Metric wave approach to flexoelectricity within density functional perturbation theory

Within the framework of density functional perturbation theory, we implement and test a “metric wave” response-function approach. It consists in the reformulation of an acoustic phonon perturbation in the curvilinear frame that is comoving with the atoms. This means that all the perturbation effects are encoded in the first-order variation of the real-space metric, while the atomic positions remain fixed. This approach can be regarded as the generalization of the uniform strain perturbation of Hamann et al. [D. R. Hamann, X. Wu, K. M. Rabe, and D. Vanderbilt, Phys. Rev. B 71, 035117 (2005)] to the case of inhomogeneous deformations, and greatly facilitates the calculation of advanced electromechanical couplings such as the flexoelectric tensor. We demonstrate the accuracy of our approach with extensive tests on model systems and on bulk crystals of Si and ${\mathrm{SrTiO}}_3$.

Figure 1

Illustration of the coordinate transformation to the comoving frame. (a) Unperturbed crystal lattice; black circles represent the atomic sites, horizontal and vertical lines represent the coordinate system. (b) Transverse acoustic phonon in the laboratory frame. (c) The same phonon in the curvilinear frame; note that the atoms do not move in this coordinate system: the mechanical deformation is described via the metric.

Figure 2

Representation of the Fourier space in 2D. The small black crosses are the $\mathbf{G}$ vectors; the black dotted circle identifies the cutoff sphere centered on $\mathrm{\Gamma }$; the blue continuous circle identifies nonzero Fourier coefficients of $u_{\mathbf{k}}^{\left(0\right)}$; and the dashed green circle identifies the Fourier coefficients of ${\tilde{u}}_{\mathbf{k}}^{\left(0\right)}$.

Figure 3

Plot of $d(n_{\mathbf{q}}^{u^{\beta }}\left(\mathbf{r}\right),n_{\mathbf{q}}^{\left(\beta \right)}\left(\mathbf{r}\right)+\mathrm{\Delta }{n}_{\mathbf{q}}^{\beta }\left(\mathbf{r}\right))$ [cf. Eq. (54)] as a function of wave vector $\mathbf{q}$ (reduced coordinates), for different cutoffs. All the results refer to longitudinal perturbations. From top to bottom: He, Ne, and Kr.

Figure 4

Cell-integrated difference $d(-i{n}_{\alpha }^{1,\beta }\left(\mathbf{r}\right),n_{\alpha \beta }^{\mathrm{strain}}\left(\mathbf{r}\right))$ between the first-order term of the long-wave expansion applied to the metric response density and the uniform strain response, calculated as in Ref. [8], for a He atom in a box undergoing a longitudinal mechanical perturbation. The two curves refer to two different values of the finite-difference increment used for obtaining $n_{\alpha }^{1,\beta }\left(\mathbf{r}\right)$.

Figure 5

Convergence of the FxE coefficients of ${\mathrm{SrTiO}}_3$ and Si with $k$-point mesh for the phonon and metric implementations.