First-Principles Theory of Spatial Dispersion: Dynamical Quadrupoles and Flexoelectricity

Abstract

Density-functional perturbation theory (DFPT) is nowadays the method of choice for the accurate computation of linear and nonlinear-response properties of materials from first principles. A notable advantage of DFPT over alternative approaches is the possibility of treating incommensurate lattice distortions with an arbitrary wave vector q at essentially the same computational cost as the lattice-periodic case. Here we show that q can be formally treated as a perturbation parameter and used in conjunction with the established results of perturbation theory (e.g., the “ 2 n + 1 ” theorem) to perform a long-wave expansion of an arbitrary response function in powers of the wave-vector components. This procedure provides a powerful general framework to access a wide range of spatial dispersion effects that were formerly difficult to calculate by means of first-principles electronic structure methods. In particular, the physical response to the spatial gradient of any external field can now be calculated at negligible cost by using the response functions to uniform perturbations (electric, magnetic, or strain fields) as the only input. We demonstrate our method by calculating the flexoelectric and dynamical quadrupole tensors of selected crystalline insulators and model systems.