Metric-wave approach to flexoelectricity within density-functional perturbation theory

Within the framework of density functional perturbation theory (DFPT), we implement and test a novel "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 SrTiO$_3$.