Flexoelectricity describes the electric polarization that is linearly induced by a strain gradient, and is being intensely investigated as an alternative to piezoelectricity in electromechanical sensor and actuators. Strain gradients are largest at the nanoscale, where they can lead to interesting new materials functionalities, e.g. at domain boundaries in ferroics. While several breakthough experiments have been reported in the past ten years, progress on the theoretical front has been comparatively slow. The theory and calculation of flexoelectricity from a quantum-mechanical standpoint is riddled with subtleties and presents considerable technical and conceptual challenges. The main difficulty consists in the breakdown of translational periodicity that a strain gradient entails, which at first sight questions the very applicability of traditional plane-wave pseudopotential methods.
In this talk I will show how these obstacles can be overcome by combining density-functional perturbation theory with generalized coordinate transformations [1,2], gaining access to the full microscopic response (in terms of electronic charge density, polarization and atomic displacements) of a crystal or nanostructure to an arbitrary deformation field. As a practical demonstration, I will present results on the full flexoelectric response of SrTiO3, including atomic relaxations and surface effects.  I will show that, upon bending a SrTiO3 slab, one obtains a positive voltage if the crystal lattice is terminated by a TiO2 layer, a negative voltage if the termination is of the SrO type. This points to a dramatic dependence of the flexoelectric effect on the details of the surface: an atomically thin termination layer can affect the magnitude, and even the sign, of the response of a macroscopically thick object.
 M. Stengel, Phys. Rev. B 88, 174106 (2013).
 M. Stengel, Nature Communications 4, 2693 (2013).
 M. Stengel, Phys. Rev. B 90, 201112(R) (2014).