First-principles theory and calculation of flexoelectricity

We develop a general and unified first-principles theory of piezoelectric and flexoelectric tensor, formulated in such a way that the tensor elements can be computed directly in the context of density-functional calculations, including electronic and lattice contributions. We introduce practical supercell-based methods for calculating the flexoelectric coefficients from first principles, and demonstrate them by computing the coefficients for a variety of cubic insulating materials including C, Si, MgO, NaCl, CsCl, BaZrO${}_3$, BaTiO${}_3$, PbTiO${}_3$, and SrTiO${}_3$.

Figure 1

Original (a) and 45${}^{\circ }$-rotated (b) supercells used for calculations on $AB$O${}_3$ perovskites. Arrows (red) denote the displacement of $A$ atoms consistent with fixed-$D$ boundary conditions. Large ball is $A$ atom, medium is $B$ atom, and small is O atom.

Figure 2

Change of charge-density distribution (a) and force distribution (b) in SrTiO${}_3$ supercell (original frame) at fixed $D$.

Figure 3

$AB$O${}_3$ perovskite atomic geometry in (a), (b) original Cartesian frame, and (c), (d) 45${}^{\circ }$-rotated frame, as appropriate to the two supercells of Fig. 1, respectively. (a), (c) slice at $z=0$; filled squares (red) are $A$ and filled circles (green) are O${}_3$ atoms. (b), (d) slice at $z=c/2$; filled diamonds (blue) are $B$, filled circles (green) are O${}_1$, and open circles (green) are O${}_2$ atoms.