In-plane flexoelectricity in two-dimensional \(D_{3d}\) crystals

We predict a large in-plane polarization response to bending in a broad class of trigonal two-dimensional crystals. We define and compute the relevant flexoelectric coefficients from first principles as linear-response properties of the undistorted layer, by using the primitive crystal cell. The ensuing response (evaluated for SnS\(_{2}\), silicene, phosphorene and RhI\(_{3}\) monolayers and for a hexagonal BN bilayer) is up to one order of magnitude larger than the out-of-plane components in the same material. We illustrate the topological implications of our findings by calculating the polarization textures that are associated with a variety of rippled and bent structures. We also determine the longitudinal electric fields induced by a flexural phonon at leading order in amplitude and momentum.