Theory of structural response to macroscopic electric fields in ferroelectric systems

We have developed and implemented a formalism for computing the structural response of a periodic insulating system to a homogeneous static electric field within density-functional perturbation theory (DFPT). We consider the thermodynamic potentials E(R,η,E) and F(R,η,P), whose minimization with respect to the internal structural parameters R and unit cell strain η yields the equilibrium structure at fixed electric field E and polarization P, respectively. First-order expansion of E(R,η,E) in E leads to a useful approximation in which R(P) and η(P) can be obtained by simply minimizing the zero-field internal energy with respect to structural coordinates subject to the constraint of a fixed spontaneous polarization P. To facilitate this minimization, we formulate a modified DFPT scheme such that the computed derivatives of the polarization are consistent with the discretized form of the Berry-phase expression. We then describe the application of this approach to several problems associated with bulk and short-period superlattice structures of ferroelectric materials such as BaTiO₃ and PbTiO₃. These include the effects of compositionally broken inversion symmetry, the equilibrium structure for high values of polarization, field-induced structural phase transitions, and the lattice contributions to the linear and the nonlinear dielectric constants.