Current-density implementation for calculating flexoelectric coefficients

The flexoelectric effect refers to polarization induced in an insulator when a strain gradient is applied. We have developed a first-principles methodology based on density-functional perturbation theory to calculate the elements of the bulk clamped-ion flexoelectric tensor. In order to determine the transverse and shear components directly from a unit-cell calculation, we calculate the current density induced by the adiabatic atomic displacements of a long-wavelength acoustic phonon. Previous implementations based on the charge-density response required supercells to capture these components. At the heart of our approach is the development of an expression for the current-density response to a generic long-wavelength phonon perturbation that is valid for the case of nonlocal pseudopotentials. We benchmark our methodology on simple systems of isolated noble gas atoms, and apply it to calculate the clamped-ion flexoelectric constants for a variety of technologically important cubic oxides.