Theory of structural response to macroscopic electric fields in ferroelectric systems

Abstract

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 $E(\mathbf{R},\ensuremath{\eta},\mathcal{E})$ and $F(\mathbf{R},\ensuremath{\eta},\mathbf{P}),$ whose minimization with respect to the internal structural parameters $\mathbf{R}$ and unit cell strain $\ensuremath{\eta}$ yields the equilibrium structure at fixed electric field $\mathcal{E}$ and polarization $\mathbf{P},$ respectively. First-order expansion of $E(\mathbf{R},\ensuremath{\eta},\mathcal{E})$ in $\mathcal{E}$ leads to a useful approximation in which $\mathbf{R}(\mathbf{P})$ and $\ensuremath{\eta}(\mathbf{P})$ 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 $\mathbf{P}.$ 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 ${\mathrm{BaTiO}}_{3}$ and ${\mathrm{PbTiO}}_{3}.$ 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.