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.

Figure 1

Planar average of the local [Eq. (70), green dot-dashed curve], nonlocal [Eq. (71), blue dashed], and total [Eq. (69), red solid] first-order charge for noble gas atoms displaced in the $x$ direction by a $\mathbf{q}=0$ phonon. The black circles correspond to the first-order charge calculated using a conventional, static, DFPT calculation. The box size is $16\times 16\times 16$ Bohr, but zoomed in to only show $\pm 5$ Bohr.

Figure 2

The longitudinal (red squares) and shear (blue diamonds) FxE coefficients, as well as the diamagnetic susceptibility correction (black circles) and $Q_{\text{NG}}/2\mathrm{\Omega }$, for (a) He, (b) Ne, (c) Ar, and (d) Kr atoms in cells with various lattice constants. All quantities are multiplied by the cell volume $\mathrm{\Omega }$.

Figure 3

Induced polarization vs $\mathbf{q}=(q_x,0,0)$ for cubic ${\mathrm{SrTiO}}_3$. The black (red) points correspond to the $x\left(y\right)$ component of the polarization for atomic displacements of the atoms in the $x\left(y\right)$ direction. Dashed curves are quadratic fits. Units are with respect to the calculated ${\mathrm{SrTiO}}_3$ lattice constant $a=7.435$ Bohr.