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 wavevector, 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 further-neighbor k-point couplings determined by the wavevector 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 {\tt ABINIT} code and perform illustrative calculations of the field-dependent phonon frequencies for III-V semiconductors. 
Xinjie Wang and David Vanderbilt