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The ferroelectric and cubic phases in BaTiO_{3} ferroelectrics are also antiferroelectric

Contributed by William A. Goddard III, August 1, 2006
Abstract
Using quantum mechanics (QM, Density Functional Theory) we show that all four phases of barium titanate (BaTiO_{3}) have local Ti distortions toward 〈111〉 (an octahedral face). The stable rhombohedral phase has all distortions in phase (ferroelectric, FE), whereas higher temperature phases have antiferroelectric coupling (AFE) in one, two, or three dimensions (orthorhombic, tetragonal, cubic). This FE–AFE model from QM explains such puzzling aspects of these systems as the allowed Raman excitation observed for the cubic phase, the distortions toward 〈111〉 observed in the cubic phase using xray fine structure, the small transition entropies, the heavily damped soft phonon modes, and the strong diffuse xray scattering in all but the rhombohedral phase. In addition, we expect to see additional weak Bragg peaks at the face centers of the reciprocal lattice for the cubic phase. Similar FE–AFE descriptions are expected to occur for other FE materials. Accounting for this FE–AFE nature of these phases is expected to be important in accurately simulating the domain wall structures, energetics, and dynamics, which in turn may lead to the design of improved materials.
Ferroelectric (FE) materials have broad applications in transducers, actuators, capacitors, and memories. Particularly well studied is BaTiO_{3}, which has rhombohedral (R), orthorhombic (O), tetragonal (T), and cubic (C) phases. However, the microscopic nature of the phases and transitions in BaTiO_{3} remains uncertain. A popular model in history is the displacive model (1, 2), in which the equilibrium position of each Ti atom is in the middle of the oxygen octahedron for C, but displaced microscopically in the 〈111〉, 〈011〉, or 〈001〉 macroscopic polarization directions for the R, O, and T ferroelectric phases, respectively. However, this displacive model

Predicts that firstorder Raman excitation should vanish in the cubic phase due to the microscopic inversion symmetry, contradicting experiments (3).

Contradicts the xray fine structure (XAFS) experiments (4) that show that the Ti atoms are displaced along various of the eight possible 〈111〉 directions in all phases.

Does not explain the heavily damped but nonzero frequency for the soft modes at the C to T transition observed by infrared reflectivity (IR) experiments (5).

Does not explain the strong diffuse xray scattering observed in all but the rhombohedral phase.
To explain these failures of the displacive model, a spontaneous symmetry breaking has been hypothesized to occur in these systems (6). Thus Berserker (7), Comes et al. (8) and Chaves et al. (9) introduced the order–disorder eightsite model (8OD), in which all Ti atoms are microscopically located in one of eight potential minima along the 〈111〉 directions for all crystal phases. As in the displacive model, 8OD assumes that the low temperature R phase has all Ti atoms distorted in the same direction. As temperature increases, 8OD assumes disorder in the polarization along one or more crystal directions, leading to greatly increased configurational entropy. However, 8OD

Fails to predict the heavily damped low frequency phonon modes (soft modes) observed near the phase transitions by neutron scattering (10, 11), by IR (5) and by hyperRaman scattering (12).

Would lead to the entropy changes of ≈Rln2 = 5.76 J/mol for the phase transitions, far larger than the observed entropy changes (≤0.52 J/mol) for each transition (8, 9). This observation forced the proponents of this model to postulate a 50 to 100Å correlation length to reduce the entropy changes down to observed values.
Girshberg and Yacoby (13) proposed a model combining the displacive and order–disorder models. They developed this to explain mode softening near the FE phase transitions, and it was applied recently to BaTiO_{3} (14) to explain the over damping (12) of the soft mode. The Girshberg–Yacoby theory showed that the existence of offcenter displacements and their interaction with the soft modes could play a crucial role in explaining the properties of ferroelectric phase transitions.
Using the fundamental unit cell with one molecular unit, firstprinciples quantum mechanics (QM) calculations (15, 16) using the local density approximation to density functional theory (DFT) showed clearly that the Ti atoms are not in the central position in the cubic phase. These calculations also showed large volume dependence of the softmode potential surface and revealed the expected increase in total energies for the R, O, T, and C phases of BaTiO_{3}. Further phonon analysis (17, 18) on BaTiO_{3} revealed twodimensional character in the Brillouin zone, which corresponds to chains oriented along 〈100〉 directions of displaced Ti atoms. Such character had been found in KNbO_{3} (19), a material with the same phase diagram as BaTiO_{3}. In addition, the BaTiO_{3} phonon analysis showed that X point instability is slightly less unstable than Gamma point in BaTiO_{3}, implying a ferroelectric relaxation could lead to a lowerenergy structure than an antiferroelectric relaxation.
On the other hand, the local structures and dynamics have also been studied in large supercells. Monte Carlo simulations (20) using an effective Hamiltonian based on the local density approximation DFT calculations found the correct sequence of transitions, leading to the suggestion that the phases have character intermediate between displacive and order–disorder. The crossover from a soft mode to an order–disorder dynamics was also found by a molecular dynamics (MD) study (21) using a nonlinear oxygen polarizability model. MD simulations (22, 23) using an effective Hamiltonian revealed the prevalence of local polar distortions with shortrange chainlike correlations, present even in the paraelectric phase far above Tc. These correlations were also discovered in a MD study (24) using the nonlinear oxygen polarizability model.
Despite all these progresses, the groundstate structures of the hightemperature phases were not identified in the previous studies. Knowing the groundstate structures is necessary to obtain an accurate description of domains, interface, and vacancies.
Results and Discussion
We report here QM calculations using the PW91 (25) generalized gradient approximation flavor of DFT. In contrast to previous QM studies, we use unit cells extended to allow the Ti atoms to displace along various 〈111〉 directions while retaining the proper macroscopic symmetries. This method leads to the normal R3m space group for the R phase, Pmn21 for the O phase (this was Amm2 in the displacive model), I4 cm for the T phase (this was P4 mm in the displacive model), and I43m for the C phase (this was Pm3m in the displacive model). The primitive unit cells have 1, 2, 4, and 4 molecular units for the R, O, T, and C phases (Fig. 1), respectively.
We find that for C, the optimum position of each Ti is displaced by 0.14 Å along the 〈111〉 direction, leading to an energy 2.981 kJ/mol lower than if the Ti had been kept in the center of the octahedron (as in previous calculations using a single molecular unit per cell). [Although the average accuracy of PW91/DFT for cohesive energies is only ≈17 kcal/mol (26, 27), we expect much greater accuracy along a reaction path since the distances between the various atoms change little from point to point.] Thus, the extended unit cell for the C phase has antiferroelectric (AFE) couplings in the x, y, and z directions (Fig. 1). The size of the unit cell changes little (increasing by 2% from the Pm3m cell to the I43m cell).
For T, we also find AFE coupling in the x and y directions, with displacements of 0.13 Å and an energy lowering of 0.373 kJ/mol with respect to the P4 mm cell. In addition, the QMderived FE–AFE model (space group I4 cm ) leads to c/a = 1.0104 in excellent agreement with experiment (1.010), whereas assuming the displacive model (space group P4 mm ) leads to 1.0419. Thus, the FE–AFE description resolves the difficulty in correctly predicting the c/a ratio, which long plagued the QM calculations (28).
For the O phase, the QM FE–AFE model ( Pmn21 ) leads to a displacement of 0.09 Å and an energy lowering of 0.076 kJ/mol with respect to the displacive phase ( Amm2 ).
Thus, the QM calculations lead directly to an ordered FE–AFE microscopic structure as the ground state of the O, T, and C phases. The relative energies of these phases are 0.000 (R), 0.063 (O), 0.561(T), and 1.020(C) kJ/mol per molecular unit. In the FE–AFE phase structures, each Ti atom is displaced along one of the 〈111〉 directions in such a way that inside each (TiO)_{n} chain, all Ti atoms are displaced head to tail, whereas adjacent chains are either FE coupled or AFE coupled. In each case FE coupling introduces net polarization, whereas AFE coupling leads to zero macroscopic polarization. In contrast to the displacive structures for O, T, and C, all phonon frequencies are real so that the FE–AFE phase structures are all dynamically stable. We show below that this FE–AFE model derived directly from QM explains all previous inconsistencies with experiment in the properties of BaTiO_{3}.
The TiO lattice mode near the zone center is observed to be both IR and Raman active, which is not possible for the displacive model due to the center of inversion. The FE–AFE cubic phase does not have inversion symmetry, allowing both Raman and IR transitions (3), as observed experimentally.
A strong test of the FE–AFE model is for the T phase, where XAFS (4) finds that each titanium atom displaces from its closest 〈111〉 direction by 11.7 ± 1.1° toward the c axis, whereas the displacive model would lead to 54.7°. The QM for the FE–AFE model predicts that the displacement angle is 10.8°.
The AFE coupling leads to a much softer phonon structure than the FE coupling. Thus the Γ point TiO vibration is 239 cm^{−1} for R (three degenerate values), and 60 cm^{−1} for C (three degenerate values). For the O and T phases, there are one and two lowfrequency modes, respectively, corresponding to the AFE couplings. These 0 K results agree well with the 400 K experiments: three modes at 63 cm^{−1} for C and two at 55 cm^{−1} for T. In contrast, the displacive model leads to unstable modes (negative eigenvalues in the Hessian) for the C phase (17) and also for T and O. Thus, in the QMderived FE–AFE model at each phase transition, a highenergy polar mode (>200 cm^{−1}) parallel to the FE coupled chains transforms to a lowenergy polar mode (soft mode <100 cm^{−1}) parallel to the AFE coupled chains. Such heavily damped lowenergy soft modes have long been observed (5, 10–12).
Our QM calculations of the AFE C phase (using the frozen phonon model) predict a giant volume dependence of the soft mode frequency near the transition. Thus we calculate the soft mode Gruneisen parameter to be showing the stabilization soft mode from volume expansion. This finding is in good agreement with the experimental value of −43 derived from infrared reflectivity measurements (5). This giant volume dependence of the soft mode indicates a strong coupling between the soft mode and the longitudinal acoustic mode at Γ point, leading to a large damping constant near the transition (as observed experimentally in ref. 5, ≈100 cm^{−1}).
These low frequencies for the AFE directions lead to low zero point quantum corrections and large contributions to the entropy changes upon transition. Thus, using the calculated frozen phonon spectrum to evaluate the entropy, we calculate 0.50 J/mol for T to C in excellent agreement with the value of 0.52 from experiment; ref. 29). It is this increase in entropy for the FE to AFE transitions that leads to phase transitions from R to O to T to C with increasing temperature despite the increase in enthalpy. Our DFT calculations lead to lattice parameter of 4.033 Å at 0 K for the C phase, which is 0.5% larger than the experimental value (30) of 4.012 Å at 400 K. This overestimated lattice parameter leads to a higher transition temperature and transition entropies. In addition, we ignored Ti disorder in the polarization directions. This simplification would overestimate the free energy of the C phase more than the free energy of the T phase, leading to an higher transition temperature for T to C (Fig. 2).
Another puzzling property of BaTiO_{3} has been the diffuse xray diffraction (6) observed in the O, T, and C phases (but not R). We find that the soft AFE modes of the QM FE–AFE model leads to xray thermal scattering factors (Fig. 3) that explain the observed diffuse lines in experiment (6). Girshberg and Yacoby (31) suggested that the ferroelectric phase transition results from strong intraband electron–phonon interaction, but did not provide underlying atomistic structures to explain the postulated interactions. Chapman et al. (32) found that the TA mode is softer than the TO mode for all k and the TO mode contribution to diffuse scattering sheets is negligible for BaTiO_{3}. They further suggested that disordering of the local displacements leads to the diffuse scattering.
Our study confirms that the TO mode contribution to sheets in BaTiO3 is negligible. In addition, we found that the diffuse pattern is caused by the anisotropic soft TA mode. In the following, we show that the TA mode is always softer than the TO mode due to the symmetry reduction in BaTiO_{3} The TA mode in the cubic phase ( I43m in the FE–AFE model) is closely related to the soft TO mode in the simple cubic phase ( Pm3m ). To show this relation, we constructed a series of intermediate phase structures between Pm3m and I43m . Each transitional phase structure is given by where X is the lattice parameter or atom fractional coordinates, and L ranges from 0 (leading to Pm3m ) to 1 (leading to I43m ). Then, we calculated the phonon dispersion of TO and TA modes as a function of L. Fig. 4 shows the phonon dispersion variations along ΓX, Γ–M, and Γ–R during the symmetry reduction. Certain TO modes in Pm3m for BaTiO3 transform to soft TA modes in I43m . Along Γ–X, the two TO modes in Pm3m transform into two TA modes in I43m (Fig. 4 A). Along Γ–M, one soft TO mode in Pm3m transforms into one TA in I43m and the hard TO mode with higher frequency remains a TO mode of I43m (Fig. 4 B). Along Γ–G, the two TO modes in Pm3m are not soft and they remain TO modes in I43m (Fig. 4 C). After these mode transformations, TA modes are always softer than TO modes for all k points.
Furthermore, we project the eigenvectors of the TA and TO modes in I43m to the TA and TO modes in Pm3m . Thus, we define where e is the phonon eigenvector, Q is the wave vector, and I, j = TO or TA. Fig. 5 below shows clearly that the TA mode in I43m is related to the TO mode of Pm3m ; the major vibration of TA mode in I43m is the soft TiO motion. By this TOtoTA transformation, the TA mode picks up soft TiO vibrations, leading to lower energy than the TO mode. Because the TO mode in Pm3m is quite anisotropic (18), the TA mode in I43m is very anisotropic due to this TOtoTA transformation. Thus, it is the anisotropic soft TA mode that leads to diffuse scattering in BaTO_{3}. This finding agrees with the synchrotron radiation experiment of Takesue et al. (33).
The I43m symmetries give the diffraction peaks at (h/2, k/2, l/2) if h + k + l = even. This observation leads to additional Bragg peaks at the face centers of the reciprocal lattice of Pm3m. However, because the atomic distortions from Pm3m to I43m in BaTiO_{3} are small (≈0.14 Å), the intensities of these additional points are very weak. The maximum intensity of the additional points in the (001) zone is only 0.025% of the (400) intensity (See Table 1, which is published as supporting information on the PNAS web site). These weak Bragg points are further obscured by the background of thermal diffuse scattering. Nevertheless, it should be possible to eventually refine the experiments to test the QM predictions.
The electric field gradient analysis of the NMR experiments (34, 35) for T leads to distortions along 〈100〉, in disagreement with the analysis of XAFS experiments (4), which indicate that all Ti distort nearly along 〈111〉 directions. Stern (36) has explained that this difference arises from the time averaging implicit in the NMR, so that the correct instantaneous structure is in the 〈111〉 direction found in XAFS (4) and now in QM. The AFE coupling has softer TO modes than the FE coupling, leading to diffuse xray scattering and phase transitions. Each AFE coupling is doubly degenerate, ↑↓↑↓ … ↑↓ and ↓↑↓↑ … ↓↑. The transition between the two states involves a soft TO mode (<200 cm^{−1}) at the zone boundary. The transition time between the two states is longer than that of XAFS but shorter than that for NMR, leading to 〈111〉 detection from XAFS and 〈001〉 in the tetragonal phase from NMR. In contrast, the transition between two FE coupling states, ↑↑↑↑ … ↑↑ and ↓↓↓↓ … ↓↓, requires excitation of a LO mode (≈700 cm^{−1}), which is much more difficult. This giant LO–TO splitting results from the large Born effective charge of Ti as found in the DFT calculations (20).
The QM FE–AFE model leads to many properties qualitatively similar to the 8OD model. The main difference concerns whether adjacent TiO chains have AFE coupling or are completely disordered. Our FE–AFE model predicts perfect correlation at the lowest temperatures with a gradual increase in disorder with temperature. In the 8OD model, the entropy increase at each transition would be huge, 5.76 J/mol, in complete disagreement with experiments (29) (0.17–0.52 J/mol). Consequently, the proponents of 8OD make an assumption that there is some correlation length in each chain that greatly reduces the entropy increase. Fitting the 8OD model to experiment requires a very long correlation length of 50–100 Å (8). The FE–AFE leads directly to modest entropy increases at each transition due to the soft phonon structure, without the need for a correlation length assumption.
Our QM calculations show that, for the cubic space group, the I43m structure is 2.981 kJ/mol more stable than Pm3m . This finding shows the importance of the shortrange AFE ordering. However, in such calculations, we impose the periodic symmetry of the doubled unit cell; this leaves open the question of whether the AFE ordering is longrange. In the lowtemperature rhombohedral ferroelectric phase, it is the strong longrange interactions that keep the local distortions in an ordered pattern. Even above the phase transition temperature, these longrange interactions are expected to be sufficiently strong to retain this ordered pattern characteristic of the lower symmetry group. The difference between longrange interactions and shortrange interactions can be illustrated by comparing the frequencies of the LO and TO branches of the same modes. In the Pm3m phase of BaTiO_{3}, the LO and TO frequencies of the TiO interactions were calculated in ref. 37 to be −178 and 738 cm^{−1}. This finding compares to the values of 60 and 751 cm^{−1} that we calculate by using the reduced symmetry, I43m .
The Girshberg–Yacoby theory (13) emphasizes the interaction between local displacements and phonons, which is consistent with our model: AFE coupling leads to softer phonon structures than FE. Our FE–AFE model provides a fundamental foundation for describing these interactions and can be interpreted within the framework of the Girshberg–Yacoby theory (13).
In summary, our QM studies lead directly to the FE–AFE model for all four phases of BaTiO_{3} that explains all of the puzzling properties in terms of a competition between the lowenergy FE coupling and the highentropy AFE coupling. These FE–AFE phase structures are the origin of the apparent coexistence of both displacive and order–disorder properties in BaTiO_{3}. Similar behavior has been observed for other perovskite ferroelectrics, such as KNbO_{3}, which we expect will also lead to the FE–AFE model. This new understanding of the fundamental nature of these FE materials should be essential in simulating the dynamical properties of the domain structures important in the macroscopic description of these systems and might be important in optimization the properties of such systems.
Materials and Methods
Our QM calculations (performed with CASTEP 3.8; ref. 38) used the PW91 (25) flavor of DFT, including a generalized gradient approximation for the exchangecorrelation interactions. We also used the Vanderbilttype ultrasoft pseudopotentials (39) for electron–ion interaction and plane wave basis sets (kinetic energy cutoff of 700 eV). The Brillouin zones for the electronic states were sampled by using k points translated from the original 4 × 4 × 4 Monkhorst–Pack (40) mesh for the simple unit cell.
Direct firstprinciples phonon structure calculations of the FE–AFE cubic phase are expensive and suffer numerical errors in calculating the Hessian. Consequently, we developed the PQEq force field (FF) optimized to reproduce the phase structures and energies from the firstprinciple calculations and then used this FF to calculate the phonon structures. PQEq is an extended shell model (41) using Gaussian shaped shell and core charges (rather than point charges) and including core–core shortrange interactions. The dynamic matrices were calculated including the longwave correction (42, 43) at the Gamma point. For consistency, the phonon Brillouin zone was sampled with Qpoints translated from the original 16 × 16 × 16 mesh for the simple unit cell.
The free energy and entropy of each phase were calculated from the phonon density of states (44) using the QM optimized FE–AFE structures (0 K).
The partial differential cross sections were calculated at the level of singlephonon scattering (45) from the phonon structures. The C, T, and O phases were compressed to match their experimental soft mode frequencies (5) at their calculation temperatures. The R phase was compressed to it experimental volume (30). The scattering factors of Ba and O atoms were ignored in these calculations to more clearly identify the physical origin of the diffuse scattering: softer TO modes for AFE coupling than for FE coupling.
Acknowledgments
This work was initiated with funding by the Army Research Office (ARO, MURIDAAD190110517) and by the National Science Foundation (MRSECCSEMDMR0080065) and completed with funding from Defense Advanced Research Planning Agency (Predicting Real Optimized Materials) through the Office of Naval Research (N000140210839). The Molecular Simulation Center computational facilities used in these calculations were provided by grants from Defense University Research Instrumentation Program (DURIP)–ARO, DURIP–Office of Naval Research, and National Science Foundation Major Research Instrumentation.
Footnotes
 ^{†}To whom correspondence should be addressed. Email: wag{at}wag.caltech.edu

Author contributions: W.A.G. designed research; Q.Z. performed research; Q.Z. and T.C. analyzed data; and Q.Z. wrote the paper.

↵*Present address: 241 Jack East Brown Engineering Building, 3122, Texas A&M University, College Station, TX 778433122.

The authors declare no conflict of interest.
 Abbreviations:
 FE,
 ferroelectric;
 R,
 rhombohedral;
 O,
 orthorhombic;
 T,
 tetragonal;
 C,
 cubic;
 XAFS,
 xray fine structure;
 IR,
 infrared reflectivity;
 8OD,
 order–disorder eightsite model;
 QM,
 quantum mechanics;
 DFT,
 density functional theory;
 AFE,
 antiferroelectric
 © 2006 by The National Academy of Sciences of the USA
References
 ↵

↵
 Cochran W
 ↵

↵
 Ravel B ,
 Stern EA ,
 Vedrinskii RI ,
 Kraizman V
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

↵
 Harada J ,
 Axe JD ,
 Shirane
 ↵
 ↵
 ↵
 ↵
 ↵

↵
 Ghosez P ,
 Gonze X ,
 Michenaud JP
 ↵
 ↵
 ↵

↵
 Sepliarsky M ,
 Migoni RL ,
 Stachiotti MG

↵
 Krakauer H ,
 Yu RC ,
 Wang CZ ,
 Lasota C
 ↵

↵
 Tinte S ,
 Stachiotti MG ,
 Sepliarsky M ,
 Migoni RL ,
 Rodriguez CO
 ↵

↵
 Xu X ,
 Goddard WA
 ↵

↵
 Uludogan M ,
 Cagin T ,
 Goddard WA

↵
 Shirane G ,
 Takeda A

↵
 Kay HF ,
 Vousden P
 ↵

↵
 Chapman BD ,
 Stern EA ,
 Han SW ,
 Cross JO ,
 Seidler GT ,
 Gavrilyatchenko V ,
 Vedrinskii RV ,
 Kraizman VL
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵

↵
 Born M ,
 Huang K
 Mott NF ,
 Bullard EC
 ↵

↵
 Dove MT
 Putnis A ,
 Liebermann RC

↵
 Lovesey SW
 Lovesey SW ,
 Springer T