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ISSN 1063-7834, Physics of the Solid State, 2009, Vol. 51, No. 4, pp. 802­809. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.I. Lebedev, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 4, pp. 757­763.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Ferroelectric Phase Transition in Orthorhombic CdTiO3: First-Principles Studies
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e-mail: swan@scon155.phys.msu.su
Received July 4, 2008

Abstract--Parameters of the crystal structure and phonon spectra for orthorhombic cadmium titanate with space group Pbnm and its two possible ferroelectrically distorted phases (with space groups Pbn21 and Pb21m) were calculated from first principles within the density functional theory. The obtained structural parameters and frequencies of Raman- and infrared-active modes are in good agreement with available experimental data for the Pbnm phase. Expansion of the total energy in a Taylor series of two order parameters showed that the ground state of the system corresponds to the Pbn21 structure into which the Pbnm phase transforms through a second-order phase transition without intermediate phases. A substantial discrepancy between calculated and experimentally observed lattice distortions and spontaneous polarization in the polar phase was explained by quantum fluctuations, as well as by existence of twins and competing long-period structures. PACS numbers: 61.50.Ah, 63.20.D-, 77.84.Dy DOI: 10.1134/S1063783409040283

1. INTRODUCTION The calculations of phonon spectra of ten titanates ATiO3 (A = Ca, Sr, Ba, Ra, Cd, Zn, Mg, Ge, Sn, Pb) with a perovskite structure [1] revealed that the orthorhombic phases (space group Pbnm) of CdTiO3, ZnTiO3, and MgTiO3 are characterized by a ferroelectric-type instability, which manifests itself in phonon spectra of the above crystals as one or two unstable modes with symmetries B1u and B2u at the point. In the present study, the characteristics of these modes are considered and the results of first-principles calculations of the crystal structure, phonon spectra, and spontaneous polarization of the parent orthorhombic CdTiO3 phase (space group Pbnm) and its two ferroelectrically distorted orthorhombic modifications (space groups Pbn21, Pb21m) are compared with available experimental data. The ferroelectric phase transition in cadmium titanate was discovered by Smolenskioe [2] in 1950. Since then, this phase transition was studied by dielectric [3­6] and X-ray [3, 4, 7] methods; Raman scattering [3, 8], IR reflectance, and submillimeter reflectance [9] spectroscopy; and pyroelectric current measurements [4]. The specific features of the phase transition in CdTiO3 are a smallness of lattice distortions and a considerable scatter in the data on the phase transition temperature (50­82 K) and spontaneous polarization (0.002­0.009 C/m2). Even the data on the structure of the low-symmetry phase are contradictory. In particular, the study of the dielectric properties [6] suggests that the 21/m axis is polar, whereas structural investigations with the use of synchrotron radiation indicate that the 21/n axis is polar [7].

In [5, 8, 9], the analysis of the temperature dependences of the dielectric constant and the intensity of Raman scattering lines allowed the authors to assume that, upon cooling, one more phase transition accompanied by a change in the polarization direction occurs in cadmium titanate at a temperature of about 50 K. The paper by Fabricius and LÑpez GarcËa [10] is the only work in which CdTiO3 was studied from first principles. In this work, the authors calculated the electric field gradient at cadmium atoms for different sets of atomic coordinates proposed in the literature for the Pbnm and Pbn21 structures and revealed that the relaxation of the latter structure transforms it into a more stable nonpolar Pbnm structure. In view of the discrepancies of the experimental data available in the literature for cadmium titanate, it is expedient to perform first-principles calculations in order to elucidate the factors responsible for these contradictions and to resolve them. 2. CALCULATION TECHNIQUE The calculations were carried out within the density functional theory with the pseudopotentials and the plane-wave expansion of wave functions as implemented in the ABINIT code [11]. The exchange­correlation interaction was described within the local-density approximation according to the procedure proposed in [12]. As pseudopotentials, we used the optimized separable nonlocal pseudopotentials [13] generated with the OPIUM code to which the local potential was added in order to improve their transfer-

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FERROELECTRIC PHASE TRANSITION IN ORTHORHOMBIC CdTiO3

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ability [14]. The parameters used for constructing the pseudopotentials, the results of their testing, and other details of calculations are described in [1]. 3. RESULTS OF THE CALCULATIONS 3.1. Phonon Spectrum of the Pbnm Phase The calculated frequencies of the "softest" phonons at the point that can be associated with the ferroelectric instability in orthorhombic crystals of CaTiO3, CdTiO3, ZnTiO3, and MgTiO3 with space group Pbnm and in tetragonal SrTiO3 with space group I4/mcm are presented in Table 1. All phonons in CaTiO3 and SrTiO3 turn out to be stable (which corresponds to experiment), whereas one or two unstable modes (the frequencies of which are imaginary) arise in the other three titanates. The strongest instability in these crystals is determined by the phonon with symmetry B1u, which can be associated with the phase transition Pbnm Pbn21. In CdTiO3 and ZnTiO3, the phonon with symmetry B2u also appears to be unstable. This phonon can be associated with the phase transition Pbnm Pb21m. It should be noted that, among the three crystals under consideration, the Pbnm structure can be obtained only for CdTiO3 (ZnTiO3 and MgTiO3 usually crystallize in an ilmenite structure). In this respect, hereafter, we will thoroughly consider only the properties of cadmium titanate. The calculated lattice parameters and equilibrium atomic coordinates for orthorhombic CdTiO3 with space group Pbnm are compared with available experimental data in Table 2. It is seen that the results of the calculations are in good agreement with experimental data. It should be noted that better agreement is observed for the experimental data obtained at lower temperatures. A small systematic underestimation of the calculated lattice parameters is a characteristic feature of the local-density approximation used. The vibrational spectrum of the crystal with space group Pbnm consists of 60 modes, including 24 modes (with symmetries Ag, B1g B2g, and B3g) active in Raman scattering spectra, 25 optical modes (with symmetries B1u, B2u, and B3u) active in IR reflectance spectra, 8 optical modes with symmetry Au, and three acoustic modes inactive in the above spectra. The calculated frequencies of the modes active in IR reflectance and Raman scattering spectra are compared with available experimental data [3, 8, 9] in Table 3. For the modes active in Raman scattering spectra, the results of the calculations agree well with the data of measurements performed for ceramic samples [3] and single crystals [8]. The typical relative discrepancy between frequencies is about 3%. A comparison of the frequencies of the modes determined from polarized Raman scattering spectra [8] with the results of our calculations demonstrates that the peaks observed at 303 and 392 cm­1 in the yy polarization (in the crystal setPHYSICS OF THE SOLID STATE Vol. 51 No. 4 2009

Table 1. Calculated frequencies (cm­1) of the softest ferroelectric modes in crystals of four titanates with the orthorhombic structure Pbnm and tetragonal strontium titanate Mode A2u Eu B3u B2u B1u 15 SrTiO3 55 39 ­ ­ ­ 68i CaTiO3 ­ ­ 82 97 82 165i CdTiO3 ­ ­ 54 81i 104i 187i ZnTiO3 MgTiO3 ­ ­ 73 54i 103i 240i ­ ­ 115 81 133i 260i

Note: The frequencies of the unstable TO phonons at the point of the Brillouin zone in the cubic praphase [1] are given in the lower row.

Table 2. Lattice parameters a, b, and c (å) and atomic coordinates for CdTiO3 with space group Pbnm Parameter a b c Cdx Cdy Cdz Tix Tiy Tiz O1x O1y O1z O2x O2y O2z
* At T = 150 K.

This work 5.2427 5.3815 7.5744 0.01017 0.04637 0.25000 0.00000 0.50000 0.00000 0.10130 0.46252 0.25000 0.69348 0.30304 0.05341

Experiment [15] 5.3053 5.4215 7.6176 0.00847 0.03873 0.25000 0.00000 0.50000 0.00000 0.0902 0.4722 0.25000 0.7008 0.2969 0.0472 [7]* 5.284 5.403 7.590 0.00891 0.03997 0.25000 0.00000 0.50000 0.00000 0.0918 0.4714 0.2500 0.70083 0.29660 0.04783

­ + + + + + + + + + + +

­ + + + + + + + + + + +

­ + + + + + + + + + + +

ting accepted in [8]) is most likely erroneously assigned to the B1g modes (in our crystal setting). The positions of these peaks are close to the positions of the peaks of the Ag modes, which should also be observed for the above polarization. The experimental peak in the yz polarization at 307 cm­1 assigned to the B2g or B3g mode in actual fact is most likely a "leak" of the Ag mode. A comparison of the calculated frequencies of the modes with the results obtained from analyzing IR reflectance spectra [9] appears to be a more complex problem. A direct comparison of the experimental frequencies with the calculated frequencies for the mode symmetries given in [9] revealed their substantial dis-


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Table 3. Frequencies i of the optical modes active in Raman scattering and IR reflectance spectra and oscillator strengths fi of the IR-active modes for CdTiO3 with space group Pbnm i, cm­1 Mode this work [3] Ag 96 128 195 295 414 449 512 B3g 140 211 353 486 683 B2g 117 280 445 488 764 B1g 111 142 205 356 448 492 738 95 123 190 295 390 461 ­ 135, 141 ­ 342 ­ ­ ­ ­ ­ ­ ­ 110 141 ­ ­ ­ ­ ­ [8] 99 125 194 299 390 465 496 144 ­ 346 479* ­ 114 307? 459* 509* ­ 115 141 ­ ­ ­ 504 ­ B1u B2u B3u 3.87 1.27 0.60 0.38 0.30 0.71 0.13 0.17 0.96 5.98 0.62 0.86 0.01 0.37 0.14 0.12 1.04 0.02 3.42 0.89 0.78 1.24 0.62 0.02 1.19
* Components of a broad weakly structured line. **A strong deviation from the calculations is a result of anharmonicity.

i, cm­1 Mode fi â 103, arb. units this work 54 104 167 275 302 379 423 445 513 81i 90 177 202 321 345 437 476 530 104i 60 128 224 396 428 481 experiment [9] 61? 111 165 284 306 383 ­ 458 525 61** 111** 177 ­ 321 349 ­ 483 525 44** 96** 143** 225 387 ­ 511

experiment

crepancy. In order to identify reliably the experimentally observed modes, the oscillator strength fi was additionally determined for each calculated mode (the oscillator strength characterizes the contribution of the mode to the complex dielectric constant) and it was assumed that the symmetries of the observed modes were incorrectly identified in [9]. This can be a result of the high twin density characteristic of CdTiO3 crystals [8, 15, 16], which can undeniably manifest itself in measurements with crystals having larger areas. The calculated frequencies and oscillator strengths for the cadmium titanate modification with space group

Pbnm are compared with the results of investigations of the IR reflectance spectra in Fig. 1. Under the assumption that the modes identified as B3u in [9] for the crystal setting Pnma correspond to the B1u modes in our setting Pbnm and that the B1u modes in [9] correspond to the B2u and B3u modes in our setting, the agreement between calculated and experimental data becomes more reasonable (Fig. 1, Table 3). A considerable shift in frequencies of three softest modes with symmetry B1u and two softest modes with symmetry B2u with respect to the calculated frequencies is explained by the manifestation of anharmonicity. The modes with calcuPHYSICS OF THE SOLID STATE Vol. 51 No. 4 2009


FERROELECTRIC PHASE TRANSITION IN ORTHORHOMBIC CdTiO3 6 2 (a) (a) z arb. units 4 3 1 y (b) (c)

805

Ti Cd O

­3

fi, 10

2 1 0

3 1 21 3 12 31 23 3322 2 3123 2

Fig. 2. (a) Projection of the structure of the orthorhombic phase of CdTiO3 with space group Pbnm onto the bc plane and (b, c) character of atomic displacements in the case of ferroelectric transitions to the (b) Pbn21 and (c) Pb21m phases.

10 8 fi, arb. units 6 4 2 0 ­ 200

(b)

0

200 , cm­1

400

600

the character of atomic displacements in the case of its distortion upon transition to the low-symmetry phases are shown in Fig. 2. The transition of the crystal to the Pb21m phase is accompanied by antiphase displacements of the titanium and oxygen atoms along the y axis, whereas the cadmium atoms are left almost in place. Upon distortion of the lattice associated with the transition to the Pbn21 phase, both metal atoms are slightly displaced along the z axis (by approximately identical distances). In this case, the displacements of the titanium and oxygen atoms contain a considerable y component in addition to the z component. The transition to the polar phases leads to the appearance of two nonequivalent sets of oxygen atoms O2 (designated by the letters a and b in Table 4). The disappearance of the n plane upon transition to the Pb21m phase results in the appearance of two nonequivalent cadmium atoms (Cd1, Cd2). The energy gain (per formula unit) associated with the distortion is equal to 6.21 meV for the Pbn21 phase and 1.38 meV for the Pb21m phase. A comparison of the calculated atomic displacements and the spontaneous lattice strains with the results of low-temperature X-ray measurements [4, 7] (Table 4) demonstrates that, in the experiments, the atomic displacements and the spontaneous lattice strains are significantly smaller. Specifically, it follows from the comparison of the calculated lattice parameters in Tables 2 and 4 that a considerable increase (by 0.045 å) in the lattice parameter along the polar c axis should be observed in the Pbn21 phase and the lattice parameter a rather than the lattice parameter b should substantially increase in the Pb21m phase. In the experiment, the largest spontaneous deformation below the phase transition temperature was observed for the lattice parameter b [4] (elongation of the order of 0.002 å), which is inconsistent with the predictions for both ferroelectric phases. The possible factors responsible for these discrepancies will be discussed below.

Fig. 1. Comparison of (a) the calculated frequencies and oscillator strengths of the IR-active modes for cadmium titanate with space group Pbnm with (b) the corresponding characteristics obtained from analysis of the IR reflectance spectra. Numerals near the points indicate the mode symmetry (B1u, B2u, B3u)

lated frequencies of 54 and 104 cm­1 are most likely indistinguishable in the experiment against the background of other modes, and four modes with the smallest oscillator strengths are not observed in the spectra at all. 3.2. Lattice Distortions upon Ferroelectric Phase Transitions As was shown above, the used technique of firstprinciples calculations describes well the properties of orthorhombic CdTiO3 with space group Pbnm. Now, let us consider the properties of the ferroelectrically distorted phases of cadmium titanate. The equilibrium atomic coordinates in the distorted phases were determined using the relaxation of the Hellmann­Feynman forces in the structures obtained from the parent nonpolar phase Pbnm by introducing a perturbation with symmetry B1u (for the Pbn21 phase) or B2u (for the Pb21m phase). The structure of the nonpolar Pbnm phase and
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Table 4. Lattice parameters a, b, and c (å) and atomic coordinates for ferroelectrically distorted phases Pbn21 and Pb21m of cadmium titanate Pbn21 Parameter a b c Cd1x Cd1y Cd1z Cd2x Cd2y Cd2z Tix Tiy Tiz O1ax O1ay O1az O1bx O1by O1bz O2ax O2ay O2az O2bx O2by O2bz this work 5.2392 5.3777 7.6192 0.01101 0.04425 0.25324 0.01101 0.04425 0.75324 0.00080 0.49548 0.00334 0.10277 0.46191 0.24044 0.10277 0.46191 0.74044 0.68779 0.31805 0.04267 0.29839 0.71471 0.06583 experiment [4]* 5.2946 5.4151 7.6029 0.0083 0.0407 0.25 0.0083 0.0407 0.75 0.004 0.493 0.004 0.091 0.473 0.241 0.091 0.473 0.741 0.723 0.308 0.047 0.323 0.710 0.045 this work 5.2498 5.3870 7.5699 0.01400 0.04583 0.25000 0.00509 0.04696 0.25000 0.00459 0.50699 0.00080 0.10313 0.45559 0.25000 0.09897 0.53144 0.25000 0.69372 0.29494 0.05436 0.30806 0.68835 0.05264 Pb21m experiment [7] 5.281 5.403 7.583 0.01106 0.04076 0.25000 0.00697 0.04076 0.25000 0.00190 0.5045 0.00214 0.0925 0.4759 0.25000 0.0911 0.5338 0.25000 0.6999 0.2951 0.0501 0.2989 0.7017 0.0457

­ + + + ­ + + + + + + + ­ ­ + + + + + + ­

­ + + + ­ + + + + + + ­ ­ + + + + + + ­

­ + + + ­ ­ + + + + + + ­ + ­ + + + + + ­

­ + + + ­ ­ + + + + + + ­ + ­ + + + + + ­

of seven modes with symmetry B1u (the acoustic mode with symmetry B1u that describes the uniform atomic displacements in the unit cell was excluded) and seven totally symmetric modes Ag (Fig. 3). For the Pb21m phase, the expansion involves the contributions of nine modes with symmetry B2u (also without acoustic mode) and seven Ag modes. In this case, the contributions of the modes with symmetries B1u and B2u with the lowest frequencies to the total energy of distortions amount to 92.8 and 95.4%, respectively. Therefore, when constructing the effective Hamiltonian, we can restrict ourselves to the expansion in powers of the amplitudes of these two modes. The total energy of the crystal was expanded in a Taylor series in powers of the distortion amplitudes and described by the normalized normal modes with symmetries B1u and B2u with the lowest frequencies. In this case, the lattice parameters were assumed to be fixed and equal to the lattice parameters for the Pbnm phase. The resulting expansion has the form E tot ( , ) = E tot ( 0, 0 ) + b 1 + c 1 + b 2 + c2 + d
4 2 2 2 4 2

(1)

* Correct signs of atomic displacements are recovered.

3.3. Parameters of the Effective Hamiltonian Torgashev et al. [8] and Gorshunov et al. [9] assumed that, below the ferroelectric phase transition temperature, one more phase transition accompanied by a change in the polarization direction occurs in cadmium titanate. In order to verify this hypothesis, we calculated the dependence of the total energy of the crystal on the distortion of the lattice with symmetries B1u and B2u. As is known, the normal lattice vibrations form a complete orthonormal basis set in which it is possible to expand any combination of atomic displacements. We used the expansion of the lattice distortions (determined in Subsection 3.2) in the Pbn21 and Pb21m phases in the basis set of eigenvectors of the dynamic matrix of CdTiO3 with space group Pbnm. For the Pbn21 phase, this expansion contains the contributions

with the coefficients b1 = ­0.4075 Ha (Hartree), b2 = ­ 0.2614 Ha, c1 = 183.49 Ha, c2 = 249.58 Ha, and d = 457.4 Ha. It turned out that the expansion is adequately described by the fourth-degree polynomials of two orders parameters and that sixth-degree invariants can be omitted in the expansion. Therefore, the phase transition in CdTiO3 is far from the tricritical point (the closeness to which was noted in [5]). Then, since we have d > 2 c 1 c 2 , the minima of the total energy correspond to the order parameters ±(, 0) and ±(0, ) and are separated from each other by the energy barriers. This means that the formation of the monoclinic phase with space group Pb and tilted polarization vectors is energetically unfavorable. The calculation from relationship (1) demonstrates that the energy minima are observed at the mode amplitudes ( = 0.03332, = 0) and ( = 0, = 0.02288) and that the energy gains associated with the lattice distortion are equal to 6.16 and 1.86 meV, respectively, which are close to the energies corresponding to the true distortions (see Subsection 3.2). Since we included only two modes with the lowest frequencies in expansion (1), it was necessary to check that the energetically most favorable phase Pbn21 remains stable with respect to small distortions with symmetry B2u with allowance made for all modes involved in the distortion. For this purpose, we calculated the phonon spectrum of the CdTiO3 crystal in the Pbn21 phase. In this spectrum, the ferroelectric mode polarized along the y axis has the lowest vibrational frequency equal to 83 cm­1. The positive values of all
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mode frequencies confirm that the ground state of the CdTiO3 crystal corresponds to the polar Pbn21 phase. Although the above analysis of the stability of the polar Pbn21 phase corresponds to the temperature T = 0, it is unlikely that a change in the coefficients c1, c2, and d in the thermodynamic potential with an increase in the temperature can lead to the violation of the condition d > 2 c 1 c 2 and the transition to the monoclinic phase. Therefore, the possibility of appearing sequential ferroelectric phase transitions in CdTiO3 with a variation in the temperature is most likely ruled out. Possibly, the specific features observed in the dielectric constant of CdTiO3 crystals near 50 K [3, 5, 6] are associated with the existence of side minima of the total energy that are separated from the main minima by low potential barriers. 3.4. Spontaneous Polarization The spontaneous polarization Ps in the orthorhombic phases Pbn21 and Pb21m of cadmium titanate was calculated by the Berry's phase method [17]. The calculated spontaneous polarizations Ps corresponding to the lattice distortions determined in Subsection 3.2 for the Pbn21 and Pb21m phases are equal to 0.29 and 0.21 C/m2, respectively. The spontaneous polarization Ps corresponding to the minima in expansion (1) is equal to 0.21 C/m2 for the Pbn21 phase and 0.16 C/m2 for the Pb21m phase. It is worth noting that both calculated values of Ps considerably exceed experimentally obtained polarizations of 0.002­0.009 C/m2 [4, 5]. 4. DISCUSSION OF THE RESULTS As was shown in [1], the ferroelectric instability is characteristic of the cubic phase of all ten titanates with a perovskite structure studied. In crystals that undergo structural phase transitions to the Pbnm and I4/mcm phases, the ferroelectric instability is weakened and observed only in three crystals with the Pbnm structure (CdTiO3, ZnTiO3, and MgTiO3) among five titanates characterized by the aforementioned phase transitions (Table 1). A comparison of the frequencies of the softest ferroelectric modes at the point in the cubic praphase and the Pbnm and I4/mcm phases shows that, upon structural phase transition, the ferroelectric instability is retained in crystals in which this instability is strongest in the cubic phase. Therefore, it is not surprising that the temperature of the ferroelectric phase transition in CdTiO3 is considerably lower than that in BaTiO3 or PbTiO3. Investigations of SrTiO3 have long revealed that, despite the existence of the ferroelectric instability in this compound, the corresponding phase transition is not observed in the crystal with a decrease in the temperature. It is believed this is associated with quantum
PHYSICS OF THE SOLID STATE Vol. 51 No. 4 2009

0.06 Ai 0.04 0.02 0 0.04 0.02 0 ­ 200 0 200 , cm­1 B2u 400 600 Ai Blu

Fig. 3. Relative contributions of different modes with symmetries B1u and B2u to the ferroelectric distortions upon transitions to the Pb21m and Pbn21 phases of cadmium titanate.

fluctuations, i.e., zero-point vibrations of atoms [18]. One more factor responsible for this behavior can be the structural phase transition that occurs in the crystals and, as was shown above, weakens the ferroelectric instability. It seems likely that the strong influence of quantum fluctuations on the physical properties of crystals should also manifest itself for CdTiO3. As was shown in [19], quantum effects most strongly affect the modes with a small "reduced mass," in particular, the ferroelectric mode to which light oxygen atoms make a significant contribution. This explains the fact that the replacement of the16O isotope by the heavier 18O isotope in strontium titanate leads to the real ferroelectric phase transition [20]. In order to take into account the quantum fluctuations, expression (1) for the "potential" energy should be complemented by the kinetic energy of nuclei. When performing first-principles calculations, this energy is ignored in order to calculate correctly the forces acting on atoms (the Born­Oppenheimer approximation). Although the approach used below requires a more detailed justification, it is possible to attempt (according to [19]) to take into account the quantum fluctuations at the level of mode motion and to add the kinetic energy of the mode in the form of the operator (- 2/2M*)2 to the potential energy (1) of the mode, where M* is some reduced mass. With the aim of determining the quantitative criterion corresponding to the complete suppression of the phase transition by quantum fluctuations, we consider the simple one-dimensional problem in which a particle with a mass M* moves in a double-well potential V(x) = ­ax2 + bx4. We assume that the phase transition is suppressed by quantum fluctuations when the kinetic energy of zero-point vibrations of the mode exceeds the well depth of the potential under consideration. The numerical solution


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Table 5. Frequencies of the unstable ferroelectric modes in the high-symmetry phases, energy gains in the case of lattice distortion [1], and ratios h/E0 for three ferroelectric phase transitions in the SrTiO3 and CdTiO3 compounds Phase transition SrTiO3, Pm3m CdTiO3, Pbnm CdTiO3, Pbnm R3m Pb21m Pbn21 , cm­1 E0, meV 68i 81i 104i 0.75 1.38 6.21 h/E0 11.2 7.28 2.08

crystals [8, 15, 16] and the presence of long-period structures competing with the Pbnm phase [16]. 5. CONCLUSIONS Thus, the first-principles calculations of the structural parameters and the phonon spectrum of orthorhombic CdTiO3 enabled us to refine the identification of Raman scattering and IR reflectance spectra. The revealed dependence of the total energy of the crystal on the amplitudes of two unstable modes indicates that the ferroelectrically distorted Pbn21 phase is the ground state of the crystal at the temperature T = 0. This phase appears to be the most stable with respect to quantum fluctuations, which are rather strong and suppress other possible lattice distortions. The quantum fluctuations have been found to be one of the main factors responsible for the discrepancy between calculated and experimentally observed values of the spontaneous polarization and the structural distortions upon phase transition. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 08-02-01436). REFERENCES
1. A. I. Lebedev, Fiz. Tverd. Tela (St. Petersburg) 51 (2), 341 (2009) [Phys. Solid State 51 (2), 362 (2009)]. 2. G. A. Smolenskioe, Dokl. Akad. Nauk SSSR 70, 405 (1950). 3. P.-H. Sun, T. Nakamura, Y. J. Shan, Y. Inaguma, and M. Itoh, Ferroelectrics 217, 137 (1998). 4. Y. J. Shan, H. Mori, R. Wang, W. Luan, H. Imoto, M. Itoh, and T. Nakamura, Ferroelectrics 259, 85 (2001). 5. M. E. Guzhva, V. V. Lemanov, and P. A. Markovin, Fiz. Tverd. Tela (St. Petersburg) 43 (11), 2058 (2001) [Phys. Solid State 43 (11), 2146 (2001)]. 6. Y. J. Shan, H. Mori, H. Imoto, and M. Itoh, Ferroelectrics 270, 381 (2002). 7. Y. J. Shan, H. Mori, K. Tezuka, H. Imoto, and M. Itoh, Ferroelectrics 284, 107 (2003). 8. V. I. Torgashev, Yu. I. Yuzyuk, V. B. Shirokov, V. V. Lemanov, and I. E. Spektor, Fiz. Tverd. Tela (St. Petersburg) 47 (2), 324 (2005) [Phys. Solid State 47 (2), 337 (2005)]. 9. B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, Fiz. Tverd. Tela (St. Petersburg) 47 (3), 527 (2005) [Phys. Solid State 47 (3), 547 (2005)]. 10. G. Fabricius and A. LÑpez GarcËa, Phys. Rev. B: Condens. Matter 66, 233 106 (2002). 11. X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-Y. Raty, and D. C. Allan, Comput. Mater. Sci. 25, 478 (2002). 12. J. P. Perdew and A. Zunger, Phys. Rev. B: Condens. Matter 23, 5048 (1981).
PHYSICS OF THE SOLID STATE Vol. 51 No. 4 2009

to the SchrÆdinger equation demonstrates that this holds true under the condition b/ M * a > 0.428. By eliminating the unknown mass M* from this condition, the criterion can be rewritten in the physically clear form h/E0 > 2.419. Here, = ­ 2 a / M * / 2 is the magnitude of the imaginary vibration frequency in the vicinity of the maximum of the potential barrier (determined within the classical approximation), and E0 = a2/4b is the depth of the potential well for the potential under consideration. For the four-minimum potential well described by relationship (1), the quantitative value of the criterion can be slightly different. Let us use the obtained criterion to evaluate the degree of influence of quantum fluctuations on the ferroelectric phase transitions in CdTiO3. The frequencies of the unstable ferroelectric modes determined from first principles under the assumption of classical motion of nuclei, the energies E0 of the ordered phases, and the ratios h/E0 for the hypothetical ferroelectric phase transition in SrTiO3 and two phase transitions under consideration in CdTiO3 are listed in Table 5. It can be seen from this table that the quantum fluctuations should suppress the ferroelectric phase transition in strontium titanate and the transition to the Pb21m phase in cadmium titanate. As regards the phase transition to the Pbn21 phase in cadmium titanate, this transition is susceptible to strong quantum fluctuations but is not completely suppressed. Therefore, the only phase transition that can be associated with the experimentally observed ferroelectric phase transition in CdTiO3 is the transition to the Pbn21 phase. The quantum fluctuations can also be responsible for the significant discrepancies in the structural positions of atoms and in the values of the spontaneous polarization Ps. In quantum-mechanical calculations of the ground state for multiwell potentials, the displacement corresponding to the most probable atomic position is always smaller than the displacement corresponding to the minimum of the potential energy. Therefore, the quantum fluctuations should lead to a decrease in the distortions upon phase transition and a decrease in the spontaneous polarization Ps. One more probable factor responsible for the decrease in the experimentally determined polarization for CdTiO3 can be associated with the manifestation of twinning in
3


FERROELECTRIC PHASE TRANSITION IN ORTHORHOMBIC CdTiO3 13. A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, Phys. Rev. B: Condens. Matter 41, 1227 (1990). 14. N. J. Ramer and A. M. Rappe, Phys. Rev. B: Condens. Matter 59, 12 471 (1999). 15. S. Sasaki, C. T. Prewitt, J. D. Bass, and W. A. Schulze, Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 43, 1668 (1987). 16. Yu. V. Kabirov, M. F. Kupriyanov, Ya. Dets, and P. Wawzala, Fiz. Tverd. Tela (St. Petersburg) 42 (7), 1291 (2000) [Phys. Solid State 42 (7), 1329 (2000)].

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17. N. Sai, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B: Condens. Matter 66, 104 108 (2002). 18. K. A. MÝller and H. Burkard, Phys. Rev. B: Condens. Matter 19, 3593 (1979). 19. W. Zhong and D. Vanderbilt, Phys. Rev. B: Condens. Matter 53, 5047 (1996). 20. M. Itoh, R. Wang, Y. Inaguma, T. Yamaguchi, Y.-J. Shan, and T. Nakamura, Phys. Rev. Lett. 82, 3540 (1999).

Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE

Vol. 51

No. 4

2009