In-Plane Flexoelectricity in Two-Dimensional D3d 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 ${\mathrm{SnS}}_{2}$, silicene, phosphorene, and ${\mathrm{RhI}}_{3}$ monolayers and for a hexagonal BN bilayer) is up to 1 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.