Spatial dispersion effects in ferroic oxides: Dynamical quadrupoles and flexoelectric tensor

In condensed-matter physics, spatial dispersion refers to the dependence of many material properties on the wavevector q at which they are probed, and is ultimately due to the nonlocality of the response to a given external field (electric, magnetic, strain). A remarkable example is the flexoelectric tensor, describing the polarization response to a gradient of applied strain, or equivalently the electrical current that is produced by an acoustic phonon at second-order in q. Density-functional perturbation theory (DFPT) appears as the ideal framework to compute these effects from first principles, but the general computational tools to deal with the long-wavelength limit are currently missing. Here we present a general formalism, based on the analytical long-wavelength expansion of the second-order DFPT energies, that enables the direct calculation of spatial dispersion quantities at a computational cost that is comparable to that of a uniform-field response calculation. We present results for the clamped-ion flexoelectric tensor in SrTiO₃ and the dynamical quadrupoles (the higher-order multipolar counterpart of the Born effective charges) in tetragonal PbTiO₃. The quadrupoles relate to the clamped-ion piezoelectric tensor as predicted by R. Martin in his 1972 seminal paper.