First-Principle Perturbative Computation of Phonon Properties of Insulators in Finite Electric Fields

The methods of density-functional perturbation theory have been shown to provide a powerful tool for realistic calculations of lattice-vibrational, dielectric, elastic, and other response properties of crystals.\footnote{S. Baroni {\it et al.}, Rev. Mod. Phys. {\bf 73}, 515 (2001).} Recently, a total-energy method for insulators in nonzero electric fields was proposed.\footnote{I. Souza, J. \'I\~niguez, and D. Vanderbilt, Phys. Rev. Lett. {\bf 89}, 117602 (2002).} However, the perturbative computation of phonon properties under a dc bias field has not previously been addressed. Here, we start from a variational total-energy functional with a field coupling term that represents the effect of the electric field on the crystal. The linear response of the field-polarized Bloch functions is obtained by minimizing the second-order derivative of the total-energy functional. Due to the presence of the electric field, the field-polarized Bloch functions at each k-point in the Brillouin zone are weakly coupled to those at the neighboring k-points. We implement the method in the {\tt ABINIT} code and perform illustrative calculations of the phonon frequencies for III-V semicondutors.