic crystals. It should also be tion, as in the Wigner theory. In [21], the stability of the noted that spatial dispersion of the dielectric constants polaronic Wigner crystal was investigated and the pos- (in the case of polarons and bipolarons of an intermedi-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTRON CORRELATIONS AND INSTABILITY 175 ate radius) causes the bipolaron binding energy to 9. N. I. Kashirina, E. V. Mozdor, E. A. Pashitskij, and decrease in absolute value, whereas any anisotropy of V. I. Sheka, Quantum Electron. Optoelectron. 2 (2), 7 ∆ (1999). the crystal increases the ratio E/2Jp [8, 9]. Interaction of carriers with acoustic phonons [27] or 10. V. D. Lakhno, Izv. Ross. Akad. Nauk, Ser. Fiz. 60 (9), 69 with short-wavelength optical phonons may lead to the (1996). formation of self-trapped states of a small radius. In this 11. E. P. Pokatilov, V. M. Fomin, J. T. Devreese, et al., Phys. case, the continuum model is inadequate. Small-radius Rev. B 61 (4), 2721 (2000). bipolarons treated within the strong-coupling approxi- 12. J. T. Devreese, V. M. Fomin, E. P. Pokatilov, et al., Phys. mation and superconductivity caused by their Bose Rev. B 63 (18), 184304 (2001). condensation in narrow-band metals were considered 13. J. Adamovski, M. Sobkowicz, and B. Szafran, Phys. Rev. in [28]. When considering such states, the binding B 62 (7), 4234 (2000). energy of the bipolaron is generally taken to be a phe- 14. S. J. Miyake, J. Phys. Soc. Jpn. 41 (3), 747 (1976). nomenological parameter. Various aspects of the large- 15. P. Zh. Baœmatov, D. Ch. Khuzhakulov, and Kh. T. Shari- bipolaron theory as applied to the HTSC problem are pov, Fiz. Tverd. Tela (St. Petersburg) 39 (2), 284 (1997) discussed in [1, 8, 9, 29]. Review [17] is concerned with [Phys. Solid State 39, 248 (1997)]. small-bipolaron states. In [19, 20], a system of large 16. H. L. Deving and E. Salje, Supercond. Sci. Technol. 5 bipolarons in a layered high-temperature superconduc- (1), 50 (1992). tor is treated as a Wigner crystal, in which plasma oscil- 17. N. F. Mott, J. Phys.: Condens. Matter 5 (22), 3487 lations occur and favor carrier pairing in the conducting (1993). layers. 18. É. G. Batyev, Pis’ma Zh. Éksp. Teor. Fiz. 73 (10), 635 (2001) [JETP Lett. 73, 566 (2001)]. ACKNOWLEDGMENTS 19. A. A. Remova and B. Ya. Shapiro, Physica C (Amster- dam) 160 (2), 202 (1989). This study was supported by the Russian Founda- 20. A. A. Remova and B. Ya. Shapiro, Physica C (Amster- tion for Basic Research, project no. 01-07-90317. dam) 172 (1&2), 105 (1990). 21. P. Quemerais and S. Fratini, Physica C (Amsterdam) 341 REFERENCES (1), 229 (2000). 1. V. L. Vinetskiœ, N. I. Kashirina, and É. A. Pashitskiœ, Ukr. 22. R. Rajeswarapalanichamy and K. Iyakutti, Int. J. Mod. Fiz. Zh. 37 (1), 77 (1992). Phys. B 15 (15), 2147 (2001). 2. S. G. Suprun and B. Ya. Moœzhes, Fiz. Tverd. Tela (Len- 23. D. S. Hirashima, J. Phys. Soc. Jpn. 70 (4), 931 (2001). ingrad) 24 (5), 1571 (1982) [Sov. Phys. Solid State 24, 24. V. D. Lakhno and É. L. Nagaev, Fiz. Tverd. Tela (Lenin- 903 (1982)]. grad) 20 (1), 82 (1978) [Sov. Phys. Solid State 20, 44 3. S. I. Pekar, Investigations on Electron Theory of Crystals (1978)]. (Gostekhizdat, Moscow, 1951). 25. V. D. Lakhno, Fiz. Tverd. Tela (Leningrad) 27 (3), 669 4. V. L. Vinetskiœ and M. Sh. Gitterman, Zh. Éksp. Teor. (1985) [Sov. Phys. Solid State 27, 414 (1985)]. Fiz. 33 (3), 730 (1957) [Sov. Phys. JETP 6, 560 (1958)]. 26. J. C. Inkson, J. Phys. C 6 (2), 181 (1973). 5. V. L. Vinetskiœ, O. Meredov, and V. A. Yanchuk, Teor. 27. M. F. Deœgen and S. I. Pekar, Zh. Éksp. Teor. Fiz. 21 (7), Éksp. Khim. 25 (6), 641 (1989). 803 (1951). 6. V. K. Mukhomorov, Opt. Spektrosk. 86 (1), 50 (1999) 28. A. Alexandrov and J. Ranninger, Phys. Rev. B 24 (3), [Opt. Spectrosc. 86, 41 (1999)]. 1164 (1981). 7. V. K. Mukhomorov, Zh. Tekh. Fiz. 67 (8), 1 (1997) 29. G. Verbist, F. M. Peeters, and J. T. Devreese, Phys. Rev. [Tech. Phys. 42, 855 (1997)]. B 43 (4), 2712 (1991). 8. N. I. Kashirina, E. M. Mozdor, É. A. Pashitskiœ, and V. A. Sheka, Izv. Ross. Akad. Nauk, Ser. Fiz. 59 (8), 127 (1995). Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Resonance Reflection of Light from Two-Dimensional Superlattice Structures M. M. Voronov and E. L. Ivchenko Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia Received February 21, 2002

Abstract—The exciton Bloch states in a quantum well with a two-dimensional periodic potential are studied theoretically. Expressions are derived for the light reﬂection coefﬁcient and the exciton oscillator strength for a structure with this type of lateral superlattice. The redistribution of the oscillator strength between exciton states with varying period and depth of the potential is analyzed. The limiting cases where the nearly free exciton and tight-binding approximations are applicable are considered. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The two-particle envelopes of electronÐhole wave functions are written in the form The first theoretical analysis of resonant optical Ψ ()ψ, ()()φ()φ() reflection from a structure with a planar quantum dot exc re rh = r F re Ð rh e ze h zh . (2) array was reported in [1]. The reflection coefficient was Here, the functions φ and φ describe single-particle calculated for two limiting cases, more specifically, for e h short-period structures and in the dc field approxima- quantum confinement of the electron (e) and the hole tion. Recently, analytical results have been obtained for (h) along the growth axis z, re, h is the electron or hole an arbitrary relation between the lateral period and the position vector in the interface plane, F is a function of wavelength of light, in addition to between the radiative relative exciton motion in an ideal quantum well with no additional lateral potential present (the 1s exciton is and nonradiative exciton damping [2]. The theory ψ developed in [1, 2], as well as the optical spectroscopy considered subsequently), and the envelope depends studies of three-dimensional quantum dot arrays per- on the position vector of the exciton center of mass r = formed in [3Ð5], neglected the overlap of exciton wave (x, y). Note that the effect of the lateral potential V(r) functions bound to different quantum dots. In the on internal exciton motion can be neglected if the effec- present study, the theory of optical reflection and trans- tive two-dimensional Bohr radius of the exciton is mission is generalized to take into account coherent smaller than the characteristic scale of variation of this exciton tunneling from one potential minimum to potential. another. Normally incident light excites only exciton states Γ with kx = ky = 0 ( point of the 2D Brillouin zone). In We consider a semiconductor quantum well with a ψν periodic two-dimensional (2D) potential, this case, the envelopes (r) labeled by a discrete index ν are periodic with the lattice period and can be Vxy(), ==Vx()+ ay, Vxy(), + a, (1) expanded in a Fourier series,

ν 1 ()ν which acts on the exciton as a whole while not affecting ψ ()r = ---∑c exp()ibr , the exciton internal state. For the sake of simplicity, the a b potential is assumed to be characterized by the point b symmetry of a square: ()ν 1 ν c = --- ψ ()r exp()Ðibr dr, (3) b a ∫ (), ()± , ± (), Ω Vxy ==Vxy Vyx. 0 This potential transforms the exciton energy spectrum over the vectors of the reciprocal two-dimensional lattice b = (2π/a)(l, m), where l and m are the integers 0, ប2 2 from a parabolic dispersion Eexc(kx, ky) = (kx + ±1, …. The functions ψν are normalized by the condi- 2 ≡ tion ky )/2M in an ideal quantum well with V 0 (M is the translational effective mass of the 2D exciton) to a ψ 2 series of two-dimensional minibands deﬁned in the ∫ dr = 1, π ≤ π Ω Brillouin zone Ð /a < (kx , ky) /a. 0

RESONANCE REFLECTION OF LIGHT 177 Ω where 0 is the unit cell, which can be chosen, for the dielectric polarization, which, in turn, is related to instance, in the form of a square Ða/2 < (x, y) < a/2. the field E and functions ψν through [1, 2, 6] Hence, the expansion coefficients c satisfy the identity ν b Ψ (), 3 exc rr 4πP ()r = εω πa ------Λν, (5) ()ν 2 exc b LT B∑ω ω Γ 0ν Ð Ð i ν ∑ cb = 1. ν b where In view of the fact that the states with ν ≠ ν' are mutu- ν* Λν = dz drΨ ()rr, Er(), ally orthogonal, we obtain ∫ ∫ exc Ω 0 ()ν ∗ ()ν' δ ω ν ∑cb cb = νν', 0ν is the resonant frequency of the exciton in state , b and integration in the (x, y) plane is performed over the Ω unit cell 0; for the sake of convenience, the cubed whence it also follows that 3 Bohr radius aB and the longitudinalÐtransverse split- ()ν ∗ ()ν ω ∑c c =.δ (4) ting LT of the three-dimensional exciton are isolated in b b' bb' an explicit form as dimension factors. ν By solving the wave equation with exciton polariza- Note that Bloch functions at the Γ point can be made tion (5), one can calculate the coefficients of reflection, real by properly choosing the phase factors. In this case, transmission, and diffraction of light similarly to as was because of the high symmetry of the V(r) potential, the done in [2] for a two-dimensional quantum-dot super- ()ν coefficients c are real and the complex-conjugation lattice. We present here only the expression for the b amplitude reflection coefficient: sign in Eq. (4) can be dropped. ()ν η Λ For the sake of convenience, we introduce a star β of ()ω ΓQW c0 0 ν r = i 0 ∑------, (6) π ω ν Ð ω Ð iΓν E the two-dimensional vector b = (2 /a)(l, m) which ν 0 0 contains vectors (2π/a)(±l, ±m) and (2π/a)(±m, ±l). For ()ν l ≠ m ≠ 0, the star consists of eight different vectors; in where c are the expansion coefficients in Eq. (3) for ≠ 0 other cases, the star includes four vectors if l = m 0, b = 0, E is the scalar amplitude of the light-wave elec- l = 0 and m ≠ 0, or l ≠ 0 and m = 0 and one vector in the 0 tric field, and the quantities Λν satisfy the coupled equa- particular case of l = m = 0. In what follows, symbol β tions denotes both the star of the reciprocal-lattice vectors ΓQW and the magnitude of these vectors (2π/a).l2 + m2 Λ Λ0 Λ 0 ν = ν + i∑ ν'------ω ω Γ- 0ν Ð Ð i ν (7) It is known that in bulk materials with a conduction ν' × []() and a valence band connected by optical transitions Aνν' + iBνν' + Cνν' . which are allowed at the Γ point, only the s excitons (1s, 2s, etc.) are optically active. For the same reason, only Here, ν states (2) with a totally symmetric ψ (r) function Λ0 ()ν η η φ ()φ() Γ ν ==E0c0 0, 0 ∫ e z h z coskzdz, (8) ( 1 representation) are optically active in the case of normal light incidence on the lateral superlattice. For ω ε k = ( /c),b such states, the coefficients cb in Eq. (3) for vectors belonging to the same star β coincide and index b on ΓQW 1 ω π 3 2()η2 these coefficients may be replaced by index β, c ≡ cβ. 0 = ---k LT aBF 0 0 (9) b 2 is radiative exciton damping in the ideal quantum well: 2. REFLECTION COEFFICIENT k β2 OF LIGHT AND OSCILLATOR STRENGTH νν νν β----- A ' + iB ' = ∑ n 1 Ð ------2 kβ FOR TWO-DIMENSIONAL EXCITONS β ∈ 2k B1 (10) The system to be considered is a quantum well with η 2 × ()ν ()ν' β a lateral potential V(r), which is placed between semi- cβ cβ -----η +,iIνβ infinite barriers. The difference between the dielectric 0 ε 2 constant b of the barrier material of the well and the k β ()ν ()ν' νν β----- β β νβ background dielectric constant ε is neglected. In this C ' = Ð ∑ n 1 Ð ------2 c c J , (11) a κβ β ∈ 2k case, the constitute equation coupling the electric B2 induction D with electric field E can be written as D = η φ ()φ() ε π β = e z h z coskβzdz, bE + 4 Pexc, where Pexc is the excitonic contribution to ∫

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 178 VORONOV, IVCHENKO

B1 and B2 are subsets of the reciprocal-lattice vectors The sum of the oscillator strengths is preserved, which satisfy the conditions |b| < k and |b| > k, respec- because, by virtue of Eq. (4) for b = b' = 0, we have β tively, nβ is the number of vectors in the star , kβ = ()ν ∑c 2 = 1. (16) 2 β2 κ β2 2 0 k Ð , β = Ð k , ν

Ð2 Iνβ = η dzzd 'S ()zz, ' sinkβ zzÐ,' 0 ∫∫ eh Γ 3. TWO-DIMENSIONAL 1 EXCITONS Ð2 Ðκβ zzÐ ' IN A LATERAL SUPERLATTICE Jνβ = η dzzd 'S ()zz, ' e , 0 ∫∫ eh Using the plane-wave formalism based on expan- ()φ, ()φ()φ()φ() sion (3), the Schrödinger equation can be reduced to a Seh zz' = e z h z e z' h z' . system of coupled linear equations: Equations (10) and (11) were derived taking into ប2 β2 2 2 ------Ð E cβ + ∑cβ' ∑ V lm, l'm' = 0, account that the quantities bx , by , and bxby averaged 2M β' ()βl',m' ∈ ' β β2 over vectors b = (bx, by) of the star are equal to /2, β2/2, and 0, respectively. 1 (), V lm, l'm' = -----2∫ ∫ Vxy Because for the function ψ(r) with symmetry other a Ω Γ 0 than 1 we have (17) 2π ()ν × cos------[]()l' Ð l xm+ ()' Ð m y dxdy, ∑ cβ = 0 a b ∈ β β2 π 2 2 2 ()ν where = (2 /a) (l + m ) and E is the energy reck- and, in particular, c0 = 0, it is only the totally symmet- oned from the excitation energy of the exciton at rest in Γ ric Bloch states 1 that contribute to reﬂection. the ideal quantum well. In what follows, we assume the lateral potential to be a periodic set of discs, so that If the separation between the resonance frequency ν of exciton and the adjacent frequency of another opti- (), (), cally active exciton is larger than the exciton damping, Vxy = ∑v xlaymaÐ Ð , only one term can be retained in sum (6) within the fre- lm quency interval near the given resonance; this term has Ðv , ρ ≤ R the form v ()xy, ≡ v ()ρ =  0 (18) ()ν  0, ρ > R, ΓQW 2 i 0 c0 rν()ω =,------(12) ω˜ ω ()Γ Γ 0ν Ð Ð i ν + 0ν where ρ = x2 + y2 . In this case, the matrix elements where of the periodic potential can be written as 2 2 η 2 2 R J ()2π ()l' Ð l + ()m' Ð m R/a QW ()ν 2 k β β v 1 Γ ν Γ β β------V lm, l'm' =,Ð 0------(19) 0 = 0 ∑ c n 1 Ð ------2 , (13) 2 2 kβ η a ()() β ∈ 0 2k l' Ð l + m' Ð m B1 where J (t) is the Bessel function. ω˜ ω ΓQW() 1 0ν Ð 0ν = 0 Bνν + Cνν . (14) One may conveniently transfer to dimensionless Γ quantities, In this case, the quantity 0ν is the total radiative damp- ω˜ v ing of the given exciton and 0ν is the resonance fre- ε E , 0, µ R ==----- u0 = ------, quency renormalized with due account of the lightÐ E0 E0 a exciton coupling. Note that for 2π/a > k, the subset B 1 where consists of only one element, b = 0, while subset B2 contains all reciprocal-lattice vectors except the zero ប2 2π 2 Γ E = ------, vector. In this case, 0ν coincides with the quantity 0 2MR ΓQW ()ν 2 0 c0 entering the numerator in the right-hand β˜ a β 2 2 member of Eq. (12). One may conveniently introduce a ==------π l + m , (20) dimensionless oscillator strength of the νth exciton in 2 the form and to coefﬁcients, ()ν 2 f ν = c0 . (15) Cβ = nβcβ. (21)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 RESONANCE REFLECTION OF LIGHT 179

0.4 1.0 (a) 5 (b) 4 0.8 1 0 3 1 2

0 0.6

E 23 4 / ν

–0.4 f E 0.4

–0.8 0.2

–1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 u0 u0

Fig. 1. (a) Dimensionless energy and (b) oscillator strength plotted vs. dimensionless depth of a potential disc for the lowest four exciton states ν = 1Ð4. The calculation was carried out for a lateral array of quantum discs of radius R = a/4. Solid curves are exact plane-wave calculations, dashed lines 1 and 2 in Fig. 1b are calculated in terms of the model of nearly free excitons, and dotted lines ε relate to a calculation of (a) 1 and (b) f1 made in the tight-binding approximation. Curve 5 plots the sum of the oscillator strengths for the above four states.

1.0 0.4 (a) 5 (b) 4 3 0.8 1 0.2 3 0.6 0 E / 0 ν f E 2 0.4 – 0.2 2 0.2 1 – 0.4 4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 R/a R/a

Fig. 2. Dependence of (a) energy and (b) oscillator strength on the ratio R/a for excitons with ν = 1Ð4 in a superlattice with a periodic array of quantum discs. The dimensionless depth of a potential disc is u0 = v0/E0 = 0.5. Dotted lines 1 and 2 in Fig. 2b plot the calculation made in the tight-binding approximation. Curve 5 plots the sum of the oscillator strengths for the above four states.

ε µ This permits one to recast Eqs. (17) in the form lation of dimensionless energy ν(u0, ) and oscillator µ strength fν(u0, ) by using the plane wave method, and ()µ2β˜ 2 ε Ð Cβ Ð0,u0∑Uββ'Cβ' = Sections 5 and 6 analyze interesting, nearly free-exci- β' ton and tight-binding approximations. 1 Uββ' = Ð------∑ V lm, l'm'. (22) 4. CALCULATION FOR A QUANTUM v 0 nβnβ' ()βlm, ∈ DISC ARRAY ()β, ∈ l' m' ' Figures 1Ð3 display energy and oscillator strength Thus, formulation of the problem in dimensionless as functions of the potential-disc depth and of the R/a µ Γ units leaves only two independent parameters, u0 and . ratio for the four lowest 1 exciton states in the two- The next section presents the results of an exact calcu- dimensional superlattice. The dimensionless energy of

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 180 VORONOV, IVCHENKO

1.0 0.2 5 4 (a) 3 (b) 0 3 0.8

– 0.2 0.6 0 E ν 4 / f

E – 0.4 2 0.4 – 0.6 0.2 – 0.8 1 1 2 – 1.0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 R/a R/a

Fig. 3. Same as in Fig. 2, but for u0 = 1.

ν the exciton in the ideal quantum well, i.e., for u0 = 0, equations (22) reduce to two equations: 2 ˜ 2 2 2 is µ β and successively takes on the values 0, µ , 2µ , 1 ()ε Ð ε C , +0,VC , = 4µ2, …, with only exciton ν = 1 being optically active, 0 00 10 which is in accord with the behavior of the curves in ()ε0 ε ≠ VC00, +0,2 Ð C10, = Figs. 1a and 1b as u0 tends to zero. For u0 0, spatial β harmonics with wave vectors belonging to different where stars mix, so that the upper exciton states also become optically active. In the interval from u = 0 to u = 0.2, 0 2 0 0 ε = Ðπµ u , the oscillator strength transfers from exciton 1 predom- 1 0 inantly to exciton 2 alone; the oscillator strength of 0 1 2 1 2 1 exciton 3 becomes noticeable for u > 0.3. The oscilla- ε = ---µ Ð ---πµ u Ð ---µu J ()4πµ 0 2 2 2 0 4 0 1 tor strength also undergoes a similar redistribution when the ratio of the disc radius to the superlattice 1 µ ()πµ Ð ------u0 J1 22 , period is varied (see Figs. 2, 3). As the period increases 2 ε ε (with R/a 0), the negative energies 1 and 2 µ ()πµ approach the energy of an exciton bound to a single V = Ð2u0 J1 2 . potential disc; because only two such bound states exist ε ν for u0 = 0.5 or 1, the energies ν with > 2 converge to Considered in terms of this approximate description, zero. Curves 5 in these ﬁgures plot the sum of the oscil- ε0 ε0 the diagonal energies 1 and 2 become equal at lator strengths for the four lowest exciton states. Because this sum is close to unity, excitons with ν > 4 µ u = ------. are practically not excited by light in the range of 0 ()πµ ()πµ parameters studied. 0.5J1 4 + 2J1 22 ε µ µ ε ε The behavior of the ν(u0, ) and fν(u0, ) curves At this value of u0, the separation between 2 and 1 is characteristic of level anticrossing is of particular inter- 2|V| and the oscillator strengths coincide. For µ = 0.25, this value for u is 0.27, whereas an exact calculation est. For instance, for u0 = 1Ð1.2 in Fig. 1 and near R/a = 0 0.25 in Fig. 3, the difference ε Ð ε reaches a minimum, suggests that the oscillator strengths f1 and f2 become 4 3 ≈ equal at a somewhat smaller value u0 0.2. while the oscillator strengths f3 and f4 depend linearly on u0 and R/a (the sum f3 + f4 being nearly constant) and become equal at a certain point. A similar pattern of 5. NEARLY FREE TWO-DIMENSIONAL mutual repulsion of states 1 and 2 is observed within EXCITON APPROXIMATION the interval 0 < u0 < 0.3. Consider this phenomenon in a simpliﬁed but revealing model where only stars (0, 0) In the free-exciton approximation, Bloch functions Γ and (1, 0) are included in expansion (3), so that coupled of symmetry 1 are obtained through symmetrization of

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 RESONANCE REFLECTION OF LIGHT 181 plane waves with wave vectors belonging to a definite K (2πρu Ð eν /R) and the values of eν satisfy the tran- star β: 0 0 scendental equation π ψβ(), 1 1 2 () ()π ()π xy = --- ∑ ------exp i------lx+ my . (23) u0 Ð eν J0 2 eν K1 2 u0 Ð eν a nβ a ()βlm, ∈ ()π ()π = eν J1 2 eν K0 2 u0 Ð eν . In this case, the state index ν can be conveniently replaced by β. Thus, in the zeroth approximation, we Representation (25) is valid for small overlaps of the β β Ð1/2 functions ϕν centered at two neighboring sites of a lat- have cβ = nβ δββ, and Cβ = 1. ' ' eral superlattice. This condition is upheld for a large µ2 β˜ 2 Ӷ enough superlattice period a or a large enough potential In first order in parameter (u0/ ) 1, admixture depth v0. According to Eqs. (3) and (15), the oscillator of the (0, 0) wave to the symmetrized combination (23) ν with β ≠ 0 is described by the coefficient strength of exciton in the tight-binding approximation is given by β u0 0 2 ˜ 2 c ==Ð-----U , β, εβ µ β . 2 0 0 0  εβ 1 f ν = -----ϕν()xy, dyxd . (27) 2∫∫ Therefore, in second order in the above parameter, one a obtains for the dimensionless oscillator strength ε The dashed lines in Figs. 1a and 1b plot the 1(u0) and 2 2 u V , , u ≈ 0 00; lm 0 ()πβ˜ µ f1(u0) dependences calculated from Eqs. (26) and (27), f β ≠ 0 ----0------= nβ ------3 J1 2 , εβ v 0 µβ˜ respectively. As u0 increases, the energy eν tends to its 2 limiting value eν(∞) = (tν/2π) , where tν are roots of the equation J (t) = 0. The first few limiting values of eν(∞) f ≈ 1 Ð ∑ f β. (24) 0 0 for levels with l = 0 are 0.1465, 0.7718, 1.8969, …. β ≠ z 0 The region of validity of the tight-binding approxima- In Fig. 1b, the dashed curves show the dependences of tion can also be estimated by comparing solid lines 1 the oscillator strength on u0 calculated for the two low- and 2 with the dotted curves calculated using Eq. (27). Γ est exciton 1 states in the nearly free exciton approximation, i.e., from Eq. (24). 7. CONCLUSION 6. TIGHT-BINDING APPROXIMATION Thus, we have developed a theory of resonance reflection of light from a structure with a two-dimen- ν The Bloch states with negative energy E can be sional lateral superlattice whose period is large com- analyzed in terms of the tight-binding approximation. ν pared to the Bohr radius of the quasi-two-dimensional Considered in this approximation, the functions ψ (x, exciton. The Bloch states of the exciton of symmetry Γ Γ 1 y) at the point can be written as excited by light propagating along the principal axis of the structure were calculated. The dependence of exci- ν ton energy in these states on the potential-relief ampli- ψ ()xy, = ∑ ϕν()xlaymaÐ , Ð , (25) tude and on the superlattice period, as well as the cor- lm responding redistribution of oscillator strength between where ϕν(x, y) are the normalized wave functions of the various states, was analyzed. While the calculation was exciton bound to a single potential disc centered at the performed for a periodic array of quantum discs, the point x = y = 0. Bloch states of symmetry Γ1 in a square theory is capable of treating potentials of a more com- lattice made up of circular potential discs can be plicated shape. The theory may also be of advantage in derived only from the states ϕν(x, y) in which the pro- a qualitative analysis of the relation between the oscillator strengths of the free exciton X and of the train XÐ jection lz of the orbital angular momentum on the z axis is either zero or a multiple of four. We denote the energy in doped quantum-well structures and is capable of of such states, reckoned from the potential disc bottom accounting for the oscillator strength redistribution in and expressed in units of E , by eν(u ), so that favor of the trion resonance with increasing concentra- 0 0 tion of free carriers which was observed experimentally ε () () ν u0 = eν u0 Ð u0. (26) in [7].

The ground level e1 is characterized by a zero angular momentum projection. For states with lz = 0, ACKNOWLEDGMENTS the solutions inside and outside a disc are propor- This study was supported by the Ministry of Science πρ tional to the Bessel functions J0(2 eν /R) and of the Russian Federation, program “Nanostructures.”

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 182 VORONOV, IVCHENKO

REFERENCES Modeling of Devices Based on Low-Dimensional Struc- tures, Aizu-Wakamatsu, Japan, 1998, p. 10. 1. E. L. Ivchenko and A. V. Kavokin, Fiz. Tverd. Tela 4. G. Ya. Slepyan, S. A. Maksimenko, V. P. Kalosha, et al., (St. Petersburg) 34, 1815 (1992) [Sov. Phys. Solid State Phys. Rev. B 59, 12275 (1999). 34, 968 (1992)]. 5. S. A. Maksimenko, G. Ya. Slepyan, N. N. Ledentsov, 2. E. L. Ivchenko, Y. Fu, and M. Willander, Fiz. Tverd. Tela et al., Semicond. Sci. Technol. 15, 491 (2000). (St. Petersburg) 42, 1707 (2000) [Phys. Solid State 42, 6. Y. Fu, M. Willander, E. L. Ivchenko, and A. A. Kiselev, 1756 (2000)]. Phys. Rev. B 55, 9872 (1997). 3. E. L. Ivchenko, Y. Fu, and M. Willander, in Proceedings 7. K. Kheng, R. T. Cox, Y. Merle d’Aubigné, et al., Phys. of 6th International Symposium “Nanostructures: Phys- Rev. Lett. 71, 1752 (1993). ics and Technology,” St. Petersburg, Russia, 1998, p. 374; M. Willander, E. L. Ivchenko, and Y. Fu, in Pro- ceedings of 2nd International Workshop on Physics and Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

POLYMERS AND LIQUID CRYSTALS

Dynamic and Dielectric Properties of Liquid Crystals A. V. Zakharov and L. V. Mirantsev Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, Bol’shoœ pr. 61, Vasil’evskiœ ostrov, St. Petersburg, 199178 Russia e-mail: [email protected] Received April 9, 2002

ε ε ω || ⊥ Abstract—The structural properties, static j and relaxation j( ) permittivities (j = , ), time correlation Φ1 Φ2 τ1 functions i0()t (i = 0, 1) and 00()t , and orientational relaxation times i0()t (i = 0, 1) of 4-n-pentyl-4'-cyanobiphenyl (5CB) molecules in the nematic phase are investigated in the framework of the statisticalÐmechanical ε theory and the molecular dynamics method. The permittivities j are calculated within a statisticalÐmechanical approach with the inclusion of translational, orientational, and mixed correlations in the description of the Φ1 Φ2 anisotropic systems. The time correlation functions i0()t and 00()t and the orientational relaxation times τ1 i0()t of 5CB molecules are calculated using the molecular dynamics method for liquid-crystal systems simulated by realistic intramolecular and intermolecular atomÐatom interactions. The results of calculations and the experimental data for 5CB are in good agreement. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION engaged in the field of liquid-crystal materials [6–11]. The characteristic feature of anisotropic systems However, direct experimental measurements of the per- such as liquid crystals is that they exhibit a long-range mittivities and self-diffusion coefficients are rather orientational order due to an anisotropic nature of inter- complicated. In this respect, theoretical studies have molecular interaction. There exist a number of theoret- assumed a new significance, because they can provide ical approaches to analyzing the relation between the answers to a number of fundamental questions. For measured macroscopic parameters and the microscopic example, how much do the microscopic parameters structure of liquid-crystal systems [1]. Among these responsible for the character of intermolecular interac- methods, the statisticalÐmechanical approach and tions in a liquid-crystal system affect measured macro- molecular dynamics calculations seem to hold the scopic characteristics (such as the self-diffusion coeffi- greatest promise. This can be explained by the fact that cient or orientational relaxation times) of real liquid these techniques make it possible, on the one hand, to crystals? In the present study, we used the statisticalÐ calculate directly the macroscopic characteristics of mechanical theory based on the method of conditional liquid crystals in the framework of reasonable approxi- distribution functions [12]. This method makes it possi- mations using model intermolecular interaction poten- ble to take into account not only translational and ori- tials and, on the other hand, to determine the averaged entational correlations but also mixed correlations of parameters on the basis of realistic interaction poten- the molecules involved. As a model intermolecular tials both between atoms inside molecules forming an interaction potential, we chose the dipolar GayÐBerne anisotropic system and between atoms of different mol- (GB) potential [13]. The dipoles were assumed to be ecules [1, 2]. aligned parallel to the long axes of the molecules form- In the present work, the above theoretical ing a liquid crystal. Earlier [5], it was experimentally approaches were used to investigate the dynamic and revealed that mesogenic molecules of cyanobiphenyl dielectric properties of the nematic phase of 4-n-pentyl- compounds consist of flexible hydrocarbon chains 4'-cyanobiphenyl (5CB). For this purpose, the time cor- attached to a rigid core. The flexibility of the hydrocar- relation functions, orientational relaxation times, and bon chains determines the physical properties of liquid orientation distribution functions were derived by the crystals in many respects. Moreover, these molecules molecular dynamics method [3, 4]. The choice of the possess a sufficiently large dipole moment (~4.5Ð5.0 D nematic phase of 5CB was made primarily for the fol- [14]), which is directed from the polar core to the lowing reasons: (i) this compound has a simple phase molecular tail. The pair correlation functions of the dis- diagram, and (ii) the nematic phase is observed in the tribution and the orientation distribution function of temperature range 295Ð307 K [5], which is convenient 5CB molecules in the temperature range corresponding for experimental investigations. Investigations into the to the nematic phase were calculated in the framework dielectric characteristics of nematic liquid crystals have of the statisticalÐmechanical theory. Moreover, the time long attracted the particular attention of researchers correlation functions were obtained from molecular

184 ZAKHAROV, MIRANTSEV dynamics calculations, which made it possible to calcu- tion functions, allows us to derive a closed integral Ψ late the orientational relaxation times with the use of equation with respect to the mean-force potential i, j(i) realistic interatomic interaction potentials [3, 4]. [12, 16]: This paper is organized as follows. Section 2 covers Ψ , ΨÐ1 () the basic principles of the statisticalÐmechanical ij, ()i = ∫Vi() j ji, ()j Fj()d j . (3) description of a system of interacting dipoles. Within j this approach, we calculated the orientation distribution functions, the pair correlation functions, and the order Equation (3) can be solved only by the numerical method described in detail in [16, 17]. With the use of parameters. Sections 3 and 4 present the results of cal- Ψ culations of the relaxation times and the static and the solution i, j(i) and Eq. (2), we can calculate the pair relaxation permittivities of a nematic liquid crystal correlation function F(i, j), the orientation distribution formed by 5CB molecules. function f (cosβ ) = Fi()dr dϕ (where ϕ is the azi- 0 i ∫i i i i muthal angle of the unit vector ei), the order parameters 2. THE PAIR CORRELATION FUNCTION of the liquid-crystal system The pair correlation functions for the nematic phase ()β () of 5CB are calculated in the framework of the equilib- P2L = ∫Fi()P2L cos i d i , (4) rium statistical mechanics [15] based on the method of i conditional distributions [12]. We consider a single- the correlators component system composed of ellipsoidal molecules σ σ of length || and width ⊥ in a volume V at a tempera- 〈〉e ⋅ e = d()i d()j Fi(), j()e ⋅ e , (5) ture T. The volume of the system is divided into N cells, i j ∫ ∫ i j each occupying a volume v = V/N. As a first approxi- i j mation, we take into account only the states of the sys- χ () ()π ()β tem for which each cell contains one molecule [16]. = ∫d i Fi()cos 2 zi/d P2 cos i , (6) The potential energy of this system can be represented i Φ , Φ and the Helmholtz free energy in the form U = ∑ij< ()ij , where (i, j) is the pair intermolecular interaction potential, i ≡ (r , e ), and r i i i F ()Ψ f ==---- ÐkBTiln∫d i().i (7) and ei are the vectors specifying the position and the N orientation of the ith molecule, respectively. Now, we i perform the integration of the quantity exp[ÐU/k T] B Here, P2L (L = 1, 2, and 3) are the Legendre polynomi- (where kB is the Boltzmann constant), which is the als, β is the polar angle formed by the long molecular probability density of finding the system at points 1, 2, axis and the director n aligned along the z axis, and d is 3, …, N at a temperature T [12, 15]. As a result, we the distance between two layers of the smectic-A phase. determine partial distribution functions, namely, the The parameter χ is the measure of the density wave one-particle distribution function F(i) (the probability amplitude of the layered structure of the phase. The density of finding a particle inside the ith cell), the pair χ ≠ distribution function F(i, j) (the probability density of nematic phase is characterized by = 0 and P2L 0. finding two particles in the ith and jth cells), etc. [12, The kernel V(i, j) of the integral equation (3) is deter- 16]. In the present work, we will restrict our consider- mined by the pair intermolecular interaction potential chosen as the sum of the GayÐBerne potential and the ation to the case of two-particle correlations. Φ Φ dipoleÐdipole interaction potential: (i, j) = GB(i, j) + The functions F(i) and F(i, j) can be expressed in Φ terms of the mean-force potentials [12, 16] DD(i, j). The former potential can be written in the Φ ε ε Ð12 Ð6 σ form GB(i, j) = 4 0 (R Ð R ), where R = (r Ð + Ψ()i σ σ σ ε Fi() = ------, (1) ⊥)/ ⊥ and r = [ri Ð rj]. The parameters and are the Ψ () width and the depth of the potential well, respectively. ∫ i()i d i These parameters depend on the orientation of the unit i | | vectors ei and ej (where e = r/ r ), the geometric param- , , ΨÐ1 ΨÐ1 eter of the molecule γ = σ||/σ⊥, and the two exponents ν Fi() j = Fi()Fj()Vi() j ij, ()i ji, (),j (2) ν µ and µ in the relationship ε = ε (e , e )(ε e , e , e). The Ψ Ψ ()≡ 1 i j 2 i j where j(j) = ∏ , ()j , d j dr de , ε ε ij≠ ji ∫j ∫v j∫α j formulas for 1 and 2 are given in [13]. The dipoleÐ Φ α Φ V(i, j) = exp[Ð (i, j)/kBT], and is the volume asso- dipole interaction potential has the form DD(i, j) = ciated with the orientation of the ith molecule. The ∆2 ∆ functions F(i) satisfy the normalizing condition ----- [(ei á ej) Ð 3(ei á e)(ej á e)], where is the dipole r3 Fi()d()i = 1, and the constraint F(i) = Fi(), jd()j , ∫i ∫j moment of the 5CB molecule (∆ ~ 5 D [14]). The inter- which relates the one-particle and two-particle distribu- molecular interaction parameters used in our calcula-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 DYNAMIC AND DIELECTRIC PROPERTIES OF LIQUID CRYSTALS 185 tions are as follows: γ = σ|| /σ⊥ = 3 (σ|| ≈ 1.8 nm and σ ≈ ν µ ε × Ð21 3 ⊥ 0.6 nm), = 2.0, = 0.98, and 0 = 2.07 10 J. In addition, we used the following dimensionless 3 parameters: the density ρ = Nσ⊥ /V ≈ 0.512 (corresponding to a density of 103 kg/m3 for 5CB), the tem- Θ ε perature = kBT/ 0, and the dimensionless dipole 2 ) ∆ i moment µ* = ------≈ 2.5. It should be noted that β ()ε σ3 1/2 0 ⊥ (cos 0 f the calculations were performed only for a cubic struc- 1 ture with six nearest neighbors and twelve next-to-nearest neighbors. In this case, we solved 18 nonlinear integral equations (3) in five-dimensional space. The 3 molecular dynamics calculations included 120 5CB 0 1 molecules enclosed in a cubic cell with an edge of 3.65 2 nm, which corresponds to a density of 103 kg/m3. The temperature was maintained at 300 K (Θ = 2.0). The 0 0.2 0.4 0.6 0.8 1.0 β equations of motion of 5CB molecules were solved cos i using the Verlet algorithm [18] with a step of 2 fs [3, 4]. The starting configuration corresponded to the smectic Fig. 1. Orientation distribution function for 5CB molecules phase of 5CB [4]. The director orientation n was deter- at T = 300 K according to calculations in the framework of νν' (1) the molecular dynamics method, (2) the statisticalÐ mined with the use of the matrix Qzz written in the mechanical theory with inclusion of the dipoleÐdipole inter- form [19] action, and (3) the statisticalÐmechanical theory without regard for the dipoleÐdipole interaction. N νν' 1 1 j j Q = ---- ∑ ---()3cosβ ν cosβ ν Ð δνν , (8) zz N 2 z z ' ' dynamics method with different parametrizations of the j = 1 potential energy of the system [3, 4, 20] and molecular j coordinate systems [21] considerably differ from each where N is the number of 5CB molecules and β ν is the z other [22]: 0.5 ≤ P ≤ 0.72 and 0.18 ≤ P ≤ 0.31. The angle between the long axis of the jth molecule and the 2 4 ν axis related to the cubic cell. The molecular coordi- interpretation of the experimental data also depends on nates of the system were constructed using the eigen- the choice of the coordinate system [23]. Compound vectors of the tensor of the moment of inertia [3, 4]. By 5CB over the entire range of the existence of the nem- νν atic phase is characterized by the parameter χ = 9.5 × diagonalizing the matrix Q ' , we obtained all the zz 10Ð2, which confirms that there is a nematic ordering in eigenvectors, of which the largest vector corresponds to the phase. It was also found that the dimensionless the director orientation n. Figure 1 depicts the orienta- ε β Helmholtz free energy f/ 0 of the system is equal to tion distribution function f0(cos i) calculated by the −15.55 with due regard only for the nearest neighbors molecular dynamics method with due regard for the and Ð15.67 with allowance made for the nearest and potential energy involving the intramolecular and inter- next-to-nearest neighbors. Such an insignificant change molecular atomÐatom contributions in the system at in this integral characteristic indicates that, in the T = 300 K [3, 4]. The orientation distribution functions framework of the statistical theory allowing for the derived with the use of the integral equation for polar µ ≈ µ translational, orientational, and mixed correlations, it is ( * 2.5) and nonpolar ( * = 0) systems at T = 300 K quite reasonable to take into account the nearest and are also displayed in Fig. 1. Making allowance for the next-to-nearest neighbors. fact that the calculations were performed using different potentials of the intermolecular interaction, the results obtained with different methods are in good 3. STATIC PERMITTIVITY OF A NEMATIC LIQUID CRYSTAL agreement. Moreover, the order parameters P2 and P4 were calculated in the framework of the statisticalÐ The static permittivity of an isotropic liquid is deter- ε mechanical theory (P = 0.78 and P = 0.35) and the mined by the scalar quantity i [24]. The dielectric 2 4 properties of uniaxial liquid-crystal systems are charac- molecular dynamics method (P = 0.504 and P = ε 2 4 terized by two permittivity tensor components ij, ε ε ε 0.188). The theoretical results were compared with the which are parallel ( || = zz) and perpendicular ( ⊥ = experimental data obtained using NMR spectroscopy ε ε xx = yy) to the director orientation n [5]. The difference (P2 = 0.61 and P4 = 0.15) [20]. It should be noted that between ε|| and ε⊥ is insignificant for nonpolar liquid the order parameters calculated by the molecular crystals but is substantial for polar liquid crystals such

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 186 ZAKHAROV, MIRANTSEV

20 mechanical theory described in Section 2. The relationships for Aj and yj have the form [25] ∗2 4πρµ ξ 1 Ð1 1/2 1 A ==------t , y|| ---- 1 Ð ------tan Ω , 15 2 j Θ j ΩΩ1/2 3 (12) 4 1 ξ Ð1 1/2 5 y⊥ = ---- Ð1 + ------tan Ω , j Ω1/2 ε 6 Ω 10 7 where Ω = ξ Ð 1 and ξ = ε||/ε⊥. Equation (10) can be solved using the iterative procedure

[]k 2 []k 1/2 5 [] B ()ε B ()ε [] ε k + 1 = ------j j - + ------j j - + D ()ε k , (13) j 2 4 j j where k is the iteration number. Figure 2 shows the tem- 0 ε || ⊥ 300 302 304 306 perature dependences of the permittivities j (j = , ) T, K calculated according to Eq. (10) with the use of the correlators tj obtained in the framework of the statisticalÐ Fig. 2. (1Ð3) Calculated and (4Ð7) experimental tempera- mechanical theory. The experimental data on the static ε || ⊥ ture dependences of the static permittivity j (j = , ) for permittivity for the nematic phase of 5CB [6, 11] are 5CB: (1, 4, 5) ε||, (2, 6, 7) ε⊥, and (3) ε . Calculations are also presented in Fig. 2. It should be noted that the cal- performed using relationships (9) and (10). Experimental culated and experimental data are in good agreement. data are taken from [6, 11].

4. DIELECTRIC RELAXATION as 5CB. For these two components, the mean permittiv- AND THE RELAXATION TIMES OF 5CB ε ity can be defined by the expression ε ω For 5CB, the permittivity relaxation tensor ik( ) = Reε (ω) Ð iImε (ω), which was measured over a wide 2ε⊥ + ε|| ik ik ε = ------. (9) frequency range (1 kHz ≤ ω/2π ≤ 13 MHz [9]), is char- 3 acterized by a Debye relaxation. In the laboratory coordinate system with the z axis coinciding with the direc- δε ε ε ε ω For 5CB, the value of NI = Ð i at the temperature tor orientation n, the tensor components ik( ) for TNI of the nematicÐisotropic liquid phase transition is uniaxial nematics can be written in the following form negative and the permittivity ε slowly decreases with a [26]: decrease in the temperature [5]. According to the ∞ ε ω molecular theory proposed by Edwards and Madden i()Ð 1 ω ()ω [25], the permittivity tensor components can be calcu------ε = 1 Ð i ∫C j()t exp Ði t dt, (14) j Ð 1 lated from the quadratic equation 0 where C (t) are the tensor components of the dipole ε 2 ε j j Ð B j j Ð0,D j = (10) autocorrelation function. These components can be represented by the relationships A + 1 Ð y y j j j || ⊥ 〈〉Φ1 where Bj = ------and Dj = ------(j = , ). The C|| ==ez()0 ez()t 00(),t (15) 1 + y j 1 + y j ε unknown coefficients Aj and yj are functions of tj and j, 〈〉〈〉Φ1 C⊥ ===ex()0 ex()t ey()0 ey()t 10().t (16) respectively. The quantity tj is given by the formula Φ1 Here, i0()t (i = 0, 1) are the first-rank time correlation ≡ t = e ⋅ ∑ e , (11) functions and eα (eα eiα) are the projections of the unit j j m vector e onto the α axes (α = x, y, z). The functions mR∈ 0 Φ1 i0()t can be written in the exponential form [2] where m ≠ j and 〈…〉 is defined by relationship (5). Φ1 Φ 1 ()τ1 Summation in formula (11) is carried out over all the 00()t = 00()0 exp Ðt/ 00 dipoles located inside the sphere of radius R0 with the (17) center at the jth molecule. All correlators of the type 12+ P2 ()τ1 〈 〉 = ------exp Ðt/ 00 , ej á em can be calculated in terms of the statisticalÐ 3

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 DYNAMIC AND DIELECTRIC PROPERTIES OF LIQUID CRYSTALS 187

1.0 16 Φ1 00(t) ) ω

( 12 j ε 0.9 Re 8 1 2 Φ1 4 10(t) 0.8 6

4 ) ω

0.7 ( 2 j Φ (t) ε 00 2 1 Im

0 0.6 2 0 40 80 120 160 6789 t, ps logω Fig. 4. Real and imaginary parts of the permittivity tensor ε ω || ⊥ Fig. 3. Time correlation functions for 5CB molecules in the j( ) (j = , ) calculated according to relationships (19) nematic phase according to molecular dynamics calcula- and (20) for (1) longitudinal and (2) transverse components ε ω ω π tions. j( ) ( /2 in Hz) at a temperature of 300 K.

Φ1 ()t = Φ 1 ()0 exp() Ðt/τ1 dynamics formalism and determined using NMR spec- 00 10 10 troscopy [29] for the nematic phase of 5CB at 300 K are (18) presented in the table. 1 Ð P2 ()τ1 = ------exp Ðt/ 10 . 3 From relationships (19) and (20) and the relaxation τ1 Next, Eq. (14) can be rewritten as follows: times i0 (i = 0, 1), we can calculate the coefficients of ε ω || ⊥ ε ω ωτ1 the complex permittivity tensor j( ) (j = , ). Figure 4 ||()Ð 1 1 i 00 ε ω Φ presents the real parts Re j( ) and the imaginary parts ------1= Ð 00()0 ------1-, (19) ε|| Ð 1 ωτ ε ω ε ω 1 + i 00 Im j( ) of the permittivity tensor components j( ) calculated for the nematic phase of 5CB at a tempera- ε ω iωτ1 ω ⊥()Ð 1 Φ1 10 ν ------1= Ð 10()0 ------1-. (20) ture of 300 K. The frequency = ------at which both lon- ε⊥ Ð 1 ωτ 2π 1 + i 10 gitudinal Imε||(ω) and transverse Imε⊥(ω) components It should be noted that different spectroscopic tech- ε ω of the permittivity tensor j( ) exhibit a maximum of niques make it possible to determine the relaxation the dielectric loss is in close agreement with the fre- τL ΦL ν ≈ times mn and the time correlation functions mn at quency exp 6 MHz determined experimentally for the specific values of L. In particular, the first-rank time nematic phase of 5CB at atmospheric pressure [9]. correlation functions (L = 1) can be obtained using IR and dielectric spectroscopy [27], whereas the second- τ1 rank time correlation functions (L = 2) can be deter- Orientational relaxation times i0 (i = 0, 1) calculated by the mined by NMR spectroscopy [28]. Three time correla- molecular dynamics method and determined using NMR Φ1 Φ2 spectroscopy for 5CB molecules in the nematic phase at a tion functions, i0()t (j = 0, 1) and 00()t , calculated in terms of the molecular dynamics method described temperature of 300 K in Section 2 are shown in Fig. 3. A technique for calcu- Molecular dynamics method NMR spectroscopy lating these functions and the orientational relaxation τ1 τ1 , nsτ1 , nsτ1 , nsτ1 , ns times i0 (i = 0, 1) was described in detail in [3, 4]. The 00 10 00 10 τ1 relaxation times i0 calculated within the molecular 38.6 3.66 28.9 2.83

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 188 ZAKHAROV, MIRANTSEV

5. CONCLUSIONS 5. W. H. de Jeu, Physical Properties of Liquid Crystal Materials (Gordon and Breach, New York, 1986), p. 240. Thus, in the present work, the dynamic and dielectric properties of the nematic liquid crystal formed by 6. D. A. Dunmur, M. R. Manterfield, W. H. Miller, and 5CB molecules were investigated in the framework of J. K. Dunleavy, Mol. Cryst. Liq. Cryst. 45, 127 (1978). the statisticalÐmechanical theory (based on the method 7. D. A. Dunmur and K. Toriyama, Liq. Cryst. 1, 169 of conditional distributions) and the molecular dynam- (1986). ics method (with the use of realistic interatomic inter- 8. J. Jadzyn, S. Czerbas, G. Czechowski, et al., Liq. Cryst. action potentials). Within these approaches, we calcu- 26, 432 (1999). lated the orientation distribution functions f (cosβ ) of 9. S. Urban, A. Wurflinger, and B. Gestblom, Phys. Chem. 0 i Chem. Phys. 11, 2787 (1999). 5CB molecules, the order parameters P2L (L = 1 and 2), 10. D. Demus and T. Inukai, Liq. Cryst. 26, 1257 (1999). 〈 〉 ε the orientational correlators ei á ej , the static ( j) and 11. S. R. Sarma, Mol. Phys. 78, 733 (1993). ε ω || ⊥ relaxation [ j( )] permittivities (j = , ), the orienta- 12. L. A. Rott, The Statistical Theory of Molecular Systems τ1 (Nauka, Moscow, 1978). tional relaxation times i0 (i = 0, 1), and the time cor- 13. J. G. Gay and B. J. Berne, J. Chem. Phys. 74, 3316 Φ1 Φ2 relation functions i0()t (i = 0, 1) and 00()t . The (1981). results obtained were compared with the available 14. K. P. Gueu, E. Magnassau, and A. Proutiere, Mol. Cryst. experimental data and with the results derived within Liq. Cryst. 132, 303 (1986). other independent theoretical approaches. In particular, 15. R. Balescu, Equilibrium and Nonequilibrium Statistical the relaxation times τ1 (i = 0, 1) and the order param- Mechanics (Wiley, New York, 1978), p. 440. i0 16. A. V. Zakharov, Physica A (Amsterdam) 166, 540 eter P2 = 0.504 (T = 300 K) determined in our work (1990); 175, 327 (1991); Phys. Rev. E 51, 5880 (1995). permitted us to calculate the rotational self-diffusion 17. A. V. Zakharov and S. Romano, Phys. Rev. E 58, 7428 coefficient D⊥ for 5CB molecules according to the (1998). τ1 Ð1 18. M. P. Allen and D. J. Tildesley, Computer Simulation of equation 10 = [D⊥(2 + P2 )/(1 Ð P2 )] [27]. The result Liquids (Clarendon, Oxford, 1989), p. 350. 8 Ð1 obtained (D⊥ = 1.4 × 10 s ) was compared with the 19. R. Eppenga and D. Frenkel, Mol. Phys. 52, 1303 (1984). 8 Ð1 self-diffusion coefficient (D⊥ = 5.32 × 10 s ) deter- 20. R. Y. Dong, Phys. Rev. E 57, 4316 (1998). mined from the experimental NMR data [29]. The 21. A. V. Komolkin, A. Laaksonen, and A. Maliniak, above results give grounds to make the inference that J. Chem. Phys. 101, 4103 (1994). the molecular dynamics method used in combination 22. C. W. Cross and B. M. Fung, J. Chem. Phys. 101, 6839 with the statisticalÐmechanical theory is a valuable tool (1994). for studying the macroscopic and microscopic proper- 23. T. Koboyashi, H. Yoshida, A. D. L. Chandani, et al., Mol. ties of real liquid-crystal materials. Cryst. Liq. Cryst. 136, 267 (1986). 24. G. S. Rushbrooke, Introduction to Statistical Mechanics ACKNOWLEDGMENTS (Clarendon, Oxford, 1949), p. 310. 25. D. M. F. Edwards and P. A. Madden, Mol. Phys. 48, 471 This work was supported by the Russian Foundation (1983). for Basic Research, project no. 01-03-32084. 26. G. R. Luckhurst and C. Zannoni, Proc. R. Soc. London, Ser. A 343, 389 (1975). REFERENCES 27. I. Dozov, N. Kirov, and M. P. Fontana, J. Chem. Phys. 81, 2585 (1984). 1. P. G. de Gennes and J. Prost, The Physics of Liquid Crys- tals (Oxford Univ. Press, Oxford, 1995, 2nd ed.), p. 360. 28. R. Y. Dong, Nuclear Magnetic Resonance in Liquid Crystals (Springer-Verlag, New York, 1997, 2nd ed.), 2. G. R. Luckhurst and C. A. Veracini, Molecular Dynam- p. 225. ics of Liquid Crystals (Kluwer, Dordrecht, 1994), p. 280. 29. A. V. Zakharov and R. Y. Dong, Phys. Rev. E 64, 031701 3. A. V. Zakharov, A. V. Komolkin, and A. Maliniak, Phys. (2001). Rev. E 59, 6802 (1999). 4. A. V. Zakharov and A. Maliniak, Eur. Phys. J. E 4, 435 (2001). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

FULLERENES AND ATOMIC CLUSTERS

Equilibrium State of C60, C70, and C72 Nanoclusters and Local Defects of the Molecular Skeleton O. E. Glukhova and A. I. Zhbanov Saratov State University, ul. Astrakhanskaya 83, Saratov, 410012 Russia e-mail: [email protected]; [email protected] Received February 19, 2002

Abstract—The stability of C60 and C70 fullerenes and C60 and C72 nanotubes devoid of 2Ð12 atoms of the cluster skeleton was theoretically studied. It was established that Cn molecules with an even number of atoms remain stable, which was conﬁrmed by experimental studies of monomolecular decay of clusters with the number of atoms n ≥ 30. The change in the internuclear distances and in the ionization potential of nanoclusters was determined depending on the number of eliminated atoms. Such defects were shown to decrease the ionization potential of nanoclusters by 0.5Ð0.8 eV. The electron spectrum was calculated within the Harrison semiempirical tight-binding model in the Goodwin modiﬁcation. A new parametrization of interatomic matrix elements of the Hamiltonian and atomic terms for carbon nanoclusters was suggested. © 2003 MAIK “Nauka/Interperi- odica”.

1. INTRODUCTION (nanotubes) represent a promising material for designing subminiature electrovacuum devices [11]. Since the discovery of carbon nanoclusters, interest in their properties has significantly increased from year Due to the wide application of nanoclusters in vari- to year. The reason for this is the technological ous branches of chemistry and physics and continuous advancement made in the field of cluster synthesis in a advancement in the synthesis technologies of nanoclus- macroscopic volume with significant potential for prac- ters and nanocluster-based complex compounds, inter- tical applications. est expressed in the properties of these many-particle molecules remains pertinent. As is known, C60 Currently, many efficient methods of nanocluster fullerenes are characterized by high chemical and production are being developed, among them the burn- mechanical stability. Their stability to external defor- ing of hydrocarbons using the contact arc, resistive and mations is studied. A formal estimation [12] of the high-frequency heating in inert gas atmospheres, and modulus of dilatation shows C60 fullerene to be charac- evaporation of graphite microparticles in thermal plas- terized by a smaller compressibility than diamond crys- mas at normal pressure (see [1]). The simplest, most tal. efficient, and most widely used method is arc discharge in inert-gas atmospheres with the use of graphite elec- Alongside the above-mentioned, the application of trodes [2, 3]. However, many problems of nanocluster nanoclusters calls for studies of their electronic struc- formation from hot chaotic carbon plasmas [4] remain ture, affinity to electrons, and molecule ionization to be solved. A number of studies [1, 4, 5] have been potential, all of which control the chemical properties dedicated to the discussion of these problems. of clusters (in particular, in the synthesis of organic semiconductors with the participation of fullerenes), The development of efficient technologies of nano- and of the emission properties when using films with cluster synthesis has led to the advent of novel lines of nanotube clusters as field emitting arrays. inquiry. The discovery of superconducting properties of doped fullerites [6] stimulated interest in the fullerene As has been shown in many studies dedicated to the crystalline phase [7] and gave rise to a new field of physical properties and synthesis of carbon nanoclus- physics of molecular crystals, namely, i.e., the physics ters, clusters with a skeleton devoid of some atoms are of fullerites. The possibility of producing metal-con- formed alongside typical perfect clusters with a regular taining organic compounds with the participation of molecular structure [11]. fullerenes exhibiting ferromagnetism [8] created sig- The problem of cluster stability in the case of elim- nificant prospects in the development of the metalloor- ination of several pairs of atoms and even fragments has ganic chemistry of fullerenes [9]. Another priority been theoretically studied by many scientists in the direction in cluster application is synthesis based on lines of (i) predictions of the molecular stability to the fullereneÐpolymers [10] whose molecules are bound detachment of atoms, as well as to changes in bond chemically, rather than by van der Waals forces as in lengths and in the skeleton configuration as a whole; (ii) crystalline fullerite. Films with nanotube clusters estimations of enthalpy changes due to decomposition;

190 GLUKHOVA, ZHBANOV

(iii) study of the electron shell of a molecule with bro- 2. NEW PARAMETRIZATION OF THE HARRISON ken bonds; and (iv) study of endohedral-compound for- TIGHT-BINDING MODEL IN THE GOODWIN mation by incorporating a foreign atom into the open- MODIFICATION FOR CALCULATING ing fullerene. THE ELECTRONIC STRUCTURE OF CARBON NANOCLUSTERS The studies into the cluster stability were carried out within two fragmentation models. In [13, 14], the Rice We calculated the metric characteristics of the skel- method was considered to be the mechanism of eton and the electron spectrum of carbon nanoclusters fullerene fragmentation. In this method, the prelimi- using the Harrison modified semiempirical model of nary StoneÐWales isomerization [15] is performed tight binding [19]. We chose this model for several rea- before the elimination of atom pairs, which transfers sons. An ab initio theoretical calculation of the electron the initial fullerene into an isomer with a lower symme- spectrum of polyatomic molecules requires significant ° try (C2v) as a result of 90 rotation of the CÐC bond computer resources [6], while semiempirical methods between neighboring hexagons. According to the Rice can be efficiently applied to calculate micro- and mac- method, elimination of one or more atom pairs leaves roscopic carbon systems [20]. An advantage of the Har- the electron shell closed but changes the symmetry rison scheme is that it operates with the Hamiltonian group. The authors of [13] theoretically confirmed the constructed in real space, in the basis of s and p orbitals stability of the C60 fullerene against elimination of C2, of outer electron layers of carbon atoms. This allows C4, and C6 and the stability of C62 against C2 elimina- one to calculate the skeleton metric characteristics and tion; estimated the enthalpy of StoneÐWales isomeriza- the electron energy levels at various local changes in tion and decomposition of C60; and substantiated the the molecule structure. The wave functions of valence electrons of neighboring atoms are considered to be existence of at least one stable C30 fullerene isomer. The authors of [14] studied fragmentation of carbon cluster nonoverlapping. There exist a few modifications of the Harrison scheme, for example, the Goodwin modifica- + ≤ ≤ cations Cn (3 n 60) within the Rice method and tion for calculating the band structure of diamond and showed the high thermodynamic stability of the clus- graphite [21]. ters. To calculate the metric characteristics of the skele- An alternative method of fullerene fragmentation ton and the electron spectrum of the cluster, the total was employed in [16] with the intent of studying the energy E of the cluster is minimized with respect to formation of endohedral compounds. The latter method bond lengths: is based on the tight-binding approximation of molecular dynamics. The authors of [16] confirmed the high EE= bond + Erep, (1) thermal stability of the C60 fullerene, which retains its skeleton with a bond length fluctuation of ±0.4 Å where Ebond is the band structure energy and Erep is the (“floppy phase” [17]) up to 3000 K, and studied the phenomenological energy taking into account the elec- model of fullerene decomposition with the opening of tronÐelectron and internuclear interactions. a “window” formed initially by 9 and then by 13 and The phenomenological energy is written as the sum 16 atoms as C2, C4, and C5 fragments were eliminated of pair repulsing potentials: on reaching a temperature of ~5000 K. The energy barrier for a helium atom passing inside the fullerene () Erep = ∑V rep ri Ð r j , (2) through a 9-atomic window was shown to decrease < from 8.7 to 3.5 eV in comparison with one passing ij through a hexagon ring. where i and j are indices of interacting atoms and ri and Apart from fullerenes, the physical properties of sin- rj are the Cartesian coordinates. The function Vrep is gle-walled nanotube carbon structures were theoreti- given by (see [21]) cally studied ab initio in the case where the structures 4.455 0 1.54 have no dome, as well as in the case of detachment of V ()r = V ------more extended fragments with the formation of an rep repr opened nanotube with flat and oblique edges [18]. It (3) 22 22 was shown in [18] that broken bonds become closed × r 1.54 2 exp4.455 Ð------+ ------, onto one of the bonds of the sp hybridization as 2.32 2.32 unpaired electrons appear, thus closing the electron shell of the molecule. In this case, a tight double bond 0 is formed, shortening the distance between the nuclei where V rep = 10.92 eV. The band structure energy is from 1.41 Å (for the initial molecule) to 1.27 Å. given by We studied the models of clusters devoid of several ε atoms under the assumption of the shell being closed, Ebond = 2∑ n, (4) as in [18]. n

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

EQUILIBRIUM STATE OF C60, C70, AND C72 NANOCLUSTERS 191 Table 1. Carbon atom terms and equilibrium overlap integrals (eV)

ε ε 0 0 0 0 Parametrization s p Vssσ Vspσ V ppσ V ppπ Goodwin [21] Ð5.163331 2.28887 Ð4.43338 3.78614 5.65984 Ð1.82861 His study Ð10.932 Ð5.991 Ð4.344 3.969 5.457 Ð1.938

ε where n is the energy of the occupied state with index studied carbon cluster). The carbon atomic terms were n, i.e., the Hamiltonian eigenvalue (the factor of two found to be higher by 3 eV in comparison with the Har- takes into account the electron spin). rison values (see the HarrisonÐGoodwin parametriza- Interatomic matrix elements of the Hamiltonian tion in Table 1). However, the fullerene bond lengths were determined from the formula [21] (r1 = 1.470 Å, r2 = 1.425 Å) and the energy gap (Eg = ε ε 2.796 1.64 eV) calculated for these values of s and p were 0 1.54 V α()r = V α ------significantly more different (a change for the worse) ij ij r from the experimental values in comparison with the (5) same quantities obtained theoretically with the Good- r 22 1.54 22 win parametrization [23]. × exp2.796 Ð ------+ ------, 2.32 2.32 Therefore, we attempted to improve the parametrization in another way and varied both the carbon where r is the interatomic spacing, i and j are the orbital atomic terms and the interatomic matrix elements. We quantum numbers of the wave functions, and α is the σ π modified the carbon atomic terms and the equilibrium subscript indicating the bond type ( or ). The values overlap integrals such that the bond lengths r and r , of the atomic terms ε and ε and the equilibrium over- 1 2 s p the energy gap E , and the ionization potential I corre- 0 0 0 0 g lap integrals V ssσ , V spσ , V ppσ , and V ppπ suggested by sponded to the experimental values for the C60 Goodwin are listed in Table 1. This modification was fullerene. To do this, the sum of squared deviations of used in [22] to calculate the electronic and vibrational the calculated bond lengths r1 and r2, energy gap Eg, spectra of the C60 fullerene (see Fig. 1). The bond and ionization potential I from their true (experimental) lengths (r1 = 1.463 Å and r2 = 1.418 Å for single and values (see above) was minimized. At each change of double bonds, respectively) obtained in [22] for the one of the optimized parameters (carbon atomic terms fullerene, as well as the energy gap between the last and equilibrium overlap integrals), we calculated the highest occupied and the lowest empty level in the elec- values r1, r2, Eg, and I by minimizing the total energy of tronic spectrum (Eg = 1.7 eV), were in quite good the molecule with respect to bond lengths, which takes × agreement with the experimental data from [23]: r1 = approximately 80 iterations; i.e., a 240 240 matrix ± ± 1.45 0.01 Å, r2 = 1.4 0.01 Å, and Eg = 1.7Ð1.9 eV. was diagonalized 80 times. As a result, the carbon However, the immediate application of the Goodwin parametrization to matrix elements of the Hamiltonian , eV does not allow one to determine the ionization potential I from the electronic spectrum, which somewhat narrows 7.7 the applicability range of this parametrization. 7.6 We suggest a new parametrization of interatomic 2 and diagonal matrix elements of carbon nanoclusters 1 3 8 having none of the above limitations. 7 9 12 10 The first attempt that we undertook to improve the 7.4 11 4 6 parametrization consisted in the following. We left the 5 0 0 0 equilibrium overlap integrals V ssσ , V spσ , V ppσ , and 0 7.2 V ppπ unchanged, i.e., the same as in the Goodwin modification. As a first approximation to the carbon atomic ε ε terms s and p, we took the values suggested by Harri- ε ε son [19] ( s = Ð17.52, p = Ð5.97 eV) and then varied 7.0 ε ε 0 4812 them. We increased s and p by the same amount and shifted the electronic spectrum such that the energy N position of the highest electron energy level corre- Fig. 1. Change in the ionization potential of fullerene C60 sponded to the experimental value of the ionization after elimination of twelve atoms; N is the number of potential I = 7.61 eV [6] of the C60 fullerene (the best sequentially eliminated atoms.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 192 GLUKHOVA, ZHBANOV

r7 r8 The metric characteristics of the skeleton and the electronic spectrum of the cluster are calculated as fol- r7 lows. The total energy of the molecule is calculated. The r6 internuclear distances are specified in the initial r5 r5 approximation. Then, these distances are determined r4 r4 more accurately by minimizing the molecule energy with respect to the sought-for internuclear distances. r3 r2 r2 The number of internuclear distances to be calculated is r1 σ r1 limited by the calculating time. For example, determi- h nation of the values of eight bond lengths and four skeleton diameters of the C70 fullerene takes computer time that is several tens of times longer than that taken for determination of two bond lengths of the C60 fullerene. In the case of optimization of additional internuclear distances or coordinates of separate atoms with the intent of correcting the skeleton configuration, the computing time significantly increases. In this way, we calculated the molecule skeleton configuration and the electronic spectrum. Fig. 2. C fullerene. 70 As is known, the ionization potential of a molecule can be estimated from the electronic spectrum. Accord- atomic terms and the interatomic matrix elements were ing to the Koopmans theorem, this potential represents determined by performing 1000 iterations to an accu- the total energy of an ion produced by the elimination of one electron and can be determined from the energy racy of 10Ð4. To calculate the Hamiltonian eigenvalues, spectrum of the molecule as the absolute value of the we invoked the Householder method [24], as in [22]. energy of the highest occupied level. In this case, the Minimization was carried out with the HookeÐJeeves configuration change accompanying ionization, as well method. The calculated Hamiltonian matrix elements as the fact that the electron shell of the new system are listed in Table 1. For the fullerene, we obtained r = 1 becomes open, can be neglected for large molecules 1.4495 Å, r2 = 1.4005 Å, Eg = 1.96 eV, and I = [25]. 7.6099 eV. For all three Harrison scheme modifications under consideration, according to the dimensions of the It appears of interest to study the multiplicity of the irreducible a, t, g, and h representations of the point stable state of carbon clusters. This study requires a symmetry group Y to which the C fullerene is related, group-theoretical analysis of each cluster; however, the h 60 method we applied to calculate the electronic spectrum the electron spectrum is characterized by singly, triply, does not make it possible to account for the multiplic- quadruply, and fivefold degenerate states, respectively. ity. Within the method under consideration, the ground To check the new parametrization of matrix ele- (many-electron) state of molecule Cn is controlled by ments in the Harrison modified scheme, the calculated population of the lowest 2n single-electron states by 4n metric characteristics of the C70 fullerene skeleton were valence electrons (single-electron approximation). compared to the known values numerically calculated Nevertheless, a survey of some papers on fullerene sta- from the electron energy loss spectrum for elastic scat- bility [13Ð15], nanotube electronic structure [20], phys- tering in C70 in the gaseous state [20]. ical properties of fullerenes [26], hyperfullerenes [27], and nanotubes [28, 29] shows that the cluster state multiplicity can be neglected in this case. 3. STABLE STATE OF C60 AND C70 FULLERENES AND C60 AND C72 NANOTUBES 4. C70 FULLERENE Experimental studies of C60 fullerenes demonstrate the extraordinarily high stability of these clusters [20]. The C70 fullerene belongs to the point symmetry High survival of molecules with even numbers of atoms group D5h and is a closed spheroid molecule consisting was observed in the course of monomolecular decom- of 25 hexagons and 12 pentagons. Two pentagons form position of carbon clusters with n ≥ 30 or photodissoci- opposite faces, distance with the between them being ation. This stability is explained by the structural frag- maximum for this molecule (Fig. 2). The skeleton con- ment C2 detachment [20]. figuration is characterized by four diameters, height, We theoretically studied the geometrical parameters and eight different internuclear distances [30]. One of of the molecular skeleton and the electronic spectrum the diameters, d1 (Table 2), characterizes a circle lying σ of carbon clusters in the case of elimination of frag- in the symmetry plane h perpendicular to the principal ments with even numbers of atoms. rotation axis C5 passing through the centers of the pen-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EQUILIBRIUM STATE OF C60, C70, AND C72 NANOCLUSTERS 193 tagons. The three other diameters deﬁne the circles Table 2. Some metric characteristics (Å) of the C70 fullerene σ lying in the planes parallel to h. The diameter d2 skeleton (Table 2) of the circle spaced from σ by one layer is h Parametrization smaller than d1; therefore, there is a waist in the plane Experiment [30] σ h [30]. The fullerene height h (Table 2) is understood Goodwin this study as the distance between opposite pentagons. The internuclear distances are determined by the following CÐC Bond lengths +0.03 bonds (Table 2): the bond r1 between the hexagons, r 1.41 1.466 1.454 σ 1 Ð0.01 which lies in the plane h; the bond r2 between the pen- σ r 1.39 ± 0.01 1.438 1.421 tagon and the hexagon whose one side lies in the h 2 plane; the bond r3 lying in a pentagon and also belong- r +0.01 1.448 1.433 σ 3 1.47Ð0.03 ing to the hexagon crossed by h; the bond r4, common σ r 1.46 ± 0.01 1.462 1.448 for a pentagon and the hexagon which is based on h; 4 ± the bond r5 between the irregular pentagons; the bond r5 1.37 0.01 1.414 1.396 r which lies in the pentagon and also belongs to the ± 6 r6 1.47 0.01 1.459 1.446 hexagon contiguous to the pentagon; the bond r 7 r 1.37 ± 0.01 1.419 1.401 between a pentagon and an irregular pentagon; and the 7 r 1.464 ± 0.009 1.462 1.449 bond r8 lying in a pentagon. 8 Diameters and height Using the technique described above, we calculated ± the geometrical parameters of the C fullerene with d1 6.94 0.05 7.18 7.12 70 ± various parametrizations of matrix elements, namely, d2 6.99 0.05 7.01 6.93 the Goodwin and HarrisonÐGoodwin parametrizations h 7.8 ± 0.01 8.00 7.92 and ours (Table 1). The results are listed in Table 2. From the calculations, it follows that our parametrization is adequate. The metric parameters of the fullerene one or two C6 rings alongside several C2 pairs (for C62, structure conform well to the values given in [30]. The C60, C58). This conforms to the results of similar studies electronic spectrum is controlled by singly and doubly of the C fullerene [13]. degenerate orbital multiplets of different energy 60 according to the dimensions of the irreducible a and e representations of the D5h group. As follows from a σ 5. C60 FULLERENE comparison of the circle diameter (Table 2) in h with other diameters, no waists in the fullerene symmetry Now, we consider the case when atoms that are σ plane h exist. This result is qualitatively conﬁrmed by arranged in two opposite hexagons are sequentially calculations with various parametrizations of the broken away in pairs from the molecule. The coordi- matrix elements (Table 2). nates of atoms with broken bonds were optimized, and

We studied the C70 fullerene behavior in the course of sequential elimination in two or more atoms from hexagons. The coordinates of atoms with broken bonds I, eV 7.4 were optimized, and the electronic spectra of C68, C66, 2 3 C64, C62, C60, and C58 were calculated. The sequence of 4 eliminations is shown in Fig. 3. The ﬁrst to be elimi- 1 nated from the cluster were two atoms with indices 1 6 5 and 2, then four atoms with indices 1Ð4; two fragments 7.2 containing atoms 1Ð12 were removed in the last reaction. As indicated above, the electron shell of the cluster remains closed after elimination of several atoms or a 7 8 whole fragment. Elimination of atoms from the cluster 7.0 12 9 was accompanied by an increase in the total energy, a 11 10 displacement of the highest populated electron term to the region of positive values, and an increase in the decomposition heat. The ionization potential variation is shown in Fig. 3. The reaction heat increase due to the 6.8 cluster losing more and more atoms is shown in Table 3. 0 4812 However, the enthalpy calculations show that the reac- N tion heat can be lowered by eliminating a single C6 ring Fig. 3. Variation in the C70 fullerene ionization potential in instead of several C2 pairs (for C64) or by eliminating the case of elimination of twelve atoms.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 194 GLUKHOVA, ZHBANOV

Table 3. Enthalpy of C60, C70, and C72 carbon cluster decomposition

C60 fullerene (Yh)C60 nanotube (D6h) ∆ ∆ reaction: C60 heat H, kcal/mol reaction: C60 heat H, kcal/mol

C58 + C2 392.58 (17.07 eV) C58 + C2 5947.52 (258.59 eV)

C56 + 2C2 610.42 C56 + 2C2 6116.83

C54 + 3C2 816.75 C54 + 3C2 6285.93

C54 + C6 631.48 C54 + C6 6100.66

C52 + 4C2 1235.24 C52 + 4C2 6634.69

C52 + C2 + C6 1049.97 C52 + C2 + C6 6449.40

C50 + 5C2 1440.12 C50 + 5C2 6826.15

C50 + 2C2 + C6 1049.97 C50 + 2C2 + C6 6640.87

C48 + 6C2 1640.04 C48 + 6C2 7003.26

C48 + 2C6 1269.49 C48 + 2C6 6632.71

C70 fullerene (D5h)C72 nanotube (D6d) ∆ ∆ reaction C70 heat H, kcal/mol reaction C72 heat H, kcal/mol

C68 + C2 393.23 (17.10 eV) C70 + C2 7080.84 (307.86 eV)

C66 + 2C2 599.22 C68 + 2C2 7253.89

C64 + 3C2 797.86 C66 + 3C2 7476.48

C64 + C6 612.60 C66 + C6 7256.21

C62 + 4C2 1191.43 C64 + 4C2 7894.69

C62 + C2 + C6 1006.17 C64 + C2 + C6 7676.21

C60 + 5C2 1407.85 C62 + 5C2 8112.14

C60 + 2C2 + C6 1222.59 C62 + 2C2 + C6 7901.87

C58 + 6C2 1596.83 C60 + 6C2 8334.86

C58 + 2C6 1226.30 C60 + 2C6 7908.31

the electronic spectra of C58, C56, C54, C52, C50, and C48 6. C60 AND C72 NANOTUBES molecules were calculated (Fig. 1). The atom elimina- We also studied the C60 and C72 nanotube clusters tion sequence remained as before (for the C70 (Figs. 4, 5), which have identical dome shapes and dif- fullerene). The bond lengths of the symmetric C48 mol- fer only in the number of hexagons in their skeletons. ecule were different: r1 = 1.466 Å and r2 = 1.378 Å. As The C60 and C72 nanotubes are poorly studied in com- in the case of C70 fullerene, the total energy increases, parison with the C60 and C70 fullerenes; therefore, we the ionization potential is lowered (Fig. 1), and the first determined their bond lengths. Unlike the authors decomposition enthalpy increases (Table 3) as atoms of [20], we detected five CÐC bond lengths rather than are eliminated from the cluster. The values of the reac- four: the bond in the hexagon at the “cover” (r1 = tion heat for the C58, C56, and C54 production are in gen- 1.452 Å), the bond common to two pentagons (r2 = eral agreement with the corresponding values from [13, 1.475 Å), and the bond also belonging to the hexagon 14] but do not completely coincide, since the authors of lying outside the cluster cover (r3 = 1.460 Å); this last [13, 14] used an alternative fullerene C60 decomposing hexagon is also characterized by two more bond types procedure (the Rice method) and the MNDO calcula- (r4 = 1.448 Å, r5 = 1.467 Å). The electronic spectrum of tion method. Moreover, the authors of [14] studied the the nanotubes is formed by singly and doubly degener- C+ cation rather than the neutral fullerene. As in the ate multiplets according to the dimensions of the irre- 60 ducible a, b, and e representations of the D group (for case of the C fullerene, the reaction heat tended to 6h 70 the C nanotube) and D group (for the C nanotube) decrease after elimination of one or two C rings. 60 6d 72 6 whose reducible-representation basis is formed by p orbitals. It can be concluded that the C60 and C70 fullerenes remain stable, notwithstanding some imperfections in A study of nanotubes with deformed domes (C58, the configuration of their skeleton. C56, C54, C52, C50, C48; C70, C68, C66, C64, C62, C60) and

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EQUILIBRIUM STATE OF C60, C70, AND C72 NANOCLUSTERS 195

I, eV I, eV 7.6 7.2 3 4 5 4 2 5 6 3 6 1 2 1

7.2 7.0

12 7 11 8 11 6.8 10 9 6.8 12 10 7 8 9

6.4 6.6 0 48120 4812 N N

Fig. 4. Variation in the ionization potential of the C60 nano- Fig. 5. Variation in the ionization potential of the C72 nanotube in the case of elimination of twelve atoms. tube in the case of elimination of twelve atoms.

without domes (C54, C48; C66, C60; Figs. 3, 4) showed becomes less appreciable and then the potential cluster stability to such defects of the molecular skele- increases slowly with the number of eliminated atoms. ton, a rather small change in the coordinates of atoms It has been shown that the enthalpy of fullerene with broken bonds, and an increase in the molecule fragmentation can decrease if elimination of whole energy and in the energy of the highest occupied energy fragments (one or more C6 rings), rather than of car- level. The ionization potential variation due to a bon atom pairs, is performed. A similar enthalpy ten- decrease in the number of carbon atoms in the dome is dency is observed during decomposition of carbon shown in Figs. 4 and 5 (the sequence of atom elimina- tubes. In this case, much larger fragmentation heats of tions is the same as for fullerenes). The skeleton of nan- carbon tubes are observed in comparison with those otubes whose symmetry remains unchanged was cor- for fullerenes. The reason for this effect is unclear, rected by optimizing the bond lengths. For the C54 and and, as far as we know, there are no publications on C66 nanotubes, we have r1 = 1.421, r2 = 1.476, r3, r4 = this problem. 1.428, and r = 1.441 Å. For the C and C nanotubes, 5 48 60 The results of this study allow one to predict some we have r3 = 1.410, r4 = 1.441, and r5 = 1.432 Å. The properties of molecules with a deformed skeleton reaction enthalpy variation is shown in Table 3. As in devoid of certain atoms. Such molecules are formed in the case of fullerenes, the enthalpy is lowered as one or the course of carbon nanocluster synthesis in macro- two C6 rings are eliminated alongside C atom pairs. scopic volumes. In particular, the emissivity can be qualitatively estimated from changes in the electronic spectrum and the ionization potential. This study allows 7. DISCUSSION us to conclude that a purposeful generation of certain The theoretical study of the stability of the C and defects in the course of carbon nanocluster synthesis 60 can allow one to achieve required properties. C70 fullerenes and the C60 and C72 nanotubes with defects in their molecular structure showed the following. ACKNOWLEDGMENTS In the case of elimination of an even number of This study was supported in part by the Russian atoms, the nanoclusters do not decompose. At the Foundation for Basic Research (project no. 98-02- points of local defects, a correction to the cluster skele- 17970) and the International Science and Technology ton is observed which corresponds to the recovery of Center (project no. 1024f-99). equilibrium.

The ionization potential of clusters devoid of several REFERENCES atom pairs was found to decrease. This is probably explained by a correction to the skeleton and by a large 1. D. Afanas’ev, I. Blinov, A. Bogdanov, et al., Zh. Tekh. ratio of the number of removed atoms to the total num- Fiz. 64 (10), 77 (1994) [Tech. Phys. 39, 1017 (1994)]. ber of atoms in the molecule. It has also been estab- 2. J. P. Hare, H. W. Kroto, and R. Taylor, Chem. Phys. Lett. lished that the decrease in the ionization potential 177, 394 (1991).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 196 GLUKHOVA, ZHBANOV

3. W. A. Scrivens and J. M. Tour, J. Org. Chem. 57, 6932 19. W. A. Harrison, Electronic Structure and the Properties (1992). of Solids: The Physics of the Chemical Bond (Freeman, 4. H. Kroto, Pure Appl. Chem. 62, 407 (1990). San Francisco, 1980; Mir, Moscow, 1983), Vol. 1. 5. R. F. Curl, Nature 363, 14 (1993). 20. E. G. Gal’pern, I. V. Stankevich, L. A. Chernozatonskiœ, and A. L. Chistyakov, Pis’ma Zh. Éksp. Teor. Fiz. 55 (8), 6. V. M. Loktev, Fiz. Nizk. Temp. 18 (3), 217 (1992) [Sov. 469 (1992) [JETP Lett. 55, 483 (1992)]. J. Low Temp. Phys. 18, 149 (1992)]. 21. L. Goodwin, J. Phys.: Condens. Matter 3, 3869 (1991). 7. S. V. Kozyrev and V. V. Rotkin, Fiz. Tekh. Poluprovodn. 22. N. V. Khokhryakov and S. S. Savinskiœ, Fiz. Tverd. Tela (St. Petersburg) 27 (9), 1409 (1993) [Semiconductors (St. Petersburg) 36 (12), 3524 (1994) [Phys. Solid State 27, 777 (1993)]. 36, 1872 (1994)]. 8. B. Morosin, C. Henderson, and J. E. Schrieber, Appl. 23. J. R. D. Copley, D. A. Neumann, R. L. Cappelletti, and Phys. A 59, 179 (1994). W. A. Kamitakahara, Phys. Chem. Solids 53 (1), 1353 9. P. J. Fagan, J. C. Calabrese, and B. Malone, Acc. Chem. (1992). Res. 25, 134 (1992). 24. J. H. Wilkinson and C. Reinsch, Linear Algebra 10. F. Wudl, Acc. Chem. Res. 25, 157 (1992). (Springer, Berlin, 1971; Mashinostroenie, Moscow, 1976). 11. N. I. Sinitsyn, Yu. V. Gulyaev, G. V. Torgashov, et al., 25. R. Zahradnik and R. Polak, Zaklady kvantove chemie Appl. Surf. Sci. 111, 145 (1997). (SNTL, Prague, 1976; Mir, Moscow, 1979). 12. R. S. Ruoff and A. L. Rouff, Appl. Phys. Lett. 59, 1553 26. T. L. Makarova, Fiz. Tekh. Poluprovodn. (St. Petersburg) (1991). 35 (3), 257 (2001) [Semiconductors 35, 243 (2001)]. 13. R. E. Stanton, J. Phys. Chem. 96, 111 (1992). 27. M. I. Heggie, M. Terrones, B. R. Eggen, et al., Phys. Rev. 14. J. J. Novoa and M.-H. Whangbo, New J. Chem. 18, 457 B 57 (21), 13339 (1998). (1994). 28. L. Lou, P. Nordlander, and R. E. Smalley, Phys. Rev. B 15. R. E. Stanton and D. J. Wales, Chem. Phys. Lett. 128, 52 (3), 1429 (1995). 501 (1986). 29. L. Lou and P. Nordlander, Phys. Rev. B 54 (23), 16659 16. E. Kim, D.-H. Oh, C. W. Oh, and Y. H. Lee, Synth. Met. (1996). 70, 1495 (1995). 30. A. V. Eletskiœ and B. M. Smirnov, Usp. Fiz. Nauk 163 (2), 33 (1993) [Phys. Usp. 36, 202 (1993)]. 17. S. G. Kim and D. Tomanek, Phys. Rev. Lett. 72, 2418 (1994). 18. S. Han and J. Ihm, Phys. Rev. B 61 (15), 9986 (2000). Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 197Ð200. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 187Ð190. Original Russian Text Copyright © 2003 by Golovin, Dmitrievskiœ, Nikolaev, Pushnin. FULLERENES AND ATOMIC CLUSTERS

The Influence of Ultraweak Ionizing Irradiation on the Magnetoplastic Effect in Single Crystals of a C60 Fullerite Yu. I. Golovin*, A. A. Dmitrievskiœ*, R. K. Nikolaev**, and I. A. Pushnin* * Tambov State University, Internatsional’naya ul. 33, Tambov, 392622 Russia ** Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected], [email protected] Received March 12, 2002

Abstract—The suppression of plasticity sensitivity of a C60 fullerite to the action of a magnetic ﬁeld is revealed. It is found that the C60 fullerite undergoes temporary softening due to irradiation with ultralow (<0.1 cGy) doses of β and γ radiation. © 2003 MAIK “Nauka/Interperiodica”.

Numerous magnetoplastic effects have been Investigation into optical and radiation effects under revealed in magnetically disordered solids since the simultaneous (or alternating) exposure to a magnetic first publication in this field [1]. Until presently, the field can provide additional information on the nature nontrivial nature of these effects had remained unclear of the aforementioned processes. In particular, it was in many respects and continues to be of interest in phys- found that weak x-ray irradiation [19] and light in the ical chemistry. There are some difficulties in interpret- optical range [20] affect the magnetoplastic effects in ing magnetoplastic effects. One of these difficulties is ionic crystals. The problem of low radiation doses is of ≈ associated with the fact that the magnetic energy Em current concern in itself, because, as in the case of a µ µ BB in a laboratory magnetic field B ~ 1 T ( B is the weak magnetic field, there often arise effects that have Bohr magneton) is incommensurate to any characteris- defied explanation. These are softening (instead of con- tic energy of a nonmagnetic solid at room temperature ventional radiation hardening at moderate absorbed doses) [21, 22], fast relaxation of internal stresses [23], Tr (it is at this temperature that the majority of experi- ≈ Ð2 an increase in the internal friction [24], etc. [25]. The ments were carried out). Specifically, Em/kTr 10 , α β ≈ Ð3 Ð4 ≈ Ð4 Ð5 effect of irradiation with and particles on the prop- Em/Ea 10 Ð10 , and Em/Eb 10 Ð10 . Here, k is the erties of fullerenes was considered in [26Ð28], and the ≈ Boltzmann constant, Ea 0.1Ð1 eV is the activation influence of the magnetic field on these materials was energy of overcoming the stoppers by dislocations, and analyzed in [29, 30]. However, systematic information ≈ Eb 1Ð10 eV is the binding energy in the solid. Another on the influence of low radiation doses and magnetic difficulty encountered in interpreting the magnetoplas- fields on fullerites is unavailable. tic effects is as follows. The observed macroscopic In this connection, the purpose of this work was to effect of softening (an increase in the mobility of indi- investigate the alternating effects of a pulsed magnetic vidual dislocations [2Ð5], an increase in the damping field with an induction B up to 24 T and small (<1 cGy) decrement of internal friction [6, 7], a decrease in the doses of β and γ radiation on the mechanical properties yield point [8] and the strain-hardening coefficient [9], of single-crystal fullerite C60 (>99.95%) grown from etc.) is a consequence of the multistage process of the vapor phase. Magnetic-field pulses with the ampli- structural relaxation. The result of this process mani- tude B = 24 T were generated using a capacitor bank fests itself well apart from the elementary acts in the m electron-spin subsystem, which, strictly speaking, can discharged through a solenoid with a small number of turns. The pulses had a shape close to that of a half- be affected by the magnetic field [10–13]. In most sinusoid and a duration of ~150 µs. cases, the limiting stage is likely to be the spin-dependent stage of switching of bonds in complexes of point The radioactive source used in the experiments was defects and bonds between a dislocation core and a 137 based on a 55C s radionuclide with the activity A = stopper [11, 14, 15]. It is known that even a weak mag- 4.2 MBq. The source emitted γ quanta with a maximum netic field can affect the multiplicity of short-lived energy of 0.66 MeV and a quantum yield of 0.85 per excited spin pairs, cause spin conversion in these pairs, βÐ and open (or close) some of the existing channels in decay. In addition, this source emitted particles into reactions between radicals with several possible results two bands of the energy spectrum with maximum ener- [16Ð18]. gies Em' = 0.564 MeV (the quantum yield is 0.947 per

198 GOLOVIN et al.

MF MF βγ γ βγ βγ γ

024636 38 40 42 44 74 76 124 126 Time, h

1 ~ ~ ~ ~ ~ ~ , MPa H ∆ 0

–1 1 2 –3 3

Microhardness change 4

–5

Fig. 1. Change in the microhardness H of the C60 fullerite with time due to alternating actions of irradiation and pulses of the magnetic field with the induction B = 25 T: (1) β + γ irradiation, (2) treatment with a pulsed magnetic field, (3) γ irradiation, and (4) grinding. decay) and E'' = 1.176 MeV (the quantum yield is out either in the dark or in the presence of weak red m light. 0.053) [31]. As can be seen form Fig. 1 (in which each point is βÐ Since the depth of penetration of particles with obtained by averaging 15Ð20 individual measure- energies <1 MeV into the crystal under investigation ments), the β + γ irradiation leads to a decrease in the was relatively small (tens of micrometers), their possi- microhardness H with an increase in the radiation dose ble influence on the mechanical properties of thin sur- D to saturation after a time tsat ~ 6 h. In the saturation face layers was investigated using dynamic nanoinden- state, no additional decrease in the microhardness H is tation [32]. In all the experiments, the maximum load observed in the sample exposed to a magnetic field. was 200 mN and the indentation depth was approxi- After the rest for ~30 h at room temperature in the mately 7 µm. In order to prevent rapid oxidation of the absence of irradiation and a magnetic field, the micro- surface, all manipulations (irradiation, microhardness hardness completely regains its initial value. Repeated measurements, and rest after irradiation) were carried irradiation after the rest brings about a decrease in the same quantity as after the first irradiation. A decrease in the radiation intensity leads to a decrease not only in the log∆H, MPa softening rate but also in the softening magnitude at sat- 1.0 uration. It is worth noting that the sample subjected to a magnetic-field pulse after relaxation undergoes 1 approximately the same softening as in the case of β 0.5 2 and γ irradiation. However, in this case, irradiation does not result in an additional decrease in the microhardness H. It should be noted that exposure to β + γ irradi- 0 ation and a pulsed magnetic field leads not only to close magnitudes of reversible softening but also to the same type of kinetics (the first-order reaction) and close rates –0.5 of relaxation to the initial state after these treatments (Fig. 2). –1.0 In our experiments, we examined about 20 samples. 0 10 20 30 40 50 Each sample was used repeatedly. In order to remove Time, h the oxidized layer, a 50- to 100-µm-thick surface layer was periodically ground away from the sample. The Fig. 2. Kinetics of recovery of the initial microhardness H test experiments demonstrated that, after grinding, the after the action of (1) the pulse of the magnetic field B = possible change in the microhardness H due to slow 25 T and (2) β + γ irradiation. oxidation of the surface under the aforementioned

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

THE INFLUENCE OF ULTRAWEAK IONIZING IRRADIATION 199 experimental conditions remains negligible for at least of the fullerite occurs predominantly through breaking a few days after the preparation of the surface. of weaker intermolecular bonds. A knocking-out of one The sample was screened using a 2-mm-thick alu- or two carbon atoms from a symmetric molecule of C60 minum plate, which completely absorbed the β radia- can result both in loss of molecular stability under load tion and had no effect on the intensity of the γ radiation. and in macroscopic strain not only due to slip of some This screening led to suppression of the softening spherical molecules relative to others but also due to effect. Therefore, the γ irradiation with the doses used decay of molecules themselves. This process of multi- had no noticeable effect on the microhardness H. The plying the action of a vacancy in the structure of the C60 elucidation of the role played by the γ irradiation under molecule can substantially increase the efficiency of the joint action of the β and γ radiation calls for special radiation influence. consideration. Let us now assume that the particle The reversible effect of a magnetic field and irradia- fluxes and the energies transferred by the β and γ radi- tion on the microhardness, the dependence of the satu- ation are approximately equal to each other. It is clear ration level on the irradiation intensity, repeated recov- that, under these conditions, the contribution of the β ery of softening after relaxation of treatment-induced radiation to the changes observed in the surface proper- states, and a number of other factors indicate that these ties should be substantially larger than that of the γ radi- two types of weak fields affect objects close to thermo- ation. Actually, in the case when the weighted-mean dynamic equilibrium. The pulsed magnetic field and energy of β particles in a flux emitted by the 137C s radi- radiation affect the same objects in the crystal, or, at 55 least, their actions occur through similar mechanisms. onuclide is taken as 〈Eβ〉 = 179.8 keV [12], the absorb- This can be judged from the following factors: (i) the ing layer thickness for these particles is approximately depths of softening effects caused by the magnetic field 3 γ 10 times smaller than that for quanta with an energy and irradiation are equal to each other; (ii) the time con- Eγ = 0.66 MeV and the bulk density of induced excited stants of exponential relaxation of the microhardness H states in the surface layers is higher by approximately to the initial value are also equal to each other; and the same factor. (iii) the sensitivity to one of these factors is suppressed Now, we estimate the number n of atomic defects immediately after the action of the other factor. produced by a flux of βÐ particles with a maximum flu- The mechanisms of the influence of weak magnetic ≈ × 8 Ð2 ence F = Iβtsat 2 10 cm . As a rule, the energy and radiation fields on fullerites call for further investi- required to produce atomic radiation damage is taken to gation. However, the mere fact that fullerite softening be E0 = 20Ð30 eV. In this case, the number of radiation occurs under such weak actions has stimulated a search ≈ defects formed in a surface layer of thickness h1/2 for similar effects in other cyclic carbon-containing 13 µm that absorbs one-half the electron flux with substances (for example, single crystals of an aromatic energy 〈Eβ〉 = 0.18 MeV is determined to be n = series, polymers, etc.); moreover, caution should be 〈 〉 ≈ 15 Ð3 exercised when dealing with possible consequences of F Eβ /(h1/2E0) 10 cm . For such a low concentration of structural defects, the mechanisms of their influence exposure even in very weak fields. on the plastic characteristics call for special analysis. We can suggest at least three possible reasons for an ACKNOWLEDGMENTS βÐ effective action of radiation with low doses on fuller- This work was supported by the Program ites. “Fullerenes and Atomic Clusters” (project no. 2008), (1) Plasticity of the fcc crystals is limited by the the Russian Foundation for Basic Research (project occurrence of local stoppers for glide dislocations. In no. 00-02-16094), and the Ministry of Education the C60 single crystals under investigation, these stop- (project no. E00-3.4-552). pers can be carbon molecules with a different molecular weight (mainly, C70), dimers (C60ÐC60), oxidized molecules, impurities of other elements, etc. Their total REFERENCES concentration in the surface layers amounts to ~10Ð3Ð 1. V. I. Al’shits, E. V. Darinskaya, T. M. Perekalina, and 10Ð4. By modifying even a small number of the most A. A. Urusovskaya, Fiz. Tverd. Tela (Leningrad) 29 (2), efficient stoppers under irradiation, it is possible to 467 (1987) [Sov. Phys. 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6. M. I. Molotskii, R. E. Kris, and V. Fleurov, Phys. Rev. B 20. Yu. I. Golovin, R. B. Morgunov, and S. Z. Shmurak, 51 (20), 12531 (1995). Dokl. Akad. Nauk 360 (6), 753 (1998) [Phys. Dokl. 43, 7. N. A. Tyapunina, V. L. Krasnov, and E. P. Belozerova, 340 (1998)]. Fiz. Tverd. Tela (St. Petersburg) 41 (6), 1035 (1999) 21. V. A. Makara and N. N. Novikov, Fiz. Khim. Obrab. [Phys. Solid State 41, 942 (1999)]. Mater., No. 6, 137 (1973). 8. A. A. Urusovskaya, V. I. Al’shits, A. E. Smirnov, and 22. S. Fujita, K. Maeda, and S. Hyodo, Phys. Status Solidi A N. N. Bekkauer, Pis’ma Zh. Éksp. Teor. Fiz. 65 (6), 470 109 (2), 383 (1988). (1997) [JETP Lett. 65, 497 (1997)]. 23. A. G. Lipson, D. M. Sakov, V. I. Savenko, and E. I. Sau- 9. Yu. I. Golovin and R. B. Morgunov, Pis’ma Zh. Éksp. nin, Pis’ma Zh. Éksp. Teor. Fiz. 70 (2), 118 (1999) [JETP Teor. Fiz. 61 (7), 583 (1995) [JETP Lett. 61, 596 (1995)]. Lett. 70, 123 (1999)]. 24. V. A. Lomovskoœ, A. G. Lipson, N. Yu. Lomovskaya, and 10. M. I. Molotskiœ, Fiz. Tverd. Tela (Leningrad) 33 (10), I. A. Gagina, Vysokomol. Soedin., Ser. A 42 (6), 980 3112 (1991) [Sov. Phys. Solid State 33, 1760 (1991)]. (2000). 11. V. I. Al’shits and E. V. Darinskaya, Pis’ma Zh. Éksp. 25. V. L. Indenbom, V. V. Kirsanov, and A. N. Orlov, Vopr. Teor. Fiz. 70 (11), 749 (1999) [JETP Lett. 70, 761 At. Nauki Tekh., Ser. Fiz. Radiats. Povrezhdeniœ Radiats. (1999)]. Materialov. 10 (2), 3 (1982). 12. M. I. Molotskii and V. Fleurov, Phys. Rev. B 56 (18), 26. C. E. Foerster, C. M. Lepienski, F. C. Serbena, and 10809 (1997). F. C. Zawislak, Thin Solid Films 340 (2), 201 (1999). 13. Yu. I. Golovin and R. B. Morgunov, Materialovedenie, 27. V. M. Mikushkin and V. V. Shnitov, Fiz. Tverd. Tela Nos. 3Ð6, 2 (2000). (St. Petersburg) 39 (1), 187 (1997) [Phys. Solid State 39, 14. Yu. I. Golovin, R. B. Morgunov, A. A. Dmitrievskiœ, 164 (1997)]. et al., Pis’ma Zh. Éksp. Teor. Fiz. 68 (5), 400 (1998) 28. Yu. S. Gordeev, V. M. Mikushkin, and V. V. Shnitov, Fiz. [JETP Lett. 68, 426 (1998)]. Tverd. Tela (St. Petersburg) 42 (2), 371 (2000) [Phys. 15. Yu. I. Golovin, R. B. Morgunov, A. A. Dmitrievskiœ, and Solid State 42, 381 (2000)]. V. E. Ivanov, Zh. Éksp. Teor. Fiz. 117 (6), 1080 (2000) 29. Yu. A. Osip’yan, Yu. I. Golovin, D. V. Lopatin, et al., [JETP 90, 939 (2000)]. Pis’ma Zh. Éksp. Teor. Fiz. 69 (2), 110 (1999) [JETP 16. B. Brocklehurst, Nature 221, 921 (1969). Lett. 69, 123 (1999)]. 30. Yu. A. Osip’yan, Yu. I. Golovin, R. B. Morgunov, et al., 17. B. Ya. Zel’dovich, A. L. Buchachenko, and E. L. Fran- Fiz. Tverd. Tela (St. Petersburg) 43 (7), 1333 (2001) kevich, Usp. Fiz. Nauk 155 (1), 3 (1988) [Sov. Phys. [Phys. Solid State 43, 1389 (2001)]. Usp. 31, 385 (1988)]. 31. V. F. Kozlov, Handbook on Radiation Safety (Énergoat- 18. K. M. Salikhov, Yu. N. Molin, R. Z. Sagdeev, and omizdat, Moscow, 1991). A. L. Buchachenko, Spin Polarization and Magnetic 32. Yu. I. Golovin, V. I. Ivolgin, V. V. Korenkov, and Effects in Radical Reactions (Nauka, Novosibirsk, 1978; B. Ya. Farber, Fiz. Tverd. Tela (St. Petersburg) 43 (11), Elsevier, Amsterdam, 1984). 2021 (2001) [Phys. Solid State 43, 2105 (2001)]. 19. V. I. Al’shits, E. V. Darinskaya, and O. L. Kazakova, Pis’ma Zh. Éksp. Teor. Fiz. 62 (4), 352 (1995) [JETP Lett. 62, 375 (1995)]. Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

SEMICONDUCTORS AND DIELECTRICS

Interference Effect in Combined Elastic and Inelastic Scattering of an Energetic Electron Reflected from a Disordered Medium with Generation of a Surface Plasmon B. N. Libenson St. Petersburg Scientists’ Union, St. Petersburg, 199034 Russia e-mail: [email protected] Received January 15, 2002; in ﬁnal form, February 26, 2002

Abstract—Quantum interference in combined elastic and inelastic scattering of an energetic electron with excitation of a surface plasmon leads to a change in the shape of the corresponding peak in the electron-energy- ω loss spectrum. The plasmon generation is suppressed near the frequency p/2 . The suppression increases with increasing surface-plasmon wave length, because the interference of the energetic-electron scattering processes differing in the sequential order of elastic and inelastic scattering becomes progressively more destructive. The decrease in the height of the surface-plasmon peak in the electron-energy-loss spectrum leads to a nondissipative broadening in this peak. Quantum interference also causes a speciﬁc feature to occur in the azimuth- angle dependence of the spectral intensity as the electron energy loss increases in the immediate vicinity of the surface-plasmon peak. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ence length, v is the velocity of an energetic electron, and ω is the frequency corresponding to the electron As shown in [1Ð4], the features of quantum trans- energy loss. In such a medium, elastic large-angle scat- port of energetic electrons in the inelastic-scattering tering of electrons with a high energy E (of several hun- channel differ significantly from the features of conven- dreds of electronvolts) occurs in a few near-surface tional weak localization in the elastic scattering of ener- atomic layers [5]. It is well known that bulk plasmons, getic electrons. The main coherent effect which arises which are oscillations of a longitudinal electric field, in the inelastic scattering of an electron in the bulk of a can be generated only when electrons move in the disordered medium is associated with interference in medium. Surface plasmons are not quanta of a longitu- the process of single inelastic scattering combined with dinal electric field and can be excited by electrons mov- single elastic large-angle scattering. A distinctive fea- ing both in the medium and vacuum. ture of this coherent effect is that the cross section of inelastic electron scattering is anisotropic in this case; Therefore, the coherent effect is predominantly a in particular, the cross section decreases as the scatter- result of the interference of two processes. In one of ing angle χ approaches π. This decrease in the inelastic them, an energetic electron first experiences a single electron scattering cross section occurs over a χ-value elastic noncoherent large-angle scattering from the cen- range much wider than the angular width of the back- ters located near the surface of the strongly absorbing ward peak in the case of conventional weak localization disordered medium and then passes from the medium in the elastic-scattering channel. As shown in [3], the to vacuum and excites a plasmon. In the other process, angular range in which the feature of the inelastic cross the sequential order of the electron scattering and plas- section mentioned above arises can be determined in a mon excitation is reversed. The quantum interference qualitative way if all acts of electron scattering are of these processes affects the shape of the surface-plas- assumed to occur in the bulk of the disordered medium mon peak in the characteristic energy loss spectrum: far from its boundary. this peak becomes lower in the vicinity of the energy បω /2 , because the long-wavelength part of the sur- In this paper, we study the influence of the surface p of a disordered medium on the interference correction face-plasmon generation spectrum is suppressed. to the inelastic scattering cross section in the case In an infinite medium, the interference effect in where the inelastic electron scattering is accompanied question leads to a dependence of the inelastic scatter- by a single excitation of a surface plasmon at the ing cross section on the scattering angle χ. For a finite boundary between the disordered medium and vacuum. medium, this dependence on χ transforms into a depen- α α We restrict our consideration to the case where the dence on the angle of incidence i and the exit angle f φ absorption of the electron wave field by the disordered of electrons, as well as on the azimuth angle p between medium is so strong that l Ӷ v/ω, where l is the coher- the projections of the velocities v and v' of the incident

22 LIBENSON and escaping electrons, respectively, onto the plane sur- Using the Fermi potential, the inelastic part of the face of the medium. The dependence of the inelastic electron scattering cross section can be simply calcu- α α scattering cross section on the angles i and f is basi- lated. cally not associated with the interference in question, | |2 | |2 φ For the Fermi potential, we have ubs(ka) = ubs = whereas the dependence on the azimuth angle p is due const (independent of k ) and, hence, to the quantum interference in the process of electron a transport alone. Therefore, the azimuthal dependence 〈〉U ()r U* ()r' = nu 2δ()rrÐ ' . of the electron-energy-loss spectrum can be taken as bs bs rd bs evidence of the occurrence of the coherent effect. The differential cross section of the elastic electron scattering by the potential Ubs(r) is written as 2. INCOHERENT ELASTIC SCATTERING dσ S σ OF AN ENERGETIC ELECTRON ------el- = ------bs . (3) dΩ 4πσ []()α Ð1 ()α Ð1 IN A DISORDERED MEDIUM t cos i + cos f We will describe disordered elastic-scattering cen- Here, S is the surface area of the medium, ters located in the bulk of the medium with the expres- 2 2 sion σ m ubs bs = ------4 ; N πប U ()r = ∑ V ()rrÐ bs bs j 4π j = 1 σ = ------Im f ()θ = 0 is the total cross section of ener- (1) t k N p 1 getic-electron scattering, with f (θ = 0) being the for- = ------dk u ()k ∑ exp[]ik ()rrÐ , ()π 3∫ a bs a a j ward scattering amplitude. 2 j = 1 where U (r) is the three-dimensional potential respon- bs 3. SCATTERING CROSS SECTION sible for elastic large-angle (backward) scattering of an FOR A STRONGLY ABSORBING MEDIUM energetic electron, N is the number of randomly distributed elastic-scattering centers, and ubs(ka) is the elec- The wave function of an electron interacting with a tron scattering amplitude for such a center. Upon aver- semi-infinite medium with randomly distributed elas- aging over the randomly distributed scatterers, the tic-scattering centers and exciting a surface plasmon at product U (r)(U* r') takes the form the boundary between the medium and vacuum can be bs bs found using the conventional perturbation methods. n 2 The coherent wave field and the Green’s function of the 〈〉U ()r U* ()r' = ------dk u ()k bs bs rd 3∫ a bs a energetic electron are calculated in the same way as in ()2π (2) [7]. The differential scattering cross section is deter- × exp[]ik ()rrÐ ' , mined by the second-order correction to the wave func- a tion. We have where n is the concentration of the elastic-scattering ∞ σ 2 centers. Elastic large-angle scattering of an energetic dσ d el e 1 Ð1 ------= ------dqIm------electron is characterized by a length (kp) , which is Ω ω Ω 2 ∫ Ξ (), ω d d d π ប s q (4) much smaller than atomic-scale lengths. Nevertheless, 0 the elastic large-angle scattering potential for electrons × Fq(),,v α ,α ,φ ,κ + κ ,E . can be replaced by the Fermi potential. Of course, the i f p i f q anisotropy of elastic scattering is neglected in this Here, q and បω are the wave vector and the energy, approximation. However, in the case of a strongly respectively, transferred by the electron to an excited Ξ ω absorbing medium, an electron can undergo only a sin- surface plasmon. The condition s(q, ) = 0 determines gle elastic large-angle scattering and our simplification the pole corresponding to an excited surface plasmon. α α φ κ κ is justified. Within this approximation, as shown in [1, The localization function F(q, v, i, f , p, i + f , Eq) 2, 6], the square of the elastic-scattering amplitude is given by a cumbersome expression and is presented modulus of an electron depends, to first order in in the Appendix. The velocity v of the energetic elec- បω/E Ӷ 1, only on the total momentum transfer and the tron in the initial state has components vz and vρ; the dependence on the momentum transfer in inelastic scat- components of the velocity v' of the electron in the final tering can be ignored. Therefore, the cross section of v ' v the entire process contains the elastic cross section as a state are fz and vρ ( fz > 0). The parameters involved factor; the inelastic cross section we are interested in is in the localization function are defined as the ratio of the total cross section to the elastic cross ប2q2 section and does not depend on the structure of the lat- E = ------, ter cross section. q 2m

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 INTERFERENCE EFFECT IN COMBINED ELASTIC AND INELASTIC SCATTERING 23

κ 2 σ κ 2 σ getic electron, the boundary of the disordered medium, i ==Im kz +,ikn t f Im k fz +.ikn t and the excited surface plasmon. Each of these two vac- φ uum components of the Green’s function corresponds The dependence of the quantity F on the angle p ' to the amplitude of one of the two possible realizations between vρ and vρ is determined only by the quantum of the combined elastic and inelastic scattering. The interference. zeros of the real parts of the denominators in the Equation (4) is valid for an arbitrary absorbing Green’s functions correspond to the conservation of the medium. The inelastic-scattering intensity (which also normal component of the wave vector in the processes characterizes the surface-plasmon generation) is with different sequential order of elastic and inelastic defined as scattering, while the imaginary parts of the denomina- σ Ω ω tors determine the spatial-decay frequencies of the d /d d electric field of a plasmon along the direction of motion S1inel = ------σ Ω-. (5) d el/d of the electron. Equation (6) describes the quantum interference of processes differing in the sequential For a medium strongly absorbing an electron wave order of two spatially separated acts of elastic and field, Eq. (5) is significantly simplified: inelastic electron scattering. The main consequence of e2 dq 1 this interference is suppression of surface excitations ()φ , ω 1 S1inel p = π------ប∫------ImΞ------(), ω with very long wavelengths. q s q Indeed, the square of the modulus of the sum in 1 Eq. (6) tends to zero as q 0. For this reason, the × ------(6) Ð1 dependence of the localization function on the wave ω ++qvρ ប E Ð iqv q z vector q 1 2 2π ------+ Ð1 . 1 Ðω Ð qvρ' + ប E Ð iqv Fq()= d φ------q fz ∫ ω បÐ1 ++qvρ Eq Ð iqv z In the limit of a strongly absorbing medium, 0 (7) 1 2 q Ӷ κ + κ , + ------i f ω បÐ1 Ð Ð qvρ' + Eq Ð iqv fz ω Ӷ 2()κ + κ v , ω Ӷ 2()κ + κ v , i f z i f fz has a sharp maximum, whose width and height depend α α φ the terms retained in Eq. (4) are represented in Eq. (6) on the energy E and the angles i, f , and p. An ana- as the square of the modulus of the sum of the Green’s lytical expression for the localization function F(q) can functions describing the system consisting of the ener- be found in the form

2π 1 2π 1 Fq()= ------Im------+ ------Im------v 2 v 2 q z ()ω បÐ1 ()2 q fz ()ω បÐ1 ()2 + Eq Ð iqv z Ð qv ρ Ð Eq Ð iqv fz Ð qv ρ'  1 Ð4πRe------[]()ω បÐ1 ()ω បÐ1 2 []× 2 (8) vρ + Eq Ð iqv z Ð vρ' Ð Eq Ð iqv fz Ð qvρ vρ'

2 Ð1 Ð1 2 Ð1 Ð1 v ρ()ω + ប E Ð iqv Ð vρ ⋅ vρ' ()ω + ប E Ð iqv v ρ' ()ω Ð ប E Ð iqv Ð vρ ⋅ vρ' ()ω Ð ប E Ð iqv  × ------q z q z + ------q z q fz- . ()ω បÐ1 2 ()2 ()ω បÐ1 2 ()2  + Eq Ð iqv z Ð qv ρ Ð Eq Ð iqv fz Ð qv ρ'

α α α Ӷ បω Equation (8) contains terms that are different in much simpler in the case of i = f = and Eq , nature. The terms with the sign Im are not associated which is of considerable practical importance. with interference; these terms describe the probabilities of two possible processes that differ in the sequential 1 As shown in [1, 6], the suppression of bulk plasmons with very order of elastic scattering of the energetic electron and long wavelengths caused by destructive interference does not lead of the generation of a surface plasmon. The term con- to a decrease in the intensity of the peak in the vicinity of the plasma frequency in the electron-energy-loss spectrum, because taining the sign Re describes the interference of these in an inﬁnite medium (in contrast to vacuum), the imaginary parts two processes. A direct analysis of this term involves of the denominators in the Green’s functions have constant com- intricate calculations, because the expression is fairly ponents (independent of the wave vector q) that are proportional to the imaginary part of the forward scattering amplitude of an cumbersome. However, such an analysis becomes energetic electron.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 24 LIBENSON

Let us introduce the notation z = qv/ω. It can be 4. THE DEPENDENCE OF THE ANGULAR shown that the interference term in Eq. (8) changes its STRUCTURE OF THE INTERFERENCE EFFECT sign when ON THE ELECTRON ENERGY LOSS បω 2[]2 ()α 2 () α 2 () φ It was shown in Section 3 that the interference being z cos + sin cos p/2 constructive or destructive depends on the surface-plas- gz(), α Ð zcos()α (9) mon wavelength. In order to include the spatial disper- +2zcos()α ------Ð 1 = 0. 1 + zcos()α gz(), α sion described by the surface-plasmon excitation function, we should investigate the reflection of the intrinsic Here, electrons of the medium from the boundary. We will use the results obtained in [8], where the bulk dielectric Im[] 1 Ð izcos()α 2 Ð z2 sin2 ()α properties of a medium were described within a hydro- gz(), α = ------. dynamic model. The properties of the surface of a Re[] 1 Ð izcos()α 2 Ð z2 sin2 ()α medium will be described in terms of phenomenological parameters R and P, which are the fractions of α φ Equation (9) has a root zc( , p) whose position intrinsic electrons experiencing elastic and specular determines the character of the interference. In the reflections, respectively, from the boundary of the α φ ° ω wave-vector range qv < zc( , p = 0 ) , the interfer- medium. We will restrict our consideration to the case φ ence term in Eq. (8) is negative for any value of p. where the medium strongly absorbs an electron wave Therefore, in this range, the interference of the two field and, therefore, R Ӷ 1 and P Ӷ 1. Ξ ω scattering processes of interest is destructive. In con- In this case, the expansion of the function s(q, ) α φ ° ω ω trast, when qv > zc( , p = 180 ) , the interference of in powers of the small parameter qvF/ has the form these processes is constructive for any value of φ . In p v the wave-vector range Ξ (), ω εω() βq F s q = 1 + Ð i ------ω - ω ω ----z ()αφ, = 0° <

yz()= Ð2zcos()α 2 (11) ()1 Ð R ()14++R 2R + ------+ … . ()α , α ()2 2 ()α ()2 , α 2 2 × zcos Ð gz()+ 1+ z cos 1 + g ()z ()1 + R ------()α (), α 1 + zcos gz β ε ω Here, vF is the Fermi velocity, = 4/9, and ( ) = 1 Ð and a minimum if ω2 ω ω ν ν p / ( Ð i ), with being the collision frequency of () ()α yz = Ð2zcos (12) intrinsic electrons of the medium. Substituting Eq. (13) into Eq. (6), we can find the zcos()α Ð1gz(), α Ð ()+ z2 cos2 ()α ()1 + g2()z, α φ ω × ------. function S1inel( p, ). In the figures presented below, 1 + zcos()α gz(), α this function is designated as S1. The figures also show φ ° ° the quantity S2, which is the contribution to the surface- The extrema at p = 0 and 180 , on the contrary, will plasmon generation intensity from the sum of the be minima in the former case and maxima in the latter. squared moduli of the amplitudes of two scattering pro- A detailed investigation of the dynamics of the inter- cesses differing in the sequential order of collisions of fering processes can be performed only on the basis of an energetic electron. This quantity has nothing to do φ Eq. (6), which involves not only the localization func- with the interference and is independent of the angle p. α φ φ tion F(q, , p) considered above but also the surface- The angular dependence of S1( p) presented in Ξ ω Ð1 φ plasmon excitation function Im[ s(q, )] . Fig. 1 corresponds to the localization function F( p,

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 INTERFERENCE EFFECT IN COMBINED ELASTIC AND INELASTIC SCATTERING 25 qv/ω ≈ 1). Using Eq. (10), we ﬁnd the values of the roots determining the q-value ranges of the constructive S2 α ° φ ° 372 and destructive interference: zc ( = 70 , p = 0 ) = 0.89 α ° φ ° and zc ( = 70 , p = 180 ) = 1.34. According to Eq. (13), the intensity of generation of surface plas- 2

ω ω ω ≈ S mons with = 0.707 p is maximum for qv/ 0.1 and 1, 276 plasmons are generated predominantly over the range S ω q Ӷ ----z ()αφ, = 0° . 1 v c p S 180 The suppression of generation of such long-wavelength 0 72 144 180 φ plasmons is so strong that the dominant contribution to p, deg the quantity S1 comes from plasmons with qv/ω ≈ 1. The intensity S2, which does not contain interference Fig. 1. Dependence of the intensity of electron energy loss បω បω φ terms, is nearly twice as large as the intensity S1 for any equal to = 0.707 p on the azimuth angle p for E = φ 1000 eV, E = 11.6 eV, α = α = 70°, and R = P = 0. value of the angle p. F i f φ Figure 2 shows the S1( p) and S2 curves for the case ω ω of = 0.74 p. It is seen that these curves intersect. The φ 1 range of azimuth angles where S1( p) > S2 corresponds S to the constructive interference of the scattering ampli- 202 φ ° tudes. This range includes the angle p = 0 and makes up the greater part of the entire azimuth-angle range.

Weak destructive interference of the amplitudes takes 2 φ ° S place in the vicinity of the angle p = 180 , where 1, 186 S2 φ S S1( p) < S2. The intensity of generation of plasmons ω ω ω ≈ with = 0.74 p is maximum at qv/ 1.2. The angu- φ lar dependence of S1( p) is close to that of the function 170 φ , qv ≈ 0 72 144 180 Fp ------1.2 . φ ω p, deg

A similar azimuth-angle dependence of the localization Fig. 2. Same as in Fig. 1 but for បω = 0.74បω . function was also found in [1] for the interference cor- p rection to the electron scattering cross section involving the generation of bulk plasmons. φ Figure 3 presents the S1( p) and S2 angular depen- ω ω 72.8 S1 dences for = 0.79 p. It is seen that the interference correction is positive over the entire azimuth-angle range, which is due to the fact that the dominant contri- 2 bution to the integral with respect to q in Eq. (6) comes S from plasmons with qv > ωz (α, φ = 180°). The inten- 1, 66.4 c p S ω ω sity of generation of plasmons with = 0.79 p is max- ω ≈ φ imum for qv/ 2.5. The angular dependence of S1( p) φ ° S2 exhibits a maximum at p = 110 and is similar to that φ ω of the localization function F( p, qv/ = 1.95), for 60.0 which the maximum is also located at 110°, as follows 0 72 144 180 φ from Eq. (10). The localization function F(qv/ω = 2, χ) p, deg corresponding to a new type of weak localization in the bulk-plasmon generation channel [1] also exhibits a បω បω maximum at the scattering angle χ ≈ 110°. Fig. 3. Same as in Fig. 1 but for = 0.79 p. φ A comparison of Figs. 1Ð3 shows that the S1( p) angular dependence undergoes a signiﬁcant change as dependence of the interference correction can be used the electron energy loss increases within the range of to detect the coherent effect in the electron-energy-loss បω 10% near p/2 . This speciﬁc feature of the angular spectrum associated with surface-plasmon excitation.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 26 LIBENSON

S1 S1 S2 S2 400 400

300 300 2 S 1, S 200 S1, S2 200

100 100

0 0 0.5 0.7 0.9 0.5 0.7 0.9 ω ω

Fig. 4. Effect of the interference in combined elastic and inelastic scattering on the intensity of the electron-energy- α ° φ ° α ° φ ° loss spectral peak for = 70 and p = 0 . Fig. 5. Same as in Fig. 4 but for = 70 and p = 180 .

5. INTERFERENTIAL BROADENING The characteristic energy loss spectra presented in OF THE ELECTRON ENERGY LOSS SPECTRUM Figs. 4 and 5 are calculated for a polycrystalline aluminum target. The realistic peak width is obtained without The spectral distribution of the inelastic-scattering involving an additional nondispersive mechanism of intensity without regard for the interference correction plasmon decay. The collision frequency of intrinsic can be found from Eq. (6) to be electrons of the disordered medium is assumed to be ν Ӷ ω vF p/v. e2 dq 1 S2 ()ω = ------Im------inel πប∫ q Ξ ()q, ω s 6. CONCLUSIONS  1 The following practical importance can be inferred. × ------(14) (1) The quantum interference in the process of com- []ω បÐ1 2 ()2 ++qvρ Eq + qv z bined elastic and inelastic scattering of an energetic electron causes significant additional broadening of the surface-plasmon peak in the energy loss spectrum. 1  + ------. Therefore, there is no need to search for an additional []ω បÐ1 2 ()2  + qvρ' Ð Eq + qv fz plasmon decay mechanism to explain the experimental values of the spectral-peak width. The spectra described by Eqs. (6) and (14) are com- (2) The specific features of the azimuthal depen- pared in Figs. 4 and 5. It is seen that the quantum inter- dence of the electron-energy-loss spectrum predicted in ference in combined elastic and inelastic scattering this paper can be presently observed experimentally. leads to suppression of the excitation of a plasmon with For this purpose, one needs to measure the ratio of the បω energetic-electron flux reflected from a disordered energy close to p/2 . The effect is equivalent to medium with a fixed deviation of the energy loss from broadening of the surface-plasmon peak; however, this that corresponding to the central peak to the elastically broadening is not associated with decay of the excita- reflected electron flux for different values of the azi- tion and, hence, is nondissipative. The effective broad- muth angle. ening of the surface-plasmon peak in the energy-loss spectrum can be as large as 100% of the width of the (3) The main features of the interference corrections “initial” peak described by Eq. (14). The destructive to the bulk- and surface-plasmon peaks in the energy loss spectrum of energetic electrons are identical, in បω interference in the vicinity of the energy loss p/ 2 spite of the fact that the theoretical description of this φ is dependent on the azimuth angle p. The suppression phenomenon occurring in the bulk of a disordered φ ° of excitation of a surface plasmon at p = 180 is some- medium and near its surface is different. φ ° what stronger than that at p = 0 (as can be seen from The results obtained in this paper can be used to the spectra presented in Figs. 4, 5), and the same is true experimentally detect the azimuthal dependence inher- for the broadening of the corresponding peak in the ent in the interference in combined elastic and inelastic energy-loss spectrum. scattering of energetic electrons reflected from a disor-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 INTERFERENCE EFFECT IN COMBINED ELASTIC AND INELASTIC SCATTERING 27 dered medium. The angular and energy resolutions APPENDIX attainable in the currently available characteristic- energy-loss spectroscopic methods allow one to per- The localization function for an arbitrary medium form such experiments. absorbing an electron wave ﬁeld has the form

2π e2 q + 2()κ + κ 1 Fq(),,v α ,α ,φ ,κ + κ ,E = ------dφ ------i f ------i f p i f q 2 ∫ q ()κ κ Ð1 2 2 π ប  2 i + f ()ω ប ()v 0 ++qvρ Eq + q z 1 []κq + 2()κ + κ ()+ κ Ð q + ------+ ------i f i f Ð1 2 2 ()κ κ ()κ κ ()ω ប () 2 i + f q ++i f + qvρ' Ð Eq + qv fz 1 1 × ------+ ------()ω បÐ1 2 []()κ κ 2 2 ()ω បÐ1 2 []()κ κ 2 2 ++qvρ Eq + q + 2 i + f v z + qvρ' Ð Eq + q + 2 i + f v fz v v +2()κ + κ Re ------z fz ------i f ()κ κ []()ω បÐ1 ()ω បÐ1 2 i + f v zv fz + i v fz ++qvρ Eq + v z + qvρ' Ð Eq

 1 1 1 × ------+ ------()q ++κ κ ()[]κ κ []ω បÐ1 ()[]κ κ []ω បÐ1  i f v z v fz q + 2()i + f + i + qvρ' Ð Eq v fz v z q + 2()i + f + i ++qvρ Eq 2 + ------[]()ω បÐ1 ()[]()κ κ ()ω បÐ1 qv z + i ++qvρ Eq q + 2 i + f v z + i ++qvρ Eq 2 + ------[]()ω បÐ1 ()[]()κ κ ()ω បÐ1 qv fz + i + qvρ' Ð Eq q + 2 i + f v fz + i + qvρ' Ð Eq

2   + ------ . []()ω បÐ1 []()ω បÐ1 qv z + i ++qvρ Eq qv fz + i + qvρ' Ð Eq  

REFERENCES (1975)]; Fiz. Tverd. Tela (Leningrad) 17, 2502 (1975) [Sov. Phys. Solid State 17, 1672 (1975)]. 1. B. N. Libenson, K. Yu. Platonov, and V. V. Rumyantsev, Zh. Éksp. Teor. Fiz. 101 (2), 614 (1992) [Sov. Phys. 6. V. V. Rumyantsev and B. N. Libenson, Ann. Phys. 111, JETP 74, 326 (1992)]. 152 (1978). 2. V. V. Rumyantsev and V. V. Doubov, Phys. Rev. B 49 (13), 8643 (1994). 7. E. E. Gorodnichev, S. L. Dudarev, D. B. Rogozkin, and M. I. Ryazanov, Preprint No. 025-87, MIFI (Moscow 3. V. V. Rumyantsev, E. V. Orlenko, and B. N. Libenson, Z. Engineering Physics Institute, Moscow, 1987). Phys. B 103, 53 (1997). 4. B. N. Libenson and V. V. Rumyantsev, Fiz. Tverd. Tela 8. B. N. Libenson, Fiz. Tverd. Tela (St. Petersburg) 36 (8), (St. Petersburg) 40 (8), 1413 (1998) [Phys. Solid State 2283 (1994) [Phys. Solid State 36, 1243 (1994)]. 40, 1283 (1998)]. 5. I. M. Bronshteœn and I. P. Pronin, Fiz. Tverd. Tela (Len- ingrad) 17, 2431 (1975) [Sov. Phys. Solid State 17, 1610 Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 28Ð31. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 29Ð32. Original Russian Text Copyright © 2003 by Krivolapchuk, Mezdrogina, Poletae.

SEMICONDUCTORS AND DIELECTRICS

Effect of Correlation between the Shallow and Deep Metastable Level Subsystems on the Excitonic Photoluminescence Spectra in n-GaAs V. V. Krivolapchuk, M. M. Mezdrogina, and N. K. Poletaev Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received February 28, 2002; in ﬁnal form, April 19, 2002

Abstract—The excitonic photoluminescence spectra of GaAs epitaxial layers are studied. Changes in the relative arrangement of shallow and deep centers in the tetrahedral lattice are shown to bring about changes in the decay kinetics and the shape of the (D0, x) emission line (corresponding to an exciton bound to a shallow neutral donor). This change in the excitonic photoluminescence spectra is caused by dispersion in the exciton binding energy of shallow donors ED, the dispersion being a result of the inﬂuence of the subsystem of deep metastable defects in n-GaAs crystals. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tra (studied in the reflection geometry, with the epitax- The ordering and correlation between shallow and ial layer plane normal to the spectrometer optical axis) deep defect subsystems has recently become a subject were obtained on samples immersed in liquid helium in of considerable interest. The ordering processes mani- vacuum (T = 1.8 K). The photoluminescence (PL) was excited by the radiation of a cw HeÐNe laser (λ = fest themselves both in a variation of the concentration µ of the impurity doped into a semiconductor and in fluc- 6328 Å) focused to a spot about 150 m in diameter. tuations in the composition [1, 2]. The PL decay kinetics was studied under pumping with a pulsed semiconductor laser (λ = 7980 Å). The PL It is known that at low temperatures, exciton states spectra and kinetics were investigated in the lock-in are the final stage in the electronic excitation relaxation in energy, which is affected in a large measure by exci- photon-counting mode using a DFS-52 double-grating ton interaction with defects and impurities. The charac- diffraction spectrometer. To adequately compare the teristics of exciton luminescence (the presence of spe- emission spectra of various samples, the controllable cific lines, their shape, intensity, halfwidth, decay time) experimental conditions (temperature, excitation den- provide a sufficiently correct idea of the dynamics of sity, spectral resolution, laser beam incidence angle) nonequilibrium carriers in GaAs crystals and, there- were maintained unchanged. fore, permit one to gain an understanding of the extent to which they are affected by processes involving various defects in the material. 3. RESULTS OF THE MEASUREMENTS The objective of the present work is to investigate As studied at pumping levels providing the created the connection between shallow and deep impurity sub- electronÐhole pairs in numbers comparable to N , the systems and the influence of this mutually related eff defect configuration on the specific features of the exci- emission spectra of n-GaAs epitaxial layers contain the tonic spectra in gallium arsenide epitaxial layers. following spectral components: a polariton emission of free excitons, a line due to excitons bound to shallow neutral donors (D0, x), a line originating from the 2. SAMPLES AND EXPERIMENTAL recombination of a neutral-donor electron with a ARRANGEMENT valence-band hole (D0, h), and, if the compensation by We studied epitaxial layers of n-GaAs grown by shallow acceptors is high enough, a line of excitons gas-phase epitaxy in a chloride system on gallium ars- bound to neutral acceptors (A0, x) (Fig. 1). The (D0, x) enide substrates. The shallow impurity concentration line is the strongest in all samples. The spectra of the Neff (ND Ð NA) in the samples measured using the CÐV samples investigated differ noticeably from one another technique was 1013Ð1014 cmÐ3. The concentration of in intensity (by an order of magnitude) and in the half- deep levels, as determined using DLTS, did not exceed width (FWHM = 0.15Ð0.30 meV) and position of the that of the shallow levels. The photoluminescence spec- (D0, x) line (Fig. 2).

EFFECT OF CORRELATION 29

4. DISCUSSION D0, x To understand the relation between the shallow- and deep-level subsystems, let us consider how the emission spectrum of bound and free excitons (polaritons) 30000 in a crystal depends on the presence of defects. X D0, h In general, the radiation intensity Ir at a pumping 25000 level g = f(Iex) depends on the total lifetime of non- τ equilibrium carriers (excitons) , which, in turn, is 0, τ τ 20000 A x determined by the radiative ( r) and nonradiative ( nr) τ τ τ lifetimes of excitons (carriers) as 1/ = 1/ r + 1/ nr . Therefore, the radiation intensity can be written as Ir = 15000 τ τ g/(1 + r / nr).

Because for the given material (GaAs) and the Intensity, arb. units 10000 τ experimental conditions chosen we have r = const, the τ intensity depends on nr alone. The nonradiative life- τ 5000 time nr is determined by exciton trapping (the trapping σ cross section i) into deep states (with concentration Ni) τ v σ NR 0 associated with various defects: 1/ nr = ∑i i Ni (here, v is the thermal velocity and i labels the defect 1511 1512 1513 1514 1515 1516 1517 type). It thus follows that samples with different PL E, meV intensities measured under the same experimental conditions differ primarily in the concentration of nonradi- Fig. 1. Typical n-GaAs photoluminescence spectrum in the ative recombination centers, which are associated with exciton region measured at T = 1.8 K. deep levels. The lower the concentration of the centers (deep levels) responsible for nonradiative exciton annihilation, the higher the integrated PL intensity, and vice 3500 versa.

In what follows, we focus our attention primarily on 3000 0 the emission spectrum of excitons bound to shallow D , x donors (D0, x). The above reasoning suggests that the intensity of the radiation involving shallow centers 2500 should depend on the concentrations of shallow and 0 deep centers, their mutual arrangement in the lattice, 2000 D , h and on free exciton transport to the trapping centers (both shallow and deep). One should, therefore, consider the factors governing the relative contribution of 1500 free and bound excitons to the observed emission spec- X trum. Intensity, arb. units 1000 The radiation intensity of free excitons (polaritons) can be represented in a simplified way as [3] 1 500 ()≈ ()(), IE ∑T r E FxEx = 0, 2 r 0 where T (E) is the coefficient of polariton conversion to r 1512 1513 1514 1515 1516 external radiation at the crystal boundary (transmission , meV coefficient) and F(x, E) is the polariton distribution E function in space and energy in the crystal. It thus follows that the polariton luminescence in a given material Fig. 2. Photoluminescence spectra (translated as a whole) of is determined primarily by the distribution function in two different n-GaAs samples measured under identical space and energy F(x, E). In a first approximation, this pumping and measurement conditions. function can be written in a factorized form, F(x, E) = f(E)*C(x). To be able to describe the PL spectrum, one has to determine F(x, E) in an explicit form in each spe- determined by the total exciton lifetime τ. As the life- cific case, which is a very difficult problem. For our fur- time τ increases, both the energy and the spatial distri- ther analysis of the GaAs emission spectra, however, it bution functions change. The maximum of the polari- may be sufficient to note that both f(E) and C(x) are ton energy distribution function f(E) shifts into the res-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 30 KRIVOLAPCHUK et al.

〈 〉2τ τ 1/3 lnI of free excitons, L = (1/3 v k ) , and of carriers [4]. As a result, with increasing τ (and, hence, L), the sam- 12 ple volume occupied by free excitons and the number of neutral donors D0 in this volume increase. Trapping 0 10 of free excitons by shallow neutral donors D produces an emission line (D0, x) whose intensity is determined by the neutral-donor concentration. The relative inten- 8 sity of the spectral lines produced by free and bound excitons is determined by the radiation imprisonment effect, which plays a major role in samples with long 6 lifetimes [5]. Whence it follows that samples with different total lifetimes τ differ substantially in PL inten- 4 sity from one another; therefore, the intensity of the 1 (D0, x) line (for samples with shallow donor concentrations 1013Ð1014 cmÐ3) can be considered a parameter 2 that qualitatively characterizes the concentration of 2 NR nonradiative exciton annihilation centers Ni in a sam- 0 ple. 0 5 10 15 20 0 t, µs The energy position and FWHM of the (D , x) emission line are important spectral characteristics. In the ti tf samples studied, the halfwidth of the (D0, x) line varied from 0.12 to 0.31 meV. 0 Fig. 3. Decay kinetics of the (D , x) emission line in two The (D0, x) emission line is known to be inhomoge- GaAs samples with FWHM equal to (1) 0.31 and (2) neously broadened. Its inhomogeneous broadening is 0.14 meV. The integration region bounded by ti and tf is hatched. The ﬁgures label the spectra displayed in Fig. 2. due to the fact that the excitons bound to different donors D0 emit radiation of slightly different wavelengths. This difference originates from the dispersion of thermal activation energy of shallow donors EDT 30 (and, hence, of the excitons bound to them). The reason for the EDT dispersion, in turn, lies in the different values of the local potential Vloc at donor impurity sites. 25 This means that the various defects located close to shallow donors D0 change the crystal ﬁeld. The 20 assumption that both deep nonradiative annihilation

3 centers and shallow acceptors can act as such defects

10 was not conﬁrmed. Indeed, an analysis of the spectra of

× 0 15 samples differing in the (D , x) emission line intensity

MS by more than an order of magnitude and, hence, in the B 10 concentration of deep nonradiative annihilation centers NR 0 Ni showed the (D , x) halfwidth to be independent of NR 5 the concentration Ni . The halfwidth also does not depend on the shallow acceptor concentration. [We note that the high concentration of shallow acceptors is 0 indicated by the presence of a strong (A0, x) line.] It thus 0.15 0.20 0.25 0.30 follows that the EDT dispersion is dominated not by the (D0, x) FWHM, meV nonradiative annihilation centers and shallow acceptors but rather by a subsystem of defects of another nature 0 that are responsible for the deep levels. Fig. 4. Area BMS bounded by the (D , x) decay curve as a function of the halfwidth (FWHM) of this line in different The key to understanding the origin of the EDT dis- n-GaAs samples. persion [i.e., of the (D0, x) line broadening] lies in the difference in the spectral position of this line and in the decay kinetics of the (D0, x) emission line between dif- onance region ELT and the maximum of the spatial ferent samples. It was shown in [6] that the slow decay distribution function C(x) shifts into the bulk of the (τ ≈ 10Ð6 s) of the (D0, x) line is due to the existence of crystal because of the increasing diffusion length both local metastable centers donating holes to the valence

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EFFECT OF CORRELATION 31

rahedral interstitial sites [11Ð14]. Such a reconstruction 0.30 is possible in samples with strains manifested as a spread in the (D0, x) line position. 0.25 Thus, the analysis of the variation of the exciton photoluminescence spectra combined with the investigation of their decay kinetics permits us to suggest a 0.20 consistent interpretation of coupling between the subsystems of deep and shallow defects in n-type gallium arsenide. 0.15 ) shift, meV x , 0 ACKNOWLEDGMENTS D 0.10 ( The authors are indebted to V.M. Botnaryuk and L.M. Fedorov for providing the samples. 0.05 REFERENCES 0 1. M. M. Mezdrogina, T. I. Mosina, E. I. Terukov, and I. N. Trapeznikova, Fiz. Tekh. Poluprovodn. (St. Peters- 0.120 0.160 0.200 0.240 0.280 0.320 burg) 35, 714 (2001) [Semiconductors 35, 684 (2001)]. 0 (D , x) FWHM, meV 2. V. A. Shchukin and A. N. Starodubtsev, in Proceedings of 8th International Symposium “Nanostructures: Phys- Fig. 5. Spectral position of the (D0, x) emission line plotted ics and Technology,” Ioffe Institute, St. Petersburg, as a function of its FWHM. 2000, p. 137. 3. Y. Toyozava, Prog. Theor. Phys. 20, 53 (1958). band. The number of these centers is related to the area 4. V. V. Krivolapchuk, S. A. Permogorov, and V. V. Travni- 0 kov, Fiz. Tverd. Tela (Leningrad) 23, 606 (1981) [Sov. BMS bounded by the PL decay curve of the (D , x) line Phys. Solid State 23, 343 (1981)]; V. V. Travnikov and [7, 8] (Fig. 3). It was found that there is no slow decay V. V. Krivolapchuk, Fiz. Tverd. Tela (Leningrad) 24, 961 in uncompensated samples exhibiting a (D0, x) line (1982) [Sov. Phys. Solid State 24, 547 (1982)]. halfwidth less than 0.12 meV. Figure 4 presents the val- 5. V. V. Travnikov and V. V. Krivolapchuk, Pis’ma Zh. ues of BMS and FWHM for a number of samples with a Éksp. Teor. Fiz. 37, 419 (1983) [JETP Lett. 37, 496 high emission intensity (large τ). We readily see a (1983)]. noticeable correlation between these quantities. 6. A. V. Akimov, A. A. Kaplyanskiœ, V. V. Krivolapchuk, Because the inhomogeneous halfwidth is due to the and E. S. Moskalenko, Pis’ma Zh. Éksp. Teor. Fiz. 46, 35 (1987) [JETP Lett. 46, 42 (1987)]. shallow-donor EDT dispersion and the area BMS reﬂects the number of deep metastable states, this correlation 7. A. V. Akimov, V. V. Krivolapchuk, N. K. Poletaev, and implies the existence of a connection between the shal- V. G. Shofman, Fiz. Tekh. Poluprovodn. (St. Petersburg) low- and deep-level subsystems. The scatter (1.5143Ð 27, 310 (1993) [Semiconductors 27, 171 (1993)]. 1.5158 eV) in the energy position of the maximum of 8. V. V. Krivolapchuk, N. K. Poletaev, and L. M. Fedorov, the (D0, x) line, which varies from sample to sample, Fiz. Tekh. Poluprovodn. (St. Petersburg) 28, 310 (1994) originates from the samples being differently strained [Semiconductors 28, 188 (1994)]. [9], as a result of which the PL spectra are shifted as a 9. G. L. Bir and G. E. Pikus, Symmetry and Stain-Induced whole with respect to one another (Fig. 2). Figure 5 Effects in Semiconductors (Nauka, Moscow, 1972; Wiley, New York, 1975), Chap. 7. plots the deviation of the (D0, x) spectral position from 10. D. E. Onopko, N. T. Bagraev, and A. I. Ryskin, Fiz. Tekh. a reference value in different samples as a function of Poluprovodn. (St. Petersburg) 30, 142 (1996) [Semicon- the FWHM of this line for each sample. [This relation ductors 30, 82 (1996)]. 0 was obtained in the following way: the (D , x) line posi- 11. P. M. Moony, J. Appl. Phys. 67, R1 (1990). tion in the sample with the minimum FWHM 12. D. J. Chadi and K. J. Chang, Phys. Rev. B 39, 10063 (0.12 meV) which exhibits the smallest EDT dispersion (1989). was taken as the reference.] Thus, the spectral position 13. T. N. Morgan, Mater. Sci. Forum 38Ð41, 1079 (1989). and the halfwidth of the (D0, x) emission line and the 14. D. J. Chadi and K. J. Chang, Phys. Rev. Lett. 61, 873 value of the area BMS were found to be interrelated, (1988). which may be due to reconstruction of the tetrahedral lattice [10] and the formation of deep centers as a result of impurity ions transferring from the lattice sites to tet- Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

SEMICONDUCTORS AND DIELECTRICS

On the Determination of the Debye Temperature from Experimental Data M. N. Magomedov Institute of Geothermy, Dagestan Scientiﬁc Center, Russian Academy of Sciences, pr. Kalinina 3, Makhachkala, 367003 Russia e-mail: [email protected] Received April 9, 2002

Abstract—It is shown that the Debye temperature as a function of temperature must satisfy certain equations in order for the thermodynamic functions calculated in terms of the Debye temperature to satisfy both the third law of thermodynamics and the law of equipartition of energy. A general expression for the Θ(T) function sat- isfying these thermodynamic laws is found. © 2003 MAIK “Nauka/Interperiodica”.

() In the traditional Debye theory [1Ð6], integration C/3N AnikB = CD/3N AnikB with respect to frequency is performed over the range ω × []1 Ð ()T/Θ ()dΘ/dT 2 (4) from zero to a certain value D called the Debye fre- v quency. If ω is assumed to be independent of temper- 2 2 D Ð3/8[]+ ()T/Θ D ()Θ/T Td()Θ/dT v . ature T, then the free energy F , entropy S , and the 3 D D Ӷ Θ specific heat at constant volume C of a three-dimen- At low temperatures T (T), functions (3) and (4) D must satisfy the third law of thermodynamics, which, sional crystal are found to be [1Ð6] following Planck [2, 6], can be written in the form Θ []()Θ FD/3N AnikB = 3 /8 + T ln 1 Ð exp Ð /T limS/3N AnikB == 0, limC/3N AnikB 0. (5) T/Θ → 0 T/Θ → 0 Ð ()T/3 D ()Θ/T , 3 At high temperatures T ӷ Θ(T), function (4) must S /3N n k = Ðln[]1 Ð exp()ÐΘ/T satisfy the classical law of equipartition of energy [2, 6], D A i B (1) ()()Θ a consequence of which is the DulongÐPetit law [1Ð6]: +4/3D3 /T , ()Θ limC/3N AnikB = 1. (6) CD/3N AnikB = 4D3 /T T/Θ∞→ Ð3()Θ/T /[]exp()Θ/T Ð 1 . The boundary conditions defined by Eqs. (5) and (6) impose certain restrictions on the function Θ(T) in Ӷ Θ Here, NA is the Avogadro constant, ni is the number of Eqs. (3) and (4). At low temperatures T (T), the Θ បω Debye function can be written as [1Ð6] D (Θ/T) ≅ atoms in a molecule, = D/kB is the Debye temper- 3 ប (π4/5)[T/Θ(T)]3 and Eqs. (3) and (4) take the form ature, is the Planck constant, kB is the Boltzmann constant, and D (x) is the Debye function for an n-dimen- () Ӎ () n S/3N AnikB low SD/3N AnikB low sional crystal, which has the form (7) Ð3/8()()dΘ/dT v , x ()C/3N n k Ӎ ()C /3N n k () ()n {}n[]() A i B low D A i B low Dn x = n/x ∫ t exp t Ð 1 dt. (2) (8) 2 2 2 0 × []1 Ð ()T/Θ ()dΘ/dT v Ð ()3/8 Td()Θ/dT v . Θ Here, we introduced the following designation of the For some substances, however, the parameter Θ depends on temperature. In this case, the expressions functions, which is commonly used when (T) is for the entropy and specific heat at constant volume defined at low temperatures [1Ð6]: involve derivatives of the function Θ(T) and have the () Ӎ ()π4 []Θ()3 SD/3N AnikB low 4 /15 T/ T , form (9) () Ӎ ()π4 []Θ()3 CD/3N AnikB low 4 /5 T/ T . S/3N AnikB = SD/3N AnikB (3) It is easy to see that the third law of thermodynamics []()Θ ()Θ ()Θ Ð3/8+ T/ D3 /T d /dT v , in the form of Eqs. (5) will be obeyed if the function

Θ Θs (T) does not contain terms linear in T as T 0 K. where 0 is the value determined from the experimen- This fact was pointed out by Born and von Karman [1] tal data on the specific heat without regard for the tem- and was used in [3, 5, 7]. perature dependence of Θ(T) ; that is, Θs is calcu- Ӷ Θ Θ low 0 We assume that, at T 0, the (T) dependence lated directly from Eq. (9). has the form We note that some authors have calculated the func- k tion Θ(T) using methods which give χ < 0 for k = 2 Θ()T ≅ Θ []1 Ð χ()T/Θ , (10) low low 0 0 (see, e.g., [3, Chapter 4], where this was the case for where Θ = lim Θ(T) as T 0 K. This dependence is lead). Such methods are not quite adequate. Indeed, if 0 χ < 0 for k = 2, then it follows from Eq. (12) that the a generalization of the expressions for Θ(T) used in low functions S(T) and C(T) have minima and their [1, 3, 5, 7] and reduces to them when k = 2. The factor low low minimum values are negative for 1 < k < 3. For this rea- χ is found by calculating the function Θ(T) . low son, we assume that χ ≥ 0 and k ≥ 4 in Eqs. (10) and Since Eq. (10) is commonly obtained at T Ӷ Θ0 as a (11). result of expansion in powers of the small quantity At high temperatures T ӷ Θ(T), the Debye function χ Θ k Ӷ Θ (T/ 0) 1, the function (T)low can also be repre- can be written as [1Ð6] sented as ()Θ ≅ ()Θ()()Θ()2 D3 /T 13/8Ð /T + 1/20 /T Θ() ≅ Θ []χ()Θ k T low 0 exp Ð T/ 0 . (11) and Eq. (4) reduces to Substituting Eq. (11) or (10) into Eqs. (7) and (8) for ()C/3N n k ≅ []11/20Ð ()Θ()/T 2 Ӷ Θ A i B high T 0, we find (14) 2 2 2 2 × []1 Ð ()T/Θ ()dΘ/dT v Ð ()T /Θ ()d Θ/dT v . ()S/3N n k ≅ ()4π4/15 ()T/Θ 3 A i B low 0 The function Θ(T) at high temperatures is com- +3/8()kχ()T/Θ k Ð 1, monly found by equating the experimental values of 0 Θ 2 (12) [C(T)/3NAnikB]high to the expression {1 Ð (1/20)[ (T)/T] } ()≅ ()π4 ()Θ 3 C/3N AnikB low 4 /5 T/ 0 (see [1Ð6]). Therefore, the contributions from both the first and second derivatives of the function Θ(T) to the ()()χ()Θ k Ð 1 +3/8kkÐ 1 T/ 0 . lattice specific heat of a crystal at T ӷ Θ(T) are ignored. From Eqs. (6) and (14), we obtain a differential It is seen from Eq. (12) that if k = 1 (as was assumed equation for the function Θ(T) at [T/Θ(T)] ∞: in [8, 9]), the third law of thermodynamics (5) is not 2 2 2 obeyed. Td()Θ/dT v Ð ()T/Θ []()dΘ/dT v (15) If k = 2 (as was the case in [1, 3, 5, 7]), then from +2()dΘ/dT = 0. Eq. (12) it follows that the crystal lattice entropy and v heat capacity at low temperatures follow a linear law A trivial solution to this equation is the constant Θ∞ (S, C ~ T). This contradicts the experimental data and is independent of T. A more general solution to Eq. (15) incompatible with the experimental determination of has the form the electronic specific heat, which is assumed to be Θ() Θ ()αΘ dominant (due to its linear dependence on T) over the T high = ∞ exp Ð ∞/T , lattice specific heat at T 0 K (see, e.g., [1Ð6, 8, 9]). Θ Θ( ) Θ ∞ 2 where ∞ = lim T as T/ ∞ ; the fitting param- At k = 3, we have S, C ~ T . This behavior also contra- eter α controls the rate of increase in the function dicts the experimental evidence for the cubic tempera- T/Θ as T/Θ∞ ∞. ture dependence of the lattice specific heat of three- high dimensional crystals at T 0 K. The interpolation of the low-temperature expression (11) and the high-temperature function Θ(T) over In the case of k = 4, the temperature dependence the entire temperature range can be written as given by Eq. (12) agrees with the experimental data; Θ Θ() Θ []χ()Θ k however, the value of 0 should be defined in this case T = 0 exp Ð T/ 0 not from Eq. (9) but rather from Eq. (12): (16) + Θ∞ exp()ÐαΘ∞/T . ()≅ []()π4 ()χ()Θ 3 C/3N AnikB low 4 /5 + 9/2 T/ 0 . Thus, the function Θ(T) decreases smoothly from Θ0 (at T = 0 K) to a minimum value and then increases, This definition leads to a correction to the Debye tem- approaching the asymptotic value Θ∞ as T ∞. The perature calculated from Eq. (9): function Θ(T) given by Eq. (16) is characterized by five fitting parameters Θ0, χ, k, Θ∞, and α and does not con- Θ Θs []()χ π4 1/3 0 = 0 145+ /8 , (13) tradict the laws of thermodynamics.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 34 MAGOMEDOV

For n-dimensional crystals of elementary sub- Using Eqs. (19) and (20) and performing the same Θ( ) stances (consisting of monatomic molecules, ni = 1), mathematical manipulation as in [7], the function T the function Θ(T) can be found using a procedure pro- can be found to be posed in [7] if one knows the parameters of the inter- Θ() ξ { [ ()ε ξ2 atomic interaction described by the MieÐLennardÐ T = A 0 Ð118+ + /kB A 0 Jones potential: (21) × ()]()ε 1/2 } 1 Ð kBT/4 Sn , ϕ()[] ε(){ []()b r = / baÐ ar0/ cr+ (17) where m is the mass of an atom, Ð br[]/()cr+ a } . 0 []()()()b + 2 AK= R 5knab b + 1 /144 baÐ r0/c , Here, ε and r are the depth and coordinate of the min- 0 ប2 2 imum of the potential well, respectively; b and a are K R = /kBr0m, parameters characterizing the stiffness and range of the ξ 2 () (22) potential, respectively; and c is the distance between 0 = 4n /kn n + 1 , the centers of nearest neighbor atoms in the n-dimen- ξ ()Θ Sn = Ð 0 d /dT v sional crystal. +8()n/k []1 Ð ()T/Θ ()dΘ/dT D ()Θ/T . Let us consider a model proposed by Frenkel’ [10] n v n for crystals consisting of triatomic molecules. The cen- It is easy to show that the function Θ(T) given by tral atom in a molecule is assumed to oscillate within Eq. (21) reaches a maximum at T = 0 K, which agrees the potential well formed by the “core branches” of the with the experimental data. This maximum value can pairwise interaction potential produced by the other be found from Eq. (21) to be two atoms (which are assumed to be fixed, with the separation between them being 2c) [7]: Θ ξ {}[]()ε ξ2 1/2 0 = A 0 Ð118+ + /kB A 0 . (23) ν()r = [] ε*a/2()baÐ (18) Equation (23) is a generalization of the corresponding × {}[]()b ()b []()b expression obtained for Θ in [7] and reduces to it when r0/ crÐ Ð 4 r0/c + r0/ cr+ . 0 ξ0 = 1. For three-dimensional cubic crystals, the value ε Here, * is the effective depth of potential well (18): of ξ0 is ε* =,ε Ð E /()k /2 (19) n n  0.750 for k = 12  n = 3 where En and kn are the oscillation energy and the first-  1.125 for k = 8 ξ ()() n = 3 neighbor coordination number of an atom in the 0 n = 3 = 9/kn = 3 =  (24) n-dimensional crystal, respectively; i.e., (kn/2) is the  1.500 for kn = 3 = 6  number of pairs of oppositely positioned nearest neigh- 2.250 for k = 4. bors of a given atom.  n = 3 Θ( ) Using the Debye model with the function T , the Thus, for closely packed structures, the value of ξ0 is energy En can be found to be close to unity. Θ{}[]()()Θ ()Θ Θ ӷ En = nkB n/2 n + 1 + T/ Dn /T At temperatures close to 0 K, i.e., for ( /T) 1, the (20) Debye function can be written as D (Θ/T) ≅ nn!ζ(n + × []1 Ð ()T/Θ ()dΘ/dT . n v 1)(T/Θ)n [11]. Here, ζ(n + 1) is the Riemann zeta func- Here, the first term is the zero-point energy and the section and is equal to ζ(2) = π2/6, ζ(3) = 1.202, and ond term is the thermal excitation energy of a system of ζ(4) = π4/90 for one-, two-, and three-dimensional sys- Θ harmonic oscillators characterized by a Debye fre- tems, respectively. In this case, the function (T)low has quency spectrum. the form of Eq. (10) with k = n + 1. Therefore, for three- It should be noted that in [7], when considering the dimensional crystals, we have k = 4 in Eq. (16). oscillation of an atom between its two nearest neigh- For an n-dimensional crystal of a monatomic sub- bors in an n-dimensional crystal, we assumed this oscil- stance, in the case where the zero-point energy is small lation to be one-dimensional and, instead of Eq. (19), in comparison with the depth of the potential well ε ε used the relation * = Ð En = 1, where the second term described by Eq. (17), i.e., where was expressed in terms of the one-dimensional Debye ε ӷ ξ2[]ξ2()2 function. This was the reason why, in [7], expression (10) 8 /kB A 0 0 n + 1 Ð 1 , (25) was obtained with k = 2 and χ = (4/3)π2 for an n-dimensional crystal. As will be shown below, Eq. (19) leads to the function χ has the form thermodynamically more self-consistent forms of χ ()2 ζ()()Θ ε Eqs. (10) and (11) with k = n + 1. = n /kn n! n + 1 kB 0/ . (26)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ON THE DETERMINATION OF THE DEBYE TEMPERATURE 35

We note that inequality (25) is satisfied for three- of the specific heat at constant volume for T/Θ∞ ∞ dimensional crystals of all elementary substances and can be found in many handbooks. Therefore, only except helium. the parameter α should be fitted in calculating Θ(T). Thus, for three-dimensional crystals of elementary substances, we have ACKNOWLEDGMENTS χ ()π4 ()Θ ε = 3 /5kn = 3 kB 0/ . (27) The author is grateful to K.M. Magomedov, as well as K.N. Magomedov and Z.M. Surkhaeva, for their In this case, Eq. (13) reduces to the following expres- assistance throughout this work. Θ sion, which enables one to calculate 0 from the known This study was supported by the Russian Founda- Θs tion for Basic Research, project no. 02-03-33301. values of 0 [found using Eq. (9)], of the coordination number, and of the depth of potential (17): REFERENCES Θ Θs []()()Θs ε 1/3 0 = 0 127/8+ kn = 3 kB 0/ . (28) 1. Physical Acoustics: Principles and Methods, Vol. 3, Part B: Lattice Dynamics, Ed. by W. P. Mason (Aca- Θs Numerical calculations based on the values of 0 demic, New York, 1965; Mir, Moscow, 1968). ε Θ and /kB taken from [7] showed that 0 is greater than 2. J. E. Mayer and M. Goeppert-Mayer, Statistical Θs Mechanics (Wiley, New York, 1977). 0 by 25Ð30% for He, ortho-H2, and para-D2; by 12Ð 3. A. A. Maradudin, E. W. Montroll, and G. H. Weiss, The- 16% for Ne; by 5Ð7% for Ar; and by 2Ð4% for Kr and ory of Lattice Dynamics in the Harmonic Approximation Θ Xe. Therefore, in calculating 0 for cryocrystals, one (Academic, London, 1963). should take into account the correction to Θs found 4. Ch. Kittel, Introduction to Solid State Physics (Wiley, 0 New York, 1976, 5th ed.). from Eq. (9) without regard for the temperature depen- Θ( ) 5. G. Grimvall, Thermophysical Properties of Materials dence of T . As for other crystals, the correction to (North-Holland, Amsterdam, 1986). Θs Θs ε Ӷ 0 for them is very small, since kB 0 / 1. Even for 6. H. Schilling, Statische Physik in Beispielen (VEB Fach- Θ Θs Θs ≤ buchverlag, Leipzig, 1972). Li, we have 0 Ð 0 /0 0.01. 7. M. N. Magomedov, Zh. Fiz. Khim. 61 (4), 1003 (1987). Thus, for three-dimensional fcc and bcc crystals, 8. V. Yu. Bodryakov and V. M. Zamyatin, Fiz. Met. Metall- inequality (25) holds within a large margin, the value of oved. 85 (4), 18 (1998). χ ξ is small, and 0 is close to unity. This is the reason 9. V. Yu. Bodryakov and V. M. Zamyatin, Teplofiz. Vys. why both the Debye temperature and the Grüneisen Temp. 38 (5), 724 (2000). parameter calculated in [7] were in good agreement 10. Ya. I. Frenkel’, Introduction to the Theory of Metals with the experimental data. The calculational technique (GIFML, Moscow, 1958). was then extended in [12, 13] to the case of binary ionic 11. Handbook of Mathematical Functions, Ed. by M. Abra- cubic crystals of the AB type and the agreement with the mowitz and I. A. Stegun (National Bureau of Standards, experimental data was found to be good. For the same Washington, 1964). reason, the parameters of potential (17) determined in 12. M. N. Magomedov, Teplofiz. Vys. Temp. 30 (6), 1110 [14Ð16] must differ only slightly from the values of (1992). these parameters calculated on the basis of the expres- 13. M. N. Magomedov, Zh. Fiz. Khim. 67 (11), 2280 (1993). sion for the Debye temperature derived in this paper. 14. M. N. Magomedov, Zh. Fiz. Khim. 62 (8), 2103 (1988). 15. M. N. Magomedov, Zh. Fiz. Khim. 63 (11), 2943 (1989). The quantity Θ∞ can be calculated using the methods described in the literature. It should be noted that 16. M. N. Magomedov, Teplofiz. Vys. Temp. 32 (5), 686 the high-temperature value of the Debye temperature (1994). Θ∞ is determined experimentally much more accurately Θ than is 0 because of the weak temperature dependence Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 36Ð40. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 37Ð41. Original Russian Text Copyright © 2003 by Zhuravlev, Poplavnoœ. SEMICONDUCTORS AND DIELECTRICS

The Distribution of the Valence Electron Density in Predominantly Ionic Crystals with Different Bravais Sublattices Yu. N. Zhuravlev and A. S. Poplavnoœ Kemerovo State University, ul. Krasnaya 6, Kemerovo, 650043 Russia e-mail: [email protected] Received February 20, 2002

Abstract—Based on the theory of the local-density functional, self-consistent valence electron densities are calculated for CaO with a rock-salt lattice, CaF2 with a fluorite lattice, K2O with an antifluorite lattice, and for their constituent sublattices. It is shown that in the crystals with different Bravais sublattices, the anionic sublattice is a framework with covalent bonds containing a metal sublattice inside of them. The coupling between the sublattices is characterized by the density difference, which is defined as the difference between the total electron density and the densities of the individual sublattices. The density difference is found to be an order of magnitude smaller than the crystal and sublattice densities, which is evidence of weak hybridization of the sublattices and of the predominately ionic character of the bonding between them. © 2003 MAIK “Nauka/Inter- periodica”.

1. INTRODUCTION lattice. In the K2O crystal, the fcc-sublattice sites are occupied by ions of the univalent alkali metal and the The sublattice technique for studying the electronic sites of the simple cubic sublattice, by sexivalent structure of crystals was developed by us in [1Ð3]. It has anions. In the CaO crystal, both cations and anions are been shown that some peculiarities in the electron den- located at fcc-sublattice sites, which makes the crystal sity are due to long-range interaction between identical suitable for comparison with fluorite and antifluorite. atoms of the sublattices. From this point of view, crys- In this paper, we carried out calculations in the tals with different Bravais sublattices are of particular framework of the local-density-functional theory by interest. The difference between the basis vectors of the using the pseudopotential method and the basis of sublattices in these crystals results in peculiarities in the 1 3 5 electron density which cannot be interpreted in terms of numerical atomic s p d pseudoorbitals expanded in translation symmetry of the entire crystal. Fluorides of terms of plane waves [4]. alkali-earth metals with a fluorite lattice (for example, CaF2) and oxides and sulfides of alkali metals with an 2. ROCK-SALT STRUCTURE antifluorite lattice (for example, K2O) are the most suit- Let us consider CaO as an example of a crystal with able objects for use in investigating cases where the identical fcc Bravais sublattices. The valence electron anionic and cationic Bravais sublattices are different. densities of the crystal and sublattices calculated for the For reference, a crystal with identical anionic and cat- [110] plane are shown in Fig. 1. The numerical values ionic sublattices should be considered; in our case, this of the densities are given in units of e · ÅÐ3 (e is the elec- crystal is CaO. tron charge). The valence density of the crystal ρ(CaO) The fluorite structure consists of two cubic cationic is an electron density typical of ionic compounds. The and anionic sublattices. The cations form an fcc sublat- electron density is concentrated predominantly in the tice into which a simple cubic anionic sublattice is vicinity of the anion and cation positions, where its dis- incorporated. The sublattices are arranged in such a tribution is almost spherically symmetric. The small way that the cations are placed at the centers of every density maximum between the metal ions corresponds second anionic unit cube along any coordinate axis. to the saddle point of the ρ(CaO) level surface, because This leads to such an alternation of the occupied and there is a minimum in the anion plane at this point. empty positions that the empty positions form an fcc Now, let us consider the formation of the crystal lattice. In terms of the interchange between anions and density ρ(CaO) from the densities of the anionic and cations, the antifluorite structure is reciprocal to the flu- cationic sublattices. In the oxygen sublattice, the elec- orite structure. In the CaF2 crystal, the ions of the biva- tron density ρ(O) is concentrated mainly around the lent alkali-earth metal occupy fcc-sublattice sites and anions and is partially transferred to the empty posi- the septivalent anions, the sites of the simple cubic sub- tions of the metal ions; therefore, the oxygen sublattice

THE DISTRIBUTION OF THE VALENCE ELECTRON DENSITY 37

0.3 0.35 ρ(O) ρ(CaO) 0.3 0.25 0.25 0.2

0.25 Ca 0.2 0.3 0.15 O 0.25

0.01 ρ(Ca) –0.01 0.0 ∆ρ(CaO)

0.0 0.07 0.072 –0.005 0.072 0.005 0.075

Fig. 1. Distribution of the valence crystal density ρ(CaO), the calcium and oxygen sublattice densities [ρ(Ca) and ρ(O), respectively], and the density difference ∆ρ(CaO) in the [110] plane of the calcium oxide. is an anionic framework with covalent bonds. In the terms of the overlapping of corresponding wave func- metal sublattice, the electron density is smeared quite tions of neighboring sites. From the above, it is clear uniformly over the unit cell with an average density of that if the Bravais sublattices are different, the pattern 0.073; however, a small charge is also transferred from of the formation of the total electron density will be the positions of the metal ions to the empty oxygen completely different. positions. The density difference ∆ρ(CaO) is an order of magnitude smaller than the total density ρ(CaO) in this case, which indicates that the hybridization 3. FLUORITE STRUCTURE between the sublattices is weak. As seen from the distribution of the negative (in units of e) and positive val- In [5], full-electron calculations of the band struc- ues of ∆ρ(CaO), this effect is reduced to charge transfer ture, the charge-density distribution, and the optical from the interstitials to the positions of the anions and functions of CaF2 were carried out using the ab initio cations. method of the local-density functional. The calculated total electron density is almost spherically symmetric Thus, in the case of a crystal with identical Bravais and is concentrated in the vicinity of the metal and sublattices, the main peculiarity of the sublattice elec- anion atoms. Based on the estimated values of the ion tron densities is the charge transfer from the occupied charges, it was concluded in [5] that the ionicity of the sites of one sublattice to the empty sites of the other. For bond in CaF is high. this reason, the summation of the sublattice densities 2 results in a simple pattern characterizing the ionic crys- It is well known that maps of the total electron dental. In terms of the sublattices, the hybridization of the sity are less informative than those of the valence den- wave functions of different atoms is explained by the sity. For this reason, we carried out self-consistent cal- tendency of the identical atoms of a sublattice to form culations of both the crystal and sublattice valence den- a chemical bond of covalent type by means of charge sities of CaF2 corresponding to the [110] plane (Fig. 2). ρ transfer to interstitial sites that exactly coincide with The map of crystal valence density (CaF2) shows that, the empty sites of the second sublattice. in the vicinity of the F atoms, the valence-density dis- Within the local model, the electron exchange tribution is nearly spherically symmetric. The nearest between atoms of different sublattices is explained in neighbor F atoms are surrounded by the common con-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

38 ZHURAVLEV, POPLAVNOŒ

0.3 0.25

FF0.4

ρ ρ(F) 0.4 (CaF2) 0.35 0.3 0.25 FF 0.4

0.5 Ca 0.4

FF0.0 ρ(Ca)

0.0 –0.01 ∆ρ(CaF ) 0.05 2

–0.005 0.049 FF –0.005 0.01

ρ ρ ρ Fig. 2. Distribution of the valence crystal density (CaF2), calcium and fluorine sublattice densities [ (Ca) and (F), respectively], ∆ρ and density difference (CaF2) in the [110] plane of the calcium fluoride. tour of the valence-density distribution, which is in tions of the metal ions and are equal to 0.046. The max- agreement with the data on the total electron density imum density is concentrated in the sufficiently narrow presented in [5]. At the same time, there are significant strip between the metal planes with convexities towards ρ peculiarities in the (CaF2) map (Fig. 2) which are not the atomic sites. It should be noted that the maxima of revealed in the map of the total electron density: the crystal density are also located in the region of these valence-density peaks of height 0.4 appear between the convexities; therefore, the valence density of the metal next-to-nearest neighbor anions and occupy the face- also favors the enhancement of the covalent coupling central positions. between the next-to-nearest neighbor anions. Now, let us analyze the role of the sublattices in the The density difference ∆ρ, which is defined as the formation of the crystal valence density. First, we con- difference between the electron density of the crystal sider the electron density of the fluorine sublattice ρ(F). and the densities of the separate sublattices, is positive ρ In comparing it with (CaF2), one can see that the dis- (in units of e) inside the anion region and negative in the tribution of electron density over the fluorine sublattice rest of the crystal, including the cation positions. The closely repeats the main peculiarities of the valence density difference is an order of magnitude smaller than density of the crystal. This conclusion relates to the for- the crystal and sublattice densities, which indicates that mation of the common contours “embracing” the near- the hybridization of the sublattices is weak and that the est neighbor fluorine atoms, as well as to the presence coupling between them is mainly ionic. of the maxima of the density between the next-to-near- Now, we compare these results with those for CaO. est neighbors. Thus, the origin of these peculiarities First, we note that the CaO and CaF2 sublattices are sig- becomes clear: it can be stated that the anionic sublat- nificantly different. The fluorine sublattice in CaF is a tice is the framework with covalent bonds into which 2 the cationic sublattice is inserted. simple cubic lattice, whereas the oxygen sublattice in CaO is an fcc lattice with doubled (with respect to the Now, we consider the metal sublattice. Figure 2 simple cubic lattice) period. Since the Bravais sublat- shows that the valence density ρ(Ca) is smeared quite tices in CaO are identical, the framework of the oxygen uniformly over the entire space with an average value sublattice with covalent bonds no longer arise as the of ~0.048. The density minima are located at the posi- total density ρ(CaO) is formed, because the metal sites

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

THE DISTRIBUTION OF THE VALENCE ELECTRON DENSITY 39

0.1 0.2 ρ (K2O) ρ(O)

Α Β 0.15 0.1 0.1 0.15

0.1

0.15 Κ 0.15 0.1

ρ(K) 0.005 0.031

∆ρ (K2O) 0.03 0.032

Κ 0.0 0.027 0.032 Ο

ρ ρ ρ Fig. 3. Distribution of the valence crystal density (K2O), potassium and oxygen sublattice densities [ (K) and (O), respectively], ∆ρ and density difference (K2O) in the [110] plane of the potassium oxide. are located at the positions of the bond charges of this position B in the anion plane. Hence, position B corre- framework. On the contrary, the anionic framework sponds to a saddle point of the electron-density level with covalent bonds survives upon the formation of the surface. ρ total density (CaF2) due to the difference between the Now, let us compare the electron density of the oxy- Bravais sublattices of CaF . Herein lies the principal ρ ρ 2 gen sublattice (O) with (K2O). One can see that the difference in the formation of the electron density distribution of ρ(O) closely repeats the main peculiari- between compounds with identical and different Bra- ties of the valence density of the crystal. The most sig- vais sublattices. niﬁcant fact is that electrons are transferred from the It should be noted that the metal sublattices in CaO anion ions to positions A (which are midway between and CaF2 are similar and differ only in the value of the the next-to-nearest neighbor anions), which leads to the lattice constant; this explains the differences between formation of a covalently bonded framework due to the the ρ(Ca) maps shown in Figs. 1 and 2. anionÐanion bonds. Moreover, saddle point B is also formed basically by the anion electrons. A comparison of the ρ(O) maps for CaO (Fig. 1) and 4. ANTIFLUORITE STRUCTURE K2O (Fig. 3) shows that these maps are qualitatively We illustrate the results of our calculations using the similar and that the quantitative differences between them are due to the different lattice constants of these example of K2O (Fig. 3). First, let us take a look at the map of the crystal density ρ(K O). The electron-density compounds. As in the case of ﬂuorite, the anionic 2 framework with covalent bonds (but with an fcc Bra- minima correspond to the cation positions, while the ρ density maxima are at the anion positions and in the vais lattice) survives in the total density (K2O). interstices. The main maximum in the interstices is at In the metal sublattice, the valence density is the position A midway between the next-to-nearest smeared quite uniformly with an average density of neighbor anions, and another maximum is seen at the ~0.019. Weakly pronounced maxima (equal to ~0.02) points with coordinates (1/4, 1/4, 1/2) (position B in are situated at the empty positions of oxygen atoms and Fig. 3). It should be noted that an electron density min- in the interstices midway between the empty positions imum localized between two oxygen atoms is located at of the next-to-nearest neighbor anions. At the metal

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

40 ZHURAVLEV, POPLAVNOŒ positions themselves, there are density minima equal to mum electron density in the anionic framework with 0.018. Thus, the valence density of the metal also favors covalent bonds, which can be channels for the motion the enhancement of the covalent bonds of the anionic of the alkali-metal ions. An analysis of the experimental sublattice. data [6] shows that the ionic conductivity in crystals The density difference is positive inside the anion with an antifluorite lattice is due to the transfer of the region and negative in the rest of the space, including metal ions. the cation positions. The absolute values of ∆ρ are an In the fluorite lattice, the situation is different; since order of magnitude smaller than the crystal and sublat- the volume of the unit cell of the simple cubic sublattice tice densities; this indicates that the hybridization is smaller than that of the fcc lattice, there are no tun- between the sublattices is weak and the coupling nels or wells in the anionic framework with covalent between them is mainly ionic. bonds in this lattice. However, as was previously mentioned, there are metal vacancies in the centers of the next-to-nearest cubes in this framework. This favors the 5. RESULTS AND DISCUSSION generation of anionic Frenkel defects, which can pro- Based on the calculated crystal and sublattice vide ionic conductivity. As noticed in [6], the ionic con- valence electron densities of the compounds with rock- ductivity in fluorite crystals increases when the order in salt, fluorite, and antifluorite lattices (possessing both the anionic sublattice is destroyed. The electronic struc- identical and different Bravais sublattices), one may ture of a number of alkali-earth fluorides with Frenkel draw the following conclusions. Identical sublattice defects theoretically calculated on the basis of the lin- atoms form a covalent type of chemical bond by trans- ear muffin-tin orbital method supports this model [7]. ferring a certain charge to the interstices, where max- The energies of the formation and migration of the ima in the electron density are formed. From symmetry defects estimated for two possible mechanisms of the considerations, it is obvious that these maxima possess ionic conductivity (the motion of vacancies associated the translation symmetry of the corresponding sublat- with anionic Frenkel defects and the motion of intersti- tice. If, upon summation of the sublattice densities with tial ions) indicate that the smaller migration energies the formation of the crystal density, the peaks of the correspond to the motion of anionic vacancies. electron density at interstitial sites coincide with the sublattice sites (as is the case in the rock-salt structure), then a purely ionic chemical bond is realized. However, REFERENCES this is the case only if the Bravais sublattices are iden- 1. Yu. N. Zhuravlev, Yu. M. Basalaev, and A. S. Poplavnoœ, tical. When different Bravais sublattices are super- Zh. Strukt. Khim. 42 (2), 210 (2001). posed, the electron-density maxima are necessarily left 2. Yu. N. Zhuravlev and A. S. Poplavnoœ, Zh. Strukt. Khim. at the interstitial sites, forming the covalent component 42 (5), 860 (2001). of the chemical bond. It is obvious that the electron 3. Yu. N. Zhuravlev and A. S. Poplavnoœ, Fiz. Tverd. Tela density maxima at interstitial sites are the higher, the (St. Petersburg) 43 (11), 1984 (2001) [Phys. Solid State larger the number of electrons in the sublattice where 43, 2067 (2001)]. these maxima arise. That is the reason why the frame- 4. Yu. N. Zhuravlev, Yu. M. Basalaev, and A. S. Poplavnoœ, work with covalent bonds originates due to the anions Izv. Vyssh. Uchebn. Zaved., Fiz., No. 3, 96 (2000). in the crystals under investigation. Covalent bonding of the cations can occur in metals with localized d and f 5. F. Gan, Y.-N. Xu, M.-Z. Huang, et al., Phys. Rev. B 45 (15), 8248 (1992). states in the valence region. 6. A. West, Solid State Chemistry and Its Applications The ionic conductivity of a number of crystals with (Wiley, Chichester, 1984; Mir, Moscow, 1988), Part 2. a fluorite or an antifluorite lattice is sufficiently high; that is why these crystals are classified as solid electro- 7. V. P. Zhukov and V. M. Zaœnullina, Fiz. Tverd. Tela (St. Petersburg) 40 (11), 2019 (1998) [Phys. Solid State lytes [6]. As underlined in [6], the ionic conductivity is 40, 1827 (1998)]. favored by the presence of tunnels or layers in the crystal structure. Our calculations of the electron density in K2O crystals indicate that there are wells with mini- Translated by A. Poushnov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 41Ð45. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 42Ð45. Original Russian Text Copyright © 2003 by Aleksandrov, Sorokin, Glushkov, Bezmaternykh, Burkov, Belushchenko.

SEMICONDUCTORS AND DIELECTRICS

Electromechanical Properties and Anisotropy of Acoustic Wave Propagation in CuB2O4 Copper Metaborate K. S. Aleksandrov*, B. P. Sorokin**, D. A. Glushkov**, L. N. Bezmaternykh*, S. I. Burkov**, and S. V. Belushchenko* * Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia ** Krasnoyarsk State University, Krasnoyarsk, 660041 Russia e-mail: [email protected] Received March 12, 2002

Abstract—The propagation of bulk acoustic waves in single-crystal CuB2O4 copper metaborate is studied. The elastic, piezoelectric, and dielectric constants are calculated. The anisotropy in the parameters of bulk acoustic wave propagation in this crystal are determined. © 2003 MAIK “Nauka/Interperiodica”.

1. The interest in undertaking a coordinated investi- the velocities and polarization vectors of BAWs propagation of the properties of single-crystal copper oxybo- gating in a given crystal direction. Expressions for rates has been spurred by recent studies on their low- BAW velocities, which, in general, are combinations of temperature magnetism [1Ð6]. The most comprehen- elastic, piezoelectric, and dielectric constants, are pre- sive data, including determination of the magnetic sented in Table 1 for some directions in crystals of point structure using neutron diffraction [4], were obtained symmetry group 4 2m, to which the copper metaborate for the CuB2O4 copper metaborate. This tetragonal noncentrosymmetric crystal [7], belonging to space CuB2O4 belongs. 12 group D2d = 14 2d, with the lattice parameters a = The BAW velocities in CuB2O4 were measured 11.528 Å and c = 5. 607 Å, remains paramagnetic down using the pulsed ultrasonic technique (30 MHz) [10] to 21 K. At lower temperatures, the crystal transfers to based on measuring the time of ultrasonic pulse propa- the antiferromagnetic state and its weak ferromagnetic gation in a sample. This method ensures an accuracy of moment of approximately 0.56 emu/g is accounted for not worse than 10Ð4 in absolute measurements and a by the two coupled sublattices being slightly misori- sensitivity of 10Ð6 in relative measurements. The ented [1]. Below 10 K, the compound undergoes a sec- CuB2O4 samples, shaped as polished-face rectangular ond phase transition to an incommensurate spiral struc- ≈ ture, which is due primarily to the DzyaloshinskyÐ parallelepipeds with linear dimensions 1 cm, were cut Moriya antisymmetric exchange interaction [5]. Opti- from the same boule. The crystallographic orientation cal absorption spectra of CuB O are discussed in [8]. of the samples was checked with an x-ray diffractome- 2 4 ± ′ At the same time, we are not aware of any publica- ter to within 3 . The sample orientations are shown in tions dealing with such properties as the elasticity and Fig. 1. The results of the BAW velocity measurements piezoelectric effect of this crystal. The present paper are listed in Table 2. reports on a measurement of the bulk acoustic wave (BAW) velocities and determination of the elastic, dielectric, and piezoelectric constants of this crystal, as well as on a calculation of the anisotropy in the BAW X3 X3 X3 parameters. [011] Single crystals of copper metaborate, up to 10Ð 3 15 cm in volume, were grown, as in [9], from lithium X2 X2 borate melt solutions diluted by MoO3. In such melt X2 X1 X solutions, the growing face also maintains stability with X 1 [110] a seed growing from the melt-solution surface, which 1 made possible a stable Kyropoulos process with a low- (a) (b) (c) ered temperature in the interval 920Ð850°C. 2. BAW propagation in a crystal is described by the Fig. 1. Sample orientation. (a) Sample 1, (b) sample 2, and GreenÐChristoffel equation [10], whose solution yields (c) sample 3.

42 ALEKSANDROV et al.

Table 1. BAW velocities and electromechanical constants in crystals of 4 2m symmetry

Direction Direction Type of wave of polarization ρV2 of propagation vector

E [100] L [100] C11 E S [010] C66

E S [001] C44

E [001] L [001] C33

E S C44 ()E E E [110] L [110] 1/2 C11 + C12 + C66

e2 S [001] CE + ------14- 44 εη 11 ()E E S [1 10] 1/2 C11 Ð C12

0.16()e + e 2 [011] S [100] 0.19CE ++0.81CE ------14 36 - 66 44 εη εη 0.19 11 + 0.81 33 ()E E E 1/2 0.19C11 ++0.81C33 C44 QS []()E E ()E E 2 ()E E 2 Ð 1/2 0.19 C11 Ð C44 + 0.81 C44 Ð C33 + 0.62 C44 + C13 Note: L is longitudinal, S shear, and QS quasi-shear mode.

The dielectric permittivities of mechanically free semiautomatic bridge (1 kHz). These data were used to εσ εσ determine the elastic and piezoelectric constants of samples 11 and 33 were derived from the capacity of CuB2O4 from the BAW velocities. planar capacitors prepared from X- and Z-cut plates; the capacity was measured with an E8Ð4 high-precision 3. As seen from Table 1, the signs of the piezoelectric constants e14 and e36 cannot be determined from the BAW velocities alone. The relative sign of these con- ° stants has to be found independently. It should be Table 2. BAW velocities in single-crystal CuB2O4 (20 C) pointed out that crystals belonging to point symmetry Direction Direction Type of polarization Velocity, m/s group 4 2m permit the existence of two inequivalent of propagation of wave vector sets of crystallographic coordinate systems (CCSs). In choosing the CCS, we were guided by the rules pro- ± [100] L [100] 9917.6 0.1 posed in [11], according to which the proper CCS is S [010] 4867.7 ± 0.1 that in which the condition for the piezoelectric modu- S [001] 5307.0 ± 0.5 lus d36 > 0 is met. Therefore, we chose the proper CCS [001] L [001] 8882.5 ± 1.2 and analyzed the signs of the piezoelectric constants by S 5307.0 ± 0.5 using the static direct piezoelectric effect. [110] L [110] 9227.3 ± 1.4 Consider the behavior of a sample oriented as shown S [001] 5317.3 ± 0.3 in Fig. 1c under the application of uniaxial mechanical ± compression along the [011] direction. The equation of S [1 10] 6073.4 0.2 state for this case can be written as [011] S [100] 5234.9 ± 1.1 σ QS 5471.9 ± 1.3 Di' = dikl' kl' , (1)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTROMECHANICAL PROPERTIES AND ANISOTROPY OF ACOUSTIC WAVE 43 where all quantities are deﬁned relative to the “rotated” X X3, X3' 3 coordinate system (Fig. 2a). In this coordinate system, X2' X2' the mechanical stress tensor corresponding to uniaxial X3' 63.92° compression has the form X2 X2 X1  45° X' X1, X1' Ð00p 1  (a) (b) σ' =  (2) kl 000 000 Fig. 2. Rotated coordinate systems (a) for sample 3 and (b) for sample 2.

(a) (b) in the case of compression along the X1' axis and 10000 0.08 S1  9000 000 L 0.06  X3 N σ' =  (3) 8000 kl 0 Ð0p φ 7000 X K 0.04 , m/s , 1 000 X2 V 6000 S 2 0.02 for compression along the X2' axis. The pressure is 5000 S1 assumed to be negative. Then, the electric induction S2 4000 vector should have only one nonzero component, either 0 20 40 60 80 0 20 40 60 80 φ, deg φ, deg 1 (c) (d) D' ==Ðd' p Ð---d p (4) 8 3 31 X1' 36 X1' 2 S1 L 80 or S2 6 S1 S2 1 60 , deg ' ' --- , deg 4

D ==Ðd p d p . (5) γ 3 32 X2' 36 X2' θ 2 40 2 Thus, by measuring the sign of the charges produced by 20 the direct piezoelectric effect at the sample faces per- L pendicular to the X3' axis under uniaxial mechanical 0 20 40 60 80 0 20 40 60 80 stress along X1' or X2' and assuming d36 > 0, one can φ, deg φ, deg choose the direction of axes of the original coordinate Fig. 3. Anisotropy in the BAW propagation parameters for system in a given sample. After this, because all the CuB2O4 in the (100) plane. (a) BAW phase velocities, (b) samples were prepared from the same boule and their electromechanical coupling coefficients, (c) angles between mutual orientation is known, we fix the axes of the orig- the polarization vector and direction of BAW propagation, inal CCS in sample 2 (Fig. 1b). Following the same rea- and (d) angles of BAW energy flow deviation. N is the wave soning, we find that the nonzero component of the elec- normal vector. tric induction vector in the case of uniaxial mechanical compression applied to sample 2 along X' (Fig. 2b) is where k is a positive coefficient depending on the rota- 2 tion angle of the coordinate system. Measurements performed in accordance with Eqs. (4)Ð(6) showed that in D1' = Ðkd14 pX' , (6) 2 CuB2O4 for d36 > 0 the inequality d14 > 0 is met.

° Table 3. Material constants of single-crystal CuB2O4 (20 C)

E 10 E E E E E E Cλµ , 10 Pa C11 C12 C13 C33 C44 C66 39.54 ± 0.01 9.86 ± 0.02 10.56 ± 0.02 31.72 ± 0.01 11.32 ± 0.01 9.53 ± 0.01 ρ 3 2 εσ εσ εσ , kg/m [7] eiλ, C/m e14 e36 ij 11 33 4020 0.14 ± 0.01 0.22 ± 0.01 6.09 ± 0.05 6.14 ± 0.05

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 44 ALEKSANDROV et al.

(a) (b) (a) (b) 10000 0.06 10000 0.06 L S1 L 9000 9000 S1 L X3 8000 X2 0.04 X3 0.04 8000 φ N X1 φ N , m/s

7000 K K , m/s S2 V X1 X2 V S 7000 6000 2 0.02 0.02 6000 S2 5000 S1 S1 S2 4000 5000 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 φ φ φ, deg φ, deg , deg (c) , deg (d) (c) (d) 100 24 8 S1 S2 20 S2 S 80 S S 80 1 2 1 6 S 16 2 60 60 , deg , deg , deg

θ 12 4 γ θ , deg

40 γ L 40 8 2 L 20 4 20 S L 1 L 0608020 40 60 80 0 20 40 0 20 40 60 80 0 20 40 60 80 φ φ φ φ, deg , deg , deg , deg

Fig. 4. Anisotropy in the BAW propagation parameters for Fig. 5. Anisotropy in the BAW propagation parameters for CuB2O4 in the (001) plane. (a) BAW phase velocities, (b) CuB2O4 in the (11 0) plane. (a) BAW phase velocities, (b) electromechanical coupling coefficients, (c) angles between electromechanical coupling coefficients, (c) angles between the polarization vector and direction of BAW propagation, the polarization vector and direction of BAW propagation, and (d) angles of BAW energy flow deviation. N is the wave and (d) angles of BAW energy flow deviation. N is the wave normal vector. normal vector.

The relation connecting the piezoelectric constants longitudinal waves in this plane reach a maximal value, with piezoelectric moduli presented in [11] reduces in which is relatively high (about 10000 m/s), in the X1 our case to (X2) directions. The X3 direction (fourfold inversion E axis) is the acoustic axis. Figure 3b displays the anisot- e14 = d14C44, (7) ropy of the electromechanical coupling coefﬁcient

E (ECC). Only the slow shear wave polarized along [100] e36 = d36C66. (8) is accompanied by the longitudinal piezoelectric activity. For propagation at an angle φ ≈ 44°, the ECC for Because the elastic constants are C44 > 0 and C66 > 0 this mode reaches a maximum value k ≈ 7.6%. This (see Tables 1, 2), the piezoelectric constants are e14 > 0 value characterizes the copper metaborate as a weak and e36 > 0. piezoelectric crystal. Figure 3c shows the BAW polar- The Mohs hardness of the copper metaborate is ization angles, which can be used to determine the more than seven units (a single crystal scratches directions of the “pure” modes. In this plane, pure quartz). The deep blue color, chemical stability, homo- modes propagate along crystallographic axes. Figure geneity, and fairly large dimensions of this crystal make 3d illustrates the angles of the BAW energy ﬂow devia- it attractive as material for use in jewelry; its refractive tion. Shown graphically in Fig. 4 are the results of sim- indices No = 1.69 and Ne = 1.582 were determined in ilar calculations made for the (001) plane. Note the [12]. presence of acoustic axes which do not coincide with the crystallographic directions and lie at angles φ ≈ 20° 4. The material constants thus obtained (Table 3) ° were used to calculate the anisotropy in the BAW and 70 . As one crosses the acoustic axis, the polariza- parameters for some planes of the CuB O crystal. The tion vector solutions, as it were, become replaced; more 2 4 speciﬁcally, the shear-wave polarization vectors rotate results are presented in Figs. 3Ð5. Figure 3a shows the by 90°. The shear wave polarized along [001] is an elas- anisotropy of the BAW velocities for propagation in directions lying in the (100) plane. The velocities of tically isotropic wave. As follows from Fig. 5 [the (11 0)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTROMECHANICAL PROPERTIES AND ANISOTROPY OF ACOUSTIC WAVE 45 plane], the longitudinal mode, as well as the slow shear 5. G. A. Petrakovskiœ, M. A. Popov, B. Roessli, and wave, is accompanied by the longitudinal piezoelectric B. Ouladdiaf, Zh. Éksp. Teor. Fiz. 120 (4), 926 (2001) activity. [JETP 93, 809 (2001)]. 6. H. Kageyama, K. Onizuka, T. Yamauchi, and Y. Ueda, J. Cryst. Growth 206, 65 (1999). ACKNOWLEDGMENTS 7. M. Martinez-Ripoll, S. Martinez-Carrera, and S. Garcia- Blanco, Acta Crystallogr. B 27, 677 (1971). This study was supported by the Russian Founda- 8. L. N. Bezmaternykh, A. M. Potseluœko, E. A. Erlykova, tion for Basic Research (project no. 00-15-96790) and and I. S. Édel’man, Fiz. Tverd. Tela (St. Petersburg) 43 the Federal Program “Integration” (project no. 69). (2), 297 (2001) [Phys. Solid State 43, 309 (2001)]. 9. L. N. Bezmaternykh, A. D. Vasil’ev, I. A. Gudim, et al., in Proceedings of the IX National Conference on Crystal REFERENCES Growth, Moscow, 2000, p. 457. 10. M. P. Zaœtseva, Yu. I. Kokorin, Yu. M. Sandler, 1. G. Petrakovskii, D. Velikanov, A. Vorotinov, et al., V. M. Zrazhevskiœ, B. P. Sorokin, and A. M. Sysoev, J. Magn. Magn. Mater. 205, 105 (1999). Nonlinear Electromechanical Properties of Noncen- 2. G. A. Petrakovskiœ, A. D. Balaev, and A. M. Vorotynov, trosymmetric Crystals (Nauka, Novosibirsk, 1986). Fiz. Tverd. Tela (St. Petersburg) 42 (2), 313 (2000) 11. Yu. I. Sirotin and M. P. Shaskolskaya, Fundamentals of [Phys. Solid State 42, 321 (2000)]. Crystal Physics (Nauka, Moscow, 1979; Mir, Moscow, 1982). 3. G. A. Petrakovskiœ, K. A. Sablina, D. A. Velikanov, et al., Kristallograﬁya 45 (5), 926 (2000) [Crystallogr. Rep. 45, 12. G. K. Avdulaev, P. F. Rza-Zade, and S. Kh. Mamedov, 853 (2000)]. Zh. Neorg. Khim. 27 (7), 1837 (1982). 4. B. Roessli, J. Schefer, G. Petrakovskii, et al., Phys. Rev. Lett. 86 (9), 1885 (2001). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

SEMICONDUCTORS AND DIELECTRICS

The Features of the EPR Spectrum in the Vicinity of Accidental Coincidence of Interacting Transitions V. A. Vazhenin, V. B. Guseva, and M. Yu. Artemov Institute of Physics and Applied Mathematics, Ural State University, Yekaterinburg, 620083 Russia e-mail: [email protected] Received April 2, 2002

Abstract—The mechanisms of the inhomogeneous broadening of the EPR spectra of exchange-coupled copper dimers, high-spin iron centers in lithium germanate, and off-center Tl2+ ions in potassium sulfate are analyzed. It is shown that the additional EPR signals observed for these materials when two EPR lines are nearly coincident can be due to averaging of a portion of the spin packets associated with these lines. © 2003 MAIK “Nauka/Interperiodica”.

1. It is well known that investigation of the electron 3Ð4 and 5Ð6 doublets) of the internal part of the quasi- paramagnetic resonance (EPR) spectrum near an acci- symmetrical spin packets corresponding to the two ini- dental crossing of electronic levels of a high-spin center tial EPR lines [7, 8]. The quasi-symmetrical structure allows one to detect fairly subtle effects associated with of these spin packets is due to the spread in the spin- noticeable splitting of degenerate levels due to very Hamiltonian parameters b21 and b43, which increases as weak interactions. These effects, for instance, carry the ferroelectric phase transition point is approached information on defect-induced low-symmetry distor- [9]. tions of the crystal which affect the centers under study. In a number of experimental papers [1–6], specific fea- By using computer simulation of the shape of the tures (an additional signal and a dip) were also reported experimentally observed EPR spectrum over a wide to be observed when two EPR lines were nearly coinci- temperature range around the structural phase transi- dent; these features could not be explained in terms of tion point, the temperature dependences of the inhomo- the spin Hamiltonian involving only averaged values of geneous-broadening parameters and of the relaxation the parameters. both within each doublet and between the doublets have been found. It should be noted that the feature of the In [1], for example, the dip in the absorption line of 2+ EPR spectrum observed in lead germanate is similar in two coincident EPR transitions in Ni ions in MgO nature to the cross-singular effects observed in NMR of was explained as resulting form static fluctuations of polycrystals [10], as well as to the effects that occur ε 2 the initial splitting of the form (Sz Ð 2/3), which can when relaxation without spin flipping is taken into be associated with random strains. It was shown that account between Kramers doublets originating from a fast cross relaxation between closely spaced spin pack- vibronic doublet split by a random strain field [11]. An ets of the two EPR transitions at the center of the additional signal similar in shape was observed by the absorption line can give rise to homogeneous broaden- authors of [3] in the vicinity of one of the crossings of ing of these spin packets and, as a consequence, to a dip the angular dependences of the EPR-line positions cor- in the absorption line. A similar effect has recently been responding to transitions in the triplet of dimeric observed in the EPR spectrum of axial centers Ni2+ in a exchange clusters [Cu(PU)5(ClO4)2]2 and by the single crystal of Zn(BF ) á 6H O in the case where the authors of [4] near three points of coincidence of EPR 4 2 2 transitions involving fine-structure spectral lines of initial splitting became zero under hydrostatic pressure 3+ [6]. Fe ions in the “weak”-ferroelectric lithium germanate (Li2Ge7O15, Tc = 283.5 K). Coexistence of the high- and In [2], an additional signal was observed between low-temperature EPR spectra of off-center Tl2+ ions in the two EPR lines of the 3Ð4 and 5Ð6 transitions of trig- 3+ K2SO4 near the crossover between the fast- and slow- onal centers Gd in ferroelectric lead germanate motion regimes, which can be thought of as the appear- (Pb5Ge3O11, Tc = 450 K) in the vicinity of the coinci- ance of an additional signal in the low-temperature dence of these two lines. The amplitude of the addi- EPR spectrum, was observed in [5]. tional signal increased as the distance between the initial lines decreased or as Tc was approached. The reason The objective of this paper is to interpret the effects for the appearance of this signal was found to be the observed in [3Ð5] in the vicinity of coincidence of EPR averaging (caused by relaxation transitions between the transitions in terms of the model of relaxation averag-

THE FEATURES OF THE EPR SPECTRUM IN THE VICINITY 47

10 0 ), arb. units

yy –20 D 0 , arb. units zz – dg xx / D (

res –40 d / dB res –10 dB –60 0 60 120 180 0 60 120 180 θ, deg θ, deg

Fig. 1. Angular dependence (in the zx plane) of the shift in Fig. 2. Angular dependence (in the zx plane) of the shift in the position of the EPR signal from the dimeric cluster the position of the EPR signal from the dimeric cluster caused by a change in (Dxx Ð Dyy). caused by a change in gzz. ing of part of the inhomogeneously broadened initial which may be arbitrary in character for different values lines [7, 8]. of the polar angle. 2. The angular dependences (in the zx plane) of the It was shown in [7, 8] that if the positions of spin EPR-line positions associated with the two transitions packets are approximately symmetric and a mechanism in the triplet of the centrosymmetric dimeric cluster causing a crossover between EPR transitions operates, [Cu(PU)5(ClO4)2]2 cross each other twice (at polar then an additional signal appears between the initial angles θ ≈ 36° and 107°) and are adequately described EPR lines in the vicinity of coincidence of these lines. by the spin Hamiltonian (with S = 1) To calculate the EPR spectrum, we use the following formula from [8], which is a generalization of the cor- H = βBgS⋅⋅+ ()1/2 SDS⋅⋅, (1) responding expression from [12] to the case where the β positions and widths of the EPR lines are affected by where is the Bohr magneton, B is the magnetic induc- relaxation transitions between these lines: tion, g is the g tensor, and D is the fine-structure tensor, whose parameters are defined in [3]. The additional () ()σ 2 θ ≈ IB = ∑ exp Ðmc/ 2 (anomalous) signal is observed in the vicinity of 0 ° θ θ m 36 and only for > 0. In a model where the triplet states are single-ion × ()⋅ ()Ð1 ⋅ ⋅ ()σ2 states, the observed signals can be interpreted as transi- ∑Re WAB 1 exp Ðnd/ 1 , tions of an ion subjected to the local field produced by n the spin of another ion. When the state of the other ion is changed, crossover occurs between the observed res- ) ) () Ω() ÐV V onance transitions. If the frequency of relaxation tran- A B ==B + I, I , (2) sitions is comparable to the spacing between the EPR VVÐ lines, the components of the EPR spectrum are shifted Ω() and broadened. As the relaxation rate increases further B and becomes noticeably higher than the line splitting, iα()andmcB++ Ð Ð0V the EPR spectrum is fully averaged. This motion- = 0 , induced destruction of the fine structure of the EPR 0 iα()bndmcBÐ + Ð Ð V spectrum was considered in detail in [12]. 0 Inhomogeneous electric fields and elastic strains, where W is a vector whose components are the proba- which always exist in actual crystals, give rise to a bilities of the initial transitions induced by a microwave spread in the spin-Hamiltonian parameters. Calcula- field; 1 is the unit vector; Œ is the matrix of the probabil- tions showed that a change in the components of the ities of relaxation transitions between the EPR lines; tensor D always causes symmetric shifts in the position Ω(B) is a matrix of the parameters of the initial spin of a pair of fine-structure lines, these shifts being packets (V0 is the probability of relaxation transitions); strongly dependent on the angle θ (Fig. 1). In contrast, a and b are the positions of the initial EPR lines; α = β the shifts in the pair of lines produced by variations in geff , 2n + 1, and 2m + 1 are the numbers of spin packets the diagonal components of the g tensor are of the same characterized by the symmetric and antisymmetric dis- sign (Fig. 2). A spread in the off-diagonal components tribution, respectively; d and c are the distances of the g tensor causes shifts in the EPR-line positions between neighboring spin packets; and V is the proba-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 48 VAZHENIN et al.

3. The situation in the EPR spectrum of Li2Ge7O15 : Fe3+ [4] is very similar to that for lead germanate doped with Gd3+ [2]. At room temperature, i.e., near the ferroelectric phase transition point, additional signals appear in the spectra of two of the four inequivalent high-spin 3+ θ0 + 3 Fe centers in the vicinity of three coincidences of EPR transitions (M M + 1, M + 2 M + 3); the intensity of these signals increases rapidly as a point of θ0 + 4 coincidence is approached. We note that, as in the case of lead germanate, anomalous signals were observed in the vicinity of coincidence of EPR transitions within doublets separated by an energy interval. It is obvious that the doublets within which EPR θ 0 + 4.5 transitions are observed are always coupled by relaxation transitions controlled by spinÐlattice or spinÐspin θ0 + 5 interactions. Our calculations showed that a change in any fine-structure parameter (second-rank tensor) causes symmetric shifts in the position of the EPR lines in the vicinity of their coincidence. Therefore, there 280 300 320 340 360 exist conditions for the formation of an additional sig- B, mT nal in the vicinity of coincidence of EPR transitions within different doublets. The intensity of the EPR line Fig. 3. Dependence of the first derivative of the EPR spectrum of the dimeric cluster on the polar angle in the zx plane. that forms when two initial lines exactly coincide is The parameters characterizing the shape of the initial sig- higher than the sum of the intensities of these two lines nals are presented in the text. [4], because the width of the resulting line is noticeably smaller. This last fact was indicated in [8] and is due to the averaging and to the close approach of symmetric bility of the relaxation transition between the EPR spin packets when the initial lines coincide. lines, which is equal to V0 in the case under study. 4. According to [5, 11], a thallium ion introduced into K SO (D ) occupies the potassium ion position of The calculated first derivative of I(B) is shown in 2 4 2h Fig. 3 for several values of the polar angle and demon- local symmetry Cs. The inversion and U2 operators take strates the ability of the model under consideration to one pair of the four translationally inequivalent potassium ion positions in the unit cell into the other pair; qualitatively describe the experimental data [3]. In 2+ order to calculate the EPR spectrum more accurately, therefore, the magnetic multiplicity of the Tl centers we need the experimental values of the spin-packet (effective symmetry Cs) is KM = 2. For these two types widths, as well as information on the character and of centers, the hyperfine interaction tensor A and the g tensor (which are of importance in the spin Hamilto- magnitude of the inhomogeneous broadening. In 2+ Eq. (2), the distribution of spin-packet intensity is nian for Tl ) in the system of the principal axes differ σ only in the sign of their xy components, which will lead assumed to be normal, with 1 = 4.3 mT for symmetric σ to different spectra in the xy plane for these two types spin packets and 2 = 0 for antisymmetric ones; we also of centers; the other off-diagonal components are zero. assumed that the relaxation transition probability is × 7 θ 2+ V = 9 10 Hz and that Bi( ) are linear functions in the Since the Tl ion is displaced from the symmetry range of interest. The parameters are adjusted so as to plane (off-center localization [5]), there are two types 2+ fit the experimental data on the widths and splittings of of Tl centers of symmetry C1 (related by mirror sym- the EPR signals. metry) and their spin Hamiltonians differ in the sign of the xz, zx, yz, and zy components of the A and g tensors. The absence of an additional signal for θ < θ = 36° 0 The splitting of an EPR line (corresponding to one and near the crossing of the angular dependences of the 2+ EPR-line positions at θ = 107° is most likely due to the of the Tl ion transitions within the triplet resulting 0 from hyperfine interaction with a large value of A ≈ large (relative) contribution from antisymmetric spin 115 GHz) observed in [5, 11] at temperatures below 40 packets to the inhomogeneous broadening of the EPR K is due to the contributions to the line position that are lines, because the anisotropy of the line broadening is linear in the off-diagonal g- and A-tensor components very high (Figs. 1, 2). As an illustration, Fig. 3 shows indicated above. For the orientation B || z || c, there are θ θ ° σ σ the spectrum at = 0 + 4 calculated for 1 = 2 = no such contributions and only one EPR line is σ σ 3 mT (dashed curve) and for 1 = 0, 2 = 4.3 mT (dotted observed. At θ ≠ 0 and ϕ = 0 (ϕ is the azimuth angle of curve). the magnetic field), in the coordinate system following

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 THE FEATURES OF THE EPR SPECTRUM IN THE VICINITY 49

38 K

38 K 35 K 35 K

35 K 33 K 33 K 32 K

32 K 30 K

624 628 610 620 630 B, mT Fig. 4. Temperature dependence of the EPR spectrum of the 2+ 2+ off-center Tl ion in K2SO4 for a small misorientation with Fig. 5. EPR spectrum of the off-center Tl ion in K2SO4 respect to B || z || c. for a large misorientation with respect to B || z || c. the field, the shifts in the position of the EPR line are components of the g and A tensors are small and of the found to be same sign for off-center Tl2+ positions of both types. ∆Bg∼ ± + g Bsin2θ/g , (3) Let us suppose that, at a small misorientation rela- xz zx eff tive to the orientation B || z || c, a quasi-symmetric struc- ∆ ∼ ± θ β BAzx +2Axz sin /geff ; (4) ture of spin packets with a Lorentzian intensity distribution is realized in the doublet under discussion (such these shifts for the inequivalent off-center positions of a shape is characteristic of line broadening due to ran- the Tl2+ ion differ in sign and lead to a splitting of the dom strains or chaotically distributed electric dipoles EPR line [5, 11]. When the direction of the magnetic [14]), with the distribution width being 2λ = 0.1 mT field lies in the zy plane, the components gzy, gyz, Ayz, and the spin-packet width being 0.1 mT. The relaxation and Azy are of importance in the formation of the EPR time of the off-center impurity in the double-well doublet. potential is assumed to be τ = 1/V = 5 × 10−13exp(357/T) The coexistence of the high- and low-temperature [15]. 2+ EPR spectra of Tl centers in K2SO4, mentioned in In this case, using a modified formula based on Section 1, is observed only in the case of small misori- Eq. (2), entations (θ < 5°) [5]. At a large deviation of the mag- Ð1 1 netic field direction from the z axis, only the typical IB()= ∑Re()WA⋅ ()B ⋅ 1 ------, (5) crossover between the regimes of fast and slow motion 1 + ()nd/λ 2 of the off-center ion [13] is observed in the temperature n dependence of the EPR spectrum. we obtain the EPR spectrum shown in Fig. 4 as a func- Let us consider the effect of a spread in the value of tion of temperature. This spectrum qualitatively different components of the g and A tensors on the explains the experimental data presented in [5, 13, 15]. arrangement of spin packets in the EPR spectrum of As the angle θ is increased, which corresponds to Tl2+ ions in off-center positions of both types. It is obvi- increasing the spacing between the initial EPR lines, ous that a deviation of an off-diagonal component from the peak intensity of the additional signal decreases its average value, in accordance with Eqs. (3) and (4), (Fig. 5). The values of the splitting in Figs. 4 and 5 are leads to EPR-line shifts that are different in sign for the taken to be roughly equal to the experimental values [5, two different types of off-center Tl2+ positions and, as a 13, 15]. result, a “quasi-symmetric” structure of spin packets For a large misorientation (when the lines virtually forms in the EPR doublet [7, 8]. The line shifts due to do not overlap) and a moderate hopping frequency of the terms quadratic in variations of the off-diagonal the off-center ion, only the spin packets of low intensity

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 50 VAZHENIN et al. are averaged and the additional peak is small. As the tions of the environment of a paramagnetic center can hopping frequency of the ion in the double-well poten- become fairly large under a quasi-hydrostatic pressure. tial increases with temperature, the initial lines approach each other and become broader, with the result that progressively more intense spin packets are ACKNOWLEDGMENTS averaged. The authors are grateful to V.K. Voronkova, V.N. Efimov, and A.A. Galeev for helpful discussions. Obviously, in the case where only initial lines with This study was supported by the US Civilian symmetric spin-packet structure are inhomogeneously Research and Development Foundation for the CIS broadened, the width of the additional signal will be of Countries, grant no. REC-005. the order of the spin-packet width [16]. However, as indicated in Section 2, the width of the additional peak essentially depends on the degree of inhomogeneous REFERENCES broadening characterized by antisymmetric packet 1. S. R. P. Smith, F. Dravnieks, and J. E. Wertz, Phys. Rev. structure. It is easily understood that static fluctuations 178 (2), 471 (1969). in the diagonal components of the tensors lead to EPR- 2. V. A. Vazhenin and K. M. Starichenko, Pis’ma Zh. Éksp. line shifts of the same sign for the different types of off- Teor. Fiz. 51 (8), 406 (1990) [JETP Lett. 51, 461 (1990)]. center ion positions (the shifts being proportional to 3. V. K. Voronkova, L. V. Mosina, Yu. V. Yablokov, et al., ∆ β 2θ ∆ β 2θ ∆ β 2θ ∆ 2θ gzz Bcos , gxx Bsin , gyy Bsin , Azzcos , Mol. Phys. 75 (6), 1275 (1992). ∆ 2θ ∆ 2θ Axxsin , and Ayysin ) and, therefore, give rise to 4. A. A. Galeev, N. M. Khasanova, A. V. Bykov, et al., line broadening characterized by antisymmetric Appl. Magn. Reson. 11, 61 (1996). arrangement of the spin packets in the EPR doublet. 5. G. V. Mamin and V. N. Efimov, Mod. Phys. Lett. B 12 Since the polar-angle dependence of some antisymmet- (22), 929 (1998). ric contributions (~sin2θ) is stronger than that of the 6. I. M. Krygin, A. A. Prokhorov, G. N. Neœlo, and symmetric contributions (~sinθcosθ), the former A. D. Prokhorov, Fiz. Tverd. Tela (St. Petersburg) 43 dependence can lead to a noticeable spread in the addi- (12), 2147 (2001) [Phys. Solid State 43, 2242 (2001)]. tional peak (see, e.g., the dashed curve in Fig. 5, which 7. V. A. Vazhenin, K. M. Starichenko, and A. D. Gorlov, is calculated with allowance for both the symmetric and Fiz. Tverd. Tela (St. Petersburg) 35 (9), 2449 (1993) antisymmetric broadening described by a Lorentzian [Phys. Solid State 35, 1214 (1993)]. 0.1 mT wide). This spread is likely the reason why no 8. V. A. Vazhenin, V. B. Guseva, and M. Yu. Artemov, Fiz. additional signal is observed for large misorientations Tverd. Tela (St. Petersburg) 44 (6), 1096 (2002) [Phys. relative to B || z [5, 13, 15]. Solid State 44, 1145 (2002)]. 9. V. A. Vazhenin, E. L. Rumyantsev, M. Yu. Artemov, and 5. Thus, the appearance of an additional signal in the K. M. Starichenko, Fiz. Tverd. Tela (St. Petersburg) 40 vicinity of accidental coincidence of the two EPR tran- (2), 321 (1998) [Phys. Solid State 40, 293 (1998)]. sitions considered in Sections 2Ð4 can be explained 10. É. P. Zeer, V. E. Zobov, and O. V. Falaleev, Novel Effects qualitatively in terms of selective averaging of the inter- in NMR of Polycrystals (Nauka, Novosibirsk, 1991). acting spin packets associated with these transitions. 11. J. R. Herrington, T. L. Estle, and L. A. Boatner, Phys. Computer simulation of experimental EPR spectra will Rev. B 13 (9), 2933 (1971). allow one to determine the parameters characterizing 12. A. Abragam, The Principles of Nuclear Magnetism the broadening and interaction of the EPR lines. (Clarendon, Oxford, 1961; Inostrannaya Literatura, Moscow, 1963). In closing, it is worth remarking on the anomalous 13. V. N. Efimov and G. V. Mamin, Mod. Phys. Lett. B 11 shape of an EPR signal resulting from a superposition (13), 579 (1997). of allowed and forbidden transitions of Ni2+ centers in 14. A. B. Roœtsin, in Radiospectroscopy of Solid: Collection Zn(BF4)2 á 6H2O [6]. Since these transitions involve of Scientific Works of Academy of Sciences of Ukraine accidentally degenerate states 〈+1| and 〈0|, even very (Naukova Dumka, Kiev, 1987), p. 89. weak low-symmetric distortions can affect the shape of 15. G. V. Mamin, Author’s Abstract of Candidate’s Disserta- the observed EPR line, which will make exact coinci- tion (Kazanskiœ Gos. Univ., Kazan, 1999). dence of the transitions impossible. For example, if the 16. J. E. Wertz and J. R. Bolton, Electronic Spin Resonance: spin Hamiltonian contains the term (1/3)b O with Elementary Theory and Practical Applications (McGraw- 21 21 Hill, New York, 1972; Mir, Moscow, 1975). b21 = 0.1 GHz, the resonance positions of the two transitions at the “level crossing” will differ by approximately 0.5 mT. In our opinion, low-symmetric distor- Translated by Yu. Epifanov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 51Ð56. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 51Ð55. Original Russian Text Copyright © 2003 by Ryabov, Gaœster, Zharikov. SEMICONDUCTORS AND DIELECTRICS

Electron Paramagnetic Resonance of Cr3+ÐLi+ Centers in (Cr,Li) : Mg2SiO4 Synthetic Forsterite I. D. Ryabov*, A. V. Gaœster**, and E. V. Zharikov** * Institute of the Lithosphere of Marginal Seas, Russian Academy of Sciences, Staromonetnyœ per. 22, Moscow, 119180 Russia ** Research Center of Laser Materials and Technologies, Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia Received January 14, 2002; in ﬁnal form, April 9, 2002

Abstract—Synthetic single crystals of chromium- and lithium-doped forsterite, namely, (Cr,Li) : Mg2SiO4, are studied using electron paramagnetic resonance spectroscopy. It is revealed that, apart from the known centers 3+ 3+ Cr (M1) and Cr (M2) (with local symmetries Ci and Cs, respectively), these crystals involve two new types 3+ 3+ of centers with C1 symmetry, namely, Cr (M1)' and Cr (M2)' centers. The standard parameters D and E in a zero magnetic ﬁeld [zero-ﬁeld splitting (ZFS) parameters expressed in GHz] and principal components of the g tensor are determined as follows: D = 31.35, E = 8.28, and g = (1.9797, 1.9801, 1.9759) for Cr3+(M1)' centers and D = 15.171, E = 2.283, and g = (1.9747, 1.9769, 1.9710) for Cr3+(M2)' centers. It is found that the low- symmetric effect of misalignment of the principal axes of the ZFS and g tensors most clearly manifests itself (i.e., its magnitude reaches 19°) in the case of Cr3+(M2)' centers. The structural models Cr3+(M1)ÐLi+(M2) and Cr3+(M2)ÐLi+(M1) are proposed for the Cr3+(M1)' and Cr3+(M2)' centers, respectively. The concentrations of both centers are determined. It is demonstrated that, upon the formation of Cr3+ÐLi+ ion pairs, the M1 position for chromium appears to be two times more preferable than the M2 position. Reasoning from the results 2 4 obtained, the R1 line (the E A2 transition) observed in the luminescence spectra of (Cr,Li) : Mg2SiO4 crystals in the vicinity of 699.6 nm is assigned to the Cr3+(M1)' center. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Earlier [16Ð19], it was found that the introduction of lithium dopants also brings about a considerable Chromium-doped forsterite (Cr : Mg2SiO4 or chro- change in the luminescence properties of chromium mium forsterite) has found wide use as an active forsterite. However, the origin of chromium active cen- medium in tunable near-infrared lasers [1Ð3]. Tetrava- ters in (Cr,Li) : Mg2SiO4 crystals cannot be elucidated lent chromium ions play the role of active laser centers. using optical spectroscopy alone. In the present work, a The Cr4+ ions located at tetrahedrally coordinated sili- detailed EPR investigation of two new Cr3+ centers con positions with Cs local symmetry have been formed in (Cr,Li) : Mg2SiO4 crystals was performed for recently identified using electron paramagnetic reso- the first time. Preliminary results were reported in our nance (EPR) spectroscopy [4Ð9]. According to the EPR recent work [20]. data [10, 11], chromium-doped forsterite also contains Cr3+ and Cr2+ ions in two structurally nonequivalent, octahedrally coordinated positions of Mg2+ ions, 2. SAMPLE PREPARATION namely, the M1 and M2 positions with local symmetries AND EXPERIMENTAL TECHNIQUE Ci and Cs, respectively. Bershov et al. [12Ð14] studied The experiments were performed with single crystals crystals of chromium- and aluminum-doped forsterite of (Cr,Li) : Mg2SiO4 and Cr : Mg2SiO4 (for comparison) (Cr,Al) : Mg2SiO4 and revealed that, in addition to indi- grown by the Czochralski method using a Kristall-2 vidual ions Cr3+(M1) and Cr3+(M2), these crystals con- apparatus. The batch contained MgO (OSCh 11-2) and 3+ 3+ 3+ 3+ tain Cr (M1)ÐAl and Cr (M2)ÐAl ion pairs (in this finely dispersed SiO2 (99.99+%, Wacker). Doping 3+ 4+ case, Al ions substitute for Si ions and play the role impurities in the form of Cr2O3 and Li2CO3 were intro- of charge compensators). More recently, Mass et al. duced into an iridium crucible (∅30 × 30 mm) immedi- [15] showed that the formation of these pairs in ately prior to the crystal growth. The pulling speed of (Cr,Al) : Mg2SiO4 crystals encourages quenching of crystals was 3 mm/h, and the rotational speed was Cr3+ luminescence and an increase in the relative inten- 12 rpm. The crystals were grown in a 100% Ar atmo- sity of Cr4+ luminescence in the near-IR range, which, sphere. in turn, results in an improvement of the laser properties Samples were prepared from grown crystals in the of chromium forsterite. form of cubes (5 × 5 × 5 or 3 × 3 × 3 mm in size). The

3+ (for other directions of the magnetic ﬁeld B in the crys- Cr (M2)' 3+ 0 Cr (M1) tallographic planes ab, bc, and ca) EPR lines taken 3+ 3+ 3+ Cr (M2) Cr (M2)' Cr (M2) from magnetically nonequivalent positions of paramagnetic centers with the magnetic multiplicity Km = 4. ×10 The concentration of paramagnetic centers in the Cr3+(M1)' ×10 studied samples was determined by comparing the integrated intensities of the EPR lines attributed to these centers and the DPPH reference sample. The double Cr3+( 1)' DPPH integration of the measured EPR lines was performed M according to a standard procedure described in [21]. Cr3+(M1)

3. RESULTS We performed a comparative analysis of the EPR 100 200 300 400 500 spectra recorded at different orientations of the mag- Magnetic field, mT netic field B0 in the ab, bc, and ca crystallographic 3+ planes for both Cr : Mg2SiO4 and (Cr,Li) : Mg2SiO4 Fig. 1. EPR spectrum of Cr centers in a (Cr,Li) : Mg2SiO4 || || crystals with approximately the same content of chro- crystal in the magnetic field B0 a (B1 b) at the frequency ν mium and different contents of lithium. It was revealed 0 = 9.52 GHz. The most intense narrow lines correspond- that the (Cr,Li) : Mg SiO crystals contain four types of ing to the resonance transitions 1 2 and 3 4 (indi- 2 4 cated by arrows in Fig. 3) are shown. Cr3+ paramagnetic centers (Fig. 1), namely, the two known centers Cr3+(M1) and Cr3+(M2) and two new centers that were preliminarily designated as Cr3+(M1)' edges of cubic samples were oriented parallel to the a, and Cr3+(M2)'. The number of magnetically nonequiva- b, and c crystallographic axes with an accuracy of bet- lent positions K for both new centers was found to be ter than 1.5°. Initially, the orientation of the crystals m equal to 4. Recall that, for the known centers Cr3+(M1) under investigation was determined from their growth 3+ facet and natural pleochroism (the crystals were green, and Cr (M2), these numbers are equal to 4 and 2, blue, and red in appearance in the a, b, and c directions, respectively [10]. respectively). Then, the crystal orientation was refined The angular dependences of the location of the EPR on a DRON-2 diffractometer. (In what follows, all crys- lines for 52Cr3+(M1)' and 52Cr3+(M2)' centers (Fig. 2) tallographic data will be given in the Pbnm setting.) can be adequately described by the spin Hamiltonian

The EPR spectra were recorded on a Varian E-115 HH= ZFS + HZe. (1) spectrometer operating in the X band (at a frequency of approximately 9.5 GHz) at room temperature with the Here, the first term characterizes the splitting of the 4 use of an E-231/E-232 duplex rectangular cavity (TE104 ground level ( A2) in a zero magnetic field [i.e., zero- mode) at a modulation frequency of 100 kHz. The static field splitting (ZFS)], magnetic field was determined using an IMI Sh1-1 ⋅⋅ 2 2 2 magnetic induction meter and a Ch3-38 frequency HZFS ==SDS DXSX ++DY SY DZSZ, (2) meter. The calibration was performed against a D-688 and the second term describes the electron Zeeman DPPH reference sample (Research Institute of Physi- interaction, cotechnical and Radio Engineering Measurements, × 16 β ⋅⋅ Russia) containing 4.04 10 paramagnetic centers HZe = B0 gS (3) (g = 2.0036). The crystal was rotated in the cavity about β( ,, two mutually perpendicular axes with the use of a goni- = gX' BX' SX' +).gY ' BY 'SY ' + gZ' BZ'SZ' ometer consisting of two circles, namely, a large hori- ° Table 1 presents the principal components and zontal circle with a vernier dial (0.1 ) and a small ver- directions of the principal axes of the D and g tensors tical circle with a dial. In order to obtain the angular calculated in terms of the LevenbergÐMarquardt dependences of the location of the EPR lines (in five nonlinear least-squares method [22] using 204 and degree intervals) in the ab, bc, and ca crystallographic 186 experimental points for Cr3+(M1)' and Cr3+(M2)' planes, the crystal under investigation was rotated centers, respectively. This table also lists the standard about the a, b, and c axes, respectively. The orientation ZFS parameters D = 3DZ/2 and E = (DX Ð DY)/2 and the of the crystal was controlled using the EPR spectrum so δ 4 splittings of the A2 orbital singlet into two Kramers that, if required, the sample could be slightly reoriented ε ε (to within 1.5°) with the use of the vertical circle of the doublets ±1/2 and ±3/2: goniometer through confluence of four (for the orienta- || δε ε 2 2 tion of the static magnetic field B0 a, b, or c) or two ==±3/2 Ð2±1/2 D +.3E (4)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTRON PARAMAGNETIC RESONANCE 53

ν 3+ It is known [23, 24] that, for a constant frequency 0 Cr (M1)'1–2(I) πν 3+ of the microwave ﬁeld B1cos2 0t and scanning of the Cr (M1)'1–2(II) 3+ magnetic ﬁeld B0, the integrated intensity Iint of the EPR Cr (M1)'3–4(I) 3+ lines is proportional to the concentration of paramag- Cr (M1)'3–4(II) 3+ Cr (M2)'1–2(I) netic centers in the sample; that is, 3+ 1000 Cr (M2)'1–2(II) 2 3+ NU Cr (M2)'3–4(I) ∝ ji 3+ Iint ------() -. (5) Cr (M2)'3–4(II) 2S + 1 U jj Ð Uii Theory Here, N is the concentration of paramagnetic centers in the studied sample and subscripts i and j refer to the 800 levels (with energies Ei and Ej, respectively) characterized by the resonance transition: 〈| ⋅⋅|〉 U ji = j B1 gSi /B1, (6) 600 ∂() E j Ð Ei U jj Ð Uii = ------∂β() B0 (7) ()〈| ⋅⋅|〉 〈| ⋅⋅|〉 = j B0 gSj Ð i B0 gSi /B0. 400 2

| | Magnetic field, mT The quantities Uji and Ujj Ð Uii (for the orientations of || || the magnetic ﬁeld vectors B0 a and B1 b calculated for Cr3+(M1)' and Cr3+(M2)' centers with S = 3/2 are given in Table 2. For the DPPH reference, we obtained | |2 2 200 S = 1/2, U21 = (gDPPH/2) , and U22 Ð U11 = gDPPH = 2.0036. The concentrations of Cr3+(M1)' and Cr3+(M2)' centers in (Cr,Li) : Mg2SiO4 crystals grown from melts 0 10 20 30 40 50 60 70 80 90 with different contents of lithium and approximately Angle, deg the same content of chromium were determined from the intensities of the EPR lines of the transitions 1 Fig. 2. Angular dependences of the location of the EPR 2 and 3 4 (indicated by arrows in Fig. 3) with due lines (resonance transitions 1 2 and 3 4) for two regard for the parameters given in Table 2. The results pairs (I and II) of magnetically nonequivalent positions of obtained are presented in Fig. 4. The mean values Cr3+(M1)' and Cr3+(M2)' centers in the crystallographic plane bc. Points are the experimental data, and solid lines (closed and open circles or closed and open squares ν correspond to the results of theoretical calculations ( 0 = depicted in Fig. 4) are taken into account in the approx- 9.5 GHz). imation (nonlinear regression) with the use of the relationship (represented by lines in Fig. 4)

N = A[]1 Ð exp()ÐBN , (8) (a) 4 C Li 40 3 80 (b) 4 where NC is the concentration of paramagnetic centers 0 Cr3+(M1)' 40 × 2 Cr3+(M2)' 3 of the same type (expressed in terms of N0 = 4.04 0 16 –40 2 10 ions/mg) and NLi is the lithium content in the melt 1

Energy, GHz –40 (wt %). As a result, we obtained the following parame- –80 1 ± × ± Ð1 00400 800 600 1400 ters: A = (0.093 0.003) N0 and B = 15 2 wt % for 3+ ± × Field, mT Cr (M1)' centers and A = (0.043 0.002) N0 and B = 16 ± 3 wt %Ð1 for Cr3+(M2)' centers. Fig. 3. Energy levels and resonance transitions at different magnetic ﬁeld strengths (mT): (a) 148.9 and 500.6 for 4. DISCUSSION Cr3+(M1)' and (b) 118.4, 369.5, 447.7, 666.7, 792.8, and 3+ ν || Paramagnetic centers belonging to one of the two 1318.6 for Cr (M2)'. 0 = 9.52 GHz. B0 a. new types were initially designated as Cr3+(M1)', because they are characterized by spin Hamiltonian 3+ ± parameters, directions of the principal axes of the ZFS 14] for Cr (M1) centers are given below: D = 30.6 ± ± tensor (Table 1), and angular dependences of the loca- 0.2 GHz, E = 8.48 0.05 GHz, gX = gY = 1.980 0.002, ± 3+ tion of the EPR lines (Fig. 2) that are close to the corre- and gZ = 1.974 0.002. Since the new centers Cr (M1)' sponding characteristics of Cr3+(M1) centers [10, 12Ð are observed only in chromium forsterite samples con- 14]. For comparison, the parameters taken from [12, taining lithium, it is reasonable to assume that

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 54 RYABOV et al.

Table 1. EPR parameters of Cr3+ÐLi+ centers in we were guided by similar considerations. These cen- (Cr,Li) : Mg2SiO4 forsterite ters are analogs of the known Cr3+(M2) centers with the 3+ 3+ following spin Hamiltonian parameters [12, 14]: D = Parameters Cr (M1)' Cr (M2)' ± ± ± 21.1 0.4 GHz, E = 2.60 0.05 GHz, gX = gZ = 1.970 ± DX, GHz Ð2.17(4) Ð2.774(1) 0.002, and gY = 1.979 0.002. It should be noted that, (58.1, 55.1, 51.1) (89.2, 30.2, 59.9) compared to the Cr3+(M1)ÐLi+ centers, the Cr3+(M2)Ð + DY, GHz Ð18.73(5) Ð7.340(1) Li centers are more strongly affected by the nearest + (147.8, 73.2, 63.4) (96.3, 119.9, 30.7) neighbor ions Li . This manifests itself in an appreciable decrease in the magnitude of the ZFS parameters DZ, GHz 20.90(5) 10.114(2) (Table 1), an increase in the magnetic multiplicity to (85.8, 140.2, 50.5) (6.4, 93.8, 84.9) Km = 4, and a rotation of the X and Y axes of the ZFS D, GHz 31.35(8) 15.171(3) tensor through an angle of ~30° with respect to the cor- E, GHz 8.28(3) 2.283(1) responding axes for Cr3+(M2) centers [13, 14]. δ, cmÐ1 2.300(5) 1.0459(2) It is known [30] that crystals with space group 16 gX' 1.9797(3) 1.9747(3) Pbnm(D2h ) contain no active centers other than those (54.6, 59.7, 50.3) (81.6, 15.1, 77.6) with local symmetries Ci, Cs, and C1. For symmetry Ci g 1.9801(4) 1.9769(3) 3+ Y' [as in the case of Cr (M1) centers] or C1, the number (144.6, 68.3, 63.5) (96.5, 101.5, 13.3) of magnetically nonequivalent positions of a particular

gZ' 1.9759(3) 1.9710(2) center is determined to be Km = 4 and the preferred (90.6, 141.2, 51.2) (10.7, 99.6, 85.4) direction is absent [30]. For symmetry Cs [as in the case of Cr3+(M2) and Cr3+(M2)ÐAl3+ centers], the number of Note: The sign of ZFS parameters is not determined. The directions (deg) of the principal axes of the D and g tensors (for magnetically nonequivalent positions of a particular one of four magnetically nonequivalent positions of each center in the forsterite is Km = 2 and there exists only center) are specified with respect to the crystallographic one preferred direction (the principal axis Y [13, 14]) axes a, b, and c. The root-mean-square errors in the last sig- along the c axis. It is evident that, when the Cr3+ ion nificant digit are given in parentheses. occupies the M2 position within the mirror-reflection plane in the neighborhood of an Li+ ion lying off this 3+ + Table 2. Parameters characterizing the integrated intensities plane, the newly formed center Cr (M2)ÐLi has a [see formula (5)] of EPR lines of the resonance transitions lower C1 symmetry instead of Cs symmetry. Similarly, 1 2 and 3 4 (indicated by arrows in Fig. 3) in the the stabilization of the Cr3+(M1)ÐLi+ ion pair is accom- || || ν magnetic field B0 a (B1 b) at the frequency 0 = 9.52 GHz panied by a lowering of the symmetry from Ci to C1 (in our opinion, the formation of an Li+ÐCr3+(M1)ÐLi+ ion Magnetic Transition | |2 Center Uji Ujj Ð Uii triad with C symmetry is unlikely). field B0, mT i j i Cr3+(M1)' 148.9 1 2 1.11 4.55 Our calculations (with structural data taken from [31]) of different interatomic distances and the corre- Cr3+(M1)' 500.6 3 4 4.54 1.25 sponding directions in the forsterite unit cell demon- 3+ Cr (M2)' 369.5 1 2 2.51 1.64 strated that the vector M2ÐM1 with direction cosines Cr3+(M2)' 118.4 3 4 0.22 5.74 (0.0126, 0.8842, 0.4669) and a magnitude of 0.3202 nm is very similar in orientation to the principal axis X of the D tensor for Cr3+(M2)' centers (Table 1); in this Cr3+(M1)ÐLi+ ion pairs are formed in these crystals. In case, the angle between the two directions is equal to particular, earlier EPR investigations revealed Cr3+ÐLi+ 3.1°. This gives grounds to assume that the Cr3+ and Li+ ion pairs in crystals of (Cr,Li) : Cs2CdCl4 [25] and ions are located at the adjacent positions M2 and M1 3+ (Cr,Li) : A2MF4 (A = K, Rb, or Cs and M = Zn or Cd) (Fig. 5). Hence, the hypothetical model of a Cr (M2)' [26, 27] (see also theoretical studies [28, 29]). In these center can be represented as Cr3+(M2)ÐLi+(M1). Let us crystals, two impurity ions, namely, trivalent chromium now consider a Cr3+(M1)' center. In this case also, Cr3+ and monovalent lithium Li+, substitute for two among all the vectors M1ÐMj (j = 1, 2) in the forsterite nearest neighbor bivalent ions M2+, thus retaining the unit cell, the vector M1ÐM2, having the same magni- electric charge balance. A similar situation, i.e., when tude but opposite direction, is close in orientation to the 3+ + principal axis Z of the ZFS tensor (the corresponding two ions of different valences (Cr and Li ) substitute ° for two bivalent ions Mg2+ simultaneously, occurs in angle is equal to 12.3 ) for one of the four magnetically chromium- and lithium-doped forsterite. nonequivalent positions of this center, except for the M1 and M2 positions separated by a distance of When identifying Cr3+(M2)ÐLi+ centers of the other 0.8632 nm (the angle is 9.2°). Therefore, we can make new type, which were initially designated as Cr3+(M2)', the assumption that, upon the formation of a Cr3+(M1)'

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 ELECTRON PARAMAGNETIC RESONANCE 55

0.10 Cr3+(M1)' O3 O3 M1 O3 M1 O3 M1 ions/mg

16 O2 2 O1 O2 O1 O2 M M2 Si 10

× Li + 3+ O3 O3 Y Cr (M2)' M1 O3 X O3 M1 0.05 M1 Cr3+ Cr3+(M1)'148.9 mT 3+ O2 O1 M2 O2 O1 O2 Cr (M1)'500.6 mT Si Z Cr3+(M2)'118.4 mT M2 Cr3+(M2)'369.5 mT Cr3+(M1)' O3 O3 c 3+ M1 O3 Cr (M2)' O3 M1 M1 Concentration, 4.04 0 0.1 0.2 0.3 0.4 b Li content, wt % Fig. 4. Concentration of Cr3+(M1)' and Cr3+(M2)' centers in × (Cr,Li) : Mg2SiO4 crystals (expressed in units of 4.04 1016 ions/mg) as a function of the lithium content in the melt. The chromium content in the melt is approximately equal to 0.05Ð0.06 wt %. Fig. 5. Structural model of a Cr3+(M2)ÐLi+(M1) center. center, as for the Cr3+(M2)' center, the same adjacent of Li+ ion impurities into the forsterite crystal brings Mg2+ positions (M1 and M2) are occupied by Cr3+ and about a weak disturbance of the crystal field experi- Li+ ions but in the reverse order; i.e., the structural enced by the nearest neighbor ions Cr3+(M1). As a con- model of a Cr3+(M1)' center can be represented as sequence, the coexistence of Cr3+(M1) and Cr3+(M1)' 3+ + Cr (M1)ÐLi (M2). centers in (Cr,Li) : Mg2SiO4 crystals should manifest itself in the luminescence spectra of these crystals in the Analysis of the concentrations of Cr3+(M1)' and form of two R lines (the 2E 4A transition) that are Cr3+(M2)' centers in chromium- and lithium-doped for- 1 2 δ ≈ Ð1 sterite crystals [see Fig. 4 and compare the constants A almost equally split ( 2.3 cm ) at sufficiently low for Cr3+(M1)' and Cr3+(M2)' in the text below formula temperatures. In actual fact, the luminescence spectra of (Cr,Li) : Mg SiO crystals at T = 77 K exhibit two (8)] revealed that, upon formation of Cr3+ÐLi+ ion pairs 2 4 narrow R1 lines in the vicinity of 692.7 and 699.6 nm in (Cr,Li) : Mg2SiO4 crystals, the M1 position for chromium is two times more preferable than the M2 posi- [19]. Reasoning from the results obtained in the present tion. According to different authors, this ratio for indi- work, the R1 line observed at 699.6 nm in the lumines- 3+ 3+ cence spectra of (Cr,Li) : Mg2SiO4 crystals was vidual ions Cr (M1) and Cr (M2) varies from 3 : 2 3+ [10] to 4 : 1 [32]. assigned to the Cr (M1)' center for the first time. In our consideration of the experimental results, we did not proceed a priori from the assumption that the ACKNOWLEDGMENTS principal axes of the D tensor are aligned with those of the g tensor, unlike the authors of all the works men- We would like to thank L.D. Iskhakova and tioned in the introduction and concerned with EPR A.G. Makarevich (Research Center of Fiber Optics, investigations of chromium ions in chromium-doped Institute of General Physics, Russian Academy of Sci- forsterite. Generally speaking, this assumption is incor- ences) for their assistance in preparing the oriented rect because low-symmetric effects can manifest them- samples used in our investigation. selves in the case of centers with local symmetries Ci, This work was supported by the Russian Foundation Cs, and C1 [30]. It is found that the low-symmetric for Basic Research, project nos. 00-15-96715, 00-02- effect of misalignment of the aforementioned axes is 16103, and 01-05-65348. most clearly pronounced for Cr3+(M2)ÐLi+(M1) centers (Table 1): the misalignment of the principal axes X and X' (Y and Y') amounts to approximately 19°. REFERENCES

In the luminescence spectra of Cr : Mg2SiO4 crystals 1. V. Petricevic, S. K. Gayen, and R. R. Alfano, Appl. Phys. Lett. 53 (26), 2590 (1988). at a relatively low temperature, the R1 line observed in the vicinity of 692.7 nm is attributed to Cr3+(M1) cen- 2. H. R. Verdun, L. M. Thomas, D. M. Andrauskas, et al., ters [33, 34]. It turned out that the ZFS parameters for Appl. Phys. Lett. 53 (26), 2593 (1988). centers Cr3+(M1) [12, 14] and Cr3+(M1)' (Table 1) are 3. V. G. Baryshevskiœ, M. V. Korzhik, A. E. Kimaev, et al., close in magnitude. This suggests that the introduction Zh. Prikl. Spektrosk. 53 (1), 7 (1990).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 56 RYABOV et al.

4. V. G. Baryshevski, M. V. Korzhik, M. G. Livshitz, et al., 20. I. D. Ryabov, A. V. Gaister, and E. V. Zharikov, Bull. Soc. OSA Proc. Adv. Solid-State Lasers 10, 26 (1991). Fr. Mineral. Cristallogr. 13 (3), 106 (2001). 5. M. H. Garrett, V. H. Chan, H. P. Jenssen, et al., OSA 21. T.-T. Chang, Magn. Reson. Rev. 9 (1Ð3), 65 (1984). Proc. Adv. Solid-State Lasers 10, 76 (1991). 22. W. H. Press, B. P. Flannery, S. A. Teukolsky, and 6. K. R. Hoffman, J. Casas-Gonzalez, S. M. Jacobsen, and W. T. Vetterling, Numerical Recipes: The Art of Scien- W. M. Yen, Phys. Rev. B 44 (22), 12589 (1991). tiﬁc Computing (Cambridge Univ. Press, Cambridge, 7. M. L. Meilman and M. G. Livshitz, OSA Proc. Adv. 1986), p. 523. Solid-State Lasers 13, 39 (1992). 23. T.-T. Chang, D. Foster, and A. H. Kahn, J. Res. Natl. Bur. 8. M. H. Whitmore, A. Sacra, and D. J. Singel, J. Chem. Stand. 83 (2), 133 (1978). Phys. 98 (5), 3656 (1993). 24. C. E. Forbes, J. Chem. Phys. 79 (6), 2590 (1983). 9. D. E. Budil, D. G. Park, J. M. Burlitch, et al., J. Chem. 25. D. Kay and G. L. McPherson, J. Phys. C 14 (22), 3247 Phys. 101 (5), 3538 (1994). (1981). 10. H. Rager, Phys. Chem. Miner. 1 (4), 371 (1977). 26. H. Takeuchi and M. Arakawa, J. Phys. Soc. Jpn. 52 (1), 279 (1983). 11. V. F. Tarasov, G. S. Shakurov, and A. N. Gavrilenko, Fiz. Tverd. Tela (St. Petersburg) 37 (2), 499 (1995) [Phys. 27. M. Arakawa, H. Ebisu, and H. Takeuchi, J. Phys. Soc. Solid State 37, 270 (1995)]. Jpn. 55 (8), 2853 (1986). 12. L. V. Bershov, R. M. Mineeva, A. V. Speranskiœ, and 28. M.-L. Du and M.-G. Zhao, Solid State Commun. 76 (4), S. Hafner, Dokl. Akad. Nauk SSSR 260 (1), 191 (1981). 565 (1990). 13. L. V. Bershov, R. M. Mineeva, A. V. Speranskiœ, and 29. S.-Y. Wu and W.-C. Zheng, Physica B (Amsterdam) 262 S. Hafner, Mineral. Zh., No. 3, 62 (1981). (1Ð2), 84 (1999). 14. L. V. Bershov, J.-M. Gaite, S. S. Hafner, and H. Rager, 30. M. L. Meœl’man and M. I. Samoœlovich, Introduction to Phys. Chem. Miner. 9 (3Ð4), 95 (1983). the Spectroscopy of Electron Paramagnetic Resonance of Activated Single Crystals (Atomizdat, Moscow, 15. J. L. Mass, J. M. Burlitch, D. E. Budil, et al., Chem. 1977). Mater. 7 (5), 1008 (1995). 31. J. R. Smyth and R. M. Hazen, Am. Mineral. 58 (7Ð8), 16. N. Nishide, Y. Segawa, P. H. Kim, et al., Resa Kagaku 588 (1973). Kenkyu, No. 7, 89 (1985). 32. J. L. Mass, J. M. Burlitch, D. E. Budil, et al., Chem. 17. N. Nishide, Y. Segawa, P. H. Kim, and S. Namba, Resa Mater. 7 (5), 1008 (1995). Kagaku Kenkyu, No. 8, 97 (1986). 33. W. Jia, H. Liu, S. Jaffe, et al., Phys. Rev. B 43 (7), 5234 18. A. Sugimoto, Y. Nobe, T. Yamazaki, et al., Phys. Chem. (1991). Miner. 24 (5), 333 (1997). 34. T. J. Glynn, G. F. Imbusch, and G. Walker, J. Lumin. 48Ð 19. A. V. Gaister, V. A. Smirnov, and E. V. Zharikov, in Pro- 49 (2), 541 (1991). ceedings of the Fourth International Conference “Single Crystal Growth and Heat & Mass Transfer,” Obninsk, 2001, Vol. 2, p. 272. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 57Ð60. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 56Ð59. Original Russian Text Copyright © 2003 by Kumzerov, Parfen’eva, Smirnov, Misiorek, Mucha, Jezowski.

SEMICONDUCTORS AND DIELECTRICS

Thermal Conductivity of Crystalline Chrysotile Asbestos Yu. A. Kumzerov*, L. S. Parfen’eva*, I. A. Smirnov*, H. Misiorek**, J. Mucha**, and A. Jezowski** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 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 April 9, 2002

Abstract—The thermal conductivity of crystalline chrysotile asbestos made up of hollow tubular Mg3Si2O5(OH)4 ﬁlaments is measured in the range 5–300 K. The paper discusses the possibility of using this material in studies of the thermal conductivity of thin ﬁlaments of metals and semiconductors incorporated into the channels of crystalline chrysotile asbestos tubes. © 2003 MAIK “Nauka/Interperiodica”.

Considerable effort has been recently focused in 7.32 Å, and β = 93° [14]. The a axis is directed along leading research laboratories of Europe, the USA, and the tube channels (Fig. 1d). The tube packing structure Japan, on studies of the physical properties of small is close to hexagonal [2]. metal and semiconductor particles in the form of clus- The porosity of chrysotile asbestos (the ratio of the ters, cluster crystals, ultrathin filaments, and nanocom- channel to the total sample volume) is ~5Ð6% [2]. The posites; these studies were undertaken with the inten- porous structure of chrysotile asbestos was directly tion to develop new materials for use in present-day studied using electron microscopy in [14]. technology and nanoelectronics and to investigate their fundamental physical characteristics [1, 2]. The channels of the chrysotile asbestos tubes can be filled under pressure by molten Hg, Sn, Bi, In, Pb, Se, Studying the physical properties of these objects in and Te [2, 15] to form regular systems of ultrathin par- the free state is usually impossible. Therefore, they are allel filaments, which do not interact with one another loaded in nanopores or narrow channels of various due to their large separation. These filaments are simi- dielectric porous host matrices, more specifically, lar, in many respects, to quantum wires. porous glasses, zeolites, opals, and asbestos. The physical properties of such thin metal and semi- Here, we deal with the dielectric host matrix of crys- conductor filaments that fill the channels of chrysotile talline chrysotile asbestos. asbestos tubes have been extensively studied in recent Chrysotile asbestos, the hydrated magnesium sili- years. The effect of channel size on the superconduct- cate Mg3Si2O5(OH)4, is a filamentary material of the ing transition temperature, the melting and solidifica- serpentine group with the following chemical composition points, currentÐvoltage characteristics, electrical tion (which can vary from one deposit to another): ~37Ð resistivity, heat capacity, and other properties of mate- 44% SiO2, ~39Ð44% MgO, and ~12Ð15% H2O (bound rials loaded in chrysotile asbestos have been investi- water). It may have Fe, Al, Ca, Ni, Mn, and Na impuri- gated (see, e.g., [16Ð24]). ties. The physics of quasi-one-dimensional metal and Chrysotile asbestos has a remarkable, highly semiconductor systems has aroused considerable unusual structure. It consists essentially of structural research interest, because such objects possess proper- layers confined on the inside by a framework of silica ties that are radically different from those of bulk mate- and on the outside by a framework of magnesium rials [1, 2]. hydroxide [2Ð5] (Fig. 1a). Because of the inner frame- The behavior of thermal conductivity κ of ultrathin work being smaller in size than the outer one, the filaments (quantum wires) has also attracted recent chrysotile asbestos layers tend to roll into cylinders attention (primarily that of theoreticians). However, (tubes), with the silica layer on the inner side. Such experimental data on the magnitude of κ of metals and tubes have an outer diameter d1 ~ 300–500 Å and an semiconductors incorporated in asbestos channels are inner diameter d2 ~ 20–150 Å [2, 4] (Figs. 1b, 1c). The presently lacking. There is likewise no information space between the tubes (denoted by 1 in Fig. 1b) is available on the thermal conductivity of crystalline usually filled by an amorphous mass of the tube mate- chrysotile asbestos over a broad temperature range. rial. On the whole, the crystal lattice of the asbestos lay- This prompted us to attempt to determine the magni- ers belongs to the monoclinic system [4, 6Ð14] with the tude of thermal conductivity of crystalline chrysotile following parameters: a = 5.30 Å, b = 9.10 Å, c = asbestos at temperatures from 5 to 300 K and to esti-

58 KUMZEROV et al.

(a) OH

Si O

(b) (c)

(d)

0 b c

Fig. 1. (a) Crystal layers forming the filamentary structure of chrysotile asbestos [5]; (b) system of closely packed tubular filaments of chrysotile asbestos [2]: (1) space between asbestos tubes; (c) schematic representation of a cut through the tubular filaments of chrysotile asbestos [2]; and (d) diagram illustrating the orientation of the unit-cell axes of chrysotile asbestos with respect to the curved layers of the structure [4]. mate the possibility of its use as a dielectric host matrix the sample at this temperature [25]. After the annealing, in studies of κ of thin metal and semiconductor fila- a thin layer of varnish was deposited on the sample end ments incorporated in the nanochannels of asbestos faces to prevent penetration of atmospheric water into tubes. the asbestos tubes when mounting the sample in the We measured the thermal conductivity of a sample experimental setup. The thermal conductivity of asbes- of natural “brittle” chrysotile asbestos from an Uzbeki- tos was measured within the temperature range 5Ð300 stan deposit. The sample size was 5.5 × 6.5 × 12 mm. K on a setup similar to that employed in [26]. Its inner and outer tube diameters were d2 ~ 50 Å and The experimental data obtained on the κ of crystal- d1 = 300 Å, respectively. line chrysotile asbestos are displayed in Fig. 2. Since Before the κ measurements, the sample was chrysotile asbestos is an insulator, the experimentally κ κ annealed in air at ~150°C to remove the water possibly measured is actually the lattice heat conductivity ph κ κ present in the tubes. The bound water was not lost from ( = ph).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

THERMAL CONDUCTIVITY OF CRYSTALLINE CHRYSOTILE ASBESTOS 59

8 We note two features observed in the κ(T) dependence of chrysotile asbestos: (i) the presence of a maximum in the thermal conductivity at a fairly high temperature, ~150 K, and (ii) the relatively small magnitude of κ at low temperatures (T Շ 20Ð30 K). 6 Figure 3 compares our data on κ of crystalline chrysotile asbestos (curve 1) with the literature data on κ ph of single crystals of MgO [27] (2) and SiO2 [28] (3) (the main components of the chrysotile asbestos 4 Mg3Si2O5(OH)4), as well as information on the thermal

, W/m K conductivity of asbestos wool (4) [29]. Curve 5 was κ derived from data on crystalline chrysotile asbestos calculated using a relation from [30] with account taken of the porosity of our sample (~5%). 2 The appearance of a maximum in κ(T) of crystalline chrysotile asbestos at fairly high temperatures, as compared, for instance, to those for MgO and SiO2 single crystals (Fig. 3), can be qualitatively accounted for by the size effect, whereby the phonon mean free path 0 100 200 300 , K becomes comparable to the thickness of the asbestos T tube wall. In the sample studied by us, this thickness Fig. 2. Thermal conductivity of crystalline chrysotile asbestos plotted as a function of temperature. The heat ﬂux was was ~100–125 Å. Unfortunately, we did not succeed in directed along the asbestos tubes (along the a direction). ﬁnding any literature data on the sound velocity v and

2000 3 1000 2000 500 1000 2 500 200 200 100 100 50 2 3 50 20 20 10

, W/m K 10

κ 5 , W/m K

κ 5 2 2 1 1 5 1 0.5 4 0.5 1 0.2 4 0.2 0.1 0.1

1 2 4 10 20 50 100 200 300 T, K 1 2 4 10 20 50 100 200 300 T, K Fig. 3. Temperature dependence of the thermal conductivity of (1) crystalline chrysotile asbestos, (2) single-crystal MgO [27], (3) SiO2 along the c axis [28], and (4) asbestos wool [29]; (5) thermal conductivity of crystalline chrysotile Fig. 4. Temperature dependence of the thermal conductivity asbestos calculated with inclusion of sample porosity. Ver- of (1) crystalline chrysotile asbestos, (2) single crystals of tical dashed lines on the curves specify the temperature of SiO2 [28] and (3) of NaCl [28], and (4) crystalline NaCl for the maximum for the corresponding material. l = const = 100 Å [31].

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

60 KUMZEROV et al. specific heat Cv(T) for crystalline chrysotile asbestos, 8. E. Aruya, Miner. Mag. 27, 188 (1945). and, therefore, we could not calculate the mean free 9. N. N. Padurow, Acta Crystallogr. 3, 200 (1950). path l in this material from the well-known expression 10. E. J. W. Whittaker, Acta Crystallogr. 9, 855 (1956). 3κ 11. G. W. Brindley, X-ray Identification of Crystal Struc- l =.------ph- tures of Clay Minerals (Mineralogical Society, London, Cv v 1952), Chap. 2. κ 12. H. Jagodzinski and G. Kunze, Neues Jahrb. Mineral., Figure 4 qualitatively compares the behavior of ph Monatsh., No. 7, 134 (1954). for chrysotile asbestos (1) and for an NaCl single crys- 13. J. E. W. Whittaker and J. Zussmann, Miner. Mag. 31, 107 tal experiencing the size effect, with l = const = 100 Å κ (1956). [31] (curves 3, 4). Also shown are the data on ph for a 14. K. Yada, Acta Crystallogr. 23, 704 (1967). SiO2 single crystal [28] (2). The conclusion as to the κ 15. V. N. Bogomolov, Fiz. Tverd. Tela (Leningrad) 13 (3), (T) of crystalline chrysotile asbestos being dominated 815 (1971) [Sov. Phys. Solid State 13, 672 (1971)]. by the size effect can be drawn, however, only after 16. V. N. Bogomolov, V. K. Krivosheev, and Yu. A. Kumze- experimental data are obtained on Cv(T) and v . rov, Fiz. Tverd. Tela (Leningrad) 13 (5), 3720 (1971) A comparison of our data on κ(T) of crystalline [Sov. Phys. Solid State 13, 3148 (1971)]. chrysotile asbestos with the available information on 17. V. N. Bogomolov and V. K. Krivosheev, Fiz. Tverd. Tela the thermal conductivity of pure metals and some semi- (Leningrad) 14 (4), 1238 (1972) [Sov. Phys. Solid State conductors [28] allows us to conclude that this material 14, 1059 (1972)]. can be used as a dielectric host matrix over a fairly 18. V. N. Bogomolov and Yu. A. Kumzerov, Pis’ma Zh. broad temperature range (where κ of asbestos is lower Éksp. Teor. Fiz. 21 (7), 434 (1975) [JETP Lett. 21, 198 by several orders of magnitude than that of the filler (1975)]. materials) in studies of the thermal conductivity of thin 19. V. N. Bogomolov, E. V. Kolla, Yu. A. Kumzerov, et al., filaments of metals and semiconductors incorporated Solid State Commun. 35 (4), 363 (1980). into asbestos tubes. We hope to obtain evidence sup- 20. V. N. Bogomolov, B. E. Kvyatkovskiœ, E. V. Kolla, et al., porting this conclusion in the experiments recently Fiz. Tverd. Tela (Leningrad) 23 (7), 2173 (1981) [Sov. begun at our laboratories. Phys. Solid State 23, 1271 (1981)]. 21. V. N. Bogomolov, Yu. A. Kumzerov, and V. A. Pimenov, Fiz. Tverd. Tela (Leningrad) 23 (8), 2506 (1981) [Sov. ACKNOWLEDGMENTS Phys. Solid State 23, 1471 (1981)]. 22. V. N. Bogomolov, E. V. Kolla, and Yu. A. Kumzerov, This work was conducted within bilateral agree- Solid State Commun. 46 (2), 159 (1983). ments between the Russian and Polish Academies of 23. V. N. Bogomolov, E. V. Kolla, and Yu. A. Kumzerov, Sciences and was supported by the Russian Foundation Solid State Commun. 46 (5), 383 (1983). for Basic Research, project no. 02-02-17657. 24. V. N. Bogomolov, K. V. Kolla, and Yu. A. Kumzerov, Pis’ma Zh. Éksp. Teor. Fiz. 41 (1), 28 (1985) [JETP Lett. REFERENCES 41, 34 (1985)]. 25. L. A. Drobyshev and Ya. Ya. Govorova, Kristallografiya 1. V. N. Bogomolov and Yu. A. Kumzerov, Preprint 16 (3), 544 (1971) [Sov. Phys. Crystallogr. 16, 460 No. 971, FTI im. A. F. Ioffe AN SSSR (Ioffe Physicote- (1971)]. chnical Institute, Academy of Sciences of USSR, Lenin- grad, 1985). 26. A. Jezowski, J. Mucha, and G. Pompe, J. Phys. D 7, 1247 (1974). 2. Yu. A. Kumzerov, in Nanostructured Films and Coat- 27. I. P. Morton and M. F. Lewis, Phys. Rev. B 3 (2), 552 ings, Ed. by Gan-Moog Chow, I. A. Ovid’ko, and (1971). T. Tsakalakos (Kluwer, Dordrecht, 2000), NATO ASI Series, Partnership Sub-series 3: High Technology, 28. Heat Conductivity of Solids: A Handbook, Ed. by Vol. 78, p. 63. F. S. Okhotin (Énergoatomizdat, Moscow, 1984). 3. V. V. Bekhterev and V. I. Solomonov, Neorg. Mater. 31 29. Thermophysical Properties of Materials under Low (4), 567 (1995). Temperatures: A Handbook (Mashinostroenie, Moscow, 1982). 4. W. L. Bragg and G. F. Claringbull, Crystal Structure of Minerals (Bell, London, 1965; Mir, Moscow, 1967). 30. E. Ya. Litovskiœ, Izv. Akad. Nauk SSSR, Neorg. Mater. 16 (3), 559 (1980). 5. V. N. Bogomolov, Usp. Fiz. Nauk 124 (1), 171 (1978) [Sov. Phys. Usp. 21, 77 (1978)]. 31. V. N. Bogomolov, N. F. Kartenko, D. A. Kurdyukov, et al., Fiz. Tverd. Tela (St. Petersburg) 41 (2), 348 (1999) 6. B. E. Warren and W. L. Bragg, Z. Kristallogr. 76, 201 [Phys. Solid State 41, 313 (1999)]. (1930). 7. B. E. Warren and K. W. Herring, Phys. Rev. 59, 925 (1941). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 6Ð10. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 8Ð11. Original Russian Text Copyright © 2003 by Anshukova, Bulychev, Golovashkin, Ivanova, Krynetskiœ, Rusakov. METALS AND SUPERCONDUCTORS

Low-Temperature Anomalies in MgB2 Thermal Expansion N. V. Anshukova*, B. M. Bulychev**, A. I. Golovashkin*, L. I. Ivanova***, I. B. Krynetskiœ**, and A. P. Rusakov*** * Lebedev Physical Institute, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia e-mail: [email protected] ** Moscow State University, Vorob’evy gory, Moscow, 119899 Russia *** Moscow State Institute of Steel and Alloys, Leninskiœ pr. 4, Moscow, 117936 Russia Received January 31, 2002

α Abstract—The thermal expansion coefficient (T) of MgB2 was measured at low temperatures both in a zero magnetic field and at H = 36 kOe. As in the oxide HTSCs, a region of anomalous (negative) thermal expansion and a strong effect of magnetic field on α(T) were revealed. The results obtained indicate the anomalous properties of MgB2 and the oxide HTSCs to follow a common pattern. © 2003 MAIK “Nauka/Interperiodica”.

The discovery of superconductivity with a critical 5 h under atmospheric pressure. The single-phase mate- temperature Tc = 40 K in MgB2 was recently announced rial thus obtained contained more than 98% of the main [1]. The new superconductor is highly promising for compound. MgB2 pellets were sintered under a pres- potential application. The currently available informa- sure of 50 kbar at a temperature of 950Ð1000°C. The tion on the properties of MgB2 allow only controversial density of the sintered samples was 97% of the figure conclusions to be drawn as to the nature of supercon- obtained by x-ray diffraction. The x-ray diffractogram ductivity in this compound [2Ð5]; in other words, there of a single-phase sample recorded on a DRON-4 dif- are still no grounds to unambiguously decide whether fractometer is in full agreement with the figures listed MgB2 belongs to conventional or oxide high-tempera- in the ASTM standard catalog. It is on such samples ture superconductors (HTSCs). that we measured the specific heat and other characteristics and revealed a jump in the specific heat at the The oxide HTSCs are known to exhibit a number of superconducting transition at T ≈ 39 K. characteristic anomalies in their properties. In particular, thermal expansion of high-quality oxide HTSCs The sample chosen for dilatometric measurements reveals a low-temperature anomaly, namely, a negative was a 4-mm-high cylinder 3 mm in diameter. The thermal expansion coefficient α [6]. Furthermore, mag- change in sample length ∆L/L was measured with a netic field was found to exert a strong effect on the strain pickup with a sensitivity of ~10Ð7 [7]. The mag- anomalous temperature dependence α(T) [7]. These netic field was oriented parallel to the direction of strain anomalies are not observed in conventional supercon- measurement. The setup was calibrated through ductors. Thus, measurement of these characteristics in repeated measurement of the temperature dependence MgB2 will make it possible to establish to which group of thermal expansion α(T) of rare-earth oxide samples. of superconductors this compound belongs, which is an These measurements showed α(T) of the above sam- important aspect in revealing the mechanism of its ples to follow a normal course throughout the low-tem- superconductivity. perature region covered; i.e., the thermal expansion α We report here on measurements of the temperature coefficient of these compounds is positive ( > 0) and does not reverse sign. dependence of thermal expansion of MgB2 both in a zero magnetic field and in a field H = 36 kOe. In addi- Figure 1a displays the temperature dependence of tion, we measured the dependence of thermal expan- the quantity ∆L/L (L is the sample length) obtained in ≈ sion on a magnetic field up to H 42 kOe at a fixed tem- this study on MgB2 for H = 0. Also presented for com- perature. At low temperatures for H = 0, a region of parison in Fig. 1b are data for YBa2Cu3O7 Ð x [8], negative thermal expansion characteristic of the oxide Bi2Sr2CaCu2O8 [9], La2 Ð xSrxCuO4 (x = 0.1) [7], and HTSCs was found. It was also found that the magnetic Ba1 Ð xKxBiO3 (x = 0.13) [7] obtained earlier. For MgB2, field weakens this anomaly. ∆L/L < 0 in the interval 7 ≤ T ≤ 16.5 K. The thermal α MgB2 samples were prepared by hot pressing. The expansion coefficient = (1/L)dL/dT is negative in the starting magnesium diboride was synthesized by react- temperature interval of approximately 7Ð11 K. As seen ing metallic magnesium with elemental boron using the from Fig. 1b, oxide HTSCs also exhibit negative ther- α standard technique at a temperature of 950Ð1000°C for mal expansion (T) at low temperatures; i.e., MgB2

LOW-TEMPERATURE ANOMALIES 7

(b) 1 3.8880 Å , b 3.8865 100 200 (a) T, K 1 3 2 MgB2 5 × 10–4 300 c / c 2 c0

–4 100 200 T, K , 10 L /

L 4 3 –5 ∆ 1 2 , 10 L / 0 L 10 20 ∆ –2 T, K 0 10 20 30 40 50 60 4 4 T, K –1 K 0 –7 20 40 60 80 10

, –4 T, K α –8

∆ Fig. 1. (a) Temperature dependence of thermal expansion L/L for MgB2 and (b) its comparison with the data obtained for other HTSCs. (1) YBa2Cu3O7 Ð x (b is the lattice constant along the b axis) [8]; (2) Bi2Sr2CaCu2O8 (c is the lattice constant along the c α axis, c300 is the lattice constant at T = 300 K) [9]; (3) La2 Ð xSrxCuO4 (x = 0.1, ab plane) [7]; and (4) Ba1 Ð xKxBiO3 (x = 0.13, is the thermal expansion coefficient) [7]. possesses the same anomalous property (α < 0) as the behavior cannot be explained in terms of magnetostric- oxide HTSCs whose characteristics are plotted in tion. Magnetostriction has been recently studied in con- Fig. 1b. siderable detail for T < Tc on the HTSC compound Ba K BiO [10]. It was shown in [10] that samples Figure 2a illustrates the effect of the magnetic field 0.6 0.4 3 H = 36 kOe on the temperature dependence of ∆L/L for contract as the magnetic field is increased to 5 T at a fixed temperature, which means that ∆L/L is negative MgB2. For comparison, Fig. 2b shows the mag- ∆ and grows in absolute magnitude. The magnetic-field netic-field dependence of L/L obtained earlier for ∆ Ba K BiO and La Sr CuO samples [7]. It is seen dependences of L/L obtained by us are in full agree- 0.6 0.4 3 1.9 0.1 4 ment with this behavior for temperatures 16.5 K < T < that the magnetic field H ≈ 40 kOe exerts an anoma- T . Thus, the ∆L/L vs. H dependence plotted in Fig. 3 lously strong effect on α(T) of this class of compounds c at low temperatures. for T = 18.8, 28.2, and 37.5 K can be accounted for by magnetostriction. At low temperatures, however (see, Figure 3 plots ∆L/L vs. magnetic field H depen- for instance, the curve for T = 12.1 K in Fig. 3), the sign ∆ dences obtained at different temperatures for MgB2. We of the L/L vs. H dependence is opposite to that of readily see that the dependence of ∆L/L on H measured magnetostriction; i.e., ∆L/L > 0 and grows with H. at T = 12.1 K, i.e., in the region of negative values of Thus, in addition to the magnetostriction, MgB2 exhib- ∆L/L (T ≤ 16.5 K), differs qualitatively from those its an effect of opposite sign, which becomes stronger obtained for T > 16.5 K (this case is illustrated using for T < 16.5 K. Both these effects have also been only the three curves measured at T = 18.8, 28.2, and observed in a number of other HTSCs [7]. 37.5 K). The behavior of the curves measured at T > α 16.5 K can be accounted for by the effect of magneto- For T > Tc, where MgB2 is in a normal state, the (T) striction, whereas at T = 12.1 K, the variation of ∆L/L and α(H) relations resemble those of conventional met- with magnetic field has opposite sign. Therefore, this als. Samples of the Ba1 Ð xKxBiO3 system with metallic

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 8 ANSHUKOVA et al.

(b) (a) 4 2

–5 La Sr CuO 2 1.9 0.1 4 1

MgB , 10

2 L 15 / 0 L 10 20

1 ∆ –2 T, K 10 2 –5

, 10 4 L

/ 2

L 5 ∆

–5 1 Ba K BiO 2 0.6 0.4 3 3 , 10

0 L 10 20 30 / L 0 T, K ∆ 515 1 T, K –5 –1

Fig. 2. Effect of magnetic ﬁeld on the temperature dependence of thermal expansion. (a) MgB2 [(1) H = 0, (2) H = 36 kOe]; (b) ≈ La1.9Sr0.1CuO4 [(1) H = 0, (2) H 4 T], and Ba0.6K0.4BiO3 [(3) H = 0, (4) H = 4 T] [7]. conduction for x > 0.4 are characterized by the same ative α(T) has been carried out for the HTSCs, or even properties [11]. for simpler substances of the type of tetrahedral semiconductors (Si, Ge, etc.), where one also observes α < Both in MgB2 and in the oxide HTSC systems, the anomalous (negative) thermal expansion can be 0 in the low-temperature domain. Current phenomeno- explained by the effect of charge density waves (CDW) logical calculations, which take into account only on crystal lattice stability [12]. However, no compre- anharmonicity with a large number of ﬁtting parame- hensive microscopic-scale analysis of the effect of neg- ters, fail to explain why in such materials as Si, the Debye temperature θ ≈ 600 K α is negative at liquid- helium temperatures, i.e., in the region where the har- MgB2 monic approximation is undoubtedly valid [13]. It turns 1 out that if the additional Coulomb interaction of bond- T = 12.1 K localized charges (an analog of CDW) in tetrahedral semiconductors or the CDW in HTSCs with an ionic 0 lattice are neglected, the crystal structure of these sys- 1 tems becomes unstable; i.e., the transverse acoustic ω phonon frequency TA at the Brillouin zone boundary –5 0 vanishes. As a result of the interaction of bond-local-

, 10 ized charges in tetrahedral semiconductors or of CDW

L T = 18.8 K / ω

L 1 in ionic HTSCs, the Brillouin zone edge frequency TA

∆ ω becomes positive ( TA > 0), which is a necessary con- 0 dition for crystal lattice stability [12, 13]. Thus, in compounds with α < 0 at low temperatures (T Ӷ θ), such as 1 T = 28.2 K HTSCs, tetrahedral semiconductors, and MgB2, the crystal lattice stability is provided by a nonuniform 0 electron density distribution in the crystal. 010203040T = 37.5 K H, kOe To arrive at a qualitative explanation of the anomalous temperature dependence α(T) for the above materials, including MgB , we consider a model phonon ∆ 2 Fig. 3. Thermal expansion L/L of MgB2 plotted vs. mag- spectrum of a diatomic metal, which is presented in a netic field for fixed temperatures. The measurement temper- simplified form in Fig. 4. We do not have at our disposal atures are specified in the graphs. The measurement errors ω are shown with bars. The arrows identify the direction of an experimental dependence of the frequency on magnetic-field variation. wave vector Q for MgB2. For illustration, we took typ-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 LOW-TEMPERATURE ANOMALIES 9 ω ical dispersion relations (Q) in a high symmetry ωω(a) (b) (c) direction (Fig. 4a). This scheme is appropriate for the T MgB2 spectrum only at low frequencies (acoustic LO branches). TO The first to be excited under heating from T = 0 are the low-frequency branches of the phonon spectrum, LA ω ≈ ប ω kT/ . The lowest frequency phonon branch TA ω* TA ω* T* ω near the Brillouin zone edge (frequency TA* ) is characterized by a high density of phonon states (the low- frequency peak in F(ω) in Fig. 4b). The major contribu- 000Q F(ω) α ω tion to the frequency TA* near the Brillouin zone boundary is due to the CDW interaction with the ionic Fig. 4. Schematic of the relation between (a) phonon disper- lattice [12, 13]. A charge density wave, for instance, on sion ω(Q), (b) phonon spectrum F(ω), and (c) anomalous temperature dependence of the thermal expansion coeffi- the oxygen sublattice of an HTSC system, appears cient α(T) and of the magnetic field effect on these charac- when the effect of the Brillouin zone boundaries on teristics (solid lines for H = 0, dashed lines for H ≠ 0). For electron scattering is included. ElectronÐphonon cou- the sake of simplicity, only four phonon branches are shown pling considered with inclusion of this effect results in for the diatomic-metal model in a high-symmetry direction: LO denotes longitudinal optical; TO, transverse optical; a divergence of the dielectric susceptibility, and the LA, longitudinal acoustic; and TA, transverse acoustic dielectric permittivity ε(ω, Q) becomes negative for phonons. ω* is the transverse acoustic phonon frequency at wave vectors Q connecting such boundaries [12]. the Brillouin zone boundary, and T* is the temperature cor- Therefore, when phonons with such Q and ω are responding to the maximum absolute value of negative excited, the crystal must contract, because ε(ω*, Q) < α(T). 0. To this region of frequencies ω* corresponds the temperature T* = បω*/k, and it is near this temperature that negative α values should be observed (Fig. 4c). When heating is continued further, phonons of other effect cannot be explained as being due solely to mag- branches with higher frequencies are excited. For these netostriction. All these findings imply that the anoma- phonons, we have ε(ω, Q) > 0, which accounts for the lies in the properties of MgB2 are similar in nature to normal behavior of α(T) (i.e., α > 0). those of the oxide HTSCs. The effect of a magnetic field on the thermal-expansion anomaly in MgB2 and other HTSCs [14] is easier ACKNOWLEDGMENTS to consider using a model dielectric state. A strong magnetic field will tend to destroy singlet-paired elec- The authors are indebted to Ya.G. Ponomarev for trons and holes forming a CDW and, hence, reduce the assistance in the present study. CDW amplitude. This will bring about a decrease in the This study was supported by the Russian Founda- ω phonon frequencies TA* near the Brillouin zone tion for Basic Research (project no. 01-02-16395) and boundary, as shown by the dashed line in Fig. 4a. The the Scientific Council program “Topical Problems in peak in the phonon density of states will decrease in the Physics of Condensed Matter” (subprogram magnitude and shift toward lower frequencies (dashed “Superconductivity”). line in Fig. 4b). As a result, the temperature T* [temperature of the minimum in α(T)] will decrease, the temperature region over which α < 0 will narrow, and the REFERENCES negative α will reduce in absolute value (dashed line in 1. J. Nagamatsu, N. Nakagawa, T. Nuranaka, et al., Nature Fig. 4c). 410, 63 (2001). This decrease in T* with increasing H agrees quali- 2. S. L. Li, H. H. Wen, Z. W. Zhao, et al., Phys. Rev. B 64 tatively with the measurements shown in Fig. 2a for (9), 094522 (2001). MgB2 and in Fig. 2b for other HTSC compounds. To analyze this effect on a microscopic scale quantita- 3. S. L. Bud’ko, C. Petrovic, G. Lapertot, et al., cond- tively, one would have to have experimental phonon mat/0102413 (2001). dispersion curves ω(Q), which are presently not avail- 4. N. V. Anshukova, A. I. Golovashkin, L. I. Ivanova, and able for MgB2 [15]. A. P. Rusakov, Usp. Fiz. Nauk 167 (8), 887 (1997) [Phys. Usp. 40, 843 (1997)]. Thus, it has been found that at low temperatures, α MgB2 has a negative thermal expansion coefficient, < 5. J. Kortus, I. I. Mazin, K. D. Belashchenko, et al., cond- 0, as do oxide HTSCs. As in the case of HTSC systems, mat/0101446 (2001). a magnetic field was shown to strongly affect the low- 6. N. V. Anshukova, A. I. Golovashkin, L. I. Ivanova, et al., temperature anomaly in thermal expansion α(T); this Int. J. Mod. Phys. B 12 (29Ð31), 3251 (1998).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 10 ANSHUKOVA et al.

7. N. V. Anshukova, A. I. Golovashkin, L. I. Ivanova, and 12. L. N. Bulaevskiœ, V. L. Ginzburg, G. F. Zharkov, A. P. Rusakov, Pis’ma Zh. Éksp. Teor. Fiz. 71 (9), 550 D. A. Kirzhnits, Yu. V. Kopaev, E. G. Maksimov, and (2000) [JETP Lett. 71, 377 (2000)]. D. I. Khomskiœ, Problems in High-Temperature Super- 8. H. You, U. Welp, and Y. Fang, Phys. Rev. B 43 (4), 3660 conductivity, Ed. by V. L. Ginzburg and D. A. Kirzhnits (1991). (Nauka, Moscow, 1977). 13. H. Wendel and R. M. Martin, Phys. Rev. B 19 (10), 5251 9. Z. J. Yang, M. Yewondwossen, D. W. Lawther, and (1979). S. P. Ritcey, J. Supercond. 8, 223 (1995). 14. A. I. Golovashkin and A. P. Rusakov, Usp. Fiz. Nauk 170 10. V. V. Eremenko, V. A. Sirenko, G. Shimchak, et al., Fiz. (2), 192 (2000) [Phys. Usp. 43, 184 (2000)]. Tverd. Tela (St. Petersburg) 40 (7), 1199 (1998) [Phys. 15. C. Buzea and T. Yamashita, cond-mat/0108265 (2001). Solid State 40, 1091 (1998)]. 11. N. V. Anshukova, A. I. Golovashkin, Yu. V. Bugoslavskii, et al., J. Supercond. 7 (2), 427 (1994). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 61Ð68. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 60Ð67. Original Russian Text Copyright © 2003 by Aliev, Akhmedzhanova, Krivorotov, Kholmanov, Fridman.

SEMICONDUCTORS AND DIELECTRICS

Thermal Conductivity of Opal Filled with a LiIO3 Ionic Conductor A. É. Aliev*, N. Kh. Akhmedzhanova**, V. F. Krivorotov**, I. N. Kholmanov**, and A. A. Fridman** * Institute of Technologies LG-Elite, LG-Electronics Corporation, Seoul, 137724 Korea ** Department of Thermal Physics, Academy of Sciences of Uzbekistan, ul. Katartal 28, kvartal Ts, Chilanzar, Tashkent, 700135 Uzbekistan e-mail: [email protected] Received April 28, 2002

Abstract—The temperature dependences of the thermal conductivity of a synthetic opal and opal-based nanocomposites prepared by introducing a LiIO3 superionic conductor into pores of the opal matrix from an aqueous solution or melt are measured by the hot-wire technique in the temperature range 290Ð420 K. It is demonstrated that the thermal conductivity of pure opal increases with an increase in the diameter of the SiO2 spheres forming a face-centered cubic lattice of an opal and is determined by the total thermal resistance of interfaces between the spheres. Filling of opal pores with the ionic conductor leads to an increase in the thermal conductivity. The behavior of the thermal conductivity and its magnitude in opal-based nanocomposites depend to a large extent on the method of ﬁlling the matrix pores. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION of glasses [5] and opal [6] were revealed using acoustical spectroscopy. Research into composites with a characteristic particle size of the order of several tens of nanometers, Owing to their strictly periodic three-dimensional lattice, which are composed of SiO spheres interacting including filled porous materials, has attracted the 2 attention of many scientists owing to the prospects of mechanically, opals are also of particular interest for designing new materials with unusual properties. use as structural materials of photonic crystals [7Ð9] Recent investigations of nanostructures have aroused and thermoelectric devices [1]. Taking into account that the electrical and thermal conductivities of a nanocom- particular scientific interest due to the fresh opportuni- posite based on opal filled with bismuth decrease out of ties they offer for detailed study of the size effects, sur- proportion to each other (as compared to those of bulk face phenomena, and the influence of pore geometry on bismuth), Baughman et al. [10] succeeded in increasing the physicochemical properties of materials incorpo- the thermoelectric conversion efficiency ZT = rated into porous matrices. α2σ χ χ T/( l + c) (the Ioffe criterion) by a factor of two. Among the most extensively studied nanostructures, Here, α is the Seebeck coefficient; σ is the electrical χ χ porous matrices with a regular system of interconnect- conductivity; l and c are the phonon and electron ing pores (such as zeolites, fullerenes, opals, and thermal conductivity components, respectively; and T porous glasses) have drawn special attention, because is the temperature. they are convenient model objects [1]. A great diversity Recent interest expressed in the behavior of thermal of processes—melting, crystallization, phase transi- conductivity in similar structures and a more funda- tions to superfluid and superconducting states, phase mental problem concerning the behavior of phonon separation in binary liquids, ferroelectric phase transi- spectra in a system of mechanically coupling nanopar- tions, structural transformations in solids, transitions to ticles and composites developed on their basis have lent the vitreous state, and others—have been investigated impetus to a series of investigations into the thermal in filled porous glasses. In particular, Molz et al. [2] conduction of nanocomposites based on opal and studied porous glasses filled with different inert gases porous glasses [11Ð14]. It has been found that impreg- and found that their temperatures of melting and solid- nation of opal with a saturated aqueous solution of ification are considerably lower than those for bulk NaCl does not affect the thermal properties of the nano- materials. In porous glasses filled with gallium, size composite. Arutyunyan et al. [11] explained this find- effects were observed by Bogomolov et al. [3] and a ing by the fact that the matrix pores are occupied by change in the symmetry of the gallium lattice was NaCl in the form of needles that are not in contact; con- found by Sorina et al. [4]. Hysteresis phenomena indi- sequently, heat transfer through NaCl is absent. Bogo- cating heterogeneous crystallization of gallium in pores molov et al. [13] obtained better filling upon immersing

62 ALIEV et al.

° σ × Ð4 Ω Ð1 the opal matrix in an NaCl melt at T = 900 C. These along the optic axis C [ 33(300 K) = 5.6 10 ( m) ] α authors were the first to reveal an unusual low-temper- [16Ð19]. As a rule, -LiIO3 hexagonal prisms with high ature peak in the thermal conductivity (at 35 K) due to optical quality can be grown from an aqueous solution the size effect limiting the mean free path of phonons. and are characterized by a high hygroscopicity. At a The results obtained were explained under the assump- ° α temperature of ~247 C, -LiIO3 lithium iodate trans- tion that, apart from the standard mechanism observed forms into a low-conductivity orthorhombic γ phase in a conventional composite involving a matrix and a with a hysteresis of 46°C; as the temperature increases, filler, heat transfer in an opalÐNaCl nanocomposite pro- this compounds undergoes a destructive phase transi- ceeds through an additional mechanism typical of bulk tion at a temperature of 256°C and subsequent melting crystals. The filler introduced into a cubic lattice of opal ° α at 420 C [20]. Consequently, -LiIO3 is a convenient pores forms a matrix quasi-lattice consisting of micro- model object and can be introduced into opal pores crystallites. This leads to a manifestation of the coher- from a saturated aqueous solution and a melt at rela- ent effects and properties characteristic of bulk crystals. tively low temperatures. The choice of the filler mate- A nearly perfect crystal system of pores in opal is of rial with a low melting temperature was made for two scientific interest from the standpoint of the behavior of reasons; namely, it was necessary, first, to prevent inter- a quasi-liquid ionic subsystem involved in nanocom- action between the filler and the matrix material and, posites based on opal and superionic conductors (solid second, to avoid further sintering of close-packed electrolytes). First, the large contact area between the spheres consisting of amorphous SiO2 in the course of surface of the introduced solid electrolyte and the inner filling from the melt. surface of the pores can substantially increase the ionic conductivity at the expense of a decrease in the activa- The synthetic opal was grown through deposition of tion energy of ion migration along the interface an aqueous colloidal suspension (Nissan Chemical between two media [15]. Second, the size effects in a Inc.). This compound has a close-packed face-centered bounded pore space can bring about a decrease in the cubic lattice formed by SiO2 spheres of identical diam- internal energy of the introduced material and, hence, a eter with a lattice spacing of 250Ð400 nm. Suspensions decrease in the activation energy of ions. In particular, containing spheres of different diameters (180 ± 4, Molz et al. [2] demonstrated with the use of a geometric 220 ± 5, and 300 ± 6 nm) were treated in separating col- model that an increase in the surface-to-volume ratio umns and poured into Petri dishes. The time it takes for high-quality opals to be grown through slow crystalli- π 2 4π 3 S/V = (4 r )/--- r with a decrease in the pore size to zation of a monodisperse aqueous colloidal solution is 3 approximately equal to ten months. Although the depo- several tens of nanometers can lead to a considerable sition and crystallization of opals can occur through a change in the melting temperature of the material intro- variety of processes described in the literature [21], duced, that is, these methods are not essentially different from each other. After drying in air, the resultant precipitate was ∆T = 3kϑ T /Lr, (1) mol m similar in hardness to chalk (talc). In order to obtain ϑ mechanically high-strength opal, the precipitate was where k is the surface tension coefficient, mol is the annealed initially at 120°C for 10 h and then at 750°C molar volume of the introduced material, Tm is the melting temperature of the initial bulk material introduced for 4 h. into the pores, L is the heat of melting or solidification, The thermal conductivity coefficients were mea- and r is the mean pore radius. sured using the hot-wire technique in the temperature ° We investigated the electrical and thermal properties range 290Ð420 C in an inert gas (N2) atmosphere or of a nanocomposite based on synthetic opal and a lith- under vacuum. At room temperature, the measurements ium-conducting ionic conductor LiIO3. This paper were performed for the most part under vacuum in reports only part of the results obtained in our investi- order to provide better desorption of water vapors from gation, namely, the description of the behavior of the the opal matrix. At temperatures above 150°C, the ther- thermal conductivity coefficient in pure opals with dif- mal conductivities measured in the nitrogen atmo- ferent diameters of SiO2 spheres and in opals filled with sphere and under vacuum did not differ from each lithium iodate from a saturated aqueous solution and a other. The hot-wire technique used in the thermal melt. The electrical properties of the opalÐLiIO3 nano- experiments was described in detail in [22, 23]. A thin composite will be published in separate papers. tungsten wire 12 µm in diameter was clamped between two plane-parallel opal plates 10 × 10 × 2 mm in size. A rectangular pulse (width, 0.5 s; amplitude, 0.4 V) was 2. MATERIALS AND EXPERIMENTAL applied to the probe completely embedded between the TECHNIQUE plates. Depending on the rate of heat transfer from the Lithium iodate of the α modification, namely, α- probe to the sample, the probe temperature and, hence, LiIO3 (P63), already at room temperature is a good con- the probe resistance increased as t . The slope of the ductor with a quasi-one-dimensional ionic conductivity curve characterizing an increase in the temperature

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

THERMAL CONDUCTIVITY OF OPAL FILLED 63

(a) 1.2 1

1.1 2

1.0 3 , W/m K χ

0.9

0.8 1 µm 280 300 340 380 420 ×30000 T, K

Fig. 2. Temperature dependences of the thermal conductiv- (b) χ ity coefficient for synthetic opals containing SiO2 spheres with diameters of (1) 300, (2) 220, and (3) 180 nm. R ence sample, χ(300 K) = 1.358 W/(m K)] and did not exceed 3% in the temperature range 100 < T < 300 K II and 5% in the range 300 < T < 450 K. r2 The surface structure and the specific features in the I r1 III r3 filling of internal regions of opal were examined using a JSM-6300 (JEOL, Japan) scanning electron microscope. X-ray diffraction analysis was performed on a l DRON-2 diffractometer (CuKα radiation).

3. RESULTS AND DISCUSSION The electron micrograph of the fractured surface of a synthetic opal is displayed in Fig. 1a. The observed perfect structure of hexagonal packing of SiO2 spheres corresponds to the (111) plane of a face-centered cubic lattice. Traces of the sintering of the cleaved layer are clearly seen on the surface of the spheres. The close- packed lattice of SiO2 spheres involves octahedral (I) Fig. 1. (a) Electron micrograph of the fractured surface of a and tetrahedral (II) holes, which, according to [1], can synthetic opal with SiO2 spheres 300 nm in diameter and (b) schematic diagram of the opal structure and the cavity lat- be represented as spheres connected through cylindri- tice represented as a network of connected spherical ele- cal channels (III) 30Ð40 nm in diameter (Fig. 1b). The area of the inner surface of pores in opal can be as large ments [1]. D = 160Ð300 nm, r1 = R(2 Ð 1), r2 = R(3/2 Ð as ~10 m2/g. 1), r3 = R(23 Ð 1), and l = R(2 Ð 2 ). The temperature dependences of the thermal con- χ ductivity coefficient 12 measured along the [111] (resistance) is inversely proportional to the thermal con- direction [perpendicular to the (111) plane shown in ductivity in the direction perpendicular to the probe Fig. 1a] for synthetic opals containing SiO2 spheres location. With the aim of improving the thermal contact with diameters of 180, 220, and 300 nm are plotted in between the probe and the sample plates, the plates were Fig. 2. An increase in the thermal conductivity with an pressed to each other under a pressure of ~2 kg/cm2. increase in the temperature is characteristic of the The mean temperature of the sample was measured amorphous state. As follows from the kinetic equation using a chromelÐalumel thermocouple. The instrumen- for thermal conductivity, according to which phonons tal error in measuring the thermal conductivity coeffi- are treated as an ideal quasiparticle gas, the thermal χ cient (T) was determined with the use of Al2O3, KCl, conductivity coefficient χ is proportional to the heat and SiO2 reference samples [for the fused silica refer- capacity Cv at constant volume, the velocity V of sound

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 64 ALIEV et al. (averaged over all the directions and modes), and the tance of contact regions per unit thickness of the sam- mean free path l of phonons; that is, ple. Let us consider the propagation of a heat wave along the [111] direction in opals containing SiO 1 2 χ = ---Cv Vl. (2) spheres of different diameters: 180, 220, 250, and 3 300 nm.1 The total thermal resistance of the sample is The relative increments of the thermal conductivity equal to the sum of thermal resistances of all n1 layers with an increase in the temperature virtually coincide of the spheres and thermal resistances of n2 layers of the for all three opal samples, interfaces between the spheres; that is, χ Ð χ n n ∆χ ==------413 K 293 K - 0.22Ð0.23. (3) 1 1 2 χ χ--- = χ----- + ----χ-, (4) 293 K 1 2 This suggests the same nature of heat transfer in χ where 1/ 1 is the thermal resistance of one layer of the three similar structures. χ SiO2 spheres and 1/ 2 is the thermal resistance of one However, the magnitudes of the thermal conductiv- layer of the interfaces. It is evident that the number n1 ity differ considerably. An increase in the diameter of of layers of the spheres and the number n2 of layers of SiO spheres is accompanied by a proportional increase ϕ 2 the interfaces in the direction of the heat flux 12 across in the thermal conductivity. An insignificant increase in ρ ρ the sample thickness can be considered nearly identical the opal density ( 300/ 160 = 1.05 [10]) with an increase 1/2 (n2 = n1 Ð 1) and determined to be n1, 2 = h/(d(2/3) ), in the sphere diameter cannot be responsible for the where h is the sample thickness and d is the distance observed increase in the thermal conductivity between the centers of the spheres. (χ /χ = 1.16). 300 180 The first sample (D = 180 nm), which is 2 mm thick When analyzing the heat transfer over SiO spheres, 2 with a surface area of 100 mm2, has a thermal resistance we can assume that opals of the three types under inves- Ð1 tigation differ in the mean free path of phonons. The of 24.39 K W . For the distance between the sphere averaged velocities of sound and the heat capacities in centers d = D/1.055 = 170 nm, this sample contains 14410 layers of SiO2 spheres in the direction of the heat relationship (2) should be identical for SiO2 spheres of ϕ different diameters. The sphere size can limit the mean flux 12. The second sample, which is of the same size, Ð1 free path of phonons and, correspondingly, the thermal has a thermal resistance of 22.81 K W and involves conductivity of the sample at low temperatures. The 11664 layers of SiO2 spheres with a sphere diameter of calculated mean free paths of phonons in red opal with 220 nm (d = 210 nm). For a sample with a smaller spheres of diameter D = 300 nm (light reflected from sphere diameter, 2746 additional layers of the inter- the surface has a red hue) indicate that, even at room faces bring about an increase in the thermal resistance temperature, the condition l ≤ 3 nm is satisfied; i.e., the by 1.58 K WÐ1. Therefore, irrespective of the seal diam- mean free path l is two orders of magnitude smaller eter and the number of contacts per sphere, the thermal than the sphere size. resistance of one contact layer is equal to 0.575 × − Under the given conditions of sintering, the ratio of 10 3 K WÐ1. A comparative analysis of other opals leads the sphere diameter D to the minimum distance d to a similar result. The table presents the calculated χÐ1 between the sphere centers lies in the range from 1.05 thermal resistances of interfaces (2 ) and SiO2 spheres to 1.055 [24]. In this case, the diameter of the contact χÐ1 neck between spheres in red opal is determined as a = (1 ) for all the opal samples studied. The specific ther- 2 2 1/2 χ (D Ð d ) = 90Ð95 nm. Therefore, the contact neck mal conductivity 0 of an individual SiO2 sphere was between spheres also cannot limit the mean free path of estimated (for four samples) at ~1.24 W/m K. This phonons. value is close to the thermal conductivity of fused silica On the other hand, the diameter of the contact neck (1.36 W/m K at T = 293 K) but exceeds the thermal between spheres increases with an increase in the conductivity of SiO2 ceramic materials (0.76 W/m K at sphere diameter and, seemingly, might affect the T = 293 K) [26]. Most likely, the SiO2 spheres were through heat transfer. However, an increase in the partly fused together upon heat treatment. This is indi- sphere diameter is accompanied by a proportional cated by the low porosity (P = 0.27Ð0.30) measured for increase in the cross-sectional area of the pores; as a these samples by Baughman et al. [10]. In general, the result, the specific surface of heat transfer remains temperature dependence of the thermal conductivity of unchanged. At the same time, the number of contact the aforementioned opals is similar to that of fused sil- necks per unit length in the direction of propagation of ica and ceramic materials. a heat wave depends on the sphere size. With the Figure 3 shows the dependence of the thermal con- Wiener model of successive layers [25], it can be easily ductivity on the sphere size. This dependence is nearly shown that the thermal conductivity of a composite consisting of ordered layers formed by identical parti- 1 The thermal conductivity of the opal sample with D = 250 nm cles is determined primarily by the total thermal resis- was measured only at T = 20°C.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 THERMAL CONDUCTIVITY OF OPAL FILLED 65

χÐ1 χÐ1 Thermal resistances of interfaces 2 and SiO2 spheres 1 for four types of opals with different distances between the cen- χ ° ters of SiO2 spheres according to calculations from the experimental thermal conductivities measured at 20 C

χ, W/m K χÐ1, K WÐ1 Ð1 Ð1 χ , W/m K d, nm n χ , K WÐ1 χ , K WÐ1 0 (20°C) (10 × 10 × 2 mm) 2 1 (20°C) 170 0.82 24.39 14 410 8.29 16.10 1.24 210 0.87 22.81 11664 6.7 16.11 1.24 240 0.91 21.98 10206 5.87 16.11 1.24 280 0.95 21.14 8748 5.02 16.12 1.24

linear, in accordance with the predictions of the above higher than its initial value. The next additional anneal- model. Fayette et al. [27] observed a similar behavior ing brings about an increase in the thermal conductivity of the thermal conductivity in SnO2 ceramic materials of opal by 4.2%. However, a further increase in the over a narrow range of particle sizes. It is evident that, magnitude of the thermal conductivity is less pro- as the sphere size substantially increases or decreases, nounced. The temperature behavior of the thermal con- the thermal conductivity exhibits nonlinear behavior. It ductivity remains virtually unchanged after each seems likely that there exist other mechanisms limiting annealing. The thermal conductivity as a function of the an increase in the thermal conductivity. In particular, number of sequential annealings at T = 413 K is shown β Kitayama et al. [28] studied -Si3N4 ceramics and in the inset to Fig. 4. The total change in the thermal χ experimentally derived the relationship = C1lnA + C2, conductivity (after the fourth annealing as compared to where C1 and C2 are constants and A is the mean parti- the thermal conductivity of the initial sample) is more cle size. than 10%. If this change can result only from the increase in the diameter of the contact neck between the As was shown in [27], additional annealing of spheres (by ignoring the decrease in the thickness of the ceramic samples results in a decrease in the thickness of contact layer), the resulting shrinkage of channels con- the contact region between crystal grains and an appre- necting octahedral and tetrahedral holes will appear to ciable increase in the thermal conductivity. For spherical be critical. structural units of opal, additional annealing can lead not only to a decrease in the thickness of the contact The channel radius can be estimated from the rela- neck but also to an increase in the contact area. This is tionship r = (3)Ð1/2d Ð (D/2). As was noted above, not necessarily desirable because superfluous annealing min brings about a decrease in the opal porosity. Figure 4 depicts the temperature dependences of the thermal 1.4 conductivity of red opal (D = 300 nm) after a series of 1.35 additional annealings for 1 h at 750°C. Initial sample 1 1.30 was first annealed for 4 h at 750°C and was then kept in 1.2 1.25 5 a desiccator for a month. The first additional annealing 4 , W/m K results in a change in the slope of the thermal conduc- χ 1.20 3 tivity curve. This can be associated with hydrothermal 1.15 1.1 1234 2 treatment of the inner cavities in the porous matrix. 1 Even at T > 360 K, the thermal conductivity is 1.5% , W/m K

χ 1.0 1.0

0.9 0.9 , W/m K

χ 0.8 300 340 380 420 0.8 T, K 160 180 220 260 300 Average sphere diameter, nm Fig. 4. Temperature dependences of the thermal conductivity of red opal (D = 300 nm) after a series of additional annealings for 1 h at 750°C. (1) Thermal conductivity curve Fig. 3. Dependence of the thermal conductivity coefﬁcient for the initial sample. The inset shows the dependence of the on the size of SiO2 spheres forming the face-centered cubic thermal conductivity on the number of additional annealing structure of a synthetic opal. T = 20°C. cycles at 413 K.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 66 ALIEV et al.

4.5 5 (a)

4.0 ~ 2.5 4 2.0

, W/m K 3 χ 1.5 2 1.0 1 280 300 340 380 420 T, K 1 µm Fig. 5. Temperature dependences of the thermal conductiv- ×10000 ity of (1, 3) blue (D = 180 nm) and (2, 4) red (D = 300 nm) (b) opals: (1, 2) the initial unfilled samples and (3, 4) opals filled with LiIO from an aqueous solution (60°C). (5) 3 α Dependence of the thermal conductivity of -LiIO3 measured along the [001] direction. under the initial conditions of sintering (at 750°C for 4 h), the ratio D/d of the sphere diameter to the minimum distance between the sphere centers falls in the range 1.05Ð1.055. The channels collapse at D/d = 1.155 [10]. Therefore, additional thermal annealing that becomes possible in the course of filling of opal pores with a high-temperature liquid melt is undesirable. Figure 5 displays the temperature dependences of the thermal conductivity for blue and red opals prior to 1 µm × and after filling with an LiIO3 ionic conductor. For 20000 comparison, curve 5 shows the dependence of the thermal conductivity χ for an α-LiIO bulk single crystal 33 3 Fig. 6. Electron micrographs of (a) the surface and (b) the along the [001] direction, which was measured by cleavage of the inner region of green opal (D = 220 nm) Abdulchalikova et al. [29]. Opals were filled by filled with LiIO3 from a saturated aqueous solution at room repeated immersion in a saturated aqueous solution of temperature. lithium iodate at a temperature of 60°C. After each immersion, the sample was dried first in air for 1 h and then in a vacuum furnace for 1 h at 120°C, weighed, An increase in the solution temperature makes it and immersed again in the solution for 10 min. The per- possible to increase the solution saturation and the colation of the solution through the opal pores was con- mean degree of pore filling. At a solution temperature trolled visually: as the liquid penetrates into the pores, of 60°C, the mean degree of filling reaches 25% of the opal changes its transparency. After the fifth immer- pore volume. However, even under these conditions, sion, the channels connecting the opal pores were filled, the inner regions of the sample remain not easily acces- thus preventing further incorporation of LiIO3. This can sible. Analysis of the data on the thermal conductivity be judged from the electron micrograph in Fig. 6b, of the nanocomposites filled at room temperature and which displays the image of the central region in the ° × × 60 C revealed that, even at such a low density of the cleavage of a sample (10 10 2 mm in size) filled at incorporated material, the thermal conductivity room temperature. Lithium iodate is concentrated in increases in proportion when changing over from the narrow necks of the channels and prevents further filling of the pores. The degree of filling of the sample sur- initial opal to the completely impregnated nanocom- face (Fig. 6a) is considerably higher than that of the posite. In the narrow range of sphere sizes studied in inner regions. Analysis of the experimental data on the this work, the empirical dependence of the effective density of the prepared composites demonstrated that thermal conductivity on the sphere diameter D (nm) the highest degree of pore filling, which can be and the degree of pore filling P can be approximately achieved by immersing opals in aqueous solutions at written in the following form: room temperature, does not exceed 15% of the pore χ volume. eff = DC1 ++C2 PC3/D, (5)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 THERMAL CONDUCTIVITY OF OPAL FILLED 67

60% of the pore volume across the whole thickness of the sample. Note that, in this case, the surface and central regions of the sample are filled to the same extent. Finally, the pores can be almost completely filled 1 upon filling from a melt under pressure. At a pressure of 5 kbar and a melt temperature of 450°C, we succeeded in filling 90% of the pore volume. The electron micrograph of the central region in the cleavage of green opal (D = 220 nm) filled from the melt under pressure is displayed in Fig. 7. This cleavage represents a replica of the opal characterized by a nearly complete filling of the pores. However, the thermal conductivity of this composite does not fit into the aforementioned 2 empirical dependence on the degree of filling. X-ray diffraction analysis performed on a DRON-2 diffractometer (CuKα radiation) demonstrated that the crystallization of lithium iodate in opal pores depends α Fig. 7. Electron micrograph of the cleavage of green opal on the filling method. The hexagonal phase with a (D = 220 nm) filled with LiIO3 from the melt under a pres- high ionic conductivity dominates upon filling from a sure of 5 kbar. Arrows 1 and 2 indicate the filler SiO2 saturated aqueous solution. However, this method has spheres, respectively. failed to attain a high degree of pore filling. Upon filling from a melt at 450°C, the tetrahedral β and orthorhombic γ low-conductivity phases, which are characterized × Ð3 where C1 = 1.2 10 W/m K, C2 = 0.6 W/m K, and by a lower thermal conductivity of the initial bulk mate- C3 = 750 W/m K. rial, predominantly crystallize in the opal pores. In this case, the relative integrated intensity of the peaks attrib- A further increase in the degree of pore filling can be β γ achieved by immersion in a melt. An examination of the uted to the and phases of lithium iodate (Fig. 8) is electron micrograph of the cleavage of green opal (D = no less than 75%. 220 nm) filled from a lithium iodate melt at 450°C The preparation of nanocomposites with a high revealed that lithium iodate occupies, on average, up to ionic conductivity calls for further investigation into the

100 3.49 3.44 3.82 80

2.93 2.03 2.75 1.876 60 1.850 2.37 2.16 2.28

40 2 2.27 Intensity, arb. units 2.59 1.880

20 1.790

4.76 3.51 2.75 2.37 2.15 1 0 10 20 30 40 50 60 Scattering angle, deg

α Fig. 8. Diffraction patterns of (1) an -LiIO3 single crystal and (2) a nanocomposite comprised of opal ﬁlled with LiIO3 from the melt. Intense peaks at 1.85, 2.03, 2.93, and 3.82 Å correspond to the β phase of lithium iodate.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 68 ALIEV et al. stabilization of the hexagonal α phase of lithium iodate 13. V. N. Bogomolov, N. F. Kartenko, L. S. Parfen’eva, in pores of opal ﬁlled from the melt. et al., Fiz. Tverd. Tela (St. Petersburg) 41 (2), 348 (1999) [Phys. Solid State 41, 313 (1999)]. 14. A. E. Aliev and N. Kh. Akhmedjanova, Uzb. Fiz. Zh., ACKNOWLEDGMENTS No. 1, 373 (1999). This work was supported by the Scientiﬁc Techno- 15. J. Maier, J. Phys. Chem. Solids 46, 309 (1985). logical Center (Ukraine), project no. Uzb-05. 16. A. É. Aliev and A. Sh. Akramov, Izv. Akad. Nauk Uz. SSR, No. 1, 76 (1987). 17. A. A. Abramovich, A. Sh. Akramov, A. É. Aliev, and REFERENCES L. N. Fershtat, Fiz. Tverd. Tela (Leningrad) 29 (8), 2479 (1987) [Sov. Phys. Solid State 29, 1426 (1987)]. 1. V. N. Bogomolov and T. M. Pavlova, Fiz. Tekh. Polupro- 18. A. E. Aliev, A. Sh. Akramov, L. N. Fershtat, and vodn. (St. Petersburg) 29 (5), 826 (1995) [Semiconduc- P. K. Khabibullaev, Phys. Status Solidi A 108, 189 tors 29, 428 (1995)]. (1988). 2. E. Molz, A. P. Wong, M. H. W. Chan, and J. R. Beamish, 19. A. É. Aliev, A. Sh. Akramov, and R. R. Valetov, Fiz. Phys. Rev. B 48, 5741 (1993). Tverd. Tela (Leningrad) 31 (12), 178 (1989) [Sov. Phys. 3. V. N. Bogomolov, R. Sh. Malkovich, I. A. Smirnov, Solid State 31, 2127 (1989)]. et al., Fiz. Tverd. Tela (Leningrad) 12, 1204 (1970) [Sov. 20. K. I. Avdienko, S. V. Bogdanov, S. M. Arkhirov, Phys. Solid State 12, 938 (1970)]. B. I. Kidyarov, V. V. Lebedev, Yu. E. Nevskiœ, V. I. Trunov, D. V. Sheloput, and R. M. Shklovskaya, Lithium Iodate: 4. I. G. Sorina, C. Tien, E. V. Charnaya, et al., Fiz. Tverd. Growth of Crystals, Their Properties and Applications Tela (St. Petersburg) 40 (8), 1552 (1998) [Phys. Solid (Nauka, Novosibirsk, 1980). State 40, 1407 (1998)]. 21. A. P. Philipse, J. Mater. Sci. Lett. 8, 1371 (1989). 5. B. F. Borisov, E. V. Charnaya, Yu. A. Kumzerov, et al., 22. N. R. Abdulkhalikova and A. E. Aliev, Uzb. Fiz. Zh., Solid State Commun. 92 (6), 531 (1994). No. 4, 50 (1991). 6. J. M. Dereppe, B. F. Borisov, E. V. Charnaya, et al., Fiz. 23. N. R. Abdulchalikova and A. E. Aliev, Synth. Met. 71, Tverd. Tela (St. Petersburg) 42, 184 (2000) [Phys. Solid 1929 (1995). State 42, 193 (2000)]. 24. A. A. Zakhidov, R. H. Baughman, Z. Igbal, et al., Sci- 7. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). ence 282 (5390), 897 (1998). 8. S. John, Phys. Rev. Lett. 58, 2486 (1987). 25. O. Wiener, Phys. Z. 5, 332 (1904). 9. S. John and T. Quang, Phys. Rev. Lett. 74, 3419 (1995). 26. Acoustical Crystals, Ed. by M. P. Shaskol’skaya (Nauka, Moscow, 1982). 10. R. H. Baughman, A. A. Zakhidov, I. I. Khayrullin, et al., in Proceedings of the XVII International Conference on 27. S. Fayette, D. S. Smith, A. Smith, and C. Martin, J. Eur. Thermoelectrics (ICT’98), Nagoya, Japan, 1998. Ceram. Soc. 20, 297 (2000). 28. M. Kitayama, K. Hirao, M. Toriyama, and Sh. Kanzaki, 11. L. I. Arutyunyan, V. N. Bogomolov, N. F. Kartenko, J. Am. Ceram. Soc. 82 (11), 3105 (1999). et al., Fiz. Tverd. Tela (St. Petersburg) 40 (2), 379 (1998) [Phys. Solid State 40, 348 (1998)]. 29. N. R. Abdulchalikova, A. E. Aliev, V. F. Krivorotov, and P. K. Khabibullaev, Solid State Ionics 107, 59 (1998). 12. V. N. Bogomolov, N. F. Kartenko, L. S. Parfen’eva, et al., Fiz. Tverd. Tela (St. Petersburg) 40 (3), 573 (1998) [Phys. Solid State 40, 528 (1998)]. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 69Ð72. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 68Ð70. Original Russian Text Copyright © 2003 by Sheleg, Iodkovskaya, Kurilovich.

SEMICONDUCTORS AND DIELECTRICS

Effect of Gamma Radiation on the Permittivity and Electrical Conductivity of TlGaS2 Crystals A. U. Sheleg, K. V. Iodkovskaya, and N. F. Kurilovich Institute of Solid-State and Semiconductor Physics, Belarussian National Academy of Sciences, ul. Brovki 17, Minsk, 220072 Belarus e-mail: [email protected] Received February 20, 2002; in ﬁnal form, May 13, 2002

ε σ Abstract—The effect of gamma radiation on the permittivity and the electrical conductivity of TlGaS2 crystals is investigated at frequencies of 0.1, 1, and 10 kHz and 1 MHz in the temperature range 200Ð370 K. It is shown that an increase in the temperature leads to an increase in the permittivity ε and the electrical conductiv- σ σ ity . The electrical conductivity of TlGaS2 samples irradiated with a dose of 10 MR and then measured at all frequencies and the permittivity ε of samples irradiated with a dose of 1 MR and then measured at frequencies of 10 kHz and 1 MHz increase, whereas further accumulation of the dose results in a decrease in the values of ε and σ. The parameters studied are characterized by a considerable dispersion: as the frequency increases, the permittivity ε decreases and the electrical conductivity σ increases. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION observed anomalies in the curve Cp = f(T) at temperatures T1 = 73.5, T2 = 91, T3 = 101, T4 = 114, T5 = 133.5, Crystals of TlGaS2 belong to semiconductorsÐferro- and T6 = 187 K. According to the authors’ opinion [7], III VI electrics of the TlA B2 type (A is In or Ga and B is S this indicates a sequence of phase transitions in the or Se) with a layered structure and a number of interest- TlGaS2 crystal. Mal’sagov et al. [8] analyzed the tem- ing physical properties. The layered structure of these perature dependences of the unit cell parameters of a crystals is responsible for the strong anisotropy of their TlGaS2 crystal and revealed jumps in the parameters at dynamic characteristics. Moreover, it has been demon- T = 121 K, which suggests the occurrence of a phase strated that representatives of this family, such as transition at this temperature. Aliev et al. [9] reported TlGaSe2 and TlInS2 crystals, have incommensurate the results of the dielectric measurements of a TlGaS2 phases and, correspondingly, undergo sequences of crystal at a frequency of 100 kHz in the temperature phase transitions from a highly symmetric phase to an range 80Ð300 K. These authors found no anomalies in incommensurate phase and, then, to a ferroelectric the temperature dependence of the permittivity ε = f(T). commensurate phase [1Ð3]. Although TlGaSe2 and It is quite possible that the contradictory results relating TlInS2 compounds have been thoroughly studied, there to TlGaS2, TlGaSe2, and TlInS2 crystals are due to the exists some discrepancy between the phase transition existence of TlGaS2 polytypes. Therefore, the use of temperatures determined by different authors. Earlier, different samples leads to different results. Abdullaev et al. [4] assumed that these crystals are sus- ceptible to polytypism due to their layered structure. This paper reports on the results of investigating the ε More recently [5], different polytypic modiﬁcations of effect of gamma radiation on the permittivity and the electrical conductivity σ of TlGaS crystals. The mea- TlGaSe2 and TlInS2 crystals were revealed with x-ray 2 diffraction. The above difference in the phase transition surements were carried out at different frequencies in temperatures determined by different authors can be the temperature range 200Ð370 K. explained by the fact that the samples used in these studies belong to different polytypic modiﬁcations. 2. SAMPLES AND EXPERIMENTAL As regards TlGaS2 crystals, they have been studied TECHNIQUE to a lesser degree as compared to TlGaSe2 and TlInS2 crystals. The data available in the literature on TlGaS2 The permittivity and electrical conductivity of crystals are few in number and contradictory. In partic- TlGaS2 crystals were measured on an E7-12 digital ular, Abdullaeva et al. [6] measured the heat capacity of meter at a frequency of 1 MHz and an E7-14 meter at TlGaS2 in the temperature range 3Ð300 K and did not frequencies of 0.1, 1, and 10 kHz. The measurements revealed anomalies in the heat capacity in the tempera- were performed using continuous heating at a rate of ture range covered. Krupnikov and Abutalybov [7] 0.5 K/min.

70 SHELEG et al. ε 1 Samples were prepared in the form of 0.5- to 0.7-mm-thick single-crystal plates. The surfaces of 50 these plates were (001) cleavages. The electrodes used 1 were made of a silver paste. The studied sample was placed in a special holder and immersed in nitrogen 2 vapor. The temperature of the sample was measured by 40 a chromelÐcopel thermocouple whose junction was 2 located at the sample surface. The temperature was controlled using a heater mounted in the holder of the sample. The samples were irradiated on a gamma setup 30 with a 60Co source at room temperature. The power of the 60Co source in the irradiation zone was ~180 R/s. 3 The irradiation dose was accumulated in the same sam- 3 ple through consecutive exposures and amounted to 1, 20 4 10, 50, and 100 MR. The permittivity ε and the electrical conductivity σ were measured after each irradia- 4 tion. 10 200 240 280 320 360 3. RESULTS AND DISCUSSION T, K Figures 1 and 2 show the temperature dependences Fig. 1. Temperature dependences of the permittivity ε of the of the permittivity and the electrical conductivity of a TlGaS2 samples measured at frequencies of (1) 0.1, (2) 1, TlGaS2 crystal measured at frequencies of 0.1, 1, and and (3) 10 kHz and (4) 1 MHz before (solid lines) and after 10 kHz and 1 MHz (solid lines) and the permittivity ε = (dashed lines) irradiation with a dose of 100 MR. f(T) and the electrical conductivity σ = f(T) obtained using the same sample irradiated with a dose of 100 MR (dashed lines). It can be seen that, as the tem- ε σ × 4 Ω–1 –1 perature increases, the permittivity and the electrical 10 , cm conductivity σ also increase for all the frequencies used 30 4 in the measurements. This is due to an increase in the concentration of free charge carriers (manifestation of the semiconductor properties) and an increase in the 4 20 mobility of the domain boundaries (manifestation of the ferroelectric properties) with an increase in the temperature. It can be seen from Fig. 1 that gamma radiation with 10 a dose of 100 MR results in a decrease in the permittiv- ε 1.5 3 ity over the entire temperature range studied and for all the frequencies used. This behavior of the permittiv- 3 ity of TlGaS2 can be associated with the radiation-stimulated ageing of the samples due to gamma-activated 1.0 migration of natural defects, which leads to stabiliza- 2 tion of the domain structure and a decrease in the permittivity ε [10]. It should be noted that, at high frequen- σ 1 cies, the electrical conductivity of TlGaS2 irradiated 0.5 2 samples is substantially higher than that of unirradiated samples (Fig. 2). As is known, the main gamma-radiation effect is induced by secondary Compton electrons 1 0 and secondary photoelectrons [10]. The generation of 200 240 280 320 360 secondary electrons and ionization of the medium bring T, K about an increase in the electrical conductivity of the irradiated samples. Analysis of the experimental results ε demonstrates that the permittivities of TlGaS2 sam- Fig. 2. Temperature dependences of the electrical conduc- σ ples irradiated with doses of 10 and 50 MR are interme- tivity of the TlGaS2 samples measured at frequencies of (1) 0.1, (2) 1, and (3) 10 kHz and (4) 1 MHz before (solid diate between the permittivities of the samples mea- lines) and after (dashed lines) irradiation with a dose of sured before and after irradiation with a dose of 100 MR. 100 MR.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

EFFECT OF GAMMA RADIATION 71

ε trical conductivity, these authors made the inference 40 that, in TlInS2 semiconductors, the charge transfer occurs through the hopping mechanism and electrical conduction is due to optical transitions [11]. This can 30 also explain the frequency dependence of the electrical σ ν conductivity = f( ) in TlGaS2. The electrical conduc- 2 tivity σ(ν) ﬁrst increases as a result of irradiation with 20 1 a dose of 10 MR and then decreases with a further 3 increase in the irradiation dose, as is the case with the 4 permittivity. This is associated with the fact that, when 10 102 103 104 105 106 the irradiation doses are relatively small, defects in the ν, Hz irradiated crystal undergo annealing (the so-called effect of small doses). As was already mentioned, an ε ν increase in the irradiation dose brings about radiation- Fig. 3. Dispersion curves = f( ) of TlGaS2 samples (1) before and after irradiation with doses of (2) 1, (3) 10, and stimulated ageing of the sample and an increase in the (4) 100 MR. T = 300 K. concentration of defects in the crystal. Most probably, the scattering of charge carriers by defects leads to a decrease in the electrical conductivity σ(ν). Note that, at high frequencies, the permittivity ε(ν) of the irradiated samples, except for the case of the 1-MR dose, σ × 104, Ω–1 cm–1 2 turned out to be smaller than that of the unirradiated 28 samples (Fig. 3). 3 Analysis of the results obtained demonstrates that the permittivity ε and electrical conductivity σ of the 24 TlGaS crystal under study exhibit a pronounced dis- 4 2 persion. As the temperature increases, the permittivity ε 20 drastically increases at low frequencies and weakly varies at high frequencies (Fig. 1). In actual fact, at a 1 frequency of 1 MHz, the permittivity ε only slightly increases with increasing temperature. It seems likely that the decrease in the sensitivity of the permittivity ε 1 with an increase in the frequency is due to relaxation processes occurring in the high-frequency range. It should be noted that no anomalies are found in the 0 curves ε(T) and σ(T). This indicates that the TlGaS 102 103 104 105 106 2 ν crystal does not undergo a sequence of phase transi- , Hz tions from a high symmetric phase to an incommensurate phase and then to a ferroelectric commensurate σ ν phase, which are typical of TlGaSe and TlInS crystals Fig. 4. Dispersion curves = f( ) of TlGaS2 samples (1) 2 2 before and after irradiation with doses of (2) 10, (3) 50, and belonging to the same family as TlGaS2. (4) 100 MR. T = 300 K. ACKNOWLEDGMENTS The dispersion curves ε = f(ν) and σ = f(ν) at T = This work was supported by the Belarussian Foun- 300 K for TlGaS2 crystals prior to and after irradiation dation for Basic Research, project no. F99-043. with different doses are depicted in Figs. 3 and 4, ε respectively. It is evident that the permittivity for all REFERENCES the studied samples decreases with an increase in the frequency. The permittivities ε of the samples irradiated 1. S. B. Vakhrushev, V. V. Zhdanova, B. E. Kvyatkovskiœ, et al., Pis’ma Zh. Éksp. Teor. Fiz. 39 (6), 245 (1984) with a dose of 1 MR and measured at frequencies of [JETP Lett. 39, 291 (1984)]. 10 kHz and 1 MHz are somewhat larger than those of 2. D. F. McMorrow, R. A. Cowley, P. D. Hatton, and the unirradiated samples (Fig. 3). Further irradiation I. Banys, J. Phys.: Condens. Matter 2 (16), 3699 (1990). leads to a decrease in the permittivity ε as the irradia- 3. A. U. Sheleg, O. B. Plyushch, and V. A. Aliev, Fiz. Tverd. tion dose accumulates. It can be seen from Fig. 4 that σ Tela (St. Petersburg) 36 (1), 226 (1994) [Phys. Solid the electrical conductivity increases with increasing State 36, 124 (1994)]. frequency. Mustafaev et al. [11] obtained similar exper- 4. G. B. Abdullaev, G. I. Abutalybov, A. A. Aliev, et al., imental results for the isostructural composition TlInS2. Pis’ma Zh. Éksp. Teor. Fiz. 38 (11), 525 (1983) [JETP From analyzing the frequency dependence of the elec- Lett. 38, 632 (1983)].

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 72 SHELEG et al.

5. O. B. Plyushch and A. U. Sheleg, Kristallograﬁya 44 (5), 9. R. A. Aliev, K. R. Allakhverdiev, A. I. Baranov, et al., 873 (1999) [Crystallogr. Rep. 44, 813 (1999)]. Fiz. Tverd. Tela (Leningrad) 26 (5), 1271 (1984) [Sov. 6. S. G. Abdullaeva, A. M. Abdullaev, K. K. Mamedov, and Phys. Solid State 26, 775 (1984)]. N. G. Mamedov, Fiz. Tverd. Tela (Leningrad) 26 (2), 618 10. E. V. Peshikov, Radiation Effects in Ferroelectrics (1984) [Sov. Phys. Solid State 26, 375 (1984)]. (Tashkent, 1986). 7. E. S. Krupnikov and G. I. Abutalybov, Fiz. Tverd. Tela 11. S. N. Mustafaev, M. M. Asadov, and V. A. Ramazanzade, (St. Petersburg) 34 (9), 2964 (1992) [Sov. Phys. Solid Fiz. Tverd. Tela (St. Petersburg) 38 (1), 14 (1996) [Phys. State 34, 1591 (1992)]. Solid State 38, 7 (1996)]. 8. A. U. Mal’sagov, B. S. Kul’buzhev, and B. M. Kham- khoev, Izv. Akad. Nauk SSSR, Neorg. Mater. 25 (2), 216 (1989). Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 73Ð77. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 71Ð75. Original Russian Text Copyright © 2003 by Ulanov, Zaripov, Zheglov, Eremina.

SEMICONDUCTORS AND DIELECTRICS

Trimers of Bivalent Copper Impurity Ions in CaF2 Crystals: The Structure and Mechanism of Formation V. A. Ulanov, M. M. Zaripov, E. P. Zheglov, and R. M. Eremina Kazan Physicotechnical Institute, Russian Academy of Sciences, Sibirskiœ trakt 10/7, Kazan 29, 420029 Tatarstan, Russia e-mail: [email protected] Received February 18, 2002; in ﬁnal form, May 14, 2002

Abstract—Crystals of CaF2 : Cu (with a copper impurity content higher than 0.1 at. %) grown by the Czochral- ski method from a melt in a mixed helium–ﬂuorine atmosphere are investigated using electron paramagnetic resonance (EPR) spectroscopy. It is found that the crystals contain paramagnetic centers whose magnetic prop- 6Ð erties at low temperatures are identical to those of [CuF4F4] (S = 1/2) single centers. The magnetic properties of the centers exhibit a qualitative change in the temperature range 77Ð300 K. These changes are described 6Ð within a model according to which the center is treated as a cluster composed of three [CuF4F4] impurity complexes involved in exchange interactions and interactions occurring in the ﬁeld of Jahn–Teller lattice distortions. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION dimerization of impurity ions in the host lattice of the crystal under conditions of intensive diffusion. It is known that crystals belonging to the fluorite The purpose of this work was to analyze the possi- structural family (CdF2, CaF2, SrF2, and BaF2) have a loose-packed lattice with predominantly ionic bonds bility of forming copper impurity clusters in a CaF2 between atoms. Consequently, at high temperatures, crystal and to investigate their magnetic properties. We intensive correlated diffusion of both host-lattice and believed that clusters of a new type could be formed in impurity ions occurs in response to an external electric a matrix with a unit cell of considerably smaller size. field or in the presence of an impurity concentration gradient in the lattice [1]. This diffusion of copper 2. SAMPLE PREPARATION, EXPERIMENTAL impurity ions in a BaF2 crystal brings about the forma- TECHNIQUE, AND RESULTS tion of copper stable dimers [2]. Under certain conditions, the concentration of copper stable dimers in the The crystals to be studied were grown by the Czo- crystal can turn out to be higher than that of copper sin- chralski method. The crucibles used for the crystal gle centers. Apparently, two impurity paramagnetic growth were produced from chemically pure graphite. centers of bivalent copper are bound into a thermally Impurities in the form of carefully dried CuF2 were stable dimer due to an interaction between impurity sin- introduced into a melt of chemically pure calcium fluo- gle centers, which, in turn, leads to a decrease in the lat- ride. With the aim of maintaining equilibrium between tice energy. For copper dimers in a BaF2 crystal, the the processes of thermal decomposition and formation nature of this interaction can be better understood when of copper fluoride, gaseous fluorine was added in small it is considered that, upon replacement of a Ba2+ cation amounts to the helium atmosphere used in the crystal by a bivalent copper impurity ion, this ion is located at growth. the center of a regular cube and its orbital ground state The grown crystals were examined using electron 2 is represented by the T2g triplet. In this situation, lattice paramagnetic resonance (EPR) spectroscopy on an vibrations should significantly affect the bivalent cop- E-12 Varian spectrometer operating in the X and Q per states and, particularly, result in substantial dis- bands (9.3 and 37 GHz, respectively) at liquid-helium placements of ions surrounding the copper impurity to temperature and in the temperature range 77Ð300 K. It new equilibrium positions (the JahnÐTeller effect). In was found that, in the case when the impurity concen- the case of a BaF2 : Cu crystal, the copper impurity ion tration was less than or equal to 0.5 at. % and the tem- is also appreciably displaced (by ~1 Å). This brings perature gradient in the vicinity of the crystallization about a considerable polarization of the lattice in the front was greater than or equal to 20 K/mm, the grown 6Ð vicinity of the copper impurity ion and gives rise to crystals predominantly contained [CuF4F4] orthor- large electric dipole moments. The energy of interac- hombic centers, whose structure and magnetic parame- tion between copper off-center complexes ~5 Å apart ters were described in our earlier work [3]. For the crys- with parallel electric dipole moments is roughly esti- tals grown at higher impurity concentrations and mated at ~0.1Ð0.2 eV. This energy accounts for the smaller temperature gradients (≤5 K/mm), the EPR

74 ULANOV et al.

(1) || 〈 〉 g = 2.29 to the external magnetic field, namely, B0 001 (spec- ⊥ || 〈 〉 (2) trum a) and B0 110 (spectrum b). The angular g ⊥ = 2.28 C dependences of the line position turned out to be char- (1, 2) = 2.07 a g || acteristic of transitions between states of two spin doublets with the effective spin moments S(1) = S(2) = 1/2. These dependences can be adequately described by b two orthorhombic tensors of the electron Zeeman inter- ()1 AD action with the following principal components: gx = ()1 ()1 ()1 ()2 g = g⊥ = 2.28 ± 0.01, g = 2.075 ± 0.005, g = B y z x ()2 ()2 ± ()2 ± gy = g⊥ = 2.29 0.01, and gz = 2.075 0.005. In 250 300 350 the temperature range 77Ð300 K, these components do B0, mT not depend on the temperature within the limits of experimental error. Investigations performed at fre- Fig. 1. EPR spectra of the CaF2 : Cu (0.2 at. %) sample mea- quencies of 9.3 and 37.3 GHz revealed that the electron sured in the X band at a temperature of 77 K for two princi- Zeeman interaction almost linearly depends on the || 〈 〉 || pal orientations of the crystal: (a) B0 001 and (b) B0 magnetic field (i.e., the components of the tensors g(1) 〈110〉. and g(2) are independent of the magnetic field strength). In addition to the intense structureless lines, the EPR spectra measured for the principal orientations of gy = 2.099 the crystal contain several groups of weak lines (Fig. 1). gx = 2.802 Since these lines were masked by noise arising upon rotations of the crystal through small angles, we failed to obtain the angular dependences of their positions. Figure 2 displays the EPR spectrum of the studied crystal at a low temperature (4.2 K) for the orientation || 〈 〉 B0 110 . This spectrum coincides in every detail with 6Ð the spectrum of [CuF4F4] impurity single centers, which was thoroughly examined in [3]. The sole difference resides in the fact that the line width for lightly doped samples is two or three times smaller. No signals other than those observed in the aforementioned spectra are revealed at this temperature (with the aim of searching for broad lines, the electromagnetic field power in the cavity was reduced to a level of 0.01 mW, at which 200 240 280 320 360 the possibility of saturating a resonance transition was B0, mT ruled out). An important experimental characteristic is the tem- Fig. 2. EPR spectrum of the CaF2 : Cu (0.2 at. %) sample measured in the X band at a temperature of 4.2 K for the ori- perature dependence of the intensity ratio of spectral || 〈 〉 entation B0 110 . lines of the second and first groups, which, in turn, can be characterized by the tensors g(2) and g(1), respectively. It was found that this ratio increases with spectra measured at temperatures in the range 77Ð300 increasing temperature. K exhibited intense lines attributed to paramagnetic centers with axial symmetry of the magnetic properties. We examined the crystals grown at different concen- In this case, the crystals had a red color. trations of copper impurities in the CaF2 melts. It was The EPR spectrum of a crystal arbitrarily oriented revealed that a decrease in the content of copper intro- with respect to the external dc magnetic field vector duced into the melt leads to a drastic decrease in the contains six structureless lines, each approximately concentration of the centers under investigation. How- 5 mT wide. Moreover, these lines overlap with each ever, in this case, the components of the electron Zee- other in pairs. Figure 1 shows the EPR spectra mea- man interaction tensor and the EPR line shape remain sured in the X band at a temperature of 77 K for the two unchanged. Moreover, these parameters prove to be most important orientations of the crystal with respect independent of the shape of the studied samples.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

TRIMERS OF BIVALENT COPPER IMPURITY IONS IN CaF2 CRYSTALS 75 3. DISCUSSION

In order to answer the question as to the nature of the 2 magnetic centers revealed in the crystals under investigation, we analyzed different variants of defect structures in the samples. For this purpose, we considered the characteristic temperature dependence of the intensity ratio of two groups of structureless resonance lines, their widths, the electron Zeeman interaction tensor 1 3 components characterizing the angular dependences of these lines, the dynamic properties of copper single X _ complexes, and the character of local lattice distortions Cu2+ F Y produced by these complexes. In addition, we took into account that no impurities other than copper ions were introduced into the crystal. The hypothetical model of Fig. 3. A model of the copper trimer in a CaF2 : Cu crystal. 2+ the centers under consideration should also explain The t2 orbitals occupied by an electron hole of Cu ions are their relatively high concentration. shown. The coordination cubes of Cu2+ ions are contracted along the C axis lying in the plane of the t orbital. The experimental fact that the measurements per- 2 2 formed at a temperature of 4.2 K under conditions far from saturation revealed no broad structureless lines is On this basis, we proposed a model according to of crucial importance. It turned out that the spectra 6Ð which the cluster is treated as a copper trimer formed observed are characteristic of [CuF4F4] orthorhombic by three bivalent copper single complexes with orthor- complexes. It is also important that the intensity ratio of hombic symmetry (Fig. 3). The electron Zeeman inter- the two groups of structureless resonance lines action in orthorhombic fragments of the trimer (S = 1/2) observed at temperatures ranging from 77 to 300 K is characterized by a g tensor with the following com- depends on the temperature. These findings allow us to ponents: gx = 2.099, gy = 2.147, and gz = 2.802. Figure 3 draw the inference that one of the groups of structure- corresponds to one of the molecular configurations of less resonance lines whose relative intensity increases the trimer at low temperatures of the crystal. In this with increasing temperature corresponds to EPR transi- configuration, each orthorhombic fragment of the tri- tions within a certain excited spin doublet of a para- mer is localized in one out of six equivalent wells of the magnetic center created in the crystal. As follows from adiabatic potential. It can be seen that, within the pro- analyzing the obtained temperature dependence of the posed model, orthorhombic JahnÐTeller distortions relative intensity of the structureless lines, the energy (corresponding to a contraction of the coordination separation between the ground spin state of the center cube along one of the twofold axes [3]) correlate with and this doublet is of the order of 30Ð50 cmÐ1. It is clear each other. Note that the total energy of elastic strains that, if the above assumption holds true, this group of produced by JahnÐTeller distortions in the host lattice lines should not be observed at 4.2 K. It is also evident of the crystal in the vicinity of the trimer appears to be that the ground state of this center should be repre- less than the total energy of strains arising around sented by a spin multiplet with a half-integer effective widely spaced JahnÐTeller single centers. Since the spin moment. Consequently, the other group of more copper nuclei in the trimer are located in the plane intense structureless lines is associated with the transi- aligned parallel to one of the (001) planes in the crystal, tions within this ground spin multiplet (according to it is obvious that the number of different cluster orien- our experimental data, this is a doublet). × × tations in the CaF2 crystal is equal to the product 3 4 Therefore, the hypothetical model of the clusters 2 = 24. The number 2 in this product corresponds to the under investigation must satisfy the following three number of ground nuclear configurations of each par- main requirements. ticular cluster, i.e., to the number of wells of the adia- (1) At low temperatures, the EPR spectra of the batic potential for this cluster. One of these configura- ground spin doublet of the studied cluster should be tions is depicted in Fig. 3. The other configuration can similar to the spectra of a copper impurity single com- be obtained by transferring a hole in the Cu 3d filled plex with orthorhombic symmetry [3]. shell from one orbital t2 to another orbital t2 in each of the three complexes forming the trimer. In this case, (2) In the temperature range 77Ð300 K, these spectra care should be taken to see that the JahnÐTeller distor- should transform into structureless lines whose angular (1) tions remain correlated. Taking into account that the dependences can be described by the axial tensor g . frequency of transitions between wells of the adiabatic (3) The angular dependences of the EPR spectra of potential for a copper single complex drastically the excited doublet in the temperature range 77Ð300 K increases at temperatures of 25Ð30 K [4], we believe should correspond to experimental angular depen- that similar transitions in the trimer should be intensive dences of the structureless resonance lines character- at slightly higher temperatures (40Ð60 K). At tempera- ized by the tensor g(2). tures above 60 K, the EPR spectra of the trimer should

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 76 ULANOV et al.

E the spin Hamiltonian are the energies of three spin mul- 1.0 tiplets, namely, a quartet (S = 3/2) and two doublets (S = S(3) = 3/2 1/2). The relative positions of these multiplets on the 0.5 energy axis can differ depending on the magnitudes and A B signs of the isotropic exchange constants (J0 and J0 ). 0 (1) S = 1/2 Figure 4 depicts the dependences of the energy of the A (norm) A A 2 –0.5 spin multiplets on the quantity (J0 ) = J0 /((J0 ) + B 2 1/2 A 2 –1.0 S(2) = 1/2 ()J0 ) {the energy E is expressed in terms of [(J0 ) + B 2 1/2 –1.0 –0.5 0 0.5 1.0 ()J0 ] }. A (norm) (J 0) When analyzing the electron Zeeman interaction of the trimer with an external dc magnetic field, we restrict Fig. 4. Changes in the energy diagram for spin levels of the our consideration to the approximation in which the orbital ground state of a trimer localized in one of the two operator of this interaction can be written in the form wells of the adiabatic potential as a function of the quantity A β ()⋅ ⋅ ⋅ ⋅ (norm) HZ = e S1 g1 + S2 g2 + S3 g3 B0, (2) ()J0 . where g1, g2, and g3 are the Zeeman interaction tensors for three orthorhombic copper complexes (trimer frag- transform in such a manner that they can be adequately ments) and B0 is the external dc magnetic field vector. described by the averaged perpendicular components In the framework of our approximation, the compo- of the low-temperature g tensors; that is, nents of these tensors are taken to be equal to the corre- ()i ()i sponding components for the copper single complexes. ()i gx + gy In actual fact, these components for trimer fragments g⊥ = ------, 2 can be different, because the trimer fragments interact with each other; i.e., they can be involved in the Cou- ()i ()i where gx and gy are the x and y components of the g lomb interaction and the interaction occurring in the tensors for the ground and excited spin doublets (i = 1 field of Jahn–Teller distortions. To a first approxima- and 2, respectively) of the trimer. tion, the components of the g tensor for the excited spin doublet (with an intermediate spin moment S13 = 1) of The positions of groups A and B of weak lines in the trimer can be represented by the relationships Fig. 1 coincide with those of the lines in the low-temperature spectrum. This experimental finding indicates () 2g Ð g + 2g g 2 = Ð------1x 2x 3x, that certain trimers at T = 77 K remain localized in their x 3 wells of the adiabatic potential. It seems likely that local strains in the crystal appear to be the strongest in () 2g Ð g + 2g g 2 = Ð------1y 2y 3y, (3) the vicinity of these trimers. As regards groups C and D y 3 of resonance lines in Fig. 1, they can be associated with the resonance transitions between excited spin states of () 2g Ð g + 2g g 2 = Ð------1z 2z 3z. these trimers. However, these lines can also be attrib- z 3 uted to the transitions between excited spin states of copper clusters of another type (for example, copper For the ground spin doublet (S13 = 0), we have dimers, which can be formed in the crystal with a suffi- ()1 ()1 ()1 ciently high probability). gx ===g2x, gy g2y, gz g2z. (4)

The parameters of the model proposed were calcu- Here, gjx, gjy, and gjz are the components of the electron lated within the approximation of pair exchange cou- Zeeman interaction tensors for copper single centers pling between trimer fragments. The numbering of sin- forming the trimer under investigation (the tensors are gle complexes (trimer fragments) is shown in Fig. 3. represented in the general coordinate system XYZ) and Within a rough approximation in which the anisotropic j = 1Ð3 are the numbers of these fragments of the trimer parts of the exchange interaction tensors can be according to the numbering shown in Fig. 3. Within this ignored, the spin Hamiltonian of the orbital ground approximation, the magnitudes of the components of state of the trimer can be represented as follows: the g tensor for the excited spin doublet of the trimer are as follows: ⋅⋅A ⋅⋅A ⋅⋅B HS = ÐS1 J0 S2 Ð S2 J0 S3 Ð S1 J0 S3, (1) ()2 ()2 ()2 gx ===3.03, gy 1.87, gz 2.14. where Sj are the electron spin operators for the copper A B The components of the electron Zeeman interaction single complexes forming the trimer and J0 and J0 are tensor for the ground spin doublet coincide with the the isotropic exchange constants. The eigenvalues of corresponding components of the g tensor for an impu-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 TRIMERS OF BIVALENT COPPER IMPURITY IONS IN CaF2 CRYSTALS 77 rity single center oriented in the crystal in the same the superhyperfine interaction between the electron manner as fragment 2 in the trimer; that is, spin moment of the ith copper ion and the nucleus of the () () () F g 1 ===2.802, g 1 2.099, g 1 2.147. jth ligand; and Iij is the operator of the nuclear spin of x y z the jth ligand of the ith copper impurity ion. According For a temperature of 77 K and higher (at which the fre- to the results of our calculations, the EPR spectrum of quency of internuclear transitions in the trimer becomes the ground spin doublet of the trimer coincides (to the considerably higher than the EPR frequency), the mean second order in the perturbation theory) with the spec- value of the perpendicular component of the g tensor trum of a copper single complex with orthorhombic for the excited spin doublet was calculated in the above symmetry. approximation and amounted to 2.45. For the ground doublet, we obtained approximately the same value (~2.45). The parallel component of the g tensor for both 4. CONCLUSIONS doublets was not averaged and remained equal to 2.147. Thus, the results obtained in the present work and in These components (calculated to the second order in our earlier experimental studies [2, 4] demonstrated the perturbation theory) of the electron Zeeman interac- that a large number of complex paramagnetic clusters tion tensors are appreciably larger than the experimen- that are formed by impurity ions of the iron group and tal components: have a regular molecular structure can be created in the ()1 ()1 bulk of crystals belonging to the fluorite structural fam- g⊥ ==2.28, gz 2.075, ily. The copper impurity clusters (copper trimers) ()2 ()2 formed in CaF2 crystals are adequately described in the g⊥ ==2.29, gz 2.075. framework of the model presented in Fig. 3. These clus- This discrepancy can be explained by the fact that the ters are characterized by a multiple-well adiabatic dynamic properties of the trimer were ignored in these potential and a complex system of low-lying spin calculations. Let us assume that, during internuclear energy levels. As a consequence, their physical proper- transitions in the trimer, molecular motions in its frag- ties are extremely sensitive to distortions of the host lat- ments occur not synchronously but with a certain lag. tice of the crystal and a change in the temperature. One Under this assumption, negative corrections to the com- of the main reasons for the formation of the trimer ponents of the aforementioned tensors g arise already under investigation is the interaction between the trimer within the second order in the perturbation theory and fragments in the field of Jahn–Teller distortions. lead to an improvement in the results of calculations. For the ground spin doublet, we performed a more ACKNOWLEDGMENTS detailed calculation of the hyperfine and superhyperfine This work was supported by the Russian Foundation structures of the EPR spectra of the trimer. The spin for Basic Research, project no. 01-02-17718. Hamiltonian of electronÐnucleus interactions was represented as the sum Hen = Hhfi + Hshfi. Here, REFERENCES H hfi 1. G. C. Benson and E. Dempsey, Proc. R. Soc. London, (5a) Ser. A 226, 344 (1962). ()⋅⋅Cu Cu ⋅⋅Cu βCu Cu ⋅ = ∑ Si ai Ii + Ii Qi Ii Ð ngn Ii B0 , 2. M. M. Zaripov and V. A. Ulanov, Fiz. Tverd. Tela (Len- i ingrad) 31 (10), 254 (1989) [Sov. Phys. Solid State 31, 1798 (1989)]. H = ∑∑()S ⋅⋅A IF Ð βgFIF ⋅ B . (5b) 3. M. M. Zaripov and V. A. Ulanov, Fiz. Tverd. Tela (Len- shfi i ij ij n n ij 0 ingrad) 30, 1547 (1988) [Sov. Phys. Solid State 30, 896 i j (1988)]. In Hamiltonians (5a) and (5b), subscript i refers to the 4. V. A. Ulanov, M. M. Zaripov, and E. P. Zheglov, in copper ions in the trimer; subscript j indicates fluorine Abstracts of XI Feofilov Symposium on Spectroscopy of nuclei that are ligands of the ith impurity ion; Si is the Crystals Activated by Rare-Earth and Transition-Metal Ions, Kazan, 2001, p. 52. operator of the electron spin of the ith copper ion; ai and Qi are the hyperfine and quadrupole interaction tensors for the ith copper ion, respectively; Aij is the tensor of Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

SEMICONDUCTORS AND DIELECTRICS

Ligand Hyperfine Interaction in Gd3+ Tetragonal Centers in CaF2 and SrF2 and the Structure of the Impurity Nearest Surroundings A. D. Gorlov Institute of Physics and Applied Mathematics, Ural State University, pr. Lenina 51, Yekaterinburg, 620083 Russia e-mail: [email protected] Received February 12, 2002; in ﬁnal form, May 18, 2002.

3+ Abstract—ENDOR experimental spectra of Gd tetragonal impurity centers in CaF2 and SrF2 crystals were used to determine the superhyperfine interaction (SHFI) constants of the impurity with 19F nuclear spins of its 3+ Ð first coordination sphere and the compensator ion. The distances in the Cd F9 complex were estimated within the model of isotropic SHFI constants suggested in [1]. An analysis of the data on the SHFI and spin-Hamilto- nian constants [2] in terms of the superposition model indicates significant changes in the contributions (due to the Gd3+ mixed states) to these parameters for the tetragonal centers in comparison with the corresponding contributions for the cubic and trigonal centers in the same crystals. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tional spin Hamiltonian (SH) with the parameters given in [2] in the laboratory system of coordinates, where the A specific feature of tetragonal Gd3+ impurity cen- principal symmetry axis C4 of the center is parallel to ters (ICs) in CaF2 and SrF2 crystals is that the fluorine Ð the Z and [001] axes and the X and Y axes are parallel ion (F ) in the nearest neighbor interstice along the axis to [100] and [010], respectively. k C4 is a compensator (F ) of the surplus positive charge of the impurity. The Coulomb interaction between the The ENDOR spectra in an external magnetic field H k 3+ IC and F causes opposite displacements of Gd (from directed along the crystal symmetry axes (C4, C3, C2) the 8FÐ cube center) and Fk [3]. A comparison of the and the angular dependences in the plane {001} were coordinates of 19F nuclei located in the 2ndÐ4th shells studied. The IC displacement along the C4 axis caused of the neighbor (hereinafter, the anion environment is by Fk results in the eight fluorine nuclei (with local 3+ meant) with the positions of ligands in the undistorted symmetry Cs) nearest to Gd being divided into two 3+ || ⊥ lattice and the Gd cubic centers in the same crystals groups. For the orientations H (C2, C4) and H C4, [4] shows that the most significant displacements take the ENDOR signals for both groups of 19F ions [form- place in the region close to the compensator. In this ing regular quadrangles lying above (111-type nuclei paper, we consider the displacements of atoms of the close to the compensator) and under the {001} plane nearest ligands and the compensator using the results of ENDOR studies of the superhyperfine interaction (111 -type nuclei)] had a fine structure caused by the (SHFI) between Gd3+ (electron spin S = 7/2) and 19F indirect nucleusÐnucleus interaction through the impu- nuclei (nuclear spin I = 1/2), as well as of studies of rity ion [5, 6], with the center of this structure coincid- the parameters of initial splitting and the quadrupole ing with the signal positions in the absence of this inter- and intrinsic hyperfine (HFI) interactions [2] for 157Gd action. in CaF and SrF . 2 2 The procedure of SHFI constant determination used was the same as that in [1]. The Hamiltonian Hn ade- 2. ENDOR DATA AND DISCUSSION quately describing the electronÐnucleus interaction between Gd3+ and 19F is given by CaF2 and SrF2 crystals grown by the Stockbarger method with a GdF impurity (0.015 wt % in blend) 3 H = ()A + 2A O0()S O0()I + ()A Ð A Ð A exhibited tetragonal and cubic Gd3+ ESR spectra, and n s p 1 1 s p E 1 1 1 1 SrF2 also exhibited an additional trigonal spectrum, × O ()S O ()I + ()ΩA Ð A + A ()ΩS ()I with intensity ratios of 3 : 1 and 5 : 1 : 4, respectively. 1 1 s p E 1 1 The ESP of the tetragonal centers in the range of 3 cm ()0() 0() () at temperature T = 4.2 K was described by the conven- + A1 + 4A2 O3 S O1 I + A1 Ð 3A2 (1)

LIGAND HYPERFINE INTERACTION 79

3+ Experimental SHFI constants and the angular coordinates of the nearest ligands at the Gd tetragonal centers in CaF2 and SrF2, as well as the model isotropic constants, distances, and induced dipole moments

Crystal CaF2 SrF2 Nucleus type 111111 Fk 111111 Fk

Ligand local symmetry Cs Cs C4v Cs Cs C4v

As, MHz Ð1.994(3) Ð1.315(3) Ð0.842(3) Ð2.236(4) Ð1.179(4) Ð0.522(3) Ap, MHz 4.984(3) 4.576(3) 4.391(3) 4.841(2) 4.279(3) 3.919(3) AE, kHz Ð42(4) Ð50(3) Ð Ð55(6) Ð31(4) Ð × A1 10, kHz Ð4(3) 0.9(19) Ð9(2) Ð4(3) Ð0.7(32) Ð4(2) × A2 10, kHz Ð0.8(4) 0.2(35) 0(1) Ð1.6(6) Ð0.8(6) 0(1) × A3 10, kHz 1.8(4) 0(7) Ð 1(4) 0.6(6) Ð × A4 10, kHz 5(2) 0(7) Ð 0(2) 6(3) Ð θ, deg 63.7(1) 129.1(1) 0 63.8(1) 129.4(1) 0

As, MHz, calculation Ð1.99 Ð1.27 Ð0.80 Ð2.22 Ð1.22 Ð0.47 R, Å, calculation 2.30 2.40 2.32 2.33 2.42 2.35

Di, e Å, calculation 0.14 0.08 0.2 0.125 0.085 0.21 Dcosθ, e Å, calculation 0.035 Ð0.05 0.08 0.033 Ð0.048 0.075 Note: The FÐ ion polarizability α is taken from [7]; α = 1 for the IC.

× ()1() 1() Ω1()Ω1() () O3 S O1 I + 3 S 1 I + A3 + A4 ments of the nearest 111 -type ligands are also small in comparison with those for cubic ICs and that the × ()1() 1() Ω1()Ω1() β () O3 S O1 I Ð 3 S 1 I Ð gn n HI . increase in θ is caused only by Gd3+ displacement along

The notation in Eq. (1) is conventional [5, 6]. We the C4 axis, we approximate the distances R()111 as emphasize that Hn contains only the terms whose con- 2.40 and 2.42 Å for CaF2 and SrF2, respectively. The tributions to the ENDOR frequencies exceed the exper- same result was previously obtained in [3] in theoretical imental errors. Therefore, Eq. (1) in fact corresponds to calculations of the local structure of the tetragonal rare- the higher local symmetry C2v of nuclei. Equation (1) earth impurity centers in MeF2. k yields the SH for F with local symmetry C4v at AE and The distance R for 19F in the 2ndÐ4th shells close to A2 = A4 = 0. The SHFI constants are defined in the local the compensator is shorter than that for cubic ICs in frame of reference of a separated nucleus (the z axis is these crystals. This can take place only under the con- parallel to the Gd3+Ð19F bond axis, and the x axis lies in dition that Fk is significantly displaced to Gd3+ from the the plane containing the bond axis and C3). The ligand center of the nearest neighbor interstice. Otherwise, angular coordinates θ (between the Gd3+Ð19F bond axis Coulomb repulsion of like charges would result not and axis Z) and ϕ (between the projection of the bond only in a change in the angle θ but also in an increase in axis onto plane XY and axis X) were determined as in R for these 19F nuclei in comparison with the case of [1]. Axis C is retained at the tetragonal centers; there- 3+ 4 cubic ICs even for a Gd displacement along the C4 fore, ϕ = 45° for the nearest ligands. axis. As was indicated in [3], R(111) is roughly equal to The table lists the experimentally determined SHFI the ICÐFk distance R(k). constants and the angular coordinates of the nearest neighbor anions and Fk (the IC is at the origin of coor- 3. ANALYSIS OF THE ISOTROPIC SHFI dinates). One can see that nuclei of type 111 (111 ) are AND SH CONSTANTS θ ° ° characterized by angles > 54.74 (125.24 ), which are The distances R(111) and R(k) were estimated appreciably larger than those for the cubic centers. This k within the model proposed in [1], but the equation for is obviously caused both by F forcing the 111-type the constant A defined in [1, Eq. (3)] was modified as nuclei apart and by the IC displacement to the compen- s 19 sator. The results of [4] show that the positions of F As nuclei far from the compensator (2ndÐ4th shells of the (2) []() θ ()()3 IC environment) virtually coincide (to within the exper- = As R0 + Ks' Dcos i 1 + Ks Di R0/Ri . imental error) with the coordinates of the same anions ' for cubic ICs in these crystals when the Gd3+ displace- The parameters As(R0), Ks, and Ks have the same mean- Ð 3+ ment is taken into account. Assuming that the displace- ing as in [1]. The dipole moments of F (Di) and Gd

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 80 GORLOV

m m (D) were also determined in a similar way. For rough case (An (tetr) > An (trig) [3, 6]). This appreciably estimations, we neglect the radial dependence of the changes the overlap parameters of the outer electron 3+ Ð parameter Ks' because of its small contribution (Ks' = shells of the RE ion with the electron shells of the F −50(4) MHz/e Å for the trigonal center in this approxi- ion in comparison with trigonal ICs. Optical spectra of 3+ mation). This equation is more adequate, since it takes MeF2 : RE of various local symmetries also indicate the dependence of As on the overlapping of outer elec- an increase in the CF strength when passing from trig- trons of the polarized IC with electrons of the polarized onal to tetragonal centers in these crystals [3]. ligand into account more correctly [8]. Small changes in the radial distribution of the elec- m The values of As were calculated as in [1], i.e., by tron density of outer 5p electrons (even An couple the varying R(111), R(k), Di, and D. The coordinates of the 19 4f and 5p states [5, 9, 10]) cause a reduction of the F atoms in the second and more distant shells of unpaired spin density both in the ligand region and at neighbors were taken from [4]. The positions of cations the RE3+ nucleus site, which can cause changes in both in CaF2were taken from [3], taking into account that the SHFI and HFI [5Ð8, 10Ð13]. Indeed, studies of the HFI minimum Ca2+Ð19F distance is no shorter than the sum 157 3+ in tetragonal Gd in CaF2 and SrF2 [1, 2] showed the of their ionic radii (the distances in SrF2 were increased hyperfine isotropic constants A(s) = (A + 2B)/3 to be proportionally to the lattice parameter). appreciably larger than those for trigonal and cubic ICs. Calculations of As with the model parameters from The increase in As is undeniably caused by a change in [1] showed that the values of R for the ligands nearest the unpaired spin density of outer-shell electrons at the to the IC cannot be adjusted such that the calculated Gd3+ nucleus site as they are most sensitive to the CF [5, constants As for the two groups of fluorine nuclei near- 6, 9Ð12]. It is also worth noting that the increase in A(s) est to Gd3+ and for the compensator are close to their cannot be caused by a change in the spin density of 6s experimental values. On the other hand, if the structure electrons (if this density is present), since their contri- of the surroundings of the nearest IC is taken to be close bution would decrease this constant [5, 11, 12]. The to that calculated in [3] and the contributions due to the electron density of populated 5p states is only Ð F polarization (with the model parameter Ks from [1]) decreased due to mixing; this explains the increase in are subtracted from the experimental values of As, then A(s) due to a decrease of the negative-contribution frac- the values obtained will be regularly arranged (accord- tion in A(s) caused by the interaction of 5(s, p) electron θ 3+ ing to the sign of the projections Dz = Dcos i onto the shells, which also takes place in the free Gd ion [11, Gd3+Ð19F bond axis directions) with respect to A (R) for 12]. The reduction of the unpaired spin density of 5p s electrons at ligands results in a positive contribution to Gd3+ cubic centers in fluorite-structured crystals. This || means that the model suggested for the trigonal centers As [11, 13], i.e., to a decrease in Ks' in the model we is also applicable to the tetragonal centers, but the con- suggested, since this parameter now describes two contributions to the isotropic SHFI related to the polariza- tributions: the negative one is due to the mixing of tion and mixing of the Gd3+ electron shells differ from states by the odd CF (or Gd3+ polarization), and the pos- those suggested in [1]. itive one is related to the action of the even CF on already mixed states. To determine the parameter Ks' , we used three equations (2) for A , taken in pairs under the condition that As was shown in [9, 10] in microscopic calculations s 0 R(111) = R(k) = 2.305 Å for CaF2, since it is hardly of A2 , the contribution from the mixing of the 4f and probable that R(111) is shorter than its corresponding 5p electron states is opposite in sign to the contribution value for the cubic IC. The values obtained were differ- from the 4fÐ5d mixing. A similar conclusion as to the ent and were smaller in magnitude than those in [1]. contributions to As follows from the data presented in 3+ This was also the case with the results for SrF2 deter- [14], where the SHFI of the tetragonal Ce in CaF2 was mined under the same conditions. By only having put considered. In our opinion, all these facts explain the R(111) < R(k) and varying them, a complete set of || decrease in Ks' for tetragonal ICs. experimental constants As was obtained (see table) with Presuming the values R to be correct, we calculated the same parameter Ks' = Ð32(3) MHz/e Å for both the values Z0 (Z0 = A0 , b0 , P0 , with P0 being the crystals. However, this value of K' is smaller than that 2 2 2 2 2 2 s quadrupole interaction constant) for the tetragonal for the trigonal center by a factor of 1.56 [1]. 157Gd in CaF and SrF within the superposition model In our opinion, this difference is qualitatively 2 2 7 [9, 15] with intrinsic parameters Zp and Zs taken from explained by more efficient mixing of the 4f ground 0 0 3+ state of Gd not only with unfilled 5d but also with the [2, 9]. The signs of A2 and P2 were found to be identi- filled 5p state, since the matrix elements Am of even cal to the experimental ones [2, 16]; however, the val- n 0 and odd components of the crystalline field (CF) relat- ues are three to four times larger. In the case of b2 , the ing such electron states [5, 6, 9Ð11] are larger in this signs also coincide, but the calculated values are three

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 LIGAND HYPERFINE INTERACTION 81

3+ times smaller than the experimental ones. The A0 and ence of even and odd CF components on the Gd outer 2 electron shells, manifesting itself in tetragonal ICs. 0 b2 fit to the experiment can be improved by varying Zs; 0 however, this is impossible for P2 . This does not con- ACKNOWLEDGMENTS tradict the superposition model but calls for inclusion A.D. Gorlov thanks A.I. Rokeakh and A.S. Moskvin 0 of the additional contributions to Z2 from the mixing of for the experimental data and valuable discussions. the half-filled 4f shell with other shells induced by the This study was supported in part by the Civilian CF, which was ignored in [9]. This approach can lead to Research and Development Foundation for the New changed intrinsic parameters and radial dependences. Independent States of the Former Soviet Union (grant On the other hand, it was found that the experimen- no. REC-005). 0 tal values of Z2 correlate. This is apparent if they are determined as REFERENCES 0 () () 1. A. D. Gorlov, V. B. Guseva, A. P. Potapov, and Z2 = Z pkp+ Zsks, (3) I. A. Rokeakh, Fiz. Tverd. Tela (St. Petersburg) 43 (3), 456 (2001) [Phys. Solid State 43, 473 (2001)]. where k(p) and k(s) are the effective parameters of the model [16], including the radial and angular depen- 2. A. D. Gorlov, A. P. Potapov, V. I. Levin, and V. A. Ulanov, Fiz. Tverd. Tela (Leningrad) 33 (5), 1422 dences of the point Coulomb and short-range-interac- (1991) [Sov. Phys. Solid State 33, 801 (1991)]; A. D. Gor- 3+ Ð tion contributions in the Gd F9 complex. Having lov and A. P. Potapov, Fiz. Tverd. Tela (St. Petersburg) solved the set of two equations with respect to k(p, s) 42 (1), 49 (2000) [Phys. Solid State 42, 51 (2000)]. 0 3. M. P. Davydova, B. Z. Malkin, and A. L. Stolov, in Spec- with Zp and Zs taken from [2, 9], we find the third Z2 to troscopy of Crystals (Nauka, Leningrad, 1978), p. 27; be close to the experimental value. For example, the S. M. Arkhipov and B. Z. Malkin, Fiz. Tverd. Tela (Len- values A0 calculated for CaF and SrF are 390 and ingrad) 22 (5), 1471 (1980) [Sov. Phys. Solid State 22, 2 2 2 857 (1980)]. 190 cmÐ1, while the experimental values are 339 and Ð1 4. A. I. Rokeakh, A. A. Mekhonoshin, N. V. Legkikh, and 204 cm , respectively [16]. A. M. Batin, Fiz. Tverd. Tela (St. Petersburg) 37 (10), 3135 (1995) [Phys. Solid State 37, 1728 (1995)]. 4. CONCLUSIONS 5. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon, Oxford, 1970; Thus, the basic results of this study can be summa- Mir, Moscow, 1972), Vol. 1. rized as follows. 6. S. A. Al’tshuler and B. M. Kozyrev, Electron Paramag- (i) The SHFI constants for tetragonal Gd3+ centers in netic Resonance (Nauka, Moscow, 1972). 7. C. Fainstein, M. Tovar, and C. Ramos, Phys. Rev. B 25 CaF2 and SrF2 were determined from the experimental ENDOR spectra. These spectra are well described by (5), 3039 (1982). the SH corresponding to a higher local symmetry of flu- 8. J. M. Baker, J. Phys. C 12 (19), 4093 (1979). 3+ Ð 9. L. I. Levin, Phys. Status Solidi B 134 (2), 275 (1986). orine nuclei than the actual symmetry in the Gd F9 complex. 10. M. V. Eremin, in Spectroscopy of Crystals (Nauka, Len- ingrad, 1989), p. 30. (ii) The empirical model describing the SHFI isotro- 3+ 11. R. Watson and A. J. Freeman, in Hyperfine Interactions, pic constants in the trigonal Gd center in BaF2 is also Ed. by A. J. Freeman and R. B. Frankel (Academic, New applicable to the tetragonal ICs considered in this paper York, 1967; Mir, Moscow, 1970). if one changes the contribution to the isotropic SHFI 12. J. Andriessen, D. van Ormondt, S. N. Ray, and T. P. Das, from the mixing of the outer IC electron states caused J. Phys. B 11 (15), 2601 (1978). m 0 by both even (An ) and odd (A1 ~ D) components of the 13. J. Casas, P. Stydzinski, J. Andriessen, et al., J. Phys. C 19 ligand crystalline field. (34), 6767 (1986). (iii) The distances to the nearest neighbor 19F nuclei 14. O. A. Anikienok, M. V. Eremin, and O. G. Khutsishvili, and the compensating fluorine atom were determined Fiz. Tverd. Tela (Leningrad) 28 (6), 1690 (1986) [Sov. Phys. Solid State 28, 935 (1986)]. 3+ Ð within this model. The Gd F9 complex structure 15. D. J. Newman and W. Uraban, Adv. Phys. 24 (2), 793 obtained is similar to that calculated within the (1973). exchange-charge model. 16. L. I. Levin and A. D. Gorlov, J. Phys.: Condens. Matter 0 0 0 4 (2), 1981 (1992). (iv) An analysis of the constants A2 , b2 , and P2 within the superposition model requires the intrinsic parameters be changed, which is also due to the influ- Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 82Ð86. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 80Ð83. Original Russian Text Copyright © 2003 by Bilanych, Baœsa, V. Rizak, I. Rizak, Holovey. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Internal Friction of Li2B4O7 Single Crystals V. S. Bilanych, N. D. Baœsa, V. M. Rizak, I. M. Rizak, and V. M. Holovey Uzhgorod National University, ul. Podgornaya 46, Uzhgorod, 88000 Ukraine e-mail: [email protected] Received March 26, 2002

Abstract—This paper reports on the results of measurements of the internal friction QÐ1 and the shear modulus G of Li2B4O7 single crystals along the crystallographic directions [100] and [001] in the temperature range 300Ð550 K for strain amplitudes of (2Ð10) × 10Ð5 at infralow frequencies. The anomalies observed in QÐ1 and G in the temperature range 390Ð410 K are due to thermal activation of the mobility of lithium cations and their migration from one energetically equivalent position to another. A jump in the internal friction background is Ð1 revealed in the vicinity of the Q and G anomalies for the Li2B4O7 crystal. The magnitude of this jump depends on the crystallographic direction. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION of QÐ1 and G were calculated from the obtained loops of the mechanical hysteresis according to the formulas Single crystals of Li2B4O7 lithium tetraborate are promising materials for use in piezotechnology, Ð1 sinδ because they not only have a high ionic conduction [1, Q ==tanδ ------, 2] and exhibit anomalous behavior in a number of phys- 1 Ð sin2 δ ical parameters but also possess unique elastic properties. ∆ε δ Mcr m Earlier [3, 4], the elastic properties of Li B O sin- sin ==------ε , GA------ε -, 2 4 7 m m gle crystals and their temperature behavior were studied mainly in the frequency range 0.1Ð800 MHz. Addi- ∆ε where is the residual strain at the instant when Mcr = tional information on the specific features of the behav- 0, ε is the maximum strain, M is the maximum ior of elastic moduli and the mechanisms of interaction m cr m of an acoustic wave with the crystal structure can be torque moment, and A is the coefficient determined by obtained from analyzing the influence of the amplitude the geometric parameters of the sample under study. of an external stress field on dissipative processes. The temperature dependences QÐ1(T) and G(T) were obtained upon heating at a constant rate v = 37.5 K/h. The goal of this work was to investigate the lattice H Samples in the form of parallelepipeds (cross section, dynamics of an Li2B4O7 single crystal through an anal- × ysis of the temperatureÐfrequency dependences of the 2 2 mm; length, 20 mm) were prepared from bulk sin- internal friction QÐ1 and the shear modulus G for differ- gle crystals oriented along the crystallographic direc- ent amplitudes of an external stress field. tions [100] and [001].

The initial compound Li2B4O7 was prepared by 2. EXPERIMENTAL TECHNIQUE melting boron oxide (OSCh 12-3) and lithium carbonate (OSCh 20-2) in platinum crucibles in air. The tem- The internal friction QÐ1 and the shear modulus G of perature and time parameters of the synthesis were an Li2B4O7 single crystal at different temperatures were optimized taking into account the speciﬁc features of measured under quasi-static loading in the frequency the thermal decomposition of lithium carbonate, the range 10Ð3Ð10Ð1 Hz according to an automated experi- dehydration of the initial components, and the nature of mental procedure based on the use of an inverse-type their interaction [6]. In order to compensate for the loss torsion pendulum [5]. of boron oxide due to incongruent evaporation of the In order to determine the internal friction and the melt during the growth of single crystals, an excess of shear modulus at infralow frequencies (f = 10Ð3Ð10Ð1 Hz) B2O3 (up to 0.5 mol %) was added to the reaction prod- of forced torsional vibrations, we recorded the curves uct. The single crystals were grown by the Czochralski ε = F(Mcr), where Mcr is the harmonically variable method along the crystallographic direction [001] or torque moment under whose action the sample was [100]. The pulling speed was 3Ð6 mm/day, and the rota- deformed and ε is the strain of the sample. The values tional speed was 4Ð5 minÐ1.

0.10

50 0.10 40 (a) (b) 0.08 0.10

0.10 40 0.08 35 40 50 0.06 0.08

–1 0.10 0.10 40 0.08 35 Q

–1 0.06 30 40 Q 30 0.06 50 0.10 40 0.04 0.08 35 0.08 0.04 0.06 30 25 20 30 40 0.04 0.06 0.08 35 0.04 30 0.02 0.06 0.02 25 4 20 350 400 450 500 20 , GPa 0.04 0.06 3 10 30 G 0.02 30 300 400 500

25 , GPa 0.02 20 0.04 3 350400 450 500 G 0.04 10 0.02 25 2 300 400 500 20 20 2 350 400 450 500 0.02

0.02 1 20 10 350 400 450 500 1 300 400 500 T, K T, K

Ð1 Fig. 1. Temperature dependences of the internal friction Q and the shear modulus G of the Li2B4O7 single crystal at a frequency of 30 mHz for the crystallographic directions (a) [100] and (b) [001] at different strain amplitudes ε. ε, 10Ð5: (a) (1) 4, (2) 6, (3) 8, and (4) 10 and (b) (1) 2, (2) 6, and (3) 10.

∆ (a) 430 (b) Tm, K G/G 0.05 0.05 1 400 0.09 0.04 420 0.04 1 3 m m –1 –1 0.03 , K 0.03 390 0.08 m Q Q T 0.02 2 410 0.02 2 380 0.07 0.01 0.01 400 2 4 6 8 10 2 4 6 8 10 12 –5 ε, 10–5 ε, 10

Ð1 ∆ ε Fig. 2. Dependences of the parameters of the dissipative process (1) Tm, (2) Qm , and (3) G/G on the strain of the Li2B4O7 single crystal for the crystallographic directions (a) [100] and (b) [001].

3. RESULTS AND DISCUSSION temperature range 390Ð410 K. This maximum corre- Figure 1 shows the temperature dependences of the sponds to a decrease in the shear modulus G(T). The internal friction QÐ1 and the shear modulus G of an anomaly observed in the shear modulus G(T) of the Li2B4O7 single crystal for different amplitudes of shear Li2B4O7 single crystal is similar to the anomaly found strain ε = (2Ð10) × 10Ð5. The experimental dependences in the temperature dependence of the velocity of longi- QÐ1(T) exhibit a maximum in the internal friction in the tudinal ultrasonic waves at a frequency of 150 kHz [3].

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 84 BILANYCH et al. It is evident from Fig. 2 that the parameters of the afore- ity of jumps in direct and opposite directions, respec- mentioned anomalies depend on the strain amplitude. tively. ε As the strain amplitude increases, the temperature Tm For strong mechanical stresses (vσ ӷ kT), expres- at the maximum of the internal friction somewhat sion (1) can be rewritten in the form decreases, whereas its amplitude QÐ1, the half-width ∆ ∆ dx H Ð vσ T1/2, and the defect of the shear modulus G/G ------= ν aexp Ð.------(2) ∆ 0  increase. Here, G = G0 Ð G∞, G0 is the shear modulus dt kT before the anomaly, and G∞is the shear modulus after In this case, the mobility of a kinetic particle depends ε × Ð5 the anomaly. At = 10 10 , these parameters are as on the temperature and mechanical stress through the Ð1 × Ð3 ∆ σ follows: Tm = 410 K, Qm = 20 10 , T1/2 = 33 K, and term v , which contains the activation volume. It is evi- ∆G/G = 0.12 for the [100] direction and T = 392 K, dent that, under the action of an external stress field, the m potential barrier decreases when the particle attempts to Ð1 × Ð3 ∆ ∆ Qm = 20 10 , T1/2 = 21 K, and G/G = 0.08 for the pass through it “along the field” and increases in the [001] direction. opposite case. It follows from the above considerations σ ӷ Apart from the maximum observed in the depen- that, under the condition v kT, an increase in the dence QÐ1(T), there also exists a difference in the back- mechanical stress at a constant frequency of deforma- Ð1 tion (f = const) should lead to a decrease in the temper- ground of the internal friction Q f before the anomaly ature Tm at the maximum in the internal friction, which and after it. In the temperature range T < Tm, the back- is observed in Figs. 1 and 2. According to the data pre- Ð1 sented in Figs. 1 and 2, the maximum in the internal ground of the internal friction Q f is approximately Ð1 ε × Ð5 × Ð3 friction Qm at < 6 10 sharply decreases with equal to 10 10 . At temperatures T > Tm, the back- Ð1 decreasing ε. It should be expected that, when the ground values of Q f for the [001] and [100] directions strains are significantly smaller (of the order of 10Ð6Ð increase by 12 × 10Ð3 and 4 × 10Ð3, respectively. 10Ð7), this maximum can be lower than the background The aforementioned features in the behavior of loss in the sample. The jump in the background loss Q−1(T) and G(T) at different values of ε indicate that, in decreases in a similar manner. The amplitude of the the studied ranges of frequencies and temperatures, the anomaly also noticeably decreases with a decrease in internal friction in single-crystal lithium tetraborate the amplitude of the exciting field. depends on the strain amplitude. The change in the frequency of the exciting stress As is known [7, 8], the amplitude-dependent effects field does not lead to a shift in either the temperature of of internal friction in crystals are frequently interpreted the internal friction maximum or the range of variation in the framework of a concept according to which a dif- in ∂G/∂T. The dependences QÐ1(T) and G(T) at differ- fusing particle executes a thermally activated motion ent infralow frequencies of mechanical vibrations are within a multistep potential relief that is a superposition depicted in Fig. 3. As can be seen from Fig. 3, the curve of potential wells of equal depths. On this basis, for a G(T) in the temperature range 380Ð420 K is similar to crystal subjected to a harmonically variable strain, we that of the elastic moduli in the relaxation processes. At consider an ensemble of kinetic structural units moving the same time, the location of the maxima in the afore- along a potential relief of height H whose slope varies mentioned anomalies of the strength properties and the with frequency f. The velocity of motion of a kinetic temperature range of their manifestation weakly particle (kinetic unit) in a homogeneous periodic poten- depend on the frequency. This suggests the absence of tial field can be represented by the following relation- a pronounced dispersion and makes reliable estimation ship [7]: of the activation energy from the frequency shift of the maxima difficult. This frequency behavior of the σ σ strength properties is typical of the amplitude-depen- dx ν H Ð v H + v ------= 0aexp Ð------Ð exp Ð------,(1) dent internal friction [8]. dt kT kT It is known [9] that the crystal structure of Li2B4O7 ν is based on an anion sublattice formed by a three- where 0 is the frequency of attempts made by the dimensional network consisting of boronÐoxygen tri- kinetic particle to pass through the potential barrier, a is angles and tetrahedra with channels occupied by tetra- ∂H the distance between the potential barriers, v = Ð------is hedrally coordinated lithium ions. The lithium ions are ∂σ rather weakly bound to the anion sublattice and can the rate of decrease in the height H with an increase in occupy several energetically equivalent positions; the mechanical stress σ (v is often called the activation hence, during thermal motion, they can pass from one volume), and vσ is the work done by the external field equilibrium position to another. The thermal activation on a kinetic particle with an effective activation volume of the mobility of these ions in Li2B4O7 occurs at T > to move it through a potential barrier of height H. The 350 K. This is responsible for both the dielectric loss first and second terms in Eq. (1) characterize the veloc- and ionic conduction in these single crystals. Note that

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 INTERNAL FRICTION 85

0.10 0.10 (a) (b) 45 50

0.10 40 0.10 45 40 50

35 –1 –1 Q Q 0.05 40 0.05 0.10 30 45 0.10 35 30 40 50 0.05 25 40 0.05 30 3 20 30 40 300 400 500 35 20 25 300 400 500 , GPa G 0.05 , GPa G 30 0.05 2 20 300 400 500 20 30 300 400 500 25

1 20 20 300 400 500 300 400 500 T, K T, K

Ð1 Fig. 3. Temperature dependences of the internal friction Q and the shear modulus G of the Li2B4O7 single crystal at different infralow frequencies for the crystallographic directions (a) [100] and (b) [001]. f, mHz: (a) (1) 30, (2) 50, and (3) 80 and (b) (1) 10, (2) 50, and (3) 10. the temperature range of manifestation of the anoma- we estimated the activation volume and determined the lies observed in the strength properties of Li2B4O7 sin- radius of the kinetic particle: r = 0.50 Å. This value is gle crystals at infralow frequencies (Fig. 1) virtually close to the classical radius of lithium ions [14]. coincides with the range of thermal activation of lith- It is evident from Figs. 1 and 2 that, over the entire ium ions. Therefore, it can be assumed that the temperature range under investigation, the magnitudes observed maximum in the mechanical loss is attributed of the shear modulus for the sample oriented along the to the activation of the mobility of the lithium ions. In [001] direction are, on average, 25% greater than those the temperature range 300Ð600 K, the lattice of for the sample oriented along the [100] orientation. Li2B4O7 single crystals can be considered as two struc- Moreover, the jump in the background value of the tural subsystems differing in the temperature behavior internal friction (at T > 450 K) as a result of the activa- of mechanical rigidity. In this case, the totality of lith- tion of the dissipative process for the [001] direction is ium ions can be treated as one of these subsystems. An also considerably larger. This anisotropy of the strength increase in the temperature gives rise to a peak in the properties suggests that the cation sublattice is charac- internal friction against the background of the mechan- terized by different mobilities in different crystallo- ical loss in Li2B4O7 in the temperature range 390Ð graphic directions. Since the difference between the 410 K. The results obtained are consistent with avail- background losses in the internal friction upon the pas- able data on the dielectric properties of lithium tetrabo- sage through the maximum is larger for the [001] direc- rate [10Ð12]. According to these data, lithium tetrabo- tion, it is natural to assume that the mobility of lithium rate possesses ionic conduction due to the Li+ mobility cations is also larger in this direction. of a thermoactivation nature. Judging from the intensity of the maximum in the 4. CONCLUSIONS internal friction of Li2B4O7 and its temperature location Thus, the internal friction and the shear modulus of and using the formula taken from [13], Li2B4O7 single crystals along the crystallographic directions [100] and [001] at infralow frequencies were 1 --- Ð1 2 investigated by the method of torsional vibrations. 2kTmQm v = ------, (3) Analysis of the results obtained revealed a dissipative N0G0 process occurring in the temperature range 390Ð410 K

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 86 BILANYCH et al. and anisotropy of the strength properties. The activa- 4. A. É. Aliev, Ya. V. Burak, V. V. Vorob’ev, et al., Fiz. tion volume and the radius of kinetic particles were Tverd. Tela (Leningrad) 32 (9), 2826 (1990) [Sov. Phys. determined. The radius of kinetic particles proved to be Solid State 32, 1641 (1990)]. close to the classical radius of lithium ions. 5. V. S. Bilanych, Author’s Abstract of Candidate’s Disser- tation (Uzhgorod. Univ., Uzhgorod, 1993). It was established that the maximum in the internal 6. I. I. Turok, V. M. Holovey, and P. P. Puga, Declaration friction in the range 390Ð410 K is associated with the Ukraine Patent No. 32242, MPK6 S01 V 35/12. migration of lithium ions from one energetically equiv- 7. Physical Acoustics: Principles and Methods, Vol. 3, alent position to another under the action of a periodic Part A: The Effect of Imperfections, Ed. by W. P. Mason external stress field. (Academic, New York, 1966; Mir, Moscow, 1969). 8. V. S. Postnikov, Internal Friction in Metals (Metal- In the range T > Tm, there occurs a jump in the back- lurgiya, Moscow, 1969). ground of the internal friction due to absorption of 9. S. F. Radaev, L. A. Muradyan, L. F. Malakhova, et al., mechanical energy by the mobile cation sublattice at Kristallografiya 34 (6), 1400 (1989) [Sov. Phys. Crystal- temperatures higher than the temperature of the anom- logr. 34, 842 (1989)]. Ð1 alies in Q (T) and G(T). The difference between the 10. Ya. V. Burak, I. T. Lyseœko, and I. V. Garapin, Ukr. Fiz. jumps in the background of QÐ1 for the crystallographic Zh. 34 (2), 226 (1989). directions [100] and [001] can be explained by the 11. A. É. Aliev and R. R. Valetov, Kristallografiya 36 (6), anisotropy of the mobility of lithium cations. 1507 (1991) [Sov. Phys. Crystallogr. 36, 855 (1991)]. 12. M. M. Nassar, B. F. Borisov, E. V. Charnaya, et al., Vestn. Leningr. Univ., Ser. 4: Fiz., Khim., No. 2, 82 REFERENCES (1991). 13. V. A. Bershteœn, Yu. A. Emel’yanov, and V. A. Stepanov, 1. V. M. Rizak, I. M. Rizak, N. D. Baisa, et al., in Proceed- Fiz. Tverd. Tela (St. Petersburg) 22 (2), 399 (1980) [Sov. ings of the IFM-10, Madrid, Spain, 2001, p. 219. Phys. Solid State 22, 234 (1980)]. 2. A. É. Aliev, Ya. V. Burak, and I. T. Lyseœko, Izv. Akad. 14. B. K. Vainshtein, V. M. Fridkin, and V. L. Indenbom, Nauk SSSR, Neorg. Mater. 26 (9), 1991 (1990). Modern Crystallography, Vol. 2: Structure of Crystals (Nauka, Moscow, 1979; Springer-Verlag, Berlin, 1982). 3. A. É. Aliev and R. R. Valetov, Fiz. Tverd. Tela (St. Petersburg) 34 (10), 3061 (1992) [Sov. Phys. Solid State 34, 1639 (1992)]. Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 87Ð93. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 84Ð90. Original Russian Text Copyright © 2003 by Valeeva, Tang, Gusev, Rempel’.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Observation of Structural Vacancies in Titanium Monoxide Using Transmission Electron Microscopy A. A. Valeeva*, G. Tang**, A. I. Gusev*, and A. A. Rempel’* * Institute of Solid-State Chemistry, Ural Division, Russian Academy of Sciences, ul. Pervomaœskaya 91, Yekaterinburg, 620219 Russia e-mail: [email protected] ** Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57/VI, Stuttgart, 70550 Germany Received May 8, 2002

Abstract—Structural vacancies were directly observed in nonstoichiometric ordered titanium monoxide using high-resolution transmission electron microscopy under a magnification of 4 × 106. The observation of structural vacancies became possible due to their ordering and the formation of continuous vacancy channels in certain crystallographic directions. Microdiffraction was employed to orient the sample in the direction permitting the observation of vacancy channels. Transmission electron microscopy providing a magnification of tens of thousands of times revealed that titanium monoxide grains do not contain cracks and macropores and confirmed that the free volume detected picnometrically in the titanium monoxide is concentrated in structural vacancies on the titanium and oxygen sublattices. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION vacancies have thus far not been observed, despite the fact that they exist in all crystals at any finite tempera- Structural vacancies are an essential element of ture. In this study, we propose to use the ordered state crystal structure, as they affect the diffusion, mechani- of strongly nonstoichiometric compounds for the cal, electrical, and magnetic properties of solids [1]. observation of vacancies. In this case, structural vacan- The existence of structural vacancies has thus far been cies are arranged in an orderly way, so that directions in established indirectly, for instance, through the com- a crystal which pass only through vacant sites to form bined application of three methods, namely, chemical, continuous vacancy channels can be found. When ide- picnometric, and x-ray diffraction. Structural vacancies ally ordered, the atom columns and the vacancy chan- are a fundamental element of the crystal structure of nels are arranged with a certain periodicity and the transition-metal compounds, such as the carbides, atomic rows in HTEM images should exhibit contrast at nitrides, and oxides. The concentration of structural the points where they are broken by vacancy channels vacancies in these substances may be as high as a few directed perpendicular to the image plane. tens of percent, which accounts for these compounds being termed strongly nonstoichiometric [1, 2]. Inter- Nonstoichiometric cubic titanium monoxide TiOy particle interaction forces structural vacancies in with B1-type basal structure is the most promising strongly nonstoichiometric compounds to order [3], among the nonstoichiometric compounds for direct and the appearance of superstructure reflections in the observation of vacancies. The nonstoichiometric cubic x-ray and neutron diffractograms may be considered titanium monoxide belongs to strongly nonstoichio- additional indirect evidence of the existence of vacan- metric interstitial compounds [1, 2] and exhibits a very cies. broad range of homogeneity, from TiO0.7 to TiO1.25. A In the present study, we used high-resolution trans- specific feature of the titanium monoxide is that it has mission electron microscopy (HTEM) [4] to observe heavily defected sublattices not only of the oxygen but structural vacancies directly. This method makes it pos- also of the metal [5, 6]. It is the high concentration of sible to observe atomic rows in crystals. The image thus vacancies on the metal sublattice that makes their obtained results from a superposition of tens of atomic observation by HTEM possible. The fact is that most of layers. In the disordered state, structural vacancies are the information seen in HTEM images is provided by distributed at random and, therefore, each column of metal atoms and structural vacancies of the metal sublattice sites contains both atoms and structural vacan- lattice because of the low scattering capacity of the cies. Structural vacancies are, however, not seen atoms of oxygen compared to that of titanium. For the because different columns do not differ in the relative same reason, one cannot observe structural vacancies in content of atoms and structural vacancies, thus produc- strongly nonstoichiometric carbides, because the ing no contrast. For the same reason, thermal structural vacancies exist on the carbon sublattice only.

88 VALEEVA et al. The first studies [6, 7] of titanium monoxide using The crystal structure of the titanium monoxide was the chemical, picnometric, and x-ray diffraction meth- studied using x-ray diffraction on a Siemens D-500 dif- α ods already revealed that the monoxide of equiatomic fractometer (CuK 1, 2 radiation) in the BraggÐBrentano composition, TiO1.00, has 15Ð16 at. % vacant sites on geometry in the 2θ angle interval from 10° to 160° in each sublattice. This finding was questioned in [8], steps of ∆(2θ) = 0.025°. The samples used were finely where it was assumed that the free volume available in dispersed powders of titanium monoxide. titanium monoxide derives from the presence of micro- The microscopic and crystalline structure of the tita- voids and pores rather than from metal sublattice nium monoxide was also studied using electron micros- vacancies. According to [8], picnometric determination copy and diffraction. Samples for these studies were of density entails a large inherent error, because in this prepared as follows. The sample surface was ground, case, the free volume is detected not only in the vacant and its central part was thinned with a dimple machine sites but also in such defects as microcracks and pores. (Gatan, USA). After cleaning in alcohol, the sample It is the presence of microcracks and micropores that was ion-milled to make a hole. The surface was milled accounts for the large difference of the picnometric by Ar ions at a voltage of 5 kV. The microstructure was from the so-called x-ray diffraction density, which is studied at the edge of the foil, whose thickness did not calculated using data on the composition and lattice exceed 50 nm. period of the monoxide under the assumption of the Images with a high atomic resolution were obtained absence of metal vacancies. using a JEM-4000FX electron microscope (wave- The viewpoint expressed in [8] did not receive sub- length, λ = 0.00164 nm; voltage, 400 kV). The microd- sequent support, because the results obtained in numer- iffraction and microstructure were studied with a Phil- ous structural studies could be accounted for only for ips CM-200 electron microscope (beam width, 70 nm; λ the case of metal vacancies existing in TiOy. However, wavelength, = 0.00251 nm; voltage, 200 kV). no direct evidence of the existence of metal vacancies Microdiffraction was observed at zero angle with respect to the high-resolution images. The aperture per- in the TiOy monoxide had been presented until recently. mitted the observation of [111]]B1 ND [220]B1 diffrac- This has stimulated us to attempt a direct observation spots. The bright-field image was obtained in the tion of structural vacancies in ordered Ti O oxide 5 5 central spot [000]B1; the dark-field image, in the [111]B1 using high-resolution transmission microscopy. λ reflection. The instrumental constant was L = Rdhkl = 2.33Ð2.51 mm nm, where L is the chamber length, R is the distance between the diffracted [hkl]B1 and direct 2. SAMPLES AND EXPERIMENTAL [000] beams, and d is the interplanar distance. TECHNIQUES B1 hkl

A sample of the TiO1.087 titanium monoxide was 3. RESULTS AND DISCUSSION prepared by reacting titanium with titanium dioxide powder at a high temperature in vacuum. To reach a dis- Figure 1 presents x-ray diffraction patterns of the ordered distribution of structural vacancies, the sample ordered and disordered titanium monoxide TiO1.087. thus prepared was quenched from 1330 to 300 K at a The x-ray diffractogram of the disordered sample rate of 200 K sÐ1. The measured picnometric density of obtained by quenching from 1330 K contains only the the sample (4.968 g cmÐ3) was found to be 20% less structural reflections due to the basal cubic phase. The Ð3) x-ray pattern of the sample subjected to annealing and than the theoretical density (5.962 g cm , which was subsequent slow cooling to room temperature revealed calculated taking into account the chemical composi- additional weak reflections. This indicates the formation and experimentally determined cubic-lattice tion of a superstructure in the annealed titanium mon- period (aB1 = 0.4174 nm) under the assumption of the oxide. A thorough analysis of the position and intensity metal sublattice having no defects. A comparison of the of the additional reflections showed the ordering of picnometric with theoretical density, made with due atoms and structural vacancies of titanium and oxygen account of the relative content of titanium and oxygen, in TiOy to correspond to the Ti5O5 monoclinic super- yields the formula Ti0.833O0.906. This means that 0.167 structure with space group C2/m (A12m/1). of all sites on the titanium sublattice and 0.094 of the oxygen sublattice sites are vacant. The monoclinic ordered structure Ti5O5 has been detected with x-ray [10Ð12] and electron [13] diffrac- The ordered titanium monoxide was obtained by tion in titanium monoxide annealed at a temperature annealing the synthesized sample at 1330 K for 4 h, below 1263 K. The low-temperature modification of with subsequent slow cooling to 300 K at a rate of titanium monoxide was found to be stable within the 10 K hÐ1. The technique employed and the conditions composition region 0.9 < O/Ti < 1.1. The low-tempera- of preparation and characterization of nonstoichiomet- ture ordered phase has also been observed to form in ric disordered and ordered titanium monoxide samples the phase transition from the high-temperature cubic throughout the interval in which the cubic phase exists disordered phase in titanium monoxide single crystals are described in [9]. by using transmission electron microscopy [14] and x-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

OBSERVATION OF STRUCTURAL VACANCIES 89

Ordered TiO1.087 1000

500 Counts 2000 (111)Bl Disordered TiO1.087

1500

(220) 1000 Bl

500 (311)Bl (200)Bl (222)Bl 20 30 40 50 60 70 80 2θ, deg

Fig. 1. X-ray diffractogram of disordered (the structural lines of the B1 phase are identiﬁed) and ordered titanium monoxide TiO1.087. The x-ray diffractogram of the ordered sample clearly exhibits the structural and additional superstructure lines.

ray diffraction analysis [15]. The same phase transition [220]B1 reflections from the basal cubic phase. Calcula- was detected, using electron diffraction, in polycrystal- tion of the position of the superstructure reflections line samples of cubic titanium monoxide TiO0.7, (Fig. 2) in the reciprocal-space cross section containing TiO1.03, TiO1.13, and TiO1.19 [14]. the [111]B1 and [220]B1 reflections from the basal phase The experimental microdiffraction pattern of the permitted us to assign these basal-phase reflections to ordered TiO1.087 monoxide (Fig. 2a) contains structural the [211 ]C2/m and [2 22 ]C2/m superstructure spots, reflections (high-intensity spots) and a number of addi- respectively. tional reflections. Computer modeling of microdiffraction made for the Ti5O5 titanium monoxide with space Thus, the presence of a large number of structural group C2/m identifies those reflections that are due to a vacancies on the oxygen and titanium sublattices favors superstructure (spots with an intensity weaker by a few the formation of an ordered titanium monoxide phase. times than that from the basal phase in Fig. 2). The As shown by an analysis of the position and intensity of model pattern shows the indices of the structural reflec- the superstructure reflections in the x-ray diffractogram tions. A comparison of the experimental (Fig. 2a) with and of spots in the microdiffraction pattern, the order- the model pattern (Fig. 2b) reveals that the additional ing of the oxygen and titanium vacancies corresponds reflections contain superstructure features and, in addi- to the ordered monoclinic Ti5O5 phase (space group tion, reflections which are not connected with ordering C2/m). but originate from the presence of twins, which are seen clearly in the photomicrographs (Fig. 3). The TEM microphotographs of TiO1.087 obtained The upward-pointing normal to the plane of Fig. 2 is with a magnification of 38000, 50000, and 115000 show the samples studied to be polycrystals with a min- oriented along [011 ]C2/m or [112 ]B1. The distances imum grain size of about 100 nm (Fig. 3) and a high from the central spot to the two nearest structural spots density of grain boundaries. Electron microscopy arranged at 90° are 9.98 and 15.90 mm. The interplanar revealed that monoxide grains do not contain distances dhkl calculated using the instrumental constant macropores or cracks and that the content of small ± Rdhkl = 2.33Ð2.51 mm nm are 0.238 0.012 and pores is so small (Fig. 3c) as not to affect the picnomet- ± 0.147 0.007 nm and correspond to the [111]B1 and ric density noticeably. Thus, the large free volume

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

90 VALEEVA et al.

(a) (b) – – 131 – 222

– – 220 111

– 311 000 –– 311

– – 111 – 220

– – 222 – 131

Fig. 2. (a) Experimental microdiffraction pattern of the TiO1.087 titanium monoxide and (b) computer simulation of microdiffraction ≡ for the TiO1.087 monoxide with the basal lattice parameter aB1 = 0.4174 nm (electron beam direction [011]C2/m [112]B1). The structural reﬂection indices are speciﬁed.

(a) (b) (c)

800 nm 600 nm 300 nm

Fig. 3. Photomicrographs of a TiO1.087 sample obtained using transmission electron microscopy [magniﬁcation (a) 38000, (b) 50000, and (c) 115000] show the sample to be polycrystalline with a minimum grain size of approximately 100 nm and a very low micropore content. detected in titanium monoxide is due to structural interfaces of the twins with the host matrix exhibit vacancies rather than to pores, cracks, or other defects. striped contrast in the form of rows of parallel lines; Twins are clearly seen in the photomicrographs these lines are extinction thickness contours. Judging (Figs. 3b, 3c). They are shaped as elongated rectangles from the extinction contours (Fig. 3c), the observed ﬁve and, in some cases (Fig. 3c), are interconnected. The twins can be classed into at least two families. Belong-

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 OBSERVATION OF STRUCTURAL VACANCIES 91

aB1/2 –– [112]B1 [211]B1

2aB1/4

() aB1 5/5 6aB1/3

() aB1 5/2 1 1 B B – [102] – [121]

[010]B1 Ti O – – [101]B1 Ti O

Fig. 4. Distribution of structural vacancies and titanium and Fig. 5. Projection of the atoms and vacancies of the ordered ≡ oxygen atoms (observed in projection along [101]C2/m ≡ Ti5O5 monoclinic phase in the [110]C2/m [111 ]B1 direc- [201]B1) of ordered monoclinic titanium monoxide Ti5O5 ≡ ≡ (space group C2/m) onto the (104)C2/m (201)B1 plane. The tion onto the (211 )C2/m (111 )B1 plane. The visible dis- visible distance between rows of like atoms is d = a /2. row B1 tance between rows of like atoms is drow = (2 /4)aB1. The The breaks between atoms in the same row specify vacancy ≡ vacancy channels are oriented along [110] ≡ [111 ] ; channels oriented along [101]C2/m [201]B1. The distance C2/m B1 the distance between atoms separated by a vacancy is between atoms separated by a vacancy is (5 /5)aB1. The (6 /3)aB1. The contour of the cross section of the Ti5O5 contour of the cross section of the Ti5O5 phase unit cell by ≡ ≡ the (104)C2/m (201)B1 plane is shown. phase unit cell by the (211 )C2/m (111 )B1 plane is shown.

λ ing to the first family is the twin lying slightly above the length, Vc is the unit-cell volume, is the wavelength, θ midpoint of Fig. 3c and bounded by broad contour Fg is the structural factor, and is the angle of crystal lines. The other twins are members of the second fam- deviation from the exact reflecting position. For low- ily. The rotation angles between the twins of the second order reflections in metals, the structural factor is about family are 67°Ð68°. In crystals with a structure of the 1 nm and the electron wavelength for an accelerating B1 type, such a rotation angle (~65°) can occur in twin- voltage 200 kV is 0.00251 nm; in the case where the ning along the (111)B1 family planes if the matrix zone angle between the crystal and the incident beam coin- axis is directed along [112]B1 [16]. One of the possible cides with the Bragg angle, cosθ ≈ 1 [17]. Calculation ξ ≈ ± explanations for the appearance of striped contrast lies of the extinction length yields g 22 5 nm. Taking in the existence of a displacement field at the twin– n = 2 for the number of extinction contours, we obtain matrix interface. One could conceive, however, of a an approximate estimate of t ≈ 45 nm for the foil thick- more likely reason; namely, the presence of an extinc- ness. This estimate seems realistic; indeed, the foil tion contour implies that the depth of the twin location thickness does not exceed 50 nm, because with a larger is at an angle to the surface of the sample (foil) under thickness one would not have reached the atomic reso- study. This angle ϕ can be found as the angle between lution attained in this study. the host zone axis ([112]B1 direction) and the normal to We performed computer simulation for an ordered the twinning plane. For the twinning plane (111)B1, we monoclinic titanium monoxide Ti5O5 (space group have ϕ ≈ 19.5°; for the planes (1 11) and (11 1) , ϕ ≈ C2/m) to find the directions most favorable for the B1 B1 observation of vacancy channels using high-resolution ° ° 61.9 ; and for the (11 1)B1 plane, the angle is 90 . Since electron microscopy. These are the directions in which the extinction contours of the twins of the second fam- the vacancy channels lie along the direction of propaga- ily are fairly narrow, their angle ϕ does not reach 90°; tion of the electron beam. As shown by an analysis of we may thus accept that twinning occurs along the the crystal lattice of ordered titanium monoxide, the three directions most favorable for the observation of (111)B1 or (1 11)B1 and (11 1)B1 planes. The foil thick- ≡ ξ vacancy channels are [101]C2/m [201]B1 (Fig. 4), ness t can be found as t = n g, where n is the number of ξ π θ λ ≡ ≡ extinction contours, g = Vccos / Fg is the extinction [110]C2/m [111 ]B1 (Fig. 5), and [011]C2/m [112]B1

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 92 VALEEVA et al.

(a) (b) (c) – – – – [110]B1 [111]B1 [110]B1 [111]B1 0.221 nm 3aB1/3 2aB1/2 3aB1/3 2aB1/2 1 B 1 B –– [152] –– [152]

1 nm –– – [201]B1 [201]B1 Ti O Ti

≡ Fig. 6. Distribution of structural vacancies and Ti and O atoms seen in the [011]C2/m [112]B1 direction relative to the unit cell of ≡ ordered monoclinic Ti5O5 oxide (space group C2/m) (projection of atoms and vacancies of the Ti5O5 phase in the [011]C2/m ≡ [112]B1 direction onto the (1 14)C2/m (112)B1 plane). (a) Simulation with inclusion of Ti and O atoms, (b) experimental pattern obtained with a high atomic resolution (magniﬁcation 4 × 106), and (c) simulation made neglecting O atoms. The visible model

distance between rows of like atoms is drow = 3 /3aB1 = 0.241 nm (for aB1 = 0.4174 nm), and the experimental row separation is drow = 0.221 nm. The breaks between atoms in the same row in the model patterns correspond to vacancy channels in the [011]C2/m

direction. The visible distance between atoms separated by a vacancy is 2 /2aB1 = 0.295 nm. Model patterns show the contour of

the cross section of the Ti5O5 phase unit cell by the (1 14)C2/m plane.

(Fig. 6a). Viewed in other directions, each column con- The visible distance between two atoms separated tains both vacancies and atoms, so that the HTEM con- by a vacancy is a maximum (equal to (6 /3)a or trast will be either low or altogether zero. To visualize B1 the arrangement of atoms and vacancies in a cross sec- 0.3408 nm for aB1 = 0.4174 nm) when the (211 )C2/m ≡ tion perpendicular to the chosen direction, we deter- atomic planes are observed in the [110]C2/m [111 ]B1 mined the projections of atoms and vacancies of the direction (Fig. 5). In the other two directions, the visi- Ti5O5 monoclinic phase onto the planes perpendicular ble distances between two atoms separated by a to each of the directions indicated above. The vacant vacancy is smaller: (2 /2)a = 0.2951 nm for the sites in the projections identify the vacancy channels. B1 ≡ [011]C2/m [112]B1 direction (Fig. 6a) and (5 /5)aB1 = Vacancy channels can be visualized in the ≡ 0.1867 nm for [101]C2/m [201]B1 (Fig. 4). Thus, the [101] ≡ [201] direction, i.e., along the normal to C2/m B1 ≡ the (104) plane family (Fig. 4), or along [110] ≡ [110]C2/m [111 ]B1 direction is preferable for direct C2/m C2/m observation of vacancies using high-resolution electron [111 ]B1, i.e., along the normal to the (211 )C2/m planes microscopy. (Fig. 5). In both directions, the channel passes through The visible distance between rows of like atoms alternating nonmetal and metal vacancies. The third ≡ ≡ drow = dTiÐTi dOÐO is maximum if atoms and vacancies direction of the vacancy channel is [011]C2/m [112]B1, of the Ti5O5 ordered phase are projected in the i.e., the normal to the (1 14) planes (Fig. 6a). In this C2/m [011] ≡ [112] direction onto the (1 14) ≡ case, the vacancy channel passes through vacant sites C2/m B1 C2/m (112) plane, which is perpendicular to the direction of belonging only to the metal or only to the nonmetal B1 sublattice. Figures 4, 5, and 6a specify the atomic-row projection (Fig. 6a); in this case, drow = (3 /3)aB1 or directions, the visible distances drow between rows of 0.2410 nm for aB1 = 0.4174 nm. If the atoms are pro- ≡ like atoms, and visible distances between two atoms jected in the [101]C2/m [201]B1 direction onto a plane ≡ separated by a vacancy. Also shown are the contours of of the (104)C2/m (201)B1 family, the visible distance is the cut of the unit cell made by the above planes. drow = aB1/2 = 0.2087 nm; for the projection in the

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 OBSERVATION OF STRUCTURAL VACANCIES 93

≡ ≡ ACKNOWLEDGMENTS [110]C2/m [111 ]B1 direction onto the (211 )C2/m The authors are indebted to Dr. Phillipp and (111 )B1 plane, this distance is (2 /4)aB1 = 0.1476 nm. Thus, atomic rows are most clearly visualized in the Mrs. M. Kelsch (Max Planck Institute, Stuttgart, Ger- [011] direction. Figure 6b displays an experimental many) for assistance in the experiment and to Prof. C2/m H.-E. Schaefer (Institute of Theoretical and Applied pattern obtained with a high atomic resolution (magni- × 6 Physics, Stuttgart University) and Dr. E.V. Shalaeva fication 4 10 times) for the ordered monoclinic (Institute of Solid-State Physics, Ural Division, Rus- TiO1.087 monoxide (space group C2/m). The image was sian Academy of Sciences) for valuable discussions. ≡ obtained in the (1 14)C2/m (112)B1 plane (perpendicu- ≡ lar to the [011]C2/m [112]B1 direction) and clearly reveals atomic rows. REFERENCES 1. A. I. Gusev and A. A. Rempel’, Nonstoichiometry, Dis- The sample studied is about 45 nm thick; therefore, order, and Order in Solids (Ural. Otd. Ross. Akad. Nauk, the observed pattern results from a superposition of Yekaterinburg, 2001). several tens of layers, but only in one direction within the region covered are atoms arranged in distinct rows. 2. A. I. Gusev, A. A. Rempel, and A. A. Magerl, Disorder The atomic chains in a row undergo breaks; these and Order in Strongly Non-Stoichiometric Compounds: Transition Metal Carbides, Nitrides, and Oxides breaks are associated with vacancies, but their arrange- (Springer, Berlin, 2001). ment is not fully periodic. The presence of regions with a darker background in the photomicrograph (Fig. 6b) 3. A. A. Rempel’, Ordering Effects in Non-Stoichiometric means that the sample has a larger thickness there. We Interstitial Compounds (Nauka, Yekaterinburg, 1992). do not see oxygen atoms in the experimental pattern, 4. D. Shindo and K. Hiraga, High-Resolution Electron because their scattering factor is one third that of the Microscopy for Materials Science (Springer, Berlin, titanium atoms. For comparison, Fig. 6c shows a com- 1998). puter simulation pattern; it exhibits only the titanium 5. A. A. Valeeva, A. A. Rempel, M. A. Müller, et al., Phys. atoms and, therefore, more closely approaches the Status Solidi B 224 (2), R1 (2001). experimental image of Fig. 6b. In the model pattern, 6. P. Ehrlich, Z. Elektrochem. 45 (5), 362 (1939). each five titanium atoms are separated by a metal structural vacancy. While in the experimental image there is 7. S. Andersson, B. Collen, U. Kuylenstierna, and A. Mag- no strict periodicity in the alternation of atoms and neli, Acta Chem. Scand. 11 (10), 1641 (1957). vacancies, this image is, on the whole, similar to the 8. B. F. Ormont, Structures of Inorganic Substances (Tekh- model one. The absence of exact periodicity in the teoretizdat, Moscow, 1950), p. 462. atomic rows is accounted for by the fact that the com- 9. A. A. Valeeva, A. A. Rempel’, and A. I. Gusev, Neorg. position of the experimentally studied titanium monox- Mater. 37 (6), 716 (2001). ide TiO1.087 differs from that of monoxide TiO1.00 10. D. Watanabe, J. R. Castles, A. Jostson, and A. S. Malin, (Ti0.833O0.833); the latter can form a perfect ordered Nature 210 (5039), 934 (1966). phase. 11. D. Watanabe, J. R. Castles, A. Jostson, and A. S. Malin, Acta Crystallogr. 23 (2), 307 (1967). 4. CONCLUSION 12. E. Hilti and F. Laves, Naturwissenschaften 55 (3), 131 (1968). Thus, the analysis of the microdiffraction pattern 13. D. Watanabe, O. Terasaki, A. Jostsons, and J. R. Castles, has shown the ordering of the titanium and oxygen in The Chemistry of Extended Defects in Non-Metallic atoms and vacancies to correspond to the ordered Ti5O5 Solids, Ed. by L. Eyring and M. O. Keeffe (North-Hol- phase with the monoclinic structure C2/m. Transmis- land, Amsterdam, 1970), p. 238. sion electron microscopy performed with a magnifica- 14. A. W. Vere and R. E. Smallman, Mechanism of Phase tion of tens of thousands of times revealed that titanium Transformation in Crystalline Solids (Institute of Met- monoxide grains are dense and do not contain large als, London, 1969), p. 212. pores. This provided evidence supporting the earlier 15. H. Terauchi, J. B. Cohen, and T. B. Reed, Acta Crystal- conjectures that the free volume detected in titanium logr. A 34 (4), 556 (1978). monoxide using the picnometric method is due to the 16. L. M. Utevskiœ, Electron Diffraction Microscopy in presence of a large number of structural vacancies on Physical Metallurgy (Metallurgiya, Moscow, 1973). the titanium and oxygen sublattices. High-resolution transmission electron microscopy permitted the detec- 17. Electron Microscopy of Thin Crystals, Ed. by P. B. Hir- tion and observation of structural vacancies in titanium sch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan (Plenum, New York, 1965; Mir, Moscow, monoxide. The observation of structural vacancies in 1968). × 6 TiO1.087 at a magnification of 4 10 became possible due to vacancy ordering, which gave rise to the formation of continuous vacancy channels. Translated by G. Skrebtsov

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DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

A Correlation between Manifestations of the Magnetoplastic Effect and Changes in the Electron Paramagnetic Resonance Spectra after Quenching of NaCl : Eu Single Crystals R. B. Morgunov and A. A. Baskakov Institute of Solid-State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia e-mail: [email protected] Received February 19, 2002

Abstract—The electron paramagnetic resonance (EPR) spectra of Eu2+ impurity ions in NaCl : Eu single crystals are investigated. It is found that the intensity of the EPR spectra undergoes prolonged (~200 h) multistage variations after quenching of NaCl : Eu single crystals. The variations observed in the EPR signal intensity are explained by the aggregation of impurityÐvacancy dipoles into complexes. It is revealed that the magnetoplastic effect (a change in the microhardness in a magnetic ﬁeld with an induction of 6 T) in these crystals manifests itself at an intermediate stage of impurity aggregation when all individual impurityÐvacancy dipoles are temporarily stabilized in the sample. This can be associated with the thermally activated transformation of the internal atomic structure in the majority of already existing complexes. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. SAMPLES AND EXPERIMENTAL TECHNIQUE The first reliable experimental evidence supporting the existence of the magnetoplastic effect in ionic crys- The experiments were performed with quenched tals was obtained by Al’shits et al. [1Ð5], who revealed crystals of NaCl : Eu (0.1 at. %). The choice of and thoroughly investigated the influence of a mag- europium impurities was made for the following rea- netic field with an induction B ~ 1 T on the mobility of son: the use of europium-doped crystals provided a means for observing the magnetoplastic effect [6, 7] individual dislocations [1Ð3] and the yield point [4, 5]. and monitoring the state of paramagnetic point defects At present, it has been reliably established that the through recording and analyzing the EPR spectra [11]. magnetoplastic effect in ionic crystals is caused, The crystals were quenched by heating at a temperature among other factors, by magnetic-field-induced of 770 K for 1 h followed by cooling in a copper vessel changes in both the kinetics of aggregation and the to a temperature of 293 K at a mean rate of ~5 K/s. In atomic configuration of intermediate nonequilibrium order to prevent diffusion of oxygen and hydroxyl complexes formed by paramagnetic point defects serv- groups into the crystals under investigation, their heating as stoppers for dislocations [6Ð10]. In order to gain ing in all experiments was performed in a helium or a deeper insight into the microscopic mechanism of the argon atmosphere. influence of a magnetic field on the structure of the The presence of magnetosensitive defect complexes aforementioned complexes and to elucidate their in the studied crystal was revealed as follows. The crys- atomic structure, it is necessary to answer the follow- tal was quenched and allowed to stand for a certain time ing questions. What happens to the majority of impu- (from 1 min to 200 h) followed by measurement of the rity point defects at the instant the magnetoplastic microhardness H. Then, the crystal was exposed to a effect manifests itself? How much do their magnetic magnetic field with an induction of 6 T for 10 ms and properties change with time after quenching? What are the microhardness H was measured once again. The the specific features in the behavior of spins of defects magnitude of the difference ∆H between these micro- at the stage of nucleation of magnetosensitive defect hardnesses served as a quantitative measure of the mag- configurations? In this respect, the purpose of the netoplastic effect and indirectly characterized the concentration of magnetosensitive centers generated in the present work was to elucidate how the formation of crystal. In all the experiments, the Vickers microhard- magnetosensitive states of the complexes formed by ness of the crystals was measured under an indenter point defects affects the electron paramagnetic reso- load of 0.2 N and the loading time was 10 s. The inden- nance (EPR) spectra, whose variations, in turn, reflect tation diagonals were oriented along directions of the the aggregation of Eu2+ paramagnetic impurityÐ [110] type. Each point in the graphs was obtained by vacancy dipoles into complexes. averaging over 20Ð30 measurements. As a result, the

A CORRELATION BETWEEN MANIFESTATIONS 95 error in the microhardness measurement was reduced to +3/2 +1/2 +1/2 +3/2 –5/2 –3/2 –1/2 1Ð1.5%. This is a typical error in measurements of the +5/2 microhardness in ionic crystals (see, for example, [12, 13]). It should be noted that, in all the experiments, the microhardness measurements and exposure of the stud- : –1/2 : –7/2 : –5/2 : –3/2 : –1/2 : +1/2 : +1/2 : +3/2 Z Z Z

ied crystals to a magnetic ﬁeld were carried out at a XY XY XY XY XY temperature precisely equal to 293 K and the total duration of these procedures (taken together) was considerably shorter than that of the transient processes examined in this work. 1234 The crystal contained europium impurities in the B , kOe 2+ 0 form of Eu ÐVk impurityÐvacancy dipoles in which the cation vacancy compensated for an excess positive charge of the impurity ion. According to the estimates 220 made by Ruiz-Mejia et al. [14], the free energy of defects of all other types (for example, individual ions and neutral atoms of europium) is substantially higher 2+ than that of the Eu ÐVk system; hence, the probability , MPa

H 210 of their formation is negligible. The aggregation kinetics of impurityÐvacancy dipoles after quenching was judged from the changes in the characteristics of the EPR spectra. The EPR spectra were recorded on a standard radiospectrometer operating in the X band at a 200 modulation frequency of 100 kHz in the dc magnetic 0 50t, h 100 field scan range B = 0.7Ð4 kOe. Earlier [6, 7], it was 0 Fig. 1. Dependence of the microhardness H of the NaCl : Eu demonstrated that, when the microwave magnetic field crystal on the time elapsed after quenching in the absence of power in the spectrometer is of the order of 100 mW, a magnetic field. The EPR spectra shown are measured at crystals undergo resonance-induced disordering. More- the instants of indentation. The transitions between different over, the procedure of measuring the spectra, in princi- Eu2+ electron spin states split in the crystal field are indi- ple, can affect their form. For this reason, the power cated at the upper left of the figure. used in our experiments was two orders of magnitude less (~1 mW) than that in [6, 7] (~100 mW). netic field used made it possible to observe only eight groups of lines among the above set of groups (Fig. 1). 3. RESULTS AND DISCUSSION The line splitting within a particular group is associated The EPR spectrum of Eu2+ÐV impurityÐvacancy with the hyperfine interaction of unpaired electrons k with europium nuclei. In the crystal, there are isotopes dipoles consists of 14 groups of narrow lines [11]. The 151 153 Eu2+ ion has spin S = 7/2, and the Zeeman splitting for of two sorts ( Eu and Eu) with an approximately the Eu2+ f electrons partially screened by the d electron identical abundance, the same nuclear spins I = 5/2, but shell is larger than the splitting in the crystal field. Con- different hyperfine interaction constants. Therefore, sequently, it can be expected that the EPR spectrum each group involves 2(2I + 1) = 12 lines. will contain 2S = 7 groups of lines; each group corre- As can be seen from Fig. 1, the microhardness of the sponds to transitions between 2S + 1 = 8 spin sublevels crystals nonmonotonically varies with time t after their split by the crystal field in the Eu2+ ion. However, since quenching. The EPR spectra also change significantly the impurityÐvacancy dipoles have the C2v symmetry, (Fig. 1). These changes can be caused not only by vari- which is lower than the symmetry of the crystal lattice, ations in the hyperfine interactions occurring in the the locations of the groups of lines for the dipoles ori- complexes and the associated change in the overlap ented parallel and perpendicular to the constant mag- between individual spectral lines but also by variations netic field differ from each other. Therefore, the total in the signal amplitudes corresponding to these lines. In number of groups of lines in the EPR spectrum is equal order to separate these factors and to analyze the to 14. According to Aguilar et al. [11], the principal changes in the number of free impurityÐvacancy magnetic axes of the impurityÐvacancy dipole system dipoles in a pure form, we chose several nonoverlap- are aligned parallel to the [001] direction and, in the ping individual lines of two types characterizing free perpendicular plane, along the [110] direction. In our impurityÐvacancy dipoles (not aggregated into com- experiments, the constant magnetic field was aligned plexes) oriented along the [001] direction and in the parallel to the [001] direction in order to ensure the perpendicular plane (see, for example, the inset in maximum distance from the groups of lines to the cen- Fig. 2). The intensity of the chosen lines decreases with ter of the spectrum and the possibility of processing time elapsed after the quenching of the crystal (see the individual lines. The limited scan range of the dc mag- inset in Fig. 2). The shape of these lines is described by

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

96 MORGUNOV, BASKAKOV

a concentration of free impurityÐvacancy dipole spins b c indicate that, within a few minutes after the quenching d 20 of the crystal, this concentration is approximately equal e to 1.3 × 10Ð2 at. %, which is substantially less than the total concentration of europium ions in the sample 1430 1440 1450 (~10Ð1 at. %). This suggests that the time of heat treat-

16 B , Oe 2 0 ment of the crystal at a high temperature (or the heat , 10

, MPa treatment temperature itself) is less than that required to N 1 10 H dissolve all the complexes involved in the sample. Con- 2 ∆ sequently, in principle, there is a potentiality for 3 increasing the number of magnetosensitive complexes 1 in the crystal and enhancing the magnetoplastic effect. A decrease in the intensity of the EPR spectra of 0 0 50 100 150 NaCl : Eu single crystals within a few hours after t, h quenching from a temperature of 550 K was revealed by Rubio [15], who investigated this phenomenon at Fig. 2. Dependences of (1) the number N of spins of free impurityÐvacancy dipoles (not aggregated into complexes) T = 373 K (curve 3 in Fig. 2). However, long-term evo- and (2) the magnitude of crystal softening (the difference lution in the intensity of the EPR spectra at T = 293 K between microhardnesses) ∆H upon exposure to a mag- (observations at this temperature are of prime interest netic-field pulse (amplitude, 6 T; pulse width, 10 ms) on the for revealing a correlation with the kinetics of forma- time t elapsed after quenching of the crystal. (3) Depen- dence of the number N of spins of free impurityÐvacancy tion of magnetosensitive complexes) was not examined dipoles on the time t elapsed after quenching of the crystal in [15]. The general tendency to a decrease in the EPR from 550 K according to the data taken from [15] (measure- signal intensity with time elapsed from the quenching ments were performed at a temperature of 373 K). The inset of samples can be explained in terms of the formation shows single EPR lines (associated with the electron spin of impurityÐvacancy dipole complexes that experience transitions + 3/2 +5/2 for the projection of the 153Eu the internal transformations responsible for the consid- nuclear spin onto the direction of the dc magnetic field of erable changes observed in the EPR spectra of the the spectrometer Iz = + 5/2) within (a) 1, (b) 5, (c) 100, (d) 120, and (e) 170 h after the quenching of the crystal. defect complexes as compared to the spectrum of individual impurityÐvacancy dipoles. In particular, these transformations, which take part of the spins “out of the a Gaussian function with a high accuracy (95%). By per- game,” can be associated with the following features: forming double integration of the Gaussian function and (i) a substantial broadening of the spectral line due to normalizing the result of this integration to the area of the dipoleÐdipole or exchange interactions between impu- reference Mn2+ EPR spectrum of MgO, we determined rityÐvacancy dipoles; (ii) a change in the effective g the number N of spins of free impurityÐvacancy dipoles factor; (iii) the formation of chemical bonds between (not aggregated into complexes) and obtained the depen- individual paramagnetic defects and, consequently, dence of N on the time t elapsed after the quenching of spin pairing; and (iv) an additional splitting of spin lev- the crystal. The systematic error in the determination of els due to a change in the crystal field in the complexes. N was approximately equal to 30%, and the scatter in the It should be noted that the decrease observed in the data on N from experiment to experiment (determined intensity of the spectral line with time elapsed after the for six measurements) did not exceed 6%. quenching is not attended by variations in its width and shape (see the inset in Fig. 2). Therefore, only defects The dependence N(t) is represented by a compound of the same type, namely, free impurityÐvacancy curve; it steeply falls off at t < 20 h, flattens out in the dipoles, make a contribution to the EPR signal over a range 20 h < t < 90 h, and again falls off at t > 100 h period of 200 h after the quenching of the crystal. (curve 1 in Fig. 2). Special investigations demonstrated that this dependence is identical for all individual lines The plateau observed in the dependence N(t) belonging to any group involved in the spectrum. Of directly indicates the occurrence of several different particular interest is the fact that the magnetoplastic processes responsible for the magnetic properties of the effect (characterized by a nonzero value of ∆H and cor- formed complexes and, apparently, for their atomic responding to softening of crystals) is observed in the configuration. The change in the atomic configuration range 20 h < t < 90 h, i.e., at the stage of relaxation of of these complexes can be judged from the nonmono- the point defect subsystem when all spins of free impu- tonic change in the microhardness that is sensitive to rityÐvacancy dipoles in the sample are temporarily sta- the mobility of dislocations, which, in turn, depends on bilized (curve 2 in Fig. 2). the elastic stress fields induced by impurity–vacancy dipole complexes (Fig. 1). The inhibition of spin pair- Before proceeding to the discussion of the results ing can be explained under the assumption that a num- obtained, one additional essential remark needs to be ber of defects formed at the initial stage that do not con- made. The upper estimate of the total area of the spec- tribute to the EPR signal appear to be thermally unsta- trum and a standard recalculation in terms of the total ble and unfavorable for further growth through the

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 A CORRELATION BETWEEN MANIFESTATIONS 97 attachment of new impurityÐvacancy dipoles. Conse- role played by the magnetic field, most likely, is quently, after a time, these complexes can decay into reduced to a change in the probability of spin-depen- individual impurityÐvacancy dipoles due to thermal fluc- dent transitions occurring inside the complexes and the tuations. This process competes with the formation of probability of forming their intermediate configura- favorable configurations, leads to the escape of dipoles tions differing in atomic structure. from the complexes, and, apparently, cancels out the general tendency to a decrease in N in our experiments. ACKNOWLEDGMENTS The above assumption is in good agreement with the obtained dependence of the magnitude of the microhard- This work was supported by the Russian Foundation ness H on the time elapsed after the quenching of the for Basic Research, project nos. 01-03-42501 and 01- crystal under investigation (Fig. 1). In actual fact, 02-06307. according to the universally accepted concepts, the larger the size of defect complexes, the stronger the dislocation REFERENCES motion drag [16]. On this basis, the increase observed in 1. V. I. Al’shits, E. V. Darinskaya, T. M. Perekalina, and the microhardness H in the early stage of relaxation (at A. A. Urusovskaya, Fiz. Tverd. Tela (Leningrad) 29, 467 t < 20 h) can be explained by the increase in the size of (1987) [Sov. Phys. Solid State 29, 265 (1987)]. complexes at this stage. The subsequent decrease in the 2. V. I. Al’shits, E. V. Darinskaya, and O. L. Kazakova, microhardness H in the range 20 h < t < 40 h corresponds Pis’ma Zh. Éksp. Teor. Fiz. 62, 352 (1995) [JETP Lett. to a plateau in the dependence N(t) and, hence, can be 62, 375 (1995)]. associated with the decay of a number of already existing 3. V. I. Al’shits, E. V. Darinskaya, and O. L. Kazakova, Zh. Éksp. Teor. Fiz. 111, 615 (1997) [JETP 84, 338 complexes. Finally, the increase observed in the micro- (1997)]. hardness H at t > 50 h approximately corresponds to the 4. A. A. Urusovskaya, V. I. Al’shits, A. E. Smirnov, and second descending portion in the dependence N(t) and N. N. Bekkauér, Pis’ma Zh. Éksp. Teor. Fiz. 65, 470 can be attributed to a continuation of the aggregation (1997) [JETP Lett. 65, 497 (1997)]. from favorable configurations of the complexes. 5. V. I. Al’shits, N. N. Bekkauér, A. E. Smirnov, and

Thus, the magnetoplastic effect manifests itself only A. A. Urusovskaya, Zh. Éksp. Teor. Fiz. 115, 951 (1999) at the stage of relaxation of the excited subsystem of [JETP 88, 523 (1999)]. 6. Yu. I. Golovin, R. B. Morgunov, V. E. Ivanov, and point defects in the case when individual impurityÐ A. A. Dmitrievskiœ, Zh. Éksp. Teor. Fiz. 117, 1080 vacancy dipoles already form a number of complexes (2000) [JETP 90, 939 (2000)]. capable of dissociating (possibly, in part) or undergoing 7. Yu. I. Golovin, R. B. Morgunov, and A. A. Dmitrievskii, transformations in their internal atomic configurations Mater. Sci. Eng. A 288 (2), 261 (2000). due to thermal fluctuations. Note that the stability of 8. Yu. I. Golovin and R. B. Morgunov, Fiz. Tverd. Tela complexes depends not only on the elastic potential but (St. Petersburg) 37 (5), 1352 (1995) [Phys. Solid State also on the exchange interaction and the mutual orien- 37, 734 (1995)]. tation of spins inside the complexes. Therefore, the 9. R. B. Morgunov and A. A. Baskakov, Fiz. Tverd. Tela concentration ratio of complexes with particular con- (St. Petersburg) 43 (9), 1632 (2001) [Phys. Solid State figurations can be controlled by varying the spin orien- 43, 1700 (2001)]. tation in a specified magnetic field over a short period 10. R. B. Morgunov, A. A. Baskakov, D. V. Yakunin, and when terms of different multiplicities approach each I. N. Trofimova, Fiz. Tverd. Tela (St. Petersburg) 44 (9), other. This assumption is consistent with the concepts 2525 (2002) [Phys. Solid State 44, (2002)]. regarding the influence of a magnetic field on spin- 11. G. Aguilar, E. Munoz, H. Murrieta, et al., J. Chem. Phys. dependent reactions [17]. The analogy between this 60, 4665 (1974). phenomenon and the magnetoplastic effect was first 12. M. Suszynska, P. Grau, M. Szmida, and D. Nowak- drawn by Al’shits et al. [18]. Wozny, Mater. Sci. Eng. A 234Ð236, 747 (1997). 13. Yu. S. Boyarskaya, D. Z. Grabko, and M. S. Kats, Phys- ics of Microindentation (Shtiintsa, Kishinev, 1986), 4. CONCLUSIONS pp. 10Ð12. 14. C. Ruiz-Mejia, U. Oseguera, H. Murrieta, and J. Rubio, The experimental data obtained allowed us to draw J. Chem. Phys. 73, 60 (1980). the following inferences. The multistage stepwise 15. J. Rubio, J. Phys. Chem. Solids 52, 101 (1991). decrease in the intensity of the EPR spectra during 16. E. Orozco and J. Soullard, Philos. Mag. A 50, 425 relaxation of the subsystem of point defects excited by (1984). quenching of the crystal indicates that the processes 17. K. M. Salikhov, Yu. N. Molin, R. Z. Sagdeev, and associated with the aggregation of free impurityÐ A. L. Buchachenko, in Spin Polarization and Magnetic vacancy dipoles into complexes compete with the pro- Field Effects in Radical Reactions, Ed. by Yu. N. Molin cesses of internal transformation and decay of the com- (Elsevier, Amsterdam, 1984). plexes into individual dipoles. The magnetosensitive 18. V. I. Al’shits, E. V. Darinskaya, and E. A. Petrzhik, Fiz. states of defects arise at the stage of relaxation of the Tverd. Tela (Leningrad) 33 (10), 3001 (1991) [Sov. above subsystem when the aggregation is retarded as a Phys. Solid State 33, 1694 (1991)]. result of the decay of already existing complexes. The Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

Physics of the Solid State, Vol. 45, No. 1, 2003, pp. 98Ð103. Translated from Fizika Tverdogo Tela, Vol. 45, No. 1, 2003, pp. 95Ð100. Original Russian Text Copyright © 2003 by Tyapunina, Krasnikov, Belozerova, Vinogradov.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Effect of Magnetic Field on Dislocation-Induced Unelasticity and Plasticity of LiF Crystals with Various Impurities N. A. Tyapunina*, V. L. Krasnikov**, É. P. Belozerova***, and V. N. Vinogradov*** * Moscow State University, Vorob’evy gory, Moscow, 119899 Russia ** Nekrasov State University, Kostroma, Russia *** Kostroma State Technological University, Kostroma, 156005 Russia e-mail: [email protected] Received February 20, 2002; in ﬁnal form, April 22, 2002

Abstract—The effect of a magnetic field of 0.04–0.8 T on the amplitude dependence of the internal friction in LiF crystals with various impurities is studied using a two-component resonant-oscillator method at a frequency of 80 kHz. The state of the samples was controlled in situ from the currentÐvoltage characteristics of a compound oscillator. It is found that the state of the dislocations-pinning-centers system changes in the presence of a magnetic field; this leads to an increase in the internal friction and plasticity of the samples. The effect of the magnetic field is sensitive to the impurity composition in the crystal. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. EXPERIMENTAL TECHNIQUE AND MATERIALS It is well known that inelastic properties of crystals, We studied the effect of a magnetic field on the IF such as the internal friction (IF) and elastic-modulus and the Young modulus defect in AHCs in the kilohertz defect (EMD), are determined by dislocation mecha- frequency range by using the method of a compound nisms in the kilohertz frequency range. The mechani- piezoelectric-quartz oscillator [5]. A longitudinal cal-energy loss at room temperature is associated with standing ultrasonic wave was excited at a fundamental energy transfer from vibrating dislocations to the frequency of 80 kHz over the sample length. The max- ε phonon subsystem of the crystal. Experiments show imum amplitude of strain 0 at an antinode of the stand- that the IF and EMD also respond to external agencies ing wave was varied from 10Ð7 to 10Ð3. which directly affect the electron subsystem of the The magnetic field, with induction vector B perpen- crystal. For example, inelastic properties of alkali- dicular to the direction of propagation of ultrasound, halide crystals (AHCs) change under the action of a was produced by a permanent magnet (0.04Ð0.3 T) and magnetic field [1–3]. an electromagnet (0.4Ð1 T). The state of the sample was traced in situ from the This work aims at studying the effect of a magnetic currentÐvoltage characteristics of the compound oscil- field on the unelasticity of LiF crystals with different lator [6, 7]. The density and distribution of dislocations impurity compositions. Among other things, we ana- were controlled with the help of the selective etching lyzed the effect of Ni impurity on the unelasticity of LiF method. Data concerning the initial state of the crystals crystals. According to Al’shits et al. [4], the presence of are given in Table 1. the Ni impurity considerably intensifies the magneto- The rod-shaped samples were cleaved from crystals plastic effect in AHCs. along the cleavage planes. The sample length corre-

Table 1. Data on the initial state of LiF crystals under investigation σ ρ 9 Ð2 δ Ð4 Material, labeling Main impurities Yield stress cr, MPa Dislocation Density , 10 m Internal friction 0, 10

LiFI Mg 7.8 2Ð6 0.9

LiFII Ca, Ba 3.9 3Ð8 2.4

LiFIII Mg, Ni 10.3 0.8Ð2 0.5 Note: The composition of impurities was determined spectrographically. The total molar fraction of impurities was C ~ 10Ð5.

EFFECT OF MAGNETIC FIELD ON DISLOCATION-INDUCED UNELASTICITY 99 sponded to the condition of resonance excitation of the oscillator at the fundamental frequency. After preparation, the samples were subjected to prolonged ageing 16 1 (for three years) at room temperature. No fresh disloca-

tions were introduced before measurements. 5 12 2 10 ×

3. EXPERIMENTAL RESULTS δ 8 AND DISCUSSION Let us first consider the effect of magnetic and ultra- 4 sonic fields on LiF samples. Test experiments based on repetitive selective etch- 0 1234 4 ing before and after holding of the samples in a mag- ε0 × 10 netic field of induction B < 1 T did not reveal motion or multiplication of dislocations. However, preliminary Fig. 1. Amplitude dependence of the internal friction of an LiFIII sample: (1) first test; (2) second test carried out 1.5 h long-term holding of samples in a magnetic field pro- after the first test. duced a considerable effect on the inelastic properties (IF and Young modulus defect) of LiF crystals, as described in [8, 9]. Ultrasound can produce both reversible and irre- 140 versible changes in the dislocation structure of crystals, 2 which are manifested in the currentÐvoltage characteristics of the oscillator and in the results of IF measure- 100 1 ments.

Let us analyze the dependence of IF on the strain , V p amplitude for B = 0, which is typical of LiF (Fig. 1). V 60 Curves 1 and 2 in Fig. 1 correspond to two successive tests on the LiFIII sample. It can be seen from Fig. 1 that the height and position of the IF peak appearing at the 20 ε ≈ × Ð4 amplitude 0 2 10 remain unchanged in repeated tests. 0 5101520 , mV The ascending branches of the IF peaks are linear- V ized in the GranatoÐLucke coordinates [10]: Fig. 2. CurrentÐvoltage characteristics of an LiFII sample []()εδε()δ ΓεÐ1 tested in a magnetic ﬁeld (1) B = 0.3 T and (2) B = 0. ln 0 Ð 01 0 = C0 Ð 0 , (1) δ where 01 is the amplitude-independent IF correspond- ε Γ ing to small values of 0 and C0 and are constants related to the dislocation structure parameters. In both 1.2 cases, the squares of the correlation coefﬁcients were 2 2 r = 0.97. 0.9

The values of IF on the descending branches of the , V p

εÐ2 2 V 0.6 peaks are proportional to 0 with r = 0.79 and 0.96 in 1 the first and second tests, respectively. Consequently, 0.3 the IF peaks are associated with the hysteresis mechanism of dislocation-induced internal friction and can be interpreted using the RogersÐSuprun model [11, 12]. 0 2 4 6 8 10 12 , V The simultaneous effect of ultrasound and a mag- V netic field on the unelasticity of LiF crystals was stud- Fig. 3. CurrentÐvoltage characteristics of an LiFIII sample ied on a batch of samples, as well as on the same sam- tested in a magnetic field (1) B = 0.08 T and (2) B = 0. ple. IF and EMD measurements on the same sample were made in alternating tests in the presence and in the absence of a magnetic field. of a magnetic field, we considered the quantity Vp pro- ε The influence of the magnetic field is illustrated in portional to the strain amplitude at the antinode 0 of Figs. 2 and 3, showing the currentÐvoltage characteris- the standing wave. As a magnetic field is applied, for a tics Vp(V) of the compound oscillator with LiFII and constant voltage V supplied to the quartz plates, the ε LiFIII samples, respectively. As a response to the action value of Vp (and, hence, of 0) decreases. This means

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003

100 TYAPUNINA et al.

3.5 30 1

) 3.0 5 B

( 20 0 2 ε 2.5 × 10

δ (0)/ 0

ε 2.0 10

1.5 0 1 2 3 4 5 1.0 ε × 104 0 0.2 0.4 0.6 0.8 1.0 1.2 0 Vp, V

ε ε Fig. 4. Dependence of 0(0)/ 0(B) on Vp for an LiFIII sam- Fig. 5. Amplitude dependence of the internal friction of an ε ple. Vp is proportional to the amplitude 0(0) before the LiFIII sample tested in a magnetic ﬁeld (1) B = 0.08 T and application of a magnetic ﬁeld. (2) B = 0.

80 1 2.5 60 1 2 2.0

5 2 3 10 40 1.5 10 ×

M 3 /

δ × 1.0

M 20 ∆ 0.5

0 0.4 0.8 1.2 1.6 0 0.5 1.5 2.5 3.5 4 4 ε0 × 10 ε0 × 10

Fig. 6. Amplitude dependence of the Young modulus defect Fig. 7. Effect of magnetic field on the amplitude depen- for an LiFIII sample tested in a magnetic field (1) B = 0.08 T dences of internal friction for LiFII samples for B equal to and (2) B = 0. (1) 0, (2) 0.3, and (3) 0.6 T. that the value of IF increases jumpwise upon applica- were determined by the impurity composition of the tion of the magnetic field and the sample becomes more crystal. plastic. It was shown earlier that this effect is not a con- The results of IF measurements on the LiFIII sample sequence of the emergence of a vortex electric field at in the regime of alternating tests in a magnetic field and the instants of application and removal of the magnetic in zero field are presented in Fig. 5. The points corre- field [1, 3]. sponding to curve 1 in Fig. 5 were obtained in a mag- The strain amplitude jump under the action of the netic field B = 0.08 T, while the points corresponding to magnetic field can be characterized by the ratio curve 2 were obtained for B = 0. A comparison of ε ε ε curves 1 and 2 in Fig. 5 shows that the height of the IF 0(0)/ 0(B), where 0(0) is the amplitude in zero field δ δ ε peak in a magnetic field is larger ( max(B)/ max(0) = 1.54 and 0(B) is the same in the magnetic field. This ratio in the present case) and the position of the peak is for fixed V varies depending on the magnetic induction ε ε shifted towards smaller amplitudes 0. B and the ultrasound amplitude 0. The dependence of this ratio on ε for LiF in the field B = 0.08 T is shown Figure 6 shows the amplitude dependences of the 0 III Young modulus defect for the same LiF sample for in Fig. 4. The values of V ~ ε lying on the abscissa axis III p 0 the amplitudes corresponding to ascending segments of were obtained in a zero magnetic field. It can be seen ε ε ε the IF peaks in Fig. 5. The points on curve 1 in Fig. 6 that, for a certain amplitude 0m, the ratio 0(0)/ 0(B) were obtained in a magnetic field B = 0.08 T, while the attains its maximum value and then decreases upon an ε points on curve 2 were obtained for B = 0. It can be seen increase in 0. A similar dependence of the ratio that, in the presence of a magnetic field, the amplitude ε ε ε 0(0)/ 0(B) on 0 has also been observed for an LiFII dependence of the EMD is manifested for smaller val- ε ε sample [1], but the values of the ratio 0(0)/ 0(B) and ues of ε and the EMD values attained are higher than ε 0 of the quantity 0m were different; i.e., these values for B = 0.

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EFFECT OF MAGNETIC FIELD ON DISLOCATION-INDUCED UNELASTICITY 101 δ ε The amplitude dependences ( 0) for different val- 5 ues of the magnetic induction B are shown in Figs. 7 9 and 8. The curves in Fig. 7 were obtained for three LiFII 4 samples with mirror cleavage faces for B = 0 (curve 1), 0.3 (curve 2), and 0.6 T (curve 3). Curves 1Ð5 in Fig. 8 3 correspond to the same LiFI sample. Each series of 7 2 ε 4 points corresponding to the same amplitude 0 on this curve was obtained for the induction B = 0, 0.25, 0.35, 10 1

0.6, and 0.8 T, respectively. The height of the IF peak δ × ε 5 and its position on the 0 axis depend on the magnetic induction B. This can be seen from Fig. 9, showing the δ dependences of the peak heights max (curve 1) and of ε the amplitudes 0m corresponding to these peaks 3 (curve 2) on the magnetic induction. Thus, our experiments revealed the following. (1) The magnetic ﬁeld affects the IF and EMD both 1 0 0.5 1.0 1.5 2.0 in the amplitude-independent and in the amplitude- ε × 4 dependent regions. In analogy with the magnetoplastic 0 10 and photoacoustic effects, we will refer to the changes in the inelastic properties of crystals under the com- Fig. 8. Amplitude dependence of the internal friction of an bined action of ultrasound and a magnetic ﬁeld as the LiFI sample for various values of magnetic induction B: magnetoacoustic effect (MAE). The strongest MAE is (1) 0, (2) 0.25, (3) 0.35, (4) 0.6, and (5) 0.8 T. observed for amplitudes of ultrasound corresponding to dislocation depinning from impurity centers.

(2) There exists a threshold value B0 of magnetic induction starting from which the effect of a magnetic ﬁeld on AHC unelasticity is detectable. The values of 9 1 2.2 B for LiF crystals with different impurity compositions 0 4 4

differed considerably. For example, the threshold value 10

10 1.8 B0 = 0.14 T for LiFII crystals, while B0 < 0.08 T for 8 × m × 0

LiFIII containing Ni. ε max An analysis of the IF and EMD data revealed that δ 7 1.4 the application of a magnetic field changes the parame- 2 ters characterizing the motion of dislocations: the start- τ ing stresses st, the dislocation segment length LN, the 0 0.2 0.4 0.6 0.8 number N0 of pinning points on the segment LN, the B, T maximum force Fm of interaction of a dislocation with δ a pinning center, and the binding energy U. These data Fig. 9. (1) Peak heights max and (2) corresponding ampli- ε are given in Tables 2Ð4 for LiFI, LiFII, and LiFIII sam- tudes 0m as functions of the magnetic field for a LiFI samples, respectively. ple. The data describing the variation of the dislocation structure parameters in a magnetic field for the above- fested much more clearly than that for the remaining mentioned LiF crystals with various impurities are compared in Fig. 5. crystals. The height of the IF peak increases considerably in the magnetic field B = 0.08 T, and the displace- It can be seen from Table 5 that the dislocation den- ment of the peak position towards smaller amplitudes sity ρ remained unchanged in all crystals in the ultrasound-amplitude and magnetic-induction ranges cov- for B = 0.08 T exceeds the corresponding displacement for the LiFI sample for B = 0.3 T. The length LN of a dis- ered. Let us first compare the data for LiFI and LiFII σ location loop for LiF in a magnetic field increases with those for LiFIII. The yield stress cr (Table 1) of III LiFII crystals is half the value for LiFI and is smaller more than twofold (row 4 in Table 5), while the corre- than that for LiFIII by a factor of 2.5. We may conclude sponding values for LiFI and LiFII increase by a factor that the LiFII is the “softest” crystal. Table 5 shows that of 1.66 and 1.15, respectively. The binding energy of an the magnetic field changes the dislocation structure impurity center and a dislocation, as well as the starting parameters for LiFII to the smallest extent. The values stresses, for LiFIII in a magnetic field are reduced to a σ of cr for LiFI and LiFII samples are of the same order greater extent than for crystals free of Ni impurity of magnitude. However, the MAE for LiFIII is mani- (rows 7, 8 in Table 5).

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 102 TYAPUNINA et al.

Table 2. Parameters of dislocation structure of LiFI samples Table 3. Parameters of dislocation structure of LiFII samples Reference sample Reference sample Parameter B = 0.3 T Parameter B = 0.3 T (B = 0) (B = 0) ε Ð4 ε Ð4 0m, 10 1.0 2.1 0m, 10 1.60 1.65 ρ, 109 mÐ2 33ρ, 109 mÐ2 33 Γ, 10Ð4 1.61 4.92 Γ, 10Ð4 1.81 4.58 Ð6 Ð6 LN, 10 m 1.0 0.6 LN, 10 m 0.45 0.39 N0 34N0 37 Ð10 Ð10 Fm, 10 N 5.24 6.88 Fm, 10 N 3.17 3.99 U, eV 0.93 1.22 U, eV 0.56 0.71 τ τ st, MPa 0.73 1.20 st, MPa 0.27 0.42

Table 4. Parameters of dislocation structure of LiFIII samples Table 5. Variation of dislocation structure parameters of LiF crystals in a magnetic ﬁeld Reference sample Parameter B = 0.08 T (B = 0) Parameter LiFI(Mg), LiFII(Ca, Ba), LiFIII(Mg, Ni), ratio B = 0.3 T B = 0.3 T B = 0.08 T ε , 10Ð4 1.0 2.6 0m ε (B)/ε (0) 0.48 0.97 0.38 ρ, 109 mÐ2 1.0 1.0 0m 0m ρ(B)/ρ(0) 1 1 1 Γ, 10Ð4 2.03 3.74 Γ(B)/Γ(0) 0.33 0.40 0.54 Ð6 LN, 10 m 0.87 0.36 LN(B)/LN(0) 1.66 1.15 2.40 N0 34 N0(B)/N0(0) 0.75 0.43 0.75 F , 10Ð10 N 4.39 6.62 m Fm(B)/Fm(0) 0.76 0.79 0.66 U, eV 0.78 1.18 U(B)/U(0) 0.76 0.79 0.66 τ τ τ st, MPa 0.63 1.10 st(B)/ st(0) 0.61 0.64 0.57

The changes in the macroscopically measured yield induction vary in different manners. For example, σ σ cr stress cr of the crystals in a magnetic field can be esti- decreases in the field B = 0.48 T by a factor of 2.5 for τ mated from the data on the starting stresses st. Indeed, LiFI crystals and by a factor of 1.7 for LiFII crystals. σ τ cr and st are connected through a linear dependence τ τ The LiFIII samples containing Ni were investigated [13]. Knowing the ratio st(0)/ st(B), we can find the σ σ variation of the yield stress under the action of a mag- by us for B = 0, 0.04, and 0.08 T. The ratio cr(0)/ cr(B) calculated from the currentÐvoltage characteristics was netic field. The results of such estimation for LiFI and LiF crystals are given in Table 6, which also contains equal to unity for B = 0.04 T and to 1.88 for B = 0.08 T. II A comparison of these results with the data presented in the results of direct measurements for LiF crystals with Table 6 for LiF shows that the critical value of B for Mg impurity [14]. I which the yield stress starts to decrease for LiFIII crys- It can be seen from Table 6 that our results and the tals is lower than that for LiFI crystals free of Ni. The σ σ data from [14] for samples with Mg impurity are simi- ratio cr(0)/ cr(B) for LiFIII in the field B = 0.08 T is σ lar. The yield stresses cr for crystals with different 1.3 times larger than the corresponding ratio for LiFII impurity compositions for the same values of magnetic for B = 0.3 T. The obtained results, demonstrating a

Table 6. Variation of the yield stress of LiF crystals under the action of a magnetic ﬁeld

Variation of B, T Sample yield stress 0.1 0.2 0.3 0.4 0.48 σ σ cr(0)/ cr(B) 1 1.3 2.07 2.6 2.7 LiF(Mg) [14] τ τ st(0)/ st(B) = Ð 1.44 2.2 Ð 2.5 LiFI(Mg) σ σ cr(0)/ crB 1 1.15 1.50 Ð 1.67 LiFII(Ca, Ba)

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003 EFFECT OF MAGNETIC FIELD ON DISLOCATION-INDUCED UNELASTICITY 103 considerable effect of Ni impurity on the yield stress of 4. V. I. Al’shits, E. V. Darinskaya, O. L. Kazakova, et al., AHCs in a magnetic field, are in accord with the results Izv. Akad. Nauk, Ser. Fiz. 57 (11), 2 (1993). of direct measurements [15]. 5. S. P. Nikanorov and B. K. Kardashev, Elasticity and Dis- location-Induced Inelasticity of Crystals (Nauka, Mos- cow, 1985). 4. CONCLUSIONS 6. É. P. Belozerova, Available form VINITI, No. 5059 Thus, the above results lead to the conclusion that (1984). the MAE is sensitive to the impurity composition and that the presence of Ni impurity enhances the MAE, as 7. E. K. Naimi, Available form VINITI, No. 2589 (1985). in the case of the magnetoplastic effect. Since the dis- 8. N. A. Tyapunina, É. P. Belozerova, V. L. Krasnikov, and location density remained unchanged upon the action V. N. Vinogradov, Vestn. Tambov. Univ., Ser. Estestv. of a magnetic field on a crystal, while the parameters Tekh. Nauki 5 (3), 345 (2000). characterizing the system of dislocations and their pin- 9. N. A. Tyapunina, V. L. Krasnikov, and É. P. Belozerova, ning centers varied significantly, we can conclude that Izv. Akad. Nauk, Ser. Fiz. 64 (9), 1776 (2000). the magnetic field affects the interaction of dislocations 10. A. Granato and K. Lucke, in Ultrasonic Methods for with their pinning centers, as well as the structure of the Studying Dislocations (Inostrannaya Literatura, Mos- centers themselves. It was found that magnetically sen- cow, 1963), p. 27. sitive centers containing Ni impurity are the most sen- 11. D. G. Blair, T. S. Hutchison, and D. H. Rogers, Can. J. sitive to a magnetic field (have the lowest threshold Phys. 49 (6), 635 (1971). value B0) and their structure experiences the strongest changes under the action of a magnetic field. This is 12. I. T. Suprun, Phys. Status Solidi A 120, 363 (1990). manifested in strong changes in the starting stresses and 13. N. A. Tyapunina, E. K. Naimi, and G. M. Zimenkova, the binding energy of a dislocation and an impurity cen- Ultrasound Action on Crystals with Defects (Mosk. Gos. ter in Ni-doped LiF crystals. Univ., Moscow, 1999), p. 126. 14. A. A. Urusovskaya, V. I. Al’shits, A. E. Smirnov, and N. N. Bekkauer, Pis’ma Zh. Éksp. Teor. Fiz. 65 (6), 470 REFERENCES (1997) [JETP Lett. 65, 497 (1997)]. 1. N. A. Tyapunina, É. P. Belozerova, and V. L. Krasnikov, 15. A. A. Urusovskaya, V. I. Al’shits, A. E. Smirnov, and Materialovedenie, No. 12, 21 (1999). N. N. Bekkauer, in Proceedings of the International 2. N. A. Tyapunina, É. P. Belozerova, and V. L. Krasnikov, Conference on Growth and Physics of Crystals, Moscow Materialovedenie, No. 2, 29 (2000). Institute of Steel and Alloys, Moscow, 1998, p. 190. 3. N. A. Tyapunina, V. L. Krasnikov, and É. P. Belozerova, Fiz. Tverd. Tela (St. Petersburg) 41 (6), 1035 (1999) [Phys. Solid State 41, 942 (1999)]. Translated by N. Wadhwa

PHYSICS OF THE SOLID STATE Vol. 45 No. 1 2003