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ISSN 1063-7834, Physics of the Solid State, 2009, Vol. 51, No. 2, pp. 362­372. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.I. Lebedev, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 2, pp. 341­350.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Ab initio Calculations of Phonon Spectra in ATiO3 Perovskite Crystals (A = Ca, Sr, Ba, Ra, Cd, Zn, Mg, Ge, Sn, Pb)
A. I. Lebedev
Moscow State University, Moscow, 119992 Russia e-mail: swan@scon155.phys.msu.su
Received May 25, 2008

Abstract--First-principles calculations of phonon spectra based on the density functional theory are carried out for calcium, strontium, barium, radium, cadmium, zinc, magnesium, germanium, tin, and lead titanates with a perovskite structure. By analyzing unstable modes in the phonon spectrum, the possible types of lattice distortion are determined and the energies of the corresponding phases are calculated. From analyzing the phonon spectra, force constants, and eigenvectors of TO phonons, a conclusion is drawn concerning the nature of ferroelectric phenomena in the crystals studied. It is shown that the main factors determining the possible appearance of off-center atoms in the A position are the geometric size and electronic configuration of these atoms. PACS numbers: 61.50.Ah, 63.20.Dj, 77.84.Dy DOI: 10.1134/S1063783409020279

1. INTRODUCTION Crystals of the perovskite family are well-known materials undergoing various structural distortions with decreasing temperature. When distortions are ferroelectric in character, a number of physical characteristics of the crystals (permittivity, piezoelectric coefficients, etc.) become anomalously large in magnitude. For this reason, these materials have found wide application in modern electronics. The problem of further optimization of the properties of ferroelectrics requires a deep understanding of the microscopic mechanisms responsible for ferroelectricity and the appearance of these properties. In solving this problem, it is very helpful to further carry out ab initio calculations of physical properties of crystals, which have already made a significant contribution to the understanding of the ferroelectric phenomena in perovskite crystals [1­11]. In discussing the properties of ferroelectrics, it is very important to elucidate whether these properties result from collective displacements of atoms in the lattice (displacive phase transition) or they are determined by specific features of certain constituent atoms of the crystal (order­disorder phase transition). This problem arises, in particular, in discussing the nature of phase transitions that occur in incipient ferroelectrics doped with certain impurities [12]. Earlier studies of titanates with a perovskite structure have dealt mainly with four systems (CaTiO3 [3, 4, 13, 14], SrTiO3 [3, 4, 6], BaTiO3 [1­4, 8, 9], and PbTiO3 [2­4, 7, 9]) and solid solutions based on these

compounds. However, a comparison of the obtained results is hampered by the fact that those studies differ in terms of calculation techniques and methods used to construct atomic potentials. The objective of this work is to carry out first-principles calculations of the phonon spectra of ten ATiO3 crystals with a perovskite structure and determine the structure of the stable phases for these crystals. In order to test the method used, we first apply it for calculating the properties of the four systems mentioned above and then we study the properties of poorly investigated or hypothetical perovskite crystals RaTiO3, CdTiO3, MgTiO3, ZnTiO3, SnTiO3, and GeTiO3. From comparing the results obtained in a unified way for a large number of related materials, conclusions are inferred concerning the relation of the structural distortions in the ATiO3 crystals to the size of the A atom and its electronic structure. From analyzing the on-site force-constant matrix elements and the TO-phonon eigenvectors, we draw conclusions regarding the nature of ferroelectric phenomena in these materials and find conditions under which these phenomena can be associated with off-center atoms. 2. CALCULATION TECHNIQUE The calculations are carried out using the ABINIT software [15] based on the density functional theory, pseudopotentials, and expansions of the wave functions in plane waves. The exchange-correlation interaction is described in the local-density approximation (LDA) using the technique developed in [16]. The pseudopo-

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Table 1. Electronic configurations of atoms and parameters used to construct pseudopotentials: rs, rp, and rd are the core radii of the pseudopotentials for the s, p, and d projections; qs, qp, and qd are the limiting values of wave vectors used to optimize pseudopotentials; and rmin, rmax, and Vloc are the range limits and the depth of the correcting local potential (parameter values are in atomic units, and energy is given in Ry) Atom Ca Sr Ba Ra Mg Zn Cd Ge Sn Pb Ti O Configuration 3s23p63d04s0 4s24p64d05s0 5s25p65d06s0 6s26p67s06d07p 2s22p63s03p0 3d104s04p0 4d105s05p0 3d104s1.54p0.5 4d105s25p0 5d106s26p0 3s23p63d04s0 2s22p23d0
0

rs 1.46 1.68 1.85 1.84 1.50 1.82 2.04 1.68 2.14 1.80 1.48 1.40

rp 1.68 1.74 1.78 1.73 1.88 1.82 2.18 1.68 2.08 1.72 1.72 1.55

rd 1.82 1.68 1.83 1.98 ­ 2.00 2.10 1.96 2.18 1.98 1.84 1.40

q

s

q

p

q

d

rmin 0.01 0.01 0.01 0.01 0.01 0.01 0 0.01 0.01 0.1 0.01 ­

rmax 1.40 1.52 1.68 1.68 1.0 1.60 1.88 1.58 1.90 1.43 1.41 ­

Vloc 1.6 1.5 1.95 ­1.3 ­ 0.84 2.5 ­1.6 0.48 0.64 1.6 2.65 ­

7.07 7.07 7.07 7.8 6.7 7.07 7.07 7.07 7.07 6.05 7.07 7.07

7.07 7.07 7.07 7.8 8.1 7.07 7.07 6.0 7.07 5.52 7.07 7.57

7.27 7.07 7.07 7.8 ­ 7.47 7.07 7.77 7.07 7.17 7.07 7.07

tentials used are optimized separable nonlocal pseudopotentials [17], which are constructed using the OPIUM computer program and to which a local potential is added according to [18] in order to improve their transferability. The calculations for elements with atomic numbers Z < 46 are performed without inclusion of relativistic effects; for other elements, a scalar relativistic approximation is used. Table 1 lists the parameters used for constructing pseudopotentials. The local potential is chosen to be an s potential, except for oxygen atoms, for which a local d potential is used. The parameters of pseudopotentials are finely adjusted by comparing the computed and experimental values of lattice parameters for a number of oxides and sulfides of elements. The lattice parameters and the equilibrium atomic positions in the unit cell are found by minimizing the Hellmann­Feynman forces acting on the atoms (<10-5 Ha/Bohr), with the total crystal energy being calculated self-consistently with an accuracy of better than 10­10 Ha.1 In calculations, particular attention is given to the convergence of the results with respect to the choice of the cut energy of plane waves and the fineness of the grid of wave vectors used in integration over the Brillouin zone. For all the computed quantities that are discussed below, the convergence is attained with a cut energy of 30 Ha and with a 8 â 8 â 8 grid constructed according to [19]. Effective charges Z*, optical dielectric constant , elastic moduli Cij , bulk modulus B, force-constant matrix ij, and phonon spectra are calculated using perturbation theory [20­23]. The phonon energies are
1

exactly calculated at five points of the Brillouin zone (, X, M, R, and the point located halfway between the and R points), and then the phonon spectrum is computed over the entire Brillouin zone using the interpolation technique described in [20, 24]. 3. TESTING OF THE CALCULATION TECHNIQUE The correctness of the calculation technique described above is tested by comparing the calculated lattice parameters, spontaneous polarization, and phonon spectra with the available experimental data and calculations performed by other authors for wellstudied CaTiO3, SrTiO3, BaTiO3, and PbTiO3. The lattice parameter values corresponding to a minimum total energy of the crystal are listed in Table 2. These computed values agree well with the experimental data from [25] if we take into account that the LDA systematically slightly underestimates the lattice parameters. An analysis of the relative energies (per formula unit) of the low-symmetry phases (Table 3) shows that the energetically most favorable phase for barium titanate is the rhombohedral phase and, for lead titanate, the tetragonal phase. The calculated values of the ratio c/a for tetragonal BaTiO3 and tetragonal PbTiO3 are close to the experimental values (Table 2). In CaTiO3, the energetically most favorable phase is the orthorhombic phase Pbnm and, in SrTiO3, the tetragonal phase I4/mcm. The values of the spontaneous polarization calculated by the Berry phase method [26] are 0.26, 0.31, and 0.89 C/m2 for tetragonal BaTiO3, rhombohedral BaTiO3, and PbTiO3, respectively. These values are close to experimental data (0.26, 0.33, 0.75 C/m2 [25]).

In this paper, the energy is measured in the Hartree atomic system of units (1 Ha = 27.2113845 eV) everywhere except in Tables 1 and 3. PHYSICS OF THE SOLID STATE Vol. 51 No. 2 2009


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Table 2. Comparison of calculated and experimental crystal lattice parameters of various phases of ATiO3 compounds (experimental data are obtained at 300 K, unless otherwise specified) Compound CaTiO3 SrTiO3 Space group Pbnm Pm3m I4/mcm BaTiO3 Pm3m P4mm Amm2 R3m PbTiO3 P4mm Source This work Expt. [25] This work Expt. [25] This work Expt. [25] This work Expt. [25] This work Expt. [25] This work Expt. [25] This work Expt. [25] This work Expt. [25] a a a a a a a a a a a a a a a a = = = = = = = = = = = = = = = = Lattice parameters 5.3108 å, b = 5.4459 å, c = 7.5718 å 5.3670 å, b = 5.4439 å, c = 7.6438 å 3.8898 å 3.905 å b = 5.4680, c = 7.8338 å b = 5.510, c = 7.798 å (20 K) 3.9721 å 3.996 å (393 K) 3.9650, c = 4.0070 å, c/a = 1.0106 3.9920, c = 4.0361 å (293 K) 3.9620, b = 5.6384, c = 5.6484 å 3.990, b = 5.669, c = 5.682 å (263 K) 3.9817 å, = 89.933° 4.001 å, = 89.85° (105 K) 3.8858, c = 4.1151 å, c/a = 1.0590 3.904, c = 4.152 å

The calculated optical-phonon energies are also in good agreement with available experimental data and calculations performed by other authors (Tables 4, 5). The imaginary frequencies (negative squares of the frequencies) in Tables 4 and 5 correspond to unstable modes. The good agreement of our calculations with the experimental data and the calculation results of other authors for CaTiO3, SrTiO3, BaTiO3, and PbTiO3 suggests that we can use the proposed calculation technique to predict the properties of poorly studied and hypothetical2 perovskite titanates and analyze the factors causing the appearance of ferroelectricity in these crystals. Tables 3­7 show the characteristics of these crystals calculated for the theoretical value of the lattice parameter (i.e., the value corresponding to a minimum of the total energy). Table 3 lists the energies of several low-symmetry phases of the crystals measured relative to the energy of the original cubic phase. Table 4 gives the frequencies of optical phonons (six infrared-active 15 modes and one infrared-inactive 25 mode) in the cubic phase of the same crystals, and Table 5 gives the lowest phonon energies at high-symmetry points of the Brillouin zone. Table 6 lists the values of the effective charges and optical dielectric constant for the cubic phase of the ATiO3 compounds. Finally, Table 7 gives the elastic moduli of several crystals.
2

The phonon spectra along certain directions calculated for the cubic phase of the ten titanates studied are shown in Fig. 1. The imaginary phonon energies associated with structural instability of the crystals are represented in Fig. 1 by negative numbers. 4. RESULTS It follows from Fig. 1 that the phonon spectra of all titanates studied have unstable optical modes of different symmetry. First, we discuss the phonon spectra of well-studied crystals. A specific feature of the dispersion curves of SrTiO3 is that, out of three unstable phonons at the , R, and M points, the most unstable is the phonon at the R point (R25 mode)3 and that the phonon energy depends only slightly on wave vector along the R­M line. As shown in [6], in the real space, these unstable phonons with wave vectors near the edges of the cubic Brillouin zone correspond to rotation of oxygen octahedra linked together by shared vertices, with a correlation length of this rotation being three to five lattice periods. Thus, the R25 mode and the M3 mode (which corresponds to the unstable phonon at the M point) describe structural distortions associated with rotation of the octahedra. The unstable 15 phonon mode at the point corresponds to ferroelectric distortions of the crystal structure. The R25 mode is threefold degenerate, and distortions described by three-component order parameters (, 0, 0), (, , 0), and (, , ) lead to low-symmetry
3

Our calculations show that the energy of the ilmenite phase of SnTiO3, GeTiO3, CdTiO3, ZnTiO3, and MgTiO3 at T = 0 is lower than the energy of the most stable of the distorted perovskite phases. However, only for the two last crystals is the difference between these phases large (0.30­0.33 eV).

We use the mode notation introduced in [29]. Vol. 51 No. 2 2009

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Table 3. Relative energies of various low-symmetry phases of ATiO3 compounds (energies of the most stable phases are in boldface) Compound MgTiO3 Unstable mode X '5 X '5 15 15 X '5 X '5 M '5 M '5 15 15 15 M3 M '3 X5 X5 M '3 X5 X5 X3 25 X5 15 X5 15 25 X5 X5 15 15 15, 25 X '5 ' X5 M3 R25 X '5 X '5 M3 R25 M3 R25 R25 Space group Cmcm Pmma R3m P4mm Cmcm Pmma Pmma Cmmm R3m P4mm R3m P4/mbm P4/nmm Pmma Cmcm P4/nmm Pmma Cmcm P42/mmc P 4m2 Pmma R3m Cmcm P4mm P 4 m2 Pmma Cmcm R3m P4mm Amm2 Pmma Cmcm P4/mbm I4/mcm Pmma Cmcm P4/mbm I4/mcm P4/mbm I4/mcm R 3c
Vol. 51 No. 2

Energy, meV ­304 ­500 ­ 695 ­1028 ­ 0.6 ­ 0.9 ­5.0 ­ 6.7 ­73.7 ­ 0.71 ­ 0.75 ­9.45 ­ 0.31 ­1.22 ­1.45 ­11.1 ­14.2 ­16.9 ­45 ­134 ­160 ­245 ­282 ­340 ­341 ­ 447 ­867 ­868 ­1104 ­1254 ­328 ­ 428 ­ 444 ­ 455 ­21 ­23 ­57 ­67 ­10.1 ­19.6 ­21.6
2009

Unstable mode M3 R25 R25 R25 + M3 15 M3 R25 R25 R25 + M3 R25 R25 + M3 R25 15 15 15 15 15 15 15, 25 25 R25 M3 R25 R25 + M3 R25 M3 25 R25 R25 + M3 R25 R25 + M3 15 15 R25 R25 + M3 15 15 R25 + M3 15 15

Space group P4/mbm I4/mcm R 3c Pbnm P4mm P4/mbm I4/mcm R 3c Pbnm R 3c Pbnm I4/mcm P4mm Amm2 R3m P4mm Amm2 R3m Amm2 R32 I4/mcm P4/mbm R 3c Pbnm I4/mcm P4/mbm R32 R 3c Pbnm R 3c Pbnm P4mm R3m R 3c Pbnm R3m P4mm Pbnm R3m P4mm

Energy, meV ­1107 ­1111 ­1727 ­1992 ­123 ­321 ­365 ­385 ­ 497 ­27.5 ­28.9 ­30.9 ­5.6 ­7.4 ­8.1 ­21.8 ­28.5 ­29.7 ­ 412 ­ 486 ­912 ­920 ­1197 ­1283 ­1443 ­1449 ­1486 ­2271 ­2312 ­589 ­810 ­854 ­1053 ­74 ­84 ­240 ­291 ­22.2 ­ 66.3 ­84.4

CaTiO3

SrTiO3

BaTiO3

RaTiO3

CdTiO3

ZnTiO3

GeTiO3

SnTiO3

PbTiO3

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Table 4. Frequencies of optical phonons at the point in the cubic phase of ATiO3 compounds (in cm­1) Compound MgTiO3 CaTiO3 Source This work Same Calc. [3] Calc. [13] This work Calc. [3] Calc. [6] Expt. [27] This work Calc. [3] Calc. [8] Expt. [28]a This work This work Same This work Same '' Calc. [3] Calc. [7] TO1 260i 165i 153i 140i 68i 41i 100i ­ 151i 178i 195i ­ 212i 240i 187i 247i 185i 150i 144i 182i TO2 151 176 188 200 162 165 151 175 175 177 166 181 172 76 97 122 126 116 121 63 TO3 649 607 610 625 549 546 522 544 471 468 455 487 444 645 616 583 505 499 497 447 LO1 106i 122 133 136 152 158 146 172 172 173 162 180 166 105i 34 68i 80 96 104 47 LO2 372 407 427 428 428 454 439 475 439 453 434 468 434 316 353 356 375 394 410 418 LO3 905 857 866 864 792 829 751 796 683 738 657 717 638 815 820 762 689 693 673 610 25 191i 93 ­ 130 202 ­ 219 ­ 269 ­ 266 306 287 353i 231i 49i 183 202 ­ ­

SrTiO3

BaTiO3

RaTiO3 ZnTiO3 CdTiO3 GeTiO3 SnTiO3 PbTiO3

Note: a Data for the tetragonal phase.

phases belonging to space groups I4/mcm, Imma, and R 3 c , respectively. The M3 mode is nondegenerate, and its condensation reduces the crystal symmetry to P4/mbm. The threefold degenerate 15 mode is described by space groups P4mm, Amm2, and R3m. From comparing the energies of these phases (Table 3),4 it follows that the lowest energy phase of SrTiO3 is the I4/mcm phase, which arises with decreasing temperature. The instability of the ferroelectric mode is not sufficiently strong for ferroelectricity to occur in the crystal. In CaTiO3, in addition to the instabilities indicated above, three weak antiferroelectric-type instabilities ' ' arise associated with the X5, X 5 , and M 5 modes and the R­M segment of the phonon spectrum is practically dispersionless (cf. mode energies in Table 5). In the latter case, theoretical calculations showed [14] that the simultaneous condensation of the unstable R25 and M3 modes brings about the formation of a low-temperature Pbnm phase having the lowest energy among the possible distorted phases (Table 3).5 The transition from the high-temperature Pm3m phase to the Pbnm phase can
4

occur through one of three intermediate phases (P4/mbm, I4/mcm, and R 3 c ), whose energies are 0.11­ 0.17 eV higher than the energy of the Pbnm phase. The ferroelectric P4mm and R3m phases in CaTiO3 have far higher energies and never arise. As for the weakly ' ' unstable X 5 , X5, and M 5 modes, they are twofold degenerate and distortions described by the order parameters (, 0) and (, ) lead to the Pmma, Cmcm, and Cmmm phases. However, the energy gained in the transformation into these phases does not exceed 7 meV. The phonon spectrum of BaTiO3 differs significantly from the spectra discussed above by the absence of instability at the R point and the appearance of highly ' unstable X5 and M 3 modes (at the X and M points, respectively) corresponding to ferroelectric transformations into the Pmma, Cmcm, and P4/nmm phases. Among these modes, the most unstable is the ferroelectric 15 mode and this mode determines crystal distortions (the three antiferroelectric phases are higher in energy than the polar P4mm phase). We note that, in our
5

We do not present in the table the energies of phases described by the order parameter (, , 0), because these phases arise in the rare case when the coefficient of the second fourth-order invariant constructed from the order parameter components becomes zero.

Calculations show that, in SrTiO3, despite the presence of the unstable R25 and M3 modes in the phonon spectrum, the energy of the Pbnm phase is 2 meV higher than that of the I4/mcm phase. Vol. 51 No. 2 2009

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AB INITIO CALCULATIONS OF PHONON SPECTRA IN ATiO3 PEROVSKITE CRYSTALS Table 5. Lowest phonon frequencies at high-symmetry points of the Brillouin zone in the cubic phase of ATiO3 compounds (in cm­1) Compound Source 260i 165i 140i 68i 151i 219i 212i 353ia 231ia 247i 185i 150i 182i X 190i 32i 20 98 96i 189i 182i 319i 184i 148i 56i 30 31i M 314i 215i 207i 86i 59i 167i 158i 437i 333i 254i 144i 96i 35i R 315 226 219 119 134 128 110 i i i i 246i 122i ­ 100 105 ­ 87 337i 265i 201 97 15 58 i i i i

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Table 6. Effective charges and optical dielectric constant for the cubic phase of ATiO3 compounds Compound MgTiO3 CaTiO3 SrTiO3 BaTiO3 RaTiO3 ZnTiO3 CdTiO3 GeTiO3 SnTiO3 PbTiO3 Z* A 2.537 2.579 2.561 2.738 2.764 3.233 3.040 4.460 4.255 3.931 Z* Ti 7.773 7.692 7.725 7.761 7.789 8.257 8.052 7.572 7.529 7.623 Z* O


Z* O ­ ­ ­ ­ ­

||





MgTiO3 This work CaTiO3 Same Calc. [13] SrTiO3 This work BaTiO3 Same Calc. [8] RaTiO3 This work ZnTiO3 This work CdTiO3 Same GeTiO3 This work SnTiO3 Same PbTiO3 '' Calc. [7]
Note: a 25 mode.

­2.026 ­2.085 ­2.099 ­2.186 ­2.181

6.258 6.101 6.088 6.128 6.192

7.01 6.84 6.87 7.28 7.42 11.64 9.35 10.49 10.18 9.34

­2.427 ­ 6.637 ­2.300 ­ 6.493 ­2.860 ­ 6.314 ­2.745 ­ 6.294 ­2.635 ­ 6.283

424i 328i 251 148 113 145 i i i i

The calculated elastic moduli of the cubic RaTiO3 are given in Table 7. Now, we consider the phonon spectrum of CdTiO3. This spectrum has a number of unstable modes at the X, ' ' ' M, and R points (X3, X5, X 5 , M3, M 5 , M 2 , R25, R15 modes) and two unstable modes at the point. It is surprising that the instability at the point is due to the 25 mode associated with deformation of the oxygen octahedron (see mode energies in Table 4) rather than to the ferroelectric 15 mode. Such a deformation can bring about the formation of the P 4 m 2 , Amm2, and R32 phases, depending on the number of nonzero order parameter components.6 The energy of the most stable of these phases (R32, Table 3) is lower than that of the polar phases. Due to the qualitative similarity between the phonon spectra of calcium and cadmium titanates and between the eigenvectors of their unstable modes at the R and M points, CdTiO3 can be considered an analog of CaTiO3, but characterized by a higher instability. Therefore, at room temperature, the nonpolar phase has Pbnm symmetry, as is the case for CaTiO3. This conclusion was also drawn in [30]. However, according to our data, the energy of this phase is lower than that of the cubic phase by 1.28 eV (Table 3), which is somewhat greater than the value obtained in [30] (0.8 eV) and [10] (0.91 eV). The ferroelectric instability associated with the 15 mode is not of great importance in the cubic CdTiO3, but it is known that this instability manifests itself in the Pbnm phase and leads to a ferroelectric phase transition at 80 K. The first-principles calculations of the properties of the orthorhombic CdTiO3 performed in [30] did
6

calculations, the phonons in BaTiO3 are less unstable than in the calculations performed in [9], because we used the theoretical value of the lattice parameter, whereas the calculations in [9] were performed for the experimental value of this parameter. The dependence of the phonon spectrum on the lattice parameter is illustrated in Fig. 1, wherein the dotted line shows a fragment of the phonon spectrum of BaTiO3 calculated with the same lattice parameter as that in [9]. The weak dependence of the energy of the unstable TO phonon on wave vector along the ­X­M­ line for vibrations polarized along the fourfold axes of the cubic lattice was first discovered in [5]. This dependence shows that the linear chains ...­O­Ti­O­... oriented along these axes dominate in the vibrations and that the interaction between the parallel chains is weak. A comparison of the phonon spectra of BaTiO3 and RaTiO3 shows that these spectra are very similar. In radium titanate, the most unstable is the 15 mode, whose energy depends only slightly on wave vector along the ­X­M­ line and the instability of phonons is even more pronounced than in BaTiO3. Taking into account the calculated energies of the distorted phases (Table 3), we can assume that RaTiO3 is also a ferroelectric and that, as the temperature decreases, it successively undergoes three phase transitions as barium titanate does. The temperatures of these transitions are likely to be higher than those in barium titanate. The values of the spontaneous polarization in RaTiO3 as calculated by the Berry phase method are also higher than those in BaTiO3 and are 0.36 C/m2 in the tetragonal phase and 0.41 C/m2 in the rhombohedral phase.
PHYSICS OF THE SOLID STATE Vol. 51 No. 2 2009

The fact that the lattice symmetry can be lowered to the polar Amm2 group follows from the transformation properties of the order parameter (, , 0). The spontaneous polarization in this phase is 0.018 C/m2.


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Table 7. Elastic moduli of the cubic phase of ATiO3 compounds (in GPa) Compound CaTiO3 SrTiO3 Source This work Calc. [4] This work Calc. [4] Expt. [25] This work Calc. [4] This work Same Calc. [4] Calc. [7] C11 388 407 373 388 316­348 338 329 319 336 334 320 C12 100 96 103 104 101­103 110 117 112 127 145 141 C
44

B 196 200 193 199 174­183 186 188 181 197 208 201

BaTiO3 RaTiO3 PbTiO3

91 101 108 117 119­124 123 130 126 95 100 375?

not revealed a stable ferroelectric distortion in it. In contrast to the calculations in [30], our calculations of the phonon spectrum of the orthorhombic CdTiO3 at the point revealed two unstable B1u and B2u modes causing the formation of polar Pb21m and Pbn21 phases, respectively. Such lattice distortions have been detected by X-ray studies of cadmium titanate at low temperatures [31, 32]. We will discuss the properties of these phases in a later paper. The phonon spectrum of ZnTiO3 is qualitatively similar to that of CdTiO3, but it has an additional weak ' unstable M 3 mode and is less stable. Moreover, the 25 mode is less stable that the 15 mode in it (Table 4). However, the most unstable are the R25 and M3 modes and, therefore, the Pbnm phase is energetically most favorable in ZnTiO3 as well (Table 3). The calculations of the phonon spectrum at the point of the orthorhombic zinc titanate revealed two unstable B1u and B2u modes, which cause the formation of the same polar phases as those in cadmium titanate. The phonon spectrum of MgTiO3 is intermediate between those of zinc titanate and calcium titanate. It likewise has unstable 15 and 25 modes, but the ferroelectric 15 mode is lower in energy in magnesium titanate (Table 4). However, the phonons at the R and M points have the lowest energy and, therefore, the Pbnm phase is energetically most favorable (Table 3). The calculations of the phonon spectrum of the orthorhombic magnesium titanate at the point revealed one Pbn21 unstable B1u mode that can cause a Pbnm ferroelectric phase transition. Finally, let us discuss the phonon spectra of PbTiO3, SnTiO3, and GeTiO3. The ferroelectric instability of these three crystals is associated with the 15 mode, which competes with the unstable R25 and M3 modes. From comparing the energies of various distorted phases (Table 3), it follows that, even in GeTiO3,

wherein the unstable phonons at the , R, and M points are close in frequency, the ferroelectric instability is dominant. The calculated spontaneous polarization is 1.28 C/m2 in the tetragonal SnTiO3 and 1.37 C/m2 in the rhombohedral GeTiO3. Our value of the spontaneous polarization in SnTiO3 is significantly greater than the value 0.73 C/m2 obtained in [33]. Among the perovskite crystals studied to date, GeTiO3 is likely to have the highest spontaneous polarization. It is worth noting that, in PbTiO3 and SnTiO3, the energetically most favorable is the tetragonal P4mm phase and, in GeTiO3, the rhombohedral R3m phase. In the tetragonal GeTiO3, the lattice deformation (c/a = 1.1821) is far greater than that in lead titanate (c/a = 1.0590). This result casts doubt on the conclusion inferred in [2] that the tetragonal phase stabilizes due to a large lattice deformation (large value of the ratio c/a). 5. DISCUSSION As follows from Fig. 1, the phonon spectra of all ATiO3 perovskite crystals studied have several unstable modes, among which there is always the ferroelectric 15 mode. In the case where the R25 and M3 modes competing with it have a lower energy, the crystal undergoes distortions like octahedron rotation and the lattice symmetry is lowered to I4/mcm or Pbnm. The tendency toward such structural phase transitions increases with decreasing A atom size. From analyzing the characteristics of the 15 mode, one can draw a conclusion concerning the nature of the ferroelectric instability of the crystals. As mentioned above, the dispersion law of this mode in BaTiO3 and RaTiO3 indicates strongly correlated motions of atoms along chains ...­O­Ti­O­.... An analysis of the eigenvectors corresponding to the 15 phonon (Table 8) shows that the A atoms are indeed involved in the motion only slightly and the motion is mainly due to
PHYSICS OF THE SOLID STATE Vol. 51 No. 2 2009


AB INITIO CALCULATIONS OF PHONON SPECTRA IN ATiO3 PEROVSKITE CRYSTALS 1000 800 600 , cm­1 , cm­1 400 200 0 ­ 200 800 600 , cm­1 400 200 0 ­ 200 800 , cm­1 400 0 ­ 400 800 400 0 ­ 400 800 600 , cm­1 400 200 0 ­ 200 5' X 5' 3 M 15 15 25 R M SnTiO3 , cm­1 5' 3 X 2' 5' 3 M 15 25 15 25 R M 5 X 3' M 15 CdTiO3 , cm­1 X 3 M 15 25 R BaTiO3 , cm­1 M SrTiO3 800 600 400 200 0 ­ 200 800 600 400 200 0 ­ 200 R M 800 400 0 3' 2' 5' 3 M ZnTiO3 5 X 3' M 15 R M 5 5' X 5' 3 M 25 R M CaTiO3

369



15

RaTiO3



5

­ 400 800 600

5' 5 3 X

15(LO) 15 15 25 25 R

M

MgTiO3 , cm­1

PbTiO3

, cm­1

400 200 0 15 25 R M

53 5' X

2' 5' 3 M

15(LO) 25 15

15 25 R M

­ 200 1000 800 600 400 200 0 ­ 200

X

3 M

GeTiO3

5 5' X

2' 5' 3 M

25 15(LO) 15 15 25 R M

Fig. 1. Phonon spectra of the cubic phase of ATiO3 compounds. The symmetry of the unstable modes is indicated near the curves. PHYSICS OF THE SOLID STATE Vol. 51 No. 2 2009


370

LEBEDEV Table 9. Diagonal elements xx(0, 0) of the on-site forceconstant matrix for A and Ti atoms in the cubic phase of ATiO3 compounds (in Ha/Bohr2) Compound MgTiO3 CaTiO3 SrTiO3 BaTiO3 RaTiO3 ZnTiO3 CdTiO3 GeTiO3 SnTiO3 PbTiO3 A atom ­ + + + + 0.0109 0.0163 0.0445 0.0755 0.0856 Ti atom + + + + + 0.1431 0.1370 0.1196 0.0873 0.0750

Table 8. Eigenvectors of the dynamic matrix corresponding to an unstable TO1 phonon at the point for the cubic phase of ATiO3 compounds Compound MgTiO3 CaTiO3 SrTiO3 BaTiO3 RaTiO3 ZnTiO3 CdTiO3 GeTiO3 SnTiO3 PbTiO3 + + + + + xA 0.6828 0.5693 0.3434 0.0299 0.0051 + + + + + xTi 0.1831 0.2391 0.3852 0.6734 0.6750 ­ ­ ­ ­ ­ xO 0.4800 0.5225 0.5372 0.3561 0.2841 ­ ­ ­ ­ ­ xO|| 0.1985 0.2696 0.3956 0.5404 0.6188

+ 0.5167 + 0.4012 + 0.5367 + 0.4177 + 0.2973

+ 0.1889 + 0.2358 + 0.1382 + 0.2123 + 0.2865

­ 0.5655 ­ 0.5919 ­ 0.5573 ­ 0.5670 ­ 0.5675

­ 0.2403 ­ 0.2875 ­ 0.2677 ­ 0.3709 ­ 0.4305

­ 0.0229 ­ 0.0008 ­ 0.0150 + 0.0132 + 0.0269

+ 0.1072 + 0.1113 + 0.0949 + 0.0786 + 0.0803

antiphase Ti and O|| displacements. As the A atoms decrease in size, their contribution to the motion increases and becomes dominant, whereas the contribution from the Ti atoms decreases and the antiphase motion involves not O|| but O displacements. Thus, in crystals with small A atoms, the ferroelectric mode involves antiphase displacements of A atoms and oxygen cuboctahedra. Table 9 gives the values of the diagonal elements xx(0, 0) of the "on-site" force-constant matrix for the A and Ti atoms. These matrices are defined in terms of the restoring force acting on an atom displaced from the site, with the other atoms remaining fixed at their sites. In order to determine the on-site force constants from the force constants calculated using the ABINIT software for a crystal sublattice displaced as a unit, we average the force constants found for a regular grid of wave vectors [20, 22, 24]. Positive values of the on-site force constants indicate that the position of the atom at the site is stable, and negative values indicate that an off-center atom is formed. It follows from Table 9 that off-center A atoms arise in ATiO3 perovskites for Mg, Zn, Cd, and Ge. The Sn, Ca, and Pb atoms are fairly close to the boundary of stability against the transition to an off-center position. It will be recalled that the calculations in this paper are performed for the theoretical lattice parameter (corresponding to a minimum of the total crystal energy). Since the lattice parameters are systematically underestimated in LDA, which makes the ferroelectric instability weaker, many authors perform calculations using the experimental lattice parameter values. In order to estimate the influence of this systematic error, we carried out a computer simulation, which showed that a 1% increase in the lattice parameter (which is a typical error of LDA calculations) decreases xx(0, 0) in PbTiO3 by 0.006 Ha/Bohr2 for the A atom and by 0.016 Ha/Bohr2 for the Ti atom. As a result, the atoms

positioned near the boundary of stability against the transition to an off-center position remain actually at their sites. Perhaps, this is the case in lead titanate, as indicated by extended X-ray-absorption fine-structure (EXAFS) studies [34]. The main parameter determining the tendency for the A atom in ATiO3 crystals to transfer to an off-center position is the A atomic size. As can be seen from the dependences of the diagonal element xx(0, 0) of the force-constant matrix of the A atom on its ionic radius shown in Fig. 2, these dependences for Zn and Cd atoms, as well as for Ge, Sn, and Pb atoms, differ from those for the main series of Mg­Ca­Sr­Ba­Ra. The difference is likely due to the different electronic configurations of the filled atomic shells. This configuration is d10 for Zn and Cd; d10s2 for Ge, Sn, and Pb; and s2p6 for the atoms of the main series. The difference in the properties of these groups of atoms is also clearly manifested in the values of the effective charges of the A atoms (Table 6). Indeed, for the main series, the effec* tive charge Z A differs only slightly from the nominal cationic charge (which indicates that the A­O bond is * mainly ionic); for the other two groups, the charge Z A is significantly greater, which indicates that the bonding becomes more covalent in character [3]. These results suggest that off-center impurity atoms can exist in solid solutions of titanates with a perovskite structure. Since the average interatomic distance in such crystals is determined by the matrix, one might expect, according to the dependences discussed above, that the atoms for which xx(0, 0) has negative or small positive values will be in off-center positions. Therefore, it is likely that the ferroelectric phase transition induced by Ca, Cd, and Pb impurity atoms in SrTiO3 [12] is due to the fact that these atoms are in off-center positions. According to EXAFS data, Ba impurity atoms in SrTiO3 are not in off-center positions [35], but
PHYSICS OF THE SOLID STATE Vol. 51 No. 2 2009


AB INITIO CALCULATIONS OF PHONON SPECTRA IN ATiO3 PEROVSKITE CRYSTALS Ra 0.08 xx(0, 0), Ha/Bohr2 Sr Ca 0 Ge ­ 0.04 0.6 Mg Zn 0.8 1.0 RA, å 1.2 1.4 Sn Cd Pb Ba

371

0.04

Fig. 2. Dependence of the diagonal matrix element xx(0, 0) of the A atom on its ionic radius.

Pb atoms in SrTiO3 and BaTiO3 can be in such positions [36]. The results of this study differ somewhat from the calculations carried out by Kvyatkovskioe [11], according to which an adiabatic multiwell potential arises only for Mg and Zn atoms, whereas Cd atoms remain at their sites. The discrepancy is likely due to the fact that the calculations in [11] were performed for a relatively small clusters in which the correlation of atomic motions cannot be correctly taken into account (the correlation length can be as large as 20 å [5, 8]). 6. CONCLUSIONS Pseudopotentials have been constructed and employed to calculate the phonon spectra of ATiO3 perovskite crystals using the density functional theory. We have reproduced all known results concerning the structural instability of these crystals and predicted the properties of new, poorly studied systems. By analyzing the phonon spectra, the force-constant matrix, and the eigenvectors of unstable TO phonons, we revealed the regularities of the variation in the contributions from the chain instability and off-center atoms to the appearance of ferroelectricity in these crystals. The main factors determining the possible transfer of the A atoms to an off-center position are the geometric size of these atoms and the configuration of their outer electronic shell. REFERENCES
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Translated by Yu. Epifanov

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2009