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Electromechanical behavior of BaTiO3 from first principles

Using an effective Hamiltonian parametrized from first principles, Monte Carlo simulations are performed in order to study the piezoelectric response of BaTiO3 in the ferroelectric tetragonal phase as a function of temperature. The effect of an electric field on the phase behavior is also illustrated by a simulation of the transformation of a rhombohedral domain into a tetragonal one under a strong field.

One of the earliest technological applications of ferroelectric materials was in the area of electromechanical transducers, the physical basis of which is the piezoelectric effect. The piezoelectric coefficients characterize the (linear) change in polarization in the presence of an external stress, or equivalently, a change in shape under the application of an external electric field. While a phenomenological, descriptive framework for such electromechanical effects has been available for a long time, a quantitative microscopic understanding of the response of individual materials is lacking, and further technical development has to proceed largely by trial and error. The importance of obtaining a deeper understanding has been highlighted by recent reports [1] of giant piezoelectric response in single crystals of relaxor perovskites of the form PMN-PT and PZN-PT. The structural complexity of these materials and the variety of phenomena that might be involved make it rather difficult to offer a clear understanding of their observed properties.
It is in this context that first-principles calculations can help. They can be used to selectively "turn off" features of the system and study its response in conditions that are very difficult or impossible to realize in the laboratory. Besides, they can provide a microscopic view of the materials properties. So far, the primary uses of first-principles calculations relevant to this area have been calculations of piezoelectric coefficients for simple materials at zero temperature [2]. In this work, as a step towards the theoretical treatment of the electromechanical response of complex systems, we present the first calculations of the piezoelectric response as a function of temperature for BaTiO 3 , an important member of the perovskite family of ferroelectric compounds. We also illustrate the influence of electric fields on the phase behavior of BaTiO 3 . We base our approach on a scheme which has proven very successful [4] in the description of the rich phase diagrams of perovskite oxides: an effective Hamiltonian which contains the physically relevant degrees of freedom of the structure (notably the "soft mode") is constructed on the basis of high-quality first-principles calculations, and the statistical mechanics of the system is then studied by Monte Carlo simulation. The usefulness of the method is not restricted to the calculation of piezoelectric coefficients and the study of generalized phase diagrams. Indeed, the elastic and dielectric response can be similarly computed, and it can form the basis for an analysis of non-linear effects [5].
The appropriate thermodynamic identity for a crystal in the presence of a (possibly anisotropic) stress and an electric field is [6] Here U is the internal energy of the crystal (per unit volume), T and S are the temperature and entropy, σ ν and η ν are the components of the stress and strain tensors in the Voigt notation [7], E i is the ith component of the macroscopic electric field, and D i is the corresponding component of the electrical displacement vector (in SI units, D = ǫ 0 E + P).
We parametrize the energy U of the system by means of an effective Hamiltonian H eff which represents a Taylor expansion of the energy surface around the highsymmetry cubic perovskite structure. H eff is written in terms of the dynamical variables which are relevant to the low-energy distortions: the amplitudes {u} of the local modes (three degrees of freedom per unit cell) which represent the "soft" transverse optical phonon and are directly related to the polarization of the crystal [P = (Z * /V ) u, where Z * is the mode effective charge and V is the cell volume]; a set {v} of displacement variables representing the acoustic modes; and the six components of the homogeneous strain η. The parameters of the energy expansion, including the on-site local-mode selfenergy, the interaction between local modes (both shortrange and dipole-dipole), the elastic energy, and the local mode-elastic coupling, are computed using highly accurate first-principles LDA calculations with Vanderbilt ultrasoft pseudopotentials [3]. More details about the construction of the effective Hamiltonian are given in Ref. [4], where the method is shown to provide a good account of the phase transition sequence in BaTiO 3 within the limitations of an approach based on low-energy distortions. The extension of the standard Metropolis Monte Carlo algorithm to include the effects of stress and electric field involves replacing the Boltzmann probability factor exp(−βU j ) by exp[−β(U j − σ ν η j ν − E i P j i )] in the acceptance criterion for state j. For a given temperature, stress, and field, the strain η and the mode variables are allowed to fluctuate, their average values determining the shape and net polarization of the system.
As an important application of the method, we compute the piezoelectric response of the tetragonal (ferroelectric) phase of BaTiO 3 (point group 4mm), which is stable from approximately 278K to 403K and exhibits a spontaneous polarization that we take to be along the z axis. The relevant coefficients are where ∆X = X − X and the averages are computed using the extended Boltzmann factor defined above [8].
These equations suggest three different ways to calculate the response: direct calculations of the average strain as a function of applied field, or the average polarization as a function of (anisotropic) stress, and computation of the statistical correlation between polarization and strain. The latter is conceptually the simplest, although relatively long simulations (on the order of 100000 Monte Carlo sweeps (MCS) [9]) are needed to obtain good statistics. Of the direct approaches, the field-strain calculations are the most efficient as only one series of calculations for varying E 3 is needed to compute the most common coefficients d 31 and d 33 (representing respectively the transverse contraction and the longitudinal expansion under the application of a field parallel to the ferroelectric axis). An effective Hamiltonian based on a finite Taylor expansion of the energy should not be expected to reproduce perfectly the behavior of the material at relatively high temperatures. In particular (see Ref. [4]) the theoretical transition temperatures are progressively shifted downwards with respect to the true ones [11]. In order to provide a better comparison of our results for d 31 and d 33 to experiment (Fig. 1), we have therefore rescaled linearly the theoretical temperatures so that the end points of the range of stability of the tetragonal phase coincide. The agreement with the available experimental data is very good, and the general trend of the temperature dependence (upward tendency in d 33 , downward in d 31 ) corresponds to the pseudo-divergent behavior observed experimentally near the tetragonal-to-cubic transition [12]. In the thermodynamic limit, the correlation and field-strain approaches should be completely equivalent. Since we use finite runs and finite simulation boxes (12×12×12 or 1728 unit cells for the correlation analysis and 10×10×10 or 1000 unit cells for the field-strain calculations), the results are similar but not identical.
Our method can also be applied to the study of the stability of the different phases as a function of the external field and stress. As an illustration of the influence of electric fields on the phase diagram, we performed a simulation of the effect of a strong field on the strain and electrical polarization of a BaTiO 3 sample in the rhombohedral (R) phase. This phase, with point group 3m, is stable at temperatures below 183K, and exhibits a spontaneous polarization along one of the original 111 cubic directions, and a corresponding strain deformation with respect to the cubic phase satisfying η 1 = η 2 = η 3 and η 4 = η 5 = η 6 . Choosing a simulation temperature of 100K, we applied an electric field along the cubic z axis, progressively increasing the strength of the field up to a maximum value of 5 × 10 3 kV/cm. The evolution of the strain and polarization as a function of the field is shown in Figs. 2 and 3, respectively. The rhombohedral symmetry is immediately broken by the field, as can be seen by the η 1 = η 2 = η 3 splitting (and the analogous one among the polarization components). In the approximate range 2 − 2.5 × 10 3 kV/cm, there is a noticeable split in η 1 and η 2 , all the off-diagonal components of the strain save η 4 (2η 23 in the standard tensor notation) fall to zero, and P 1 falls to nearly zero. Thus, this range corresponds to an orthorhombic phase with the polarization in the yz plane (different simulations "pick" at random between the yz and the xz planes). For larger fields, the equality between η 1 and η 2 is restored, all the off- diagonal strain components fall to zero, and P 2 follows P 1 in its earlier drop. The resulting phase is tetragonal, with P along the [001] direction. The behavior emerging from our simulation parallels that seen experimentally [1] in the PMN-PT and PZN-PT giant-response materials: individual domains are in a rhombohedral structure, and as an electric field is applied and progressively increased along z, the polarization rotates from a [111] direction to a [001] direction.
In conclusion, we have shown how the electromechanical response of a system can be computed from first principles using an effective Hamiltonian suitably augmented by terms representing the influence of applied (anisotropic) stresses and electric fields. As an application, we have presented the first calculations of the piezoelectric response of ferroelectric tetragonal BaTiO 3 as a function of temperature. We have also performed simulations illustrating the effect of electric fields on the phase behavior of this material.
This work was supported in part by the ONR Grant N00014-97-1-0048 and by the UPV research grant 060.310-EA149/95. We thank J.M. Perez-Mato and Karin Rabe for useful comments. *