First-principles perturbative computation of phonon properties of insulators in finite electric fields

We present a perturbative method for calculating phonon properties of an insulator in the presence of a finite electric field. The starting point is a variational total-energy functional with a field-coupling term that represents the effect of the electric field. This total-energy functional is expanded in small atomic displacements within the framework of density-functional perturbation theory. The linear response of field-polarized Bloch functions to atomic displacements is obtained by minimizing the second-order derivatives of the total-energy functional. In the general case of nonzero phonon wave vector, there is a subtle interplay between the couplings between neighboring k points introduced by the presence of the electric field in the reference state and farther-neighbor k point couplings determined by the wave vector of the phonon perturbation. As a result, terms arise in the perturbation expansion that take the form of four-sided loops in k space. We implement the method in the ABINIT code and perform illustrative calculations of the field-dependent phonon frequencies for III-V semiconductors.