A mechanical deformation of a continuum can be expressed as a generalized coordinate transformation of space. Consequently, the equations of electrostatics in deformable media must satisfy covariance requirements with respect to such transformations, a problem that has long been addressed in the context of general relativity. Here we show how these ideas can be incorporated into the framework of density-functional perturbation theory, providing access to the microscopic charge density and electrostatic potential response to an arbitrary deformation field. We demonstrate the power of our approach by deriving, in full generality, the surface contributions to the flexoelectric response of a finite object, a topic that has recently been a matter of controversy. The breakdown of translational periodicity produces consequences that might seem highly paradoxical at first sight: for example, the macroscopic bulk polarization does not always correspond to the physical surface charge.
FIGURE: The three main types of strain gradients producing a dipole moment normal to the surfaces are shown: longitudinal (a), transverse (b) and shear (c). The supercell boundaries are shown as dashed lines, the slabs are schematically represented as arrays of black circles.