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ISSN 1063-7834, Physics of the Solid State, 2007, Vol. 49, No. 6, pp. 11321141. © Pleiades Publishing, Ltd., 2007. Original Russian Text © A.I. Lebedev, I.A. Sluchinskaya, 2007, published in Fizika Tverdogo Tela, 2007, Vol. 49, No. 6, pp. 10771085.

MAGNETISM AND FERROELECTRICITY

Parameters of the Potential Well of an Off-Center Ge Atom in a GeTeSnTe Solid Solution
A. I. Lebedev and I. A. Sluchinskaya
Moscow State University, Vorob'evy gory, Moscow, 119992 Russia e-mail: swan@scon155.phys.msu.su
Received September 11, 2006

Abstract--An method is proposed for determining the shape of the three-dimensional multiwell potential of an off-center atom from EXAFS data. The parameters of the potential well of a Ge atom in GeTe and Sn1 - xGexTe (x 0.4) are determined in the classical and quantum-mechanical approximations. The potentialwell depth is varied in the interval 2040 meV depending on the Ge content, which indicates that the phase transition in these crystals is intermediate in character between the displacement and orderdisorder transitions. From analyzing the conditions for the applicability of the classical approximation, it follows that quantum effects must be taken into account in determining the parameters of the potential well of an off-center Ge atom in Sn1 xGexTe. Quantum-mechanical calculations show that the energy of the lower level in the vibration spectrum of the Ge atom coincides with the maximum energy in the potential well to within several millielectronvolts. The high probability of tunneling or an over-barrier transition of the off-center atom between the potential-well minima prevents dipole reorientations from being frozen at low temperatures. PACS numbers: 61.10.Ht, 61.72.-y, 77.84.Bw DOI: 10.1134/S1063783407060194

1. INTRODUCTION Considerable interest in studying off-center impurity atoms is related to the fact that the presence of these atoms in crystals can induce ferroelectric phase transitions (PTs) [1]. To explain the properties of such PTs at the microscopic level, it is necessary to know the shape of the potential wells of off-center impurity atoms. In a microscopic one-dimensional ferroelectricity model, the Hamiltonian has the well-known form [2] ^ H=

i i, j

A B 1 ------- p 2 + --- x 2 + --- x 4 2M i 2 i 4 i (1) ( xi x j ) E
2

Potential (2) consists of the following two terms: the local anharmonic potential ((A + 2C)/2)x2 + (B/4)x4 and the term describing the interaction of the atom under consideration with the other atoms and the external field. If quantum effects are neglected, the necessary condition for a PT to occur in the crystal is A < 0 in the microscopic model [2]. In this case, the local potential has two equivalent minima (in the three-dimensional case, up to eight minima, depending on the symmetry of the site occupied by the atom). Using Eq. (1), the displacement and orderdisorder PTs can be described in the framework of a unified approach. Within the microscopic model, the character of a PT (orderdisorder or displacement transition) in a simple cubic lattice is determined by the ratio between the following two characteristic energies [2]: the well depth Uw = A2/4B and the "binding energy" Ub = 6 |A |C/B, which is equal to the energy required for the atom to move from the bottom of the well to the potential-well maximum (at x = 0) under the condition that all the other atoms remain in the ordered state. The dimensionless parameter s = Uw /Ub determines the properties and dynamics of the PT; namely, the PT is of the orderdisorder type for s 1 and of the displacement type for s 1. In the microscopic model, the PT temperature is kTc (3/ln 2)|A |C/B in the limit s 1 and kTc (8/3)|A |C/B in the limit s 1 [3]. Therefore, one can assume that Ub kTc and, consequently, s Uw /kTc. Thus, we need to know the potential-well depth in order to determine the type of the PT.

C + --2

i

xi ,

where M is the atomic mass; xi and pi are the coordinate and momentum of an atom, respectively; E is the strength of an external electric field; and A, B, and C are parameters. Equation (1) can be written as a sum of independent equations describing the motion of one atom in an effective potential: ^ H=

i

12 A + 2C 2 B 4 ------- p i + ---------------- x i + --- x i 2 2M 4 xi C

(2)

ji

x j + E .

1132

PARAMETERS OF THE POTENTIAL WELL

1133

In real ferroelectrics, PTs can rarely be assigned to the displacement or orderdisorder type; more frequently, PTs in them are of the intermediate type. The problem of determining the type of a PT can be solved using its properties, such as the Curie constant, the existence of a soft mode, and the behavior of the heat capacity near Tc. However, it is often difficult to interpret the measurement results, because the soft mode decays, the behavior of the heat capacity is distorted in the presence of defects, and it is difficult to conduct dielectric measurements in samples with high conductivity. A direct determination of the parameters of microscopic Hamiltonian (1) would be a solution of this problem. This is of fundamental importance for understanding of the nature of PTs in crystals and for the prediction of the PT properties. The GeTeSnTe system is an example of a system for which studies yield contradictory information on the PT type. The components of this system form a continuous series of solid solutions with a NaCl-type structure in the high-temperature phase. In a Sn1 xGexTe solid solution, the temperature of the PT to the rhombohedral ferroelectric phase increases from ~100 to 700 K with increasing x [4]. Precision x-ray studies of lattice distortions below Tc [5] have shown that the order of the PT in these crystals changes with increasing x; namely, the transition is of the second order for x < 0.28 and of the first order for x > 0.28. The behavior of the heat capacity [6] and elastic moduli [7] near the Curie point demonstrate that the PTs are of the displacive type (in GeTe, in addition, a soft mode was observed [8]). However, extended x-ray-absorption fine-structure (EXAFS) studies [9] have shown that Ge atoms are displaced from lattice sites both below and above Tc, which indicates that these atoms are in a multiwell potential and that the PT is of the orderdisorder type. These contradictory data and the simplicity of the crystal structure (with two atoms in the primitive cell) prompted us to use EXAFS spectroscopy to study the shape of the potential well of Ge atoms in Sn1 xGexTe. The proposed method can also be useful for studying more complicated ferroelectrics. 2. DESCRIPTION OF THE METHOD EXAFS spectroscopy is a powerful method for studying the local structure of crystals. It has been used to answer the question of whether the potential in some ferroelectrics are of the single-well or multiwell type [1015]. The solution of this problem is reduced to studying the atomic motion in a strongly anharmonic crystal. To date, the parameters of single-well potential anharmonicity have been determined from the EXAFS data using the cumulant expansion method [16]. However, this method can be applied if the anharmonicity is not very strong and, hence, one can restrict oneself to only several first terms of an expansion in powers of k.
PHYSICS OF THE SOLID STATE Vol. 49 No. 6 2007

In addition, this method is inapplicable to crystals whose potential has several minima and it does not yield information on the atomic-motion anisotropy. Because the potential of an off-center impurity atom has several energetically equivalent minima, another approach is required to solve our problem. In this paper, we propose a new method for solving the problem of determining the multiwell-potential parameters for off-center impurities. This method is free from the drawbacks inherent in the cumulant expansion method. The basic special features of this approach are the following: (i) In calculating the theoretical EXAFS function, exact three-dimensional integration is performed, which makes it possible to consider potentials with an arbitrary degree of anharmonicity. (ii) A cluster is chosen with an atomic arrangement coinciding with that in the first coordination shell of an absorbing atom, which allows one to consider multiwell potentials and determine the parameters of the atomic-motion anisotropy. In the single-scattering approximation of EXAFS theory, the oscillating part of the spectrum at the K-absorption edge (the EXAFS function) can be written as follows [17]: (k) =

j

S0 ---- Im f ( k, ) e k

2

2i

1

(r j )e -------------------------- e 2 rj

2 r j /

2 ik r

j

dr j , (3)
2

where summation is over all nearest neighbors; S 0 is the factor taking into account many-electron effects and inelastic scattering; f is the backscattering complex function; 1 is the phase of the photoelectron escaping from the absorbing atom; is the photoelectron mean free path; k is the photoelectron wave vector, which is related to the absorbed photon energy E and the photoionization energy E0 by the formula k = 2 m ( E E 0 ) / ; (r) is the probability that the scattering atom is at the point r (the position of the absorbing atom is taken to be the origin of the reference system); and r = |r | is the distance between the absorbing and 2 scattering atoms. The quantities S 0 , f, 1, and characterize the absorbing and scattering atoms. The distribution function (r) contains information on the local structure and motion of atoms. Since the function (r) is determined by V(r), the potential-well parameters can be found from the EXAFS spectra. In our approach, the EXAFS spectra are processed as follows. We first parametrize the threedimensional potential V(r), taking the crystal symmetry into account. Then, we calculate the normalized distribution function (r) and substitute it into Eq. (3) in order to calculate the theoretical EXAFS spectrum and compare it with an experimental one. Varying the potential parameters and repeating the procedure for calculating the theoretical spectrum, we next minimize the root-mean-square (rms) deviation of the theoretical

1134

LEBEDEV, SLUCHINSKAYA

borhood of the origin of coordinates. For a site characterized by the point symmetry group Oh, the expansion in a power series up to fourth-order terms has the form
r
0

Te Ge

V loc ( r ) = + r + r + ( x y + x z + y z ) , r = x +y +z ,
2 2 2 2

2

4

22

22

22

(4)

where , , , and are coefficients. Because the distribution function is independent of the choice of the zero of energy, we set = 0. Then, Eq. (4) can be rewritten as V loc ( r ) = a ( 2 R
2 2 min r

a

+r )
4

4

Fig. 1. Fragment of the fcc lattice with a Ge atom moving in the octahedron formed by Te atoms.

22 22 22 r + d x y + x z + y z --- , 3

(5)

EXAFS spectrum from the experimental one and thus determine the shape of the potential well. Above the PT temperature, our crystals have a NaCl-type structure shown in Fig. 1. In this structure, each off-center Ge atom moves in the octahedron formed by six Te atoms. In ferroelectrics, the displacement of ferroelectrically active atoms from lattice sites due to thermal motion is always greater than that of surrounding atoms. Therefore, the thermal motion of Te atoms can be neglected and only the motion of the Ge atom in the rigid octahedron can be considered in a first approximation. In our experiments, the Ge atoms absorb x-ray radiation and the scattering atoms are Te atoms. Since we are interested in the motion of Ge atoms, it is appropriate to single out a cluster consisting of seven atoms (Ge + 6Te) and pass to a new reference system whose origin is located at the octahedron symmetry center. Taking into account that (r) is the pair distribution function, we can rewrite Eq. (3) in the new reference system and show that the expression for (k) remains the same except for the fact that the vector r is now the coordinate of the Ge atom and (r) is the probability that the Ge atom is at the point r. In the new reference system, Te atoms are located at the distance a0 /2 from the origin of coordinates along the 100 axes (a0 is the lattice parameter). Therefore, for the Ge atom located at the point r = (x, y, z), the contribution of the six Te atoms to EXAFS function (3) can be expressed in terms of six three-dimensional integrals in which the 2 distances rj are calculated from the formulas r 1, 2 = (a0 /2 ± x)2 + y2 + z2, r r
2 5, 6 2 3, 4

where a = + /3 > 0 is the parameter characterizing the 2 isotropic part of the fourth-order anharmonicity, R min = /(2 + 2/3) is the square of the distance to the potential minimum, and d = is the parameter describing the anisotropic part of the fourth-order anharmonicity. As mentioned above, the Ge atoms are displaced in a 111 direction (one of the eight equivalent minima) in the low-symmetry phase of Sn1 xGexTe; therefore, we have d < 0. The quantities a, R min , and d are the parameters of the local anharmonic potential in Eq. (2), which is generalized to the three-dimensional case. The experimental data discussed in this paper are obtained for samples in the ferroelectric phase. Therefore, it is necessary to take into account the existence of a preferential direction (the term Vmf and a rhombohedral distortion of the lattice in the processing of the data. We now estimate the effect of these factors and the higher order invariants on the obtained results. The inclusion of the rhombohedral lattice distortion (known from experiment) in the data processing did not have a significant effect on the agreement between the experimental and theoretical spectra and on the values of the local-potential parameters. The inclusion of the sixthorder invariants in expansion (4) shows that their influence on the results obtained at least for a low temperature is also insignificant. As for the molecular field Emf, its influence on the potential parameters can be noticeable. However, the inclusion of this field in the set of fitting parameters in the data processing had almost no effect (the agreement between the experimental and theoretical spectra depended on Emf only slightly). For this reason, we neglect the rhombohedral distortion, the sixth-order invariants, and the influence of the molecular field in this study. If the measurement temperature exceeds the Debye temperature and the classical approximation can therefore be used, then the probability of finding the atom at an arbitrary point of the crystal is determined by the potential energy at this point, (r) ~ exp[V(r/kT)]. At lower temperatures, it is necessary to take into account
PHYSICS OF THE SOLID STATE Vol. 49 No. 6 2007
2

= x2 + (a0 /2 ± y)2 + z2, and

= x2 + y2 + (a0 /2 ± z)2.

As was mentioned above, the potential V(r) consists of two parts, namely, the local anharmonic potential Vloc(r) and the term Vmf = (d · Emf), which describes the interaction of the atom with the other atoms and the external field. We expand the potential Vloc(r) in powers of the atomic displacement components in the neigh-

PARAMETERS OF THE POTENTIAL WELL

1135

the quantum character of the atomic motion. In this case, the relation between (r) and V(r) becomes more complicated (see Section 5). The classical-approximation calculations are discussed in Section 4. In Section 5, we calculate the vibration spectrum of a Ge atom moving in the obtained anisotropic multiwell potential. By comparing the distribution functions calculated in the classical approximation and using the rigorous quantum-mechanical approach, we show that these functions are essentially different and, consequently, the determination of the potential parameters requires a quantum-mechanical approach. The data obtained within the quantum-mechanical approach are given in Section 6. 3. EXPERIMENT Measurements were made on polycrystalline Sn1 - xGexTe solid solution samples with x = 0.4, 0.7, and 1.0. The samples were obtained by alloying binary compounds and subsequent homogenizing annealing at 620°C for 48 h. Immediately prior to measurements, the samples were ground into a powder, which was sieved afterwards. The powdered material was deposited on an adhesive tape. The absorbing-layer thickness optimum for recording spectra was obtained by folding this tape 8 to 16 times. The EXAFS spectra at the Ge K-absorption edge (11.103 keV) were recorded at station 7.1 of the synchrotron radiation source (SRS, Great Britain) with an electron energy of 2 GeV and a current of 240 mA. Radiation was monochromatized using a Si (111) double-crystal monochromator. The spectra were recorded in the transmission geometry. The intensities of the radiation incident on (I0) and transmitted through (It) the sample were measured by ionization chambers. The samples were placed into a nitrogen cryostat whose temperature could be varied in the temperature range from 77 to 300 K. The EXAFS function was extracted from the absorption spectra xµ(E) = ln(I0 /It ) (where E is the photon energy) using the traditional method [17, 18]. After subtracting the background due to absorption by other atoms, we extracted the monotonic part of the atom absorption xµ0(E) using splines and calculated the dependence of the quantity (k) = (xµ xµ0)/xµ0 on the photoelectron wave vector k = [2m(E E0)/ 2]1/2. The energy corresponding to the inflection point at the absorption edge was taken as the zero of energy E0. The value of the jump in xµ at the absorption edge varied from 0.19 to 0.50. Using the direct and inverse Fourier transforms with a modified Hanning window [17], we extracted the information on the first coordination shell from the experimental (k) curves. The typical range of extraction in the R space was R = 1.653.55 е and that in the k space was k = 2.812.7 е1. The subsequent data processing included varying the parameters of the probPHYSICS OF THE SOLID STATE Vol. 49 No. 6 2007

0.4 1 0.3 0.2 k (k) 0.1 0 0.1 3 5 7 k, е1 9 11 3 2

Fig. 2. Comparison of the experimental (points) and theoretical (solid lines) EXAFS spectra for a Sn0.3Ge0.7Te sample at various temperatures T: (1) 80, (2) 180, and (3) 275 K.

lem (a, d, R min , S 0 , a0, and the shift dE0 in the zero of energy [17]) to determine their values corresponding to the minimum rms deviation of the theoretical k(k) spectrum from the spectrum extracted from the experimental curve in the way mentioned above. The FEFF computer program [19] was used to find the functions f (k, ), 1(k), and (k) necessary for calculating the theoretical spectra. As is well known [17], the number of fitting parameters used to analyze the EXAFS data must not exceed the number of independent parameters Nind = (2/)Rk in the data. In our case, the number of adjustable parameters was six and Nind = 912. 4. CLASSICAL APPROXIMATION For a Sn0.3Ge0.7Te sample, the typical EXAFS spectra extracted by the method described above and the best fit to them obtained by the method described in Section 2 are shown in Fig. 2 for three different temperatures. The small discrepancy between the curves is due to the fact that the procedure used to extract the signal from the first coordination shell cannot completely suppress the contribution from the second coordination shell. Figure 3 shows a cross section of constant-energy surfaces calculated using the obtained parameters of the potential for the sample with x = 0.7 at T = 80 K. It is seen that the equal-potential curves are substantially elongated in the 111 direction and the potential energy increases most rapidly for displacements in the 100 direction. The temperature dependences of the parameters a, 2 R min , and |d | for all Sn1 xGexTe samples under study are given in Figs. 46.1 It is seen from comparing

2

2

1136

LEBEDEV, SLUCHINSKAYA 0.6 R, е E, meV 20 10 [111] 0.4 E, meV 0 0 20 40 80 0.2 2.0 1.5 1.0 0.5 [001] 0 0.2 0.4 R, е 0.6 0 50 100 150 T, K
Fig. 4. Temperature dependence of the parameter a for Sn1 - xGexTe samples with various values of x.

x = 1.0 0.7 0.4

[110]

0

a, eV/е4

200

250

300

Fig. 3. Constant-energy surfaces of the potential well in a plane perpendicular to a 110 axis for a Sn0.3Ge0.7Te sample at 80 K.

0.35 0.30
2

250 200 |d|, eV/е4 x = 1.0 0.7 0.4 100 150 T, K 200 250 300 150 100 50 0 50 x = 1.0 0.7 0.4 100 150 T, K
2 min

Rmin, е

0.25 0.20 0.15 50

200

250

300

Fig. 5. Temperature dependence of the parameter R Sn1 xGexTe samples with various values of x.

for

Fig. 6. Temperature dependence of the parameter d for Sn1 - xGexTe samples with various values of x.

Figs. 4 and 6 that the parameter |d | is two orders of magnitude greater than the parameter a. This means that the potential is characterized by strong anisotropy. Indeed, as follows from Eq. (5), the minimum value of the coefficient of r4 (equal to a) is reached for displacements of the Ge atom in the 111 direction, and the maximum value (equal to a + |d |/3), for displacements in the 100 direction. This is not surprising, because the Te atoms near a Ge site are located along the 100 direction and displacements in this direction are accompanied by a strong nonlinear repulsion. At the same time, along the 111 direction, there is a dimple
1

in the closely packed atomic plane and it is much easier for the off-center Ge atoms to be displaced in this direction (Fig. 1). As follows from Eq. (5), the potential-well depth is 4 Uw = aR min . Estimations of Uw give 40 meV for the sample with x = 1, 30 meV for the sample with x = 0.7, and 20 meV for the sample with x = 0.4. Thus, the dimensionless parameter s = Uw /kTc is about 0.65 for all samples studied. This means that the PT in a Sn1 - xGexTe solid solution is intermediate between the displacement and orderdisorder types. We note that, in earlier experimental studies, the offcenter atomic displacement discovered above the Curie temperature was associated with an orderdisorder PT. However, this conclusion is not unambiguous, because
PHYSICS OF THE SOLID STATE Vol. 49 No. 6 2007

It should be noted that we failed to reliably determine the parameters for a number of spectra recorded at high temperatures. Probably, it is necessary to take sixth-order invariants into account in processing these spectra.

PARAMETERS OF THE POTENTIAL WELL

1137 (b) (c) (d) (e)

in the microscopic model the type of a PT is determined by the parameter s, as mentioned in Section 1. Thus, in order to prove that a PT is of the orderdisorder type, it is necessary to analyze the shape of the potential well and verify that the well depth satisfies the condition Uw kTc. ters a and R min observed in this study is an unexpected result.2 It follows from Figs. 4 and 5 that, for the samples with x 0.7, the parameter a decreases and R min increases as the temperature increases, with the potential-well depth Uw = aR min remaining practically unchanged in this case. It should be noted that these sufficiently strong changes in the potential parameters do not lead to a noticeable change in the mean-square atomic displacement from a lattice site. The calculation of the temperature dependence of the quantity r =
2 4 2 2

(a) 30

E, meV 10

The strong temperature dependence of the parame-

20

kT

0
Fig. 7. Vibration spectrum of a Ge atom in potential wells obtained in the classical approximation and using the quantum-mechanical approach for a Sn0.3Ge0.7Te sample at 80 K.

( r ) r dr

2

(6) 5. VERIFICATION OF THE APPLICABILITY OF THE CLASSICAL APPROXIMATION The data processing described above was based on the use of the classical approximation. By solving the problem by a rigorous method (using quantum mechanics) and verifying that both solutions coincide, one can justify the application the classical approximation. To calculate the distribution function by the rigorous method, it is necessary to solve the SchrЖdinger equation for the strongly anisotropic and anharmonic potential V(r) obtained in the classical approximation and find its wave functions {i} and eigenvalues {Ei}. The SchrЖdinger equation was solved by a numerical variational method in one octant on a lattice with dimensions 121 в 121 в 121. The symmetry of the solution was taken into account using boundary conditions. The energy spectrum of the Ge atom in the potential well obtained in the classical approximation is shown in Fig. 7a. It is seen that the lowest energy level is 5.3 meV above zero (the value of the potential V(0) is taken as the zero of energy), i.e., approximately 30 meV above the potential minimum. This means that the atom is not localized in a minimum, although the potential has many minima. We assume that the strong "expulsion" of energy levels is a consequence of the strong anisotropy in our problem: the localization of the atomic motion in "channels" extended in the 111 directions (Fig. 3) leads to a considerable increase in the atom kinetic energy. Thus, the quantum effects in our problem turn out to be unexpectedly important. In Fig. 8, the distribution function obtained from processing the EXAFS spectrum for a Sn0.3Ge0.7Te

shows that the displacement of the "center of gravity" of the distribution function does not exceed 0.04 е (12%) as the temperature T increases from 80 to 300 K. One of the reasons for this variation is the thermal expansion of the crystal, although this reason is not likely to be the only one. As follows from Fig. 4, the extrapolation of the a(T) dependences for the samples with x 0.7 to high temperatures predicts negative values for the parameter a at the Curie temperature (Tx 700 K for the sample with x = 1 and 500 K for the sample with x = 0.7). It was shown in [3] that, for displacement-type PTs (s 1), there is a direct relation between the coefficient of the fourth power of the order parameter (Ath) in the thermodynamic potential and the coefficient a in the microscopic potential. Therefore, the negative values of a could indicate that the PT is of the first order (which is the case if Ath < 0). However, in our case (s ~ 1), the relation between the coefficients Ath and a is much more complicated [20]. Unfortunately, we cannot determine the sign of Ath on the basis of the obtained parameters of the microscopic potential, because, in order to estimate Ath, we need to know the values of the sixth-order coefficients in the expansion of Vloc(r), but they were neglected in our analysis of the data. In addition, the inclusion of the molecular field (also neglected in this study) can affect the obtained results. These problems require further studies.
2

Because of the three-dimensional character of the problem study, estimating the statistical errors in the experimental mination of the parameters is an independent complicated lem. At present, we can estimate the errors from the scatter parameters for spectra recorded under the same conditions. PHYSICS OF THE SOLID STATE Vol. 49

under deterprobof the

No. 6

2007

1138 50 40 Density, 1/е3 30 1 20 10 0 0 0.1 0.2 x = y = z, е 1A1g 1T
1u

LEBEDEV, SLUCHINSKAYA

2

1T

2g

1A2u

0.3

0.4

Fig. 8. Comparison of (1) the distribution function in a 111 direction obtained by processing the EXAFS spectrum for Sn0.3Ge0.7Te samples at 80 K in the classical approximation and (2) the quantum-mechanical distribution function obtained using twenty wave functions for the potential found in the classical approximation. The thin solid, dashed, and dotted lines correspond to the contributions of the first twelve wave functions to the quantummechanical distribution function.

sample at 80 K in the classical approximation ((r) ~ exp[V(r)/kT]) is compared with the function 1 ( r ) = -Z

i

e

E i / kT

i( r ) ,

2

Z=

i

e

E i / kT

,

(7)

calculated by quantum statistics methods using the first twenty wave functions corresponding to the same potential V(r). As follows from Fig. 8, the graphs of these distribution functions differ essentially in both the shape and the position of the maximum. This demonstrates that, in studying multiwell potentials, the classical approximation is inapplicable to the processing of EXAFS spectra, at least those obtained at low temperatures. 6. QUANTUM-MECHANICAL APPROACH Under a rigorous quantum-mechanical approach with quantum effects consistently taken into account, the problem of the EXAFS data processing becomes much more complicated. For parametrized potential (5), it is necessary first to calculate the wave functions and eigenvalues and substitute them into Eq. (7). Then, the obtained distribution function (r) can be inserted in Eq. (3) to calculate the EXAFS function (k). By varying the parameters of the potential and repeating quantum-mechanical calculations each time, one can find (using one of the minimization algorithms) the set of parameters for which the rms discrepancy between the theoretical and experimental EXAFS spectra becomes minimum.

Because the proposed approach is very time-consuming, we should perform calculations using a minimum set of wave functions to estimate the potential parameter values. If the EXAFS spectra are measured at a sufficiently low temperature, then only several lowest energy levels are occupied in the vibration spectrum. However, because the vibration spectrum depends on the potential parameters in a complicated way and because the number of wave functions to be taken into account is unknown, we solved the problem using the iteration method. We processed the same EXAFS spectra of the Sn0.3Ge0.7Te and GeTe samples (recorded at the lowest temperature, 80 K) until the energy of the previously neglected levels became approximately 2kT greater than the energy of the ground state. First, we assumed that, in order to describe the motion of the Ge atom, it suffices to take into account only the lowest 1A1g level (corresponding to the zeropoint energy).3 The vibration spectrum for the potential determined in this approximation is shown in Fig. 7b, and the values of the potential parameters, in the table. Because the splitting of the low-lying excited energy levels is only E = 0.346 meV kT and, consequently, these states are approximately equally excited, it is clear that our approximation is not good. Next, we considered the problem in which the four lowest levels (1A1g, 1T1u, 1T2g, 1A2u) are included and the threefold degeneracy of the T1u and T2g levels is taken into account. The results of these calculations are given in Fig. 7c and in the table. Because the 2A1g level (the lowest level not taken into account in this approximation) turned out to be higher than the ground level by only 2.466 meV (which is less than kT), it is clear that one should include a larger number of wave functions. Calculations with the inclusion of eight wave functions (1A1g, 1T1u, 1T2g, 1A2u, 2A1g, 2T1u, 2T2g, 2A2u) are quite acceptable, because the nearest level not taken into account (3A1g) is higher that the ground level by 6.989 meV, which is slightly greater than kT (see Fig. 7d and table). Finally, calculations with the inclusion of twelve wave functions (the 3A1g, 3T1u, 3T2g, and 3A2u states are added) yield the most reliable results; the nearest level not taken into account is 13.356 meV higher than the ground state (see Fig. 7e and table). Now, we consider the character of the variations in the potential parameters caused by the inclusion of a greater number of wave functions and compare these parameters with those obtained in the classical approximation. As is seen from the table, the agreement between the theoretical and experimental curves (char3

In the notation for wave functions, the first numeral is the principal quantum number (it determines the number of zeros in the radial part of the wave function, just as in the hydrogen atom) and the symbols following it indicate the angular symmetry of the function. Vol. 49 No. 6 2007

PHYSICS OF THE SOLID STATE

PARAMETERS OF THE POTENTIAL WELL

1139

Parameters of the potential and the potential-well depth obtained from processing the EXAFS spectra of Sn0.3Ge0.7Te and GeTe at 80 K within the classical approximation (CA) and the quantum-mechanical approach using various numbers of wave functions Number of wave functions CA 1 4 8 12 CA 1 4 8 12 a, eV/е4 R
2 min

, е2

d, eV/е4

Uw , meV

Sn0.3Ge0.7Te 0.8476 0.6924 0.5032 0.7863 0.9662 1.4864 1.0996 0.9212 1.4183 1.7146 0.1717 0.2321 0.2553 0.2266 0.2166 GeTe 0.1414 0.1858 0.1993 0.1763 0.1686 172.91 95.21 109.07 133.83 154.39 29.73 37.95 36.61 44.08 48.74 0.014853 0.015388 0.015220 0.015058 0.015041 137.19 85.60 80.21 115.28 142.77 24.91 37.30 32.80 40.36 45.32 0.016752 0.017427 0.017309 0.017153 0.017078

Density, 1/е3

acterized by the rms deviation ) improves as the number of wave functions increases. The quantum character of atomic motion most strongly affects the position of the potential minimum Rmin and the potential-well depth Uw (see table).The well depth calculated with the inclusion of twelve wave functions is 6080% greater than that obtained in the classical approximation. The values of d in both approaches turn out to be close, and the value of a is slightly greater in the quantummechanical approach. The quantum-mechanical calculation shows that, in all cases considered, the energy of the 1A1g ground state is positive; i.e., the Ge atom is not localized in a minimum. Nevertheless, the maxima of the atomic distribution function are displaced from the lattice site (Fig. 9). The transition rate from one off-center position to another can be estimated from the splitting of the 1A1g 1A2u levels, which is E = 0.387 meV for Sn0.3Ge0.7Te and 0.142 meV for GeTe and corresponds to a "lifetime" of a particle in one minimum = /E (1.7 4.6) в 1012 s. In this case, the typical transition frequency is one order of magnitude less than the characteristic frequency of the soft mode. 7. DISCUSSION As shown above, the shape of the potential well found using the classical and quantum-mechanical approximations is qualitatively the same and the well parameters do not differ significantly. Although the quantum-mechanical problem was solved only for one temperature value, it can be expected that the pattern of the temperature dependences of the parameters a and 2 R min remains qualitatively the same in both the classical and quantum-mechanical approaches. This concluPHYSICS OF THE SOLID STATE Vol. 49 No. 6 2007

sion is based on the fact that, first, the potential parameters in both approaches must coincide in the high-temperature limit and, second, the difference between the parameters at 80 K in both approaches is less than the change in these parameters over the temperature range 80300 K in the classical approach. One would think that the use of the quantummechanical approach in the data processing only refines the values of the potential parameters. However, this is not the case; namely, this approach leads to a qualitatively new result concerning the character of the particle motion. According to the classical approach to the problem of motion of off-center atoms, their reori100 80 60 2 40 20 1

0

0.1

0.2 x = y = z, е

0.3

0.4

Fig. 9. Comparison of the distribution functions along a 111 direction found by processing the EXAFS spectrum for a Sn0.3Ge0.7Te sample at 80 K (1) in the classical approximation and (2) using the quantum-mechanical approach. The potential parameters are taken from the last two rows of the table.

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LEBEDEV, SLUCHINSKAYA

entation occurs by overcoming the potential barriers separating different wells. An example of this type of motion is the motion of an off-center Li atom in KTaO3. Measurements of the relaxation time of the Li atom by the NMR and dielectric relaxation methods have shown that the dipole reorientation is thermally activated with an activation energy of 0.085 eV [21]. The results obtained in this work for the off-center Ge atom in Sn1 - xGexTe show that the lower vibration levels of this atom are located near the potential maximum at r = 0 within several millielectronvolts. Consequently, the atomic transitions from one off-center position to another occur either by tunneling or by passing over barriers. For this reason, it can be expected that, in the limit T 0, the dipole reorientation rate will remain high and, hence, the dynamics of the off-center atom will not "frozen". The distinctions between the dynamics of off-center Li atoms in KTaO3 and off-center Ge atoms in Sn1 - xGexTe indicate that these atoms are examples of static and dynamic impurities, respectively (similar to JahnTeller impurities). For a Li atom, the quantum effects are insignificant due probably to the fact that the off-center displacement of this atom is 1.26 е (which is three times greater than that of the Ge atom in Sn1 - xGexTe) and, therefore, the probability of tunneling transitions between the wells is low for this atom (the "lifetime" of the atom in a minimum is ~104 s [22]). In this connection, it would be interesting to study the intermediate case of an off-center Ge atom in PbTe, whose displacement is 0.7 е [10]. The dynamic character of the off-center Ge atom in Sn1 xGexTe explains the illusory contradiction between the EXAFS data demonstrating the existence of off-center atoms and the physical properties showing that the PTs in Sn1 xGexTe are of the displacement type. Our quantum-mechanical calculations have been performed without inclusion of the molecular field and, therefore, describe the motion of the system above the PT point. The case of a nonzero molecular field will be studied in a later paper. Nevertheless, the conclusion that it is necessary to take into account the quantum character of the atomic motion in analyzing the properties of a system with off-center atoms is an important result of this study. 8. CONCLUSIONS We have proposed an approach that makes it possible to determine the shape of the three-dimensional potential of an off-center atom and the parameters of the microscopic ferroelectricity model from EXAFS data. For Sn1 xGexTe solid solutions with x 0.4, we have found the temperature and composition dependences of the multiwell-potential parameters for a Ge atom in the classical approximation. The potential-well depth varies from 20 to 40 meV depending on the Ge

content in the sample. It has been shown that the phase transition (PT) in these crystals is intermediate between the displacement and orderdisorder types. Analyzing the applicability of the classical approximation has revealed that it is necessary to take into account quantum effects in determining the parameters of potential wells in strongly anisotropic systems, such as off-center Ge atoms in Sn1 xGexTe. An algorithm has been proposed for data processing in which the theoretical EXAFS spectrum and the distribution function are calculated from strict formulas of quantum statistics using calculated wave functions. The fact that the lower vibration energy level of the Ge atom coincides with the maximum energy in the potential well to within several millielectronvolts is an unexpected result of our quantum-mechanical calculations. The high probability of an off-center atom tunneling through or passing over the barrier between the potential-well minima prevents dipole reorientations from being frozen at low temperatures. This special feature explains the illusory contradiction between the physical properties showing the closeness of the PTs in Sn1 xGexTe to the displacement type and the EXAFS data demonstrating the large displacements of Ge atoms from the lattice sites above the Curie temperature. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, project no. 03-02-16523. REFERENCES
1. B. E. Vugmeister and M. D. Glinchuk, Rev. Mod. Phys. 62, 993 (1990). 2. A. Brause and R. Cowley, Structural Phase Transitions (Taylor and Francis, London, 1981; Mir, Moscow, 1984). 3. B. A. Strukov and A. P. Levanyuk, Ferroelectric Phenomena in Crystals (Nauka, Moscow, 1995; Springer, Berlin, 1998). 4. J. N. Bierly, L. Muldawer, and O. Beckman, Acta Metall. 11, 447 (1963). 5. R. Clarke, Phys. Rev. B: Condens. Matter 18, 4920 (1978). 6. I. Hatta and W. Rehwald, J. Phys. C: Solid State Phys. 10, 2075 (1977). 7. W. Rehwald and G. K. Lang, J. Phys. C: Solid State Phys. 8, 3287 (1975). 8. E. F. Steigmeier and G. Harbeke, Solid State Commun. 8, 1275 (1970). 9. B. A. Bunker, Q. T. Islam, and W.-F. Pong, Physica B (Amsterdam) 158, 578 (1989). 10. Q. T. Islam and B. A. Bunker, Phys. Rev. Lett. 59, 2701 (1987). 11. O. Hanske-Petitpierre, Y. Yacoby, J. Mustre de Leon, E. A. Stern, and J. J. Rehr, Phys. Rev. B: Condens. Matter 44, 6700 (1991).
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PARAMETERS OF THE POTENTIAL WELL 12. N. Sicron, B. Ravel, Y. Yacoby, E. A. Stern, F. Dogan, and J. J. Rehr, Phys. Rev. B: Condens. Matter 50, 13168 (1994). 13. N. Sicron, Y. Yacoby, E. A. Stern, and F. Dogan, J. Phys. IV 7, 1047 (1997). 14. B. Ravel, E. Cockayne, M. Newville, and K. M. Rabe, Phys. Rev. B: Condens. Matter 60, 14632 (1999). 15. V. Shuvaeva, Y. Azuma, K. Yagi, H. Terauchi, R. Vedrinski, V. Komarov, and H. Kasatani, Phys. Rev. B: Condens. Matter 62, 2969 (2000). 16. G. Bunker, Nucl. Instrum. Methods Phys. Res. 207, 437 (1983). 17. P. A. Lee, P. H. Citrin, P. Eisenberger, and B. M. Kincaid, Rev. Mod. Phys. 53, 769 (1981).

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18. A. I. Lebedev, I. A. Slichinskaya, V. N. Demin, and I. H. Munro, Phys. Rev. B: Condens. Matter 55, 14770 (1997). 19. J. Mustre de Leon, J. J. Rehr, S. I. Zabinsky, and R. C. Albers, Phys. Rev. B: Condens. Matter 44, 4146 (1991). 20. M. E. Lines, Phys. Rev. 177, 797 (1969). 21. F. Borsa, U. T. HЖchli, J. J. van der Klink, and D. Rytz, Phys. Rev. Lett. 45, 1884 (1980). 22. W. Kleemann, V. SchЖnknecht, D. Sommer, and D. Rytz, Phys. Rev. Lett. 66, 762 (1991).

Translated by L. Kul'man

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2007