First-principles theory of frozen-ion flexoelectricity

We demonstrate that the frozen-ion contribution to the flexoelectric coefficient is given solely in terms of the sum of third moments of the charge-density distortions induced by atomic displacements, even for ferroelectric or piezoelectric materials. We introduce several practical supercell-based methods for calculating these coefficients from first principles, and demonstrate them by computing the coefficients for C, Si, MgO, NaCl, SrTiO${}_3$, BaTiO${}_3$, and PbTiO${}_3$. Three important subtleties associated with pseudopotentials, the treatment of surfaces, and the calculation of transverse components are also discussed.

Figure 1

Supercell geometries. Large (red/dark gray) and small (green/light gray) dots are two species of atoms; open dots are atoms before being displaced as shown by arrows. Rectangles indicate supercells. (a) Bulk supercell for method A. (b) Slab supercell for method B; vertical lines indicate dipole correction layers in vacuum. (c) Bulk supercell for method C.