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 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.

Direct and Converse Flexoelectricity in Two-Dimensional Materials.

Building on recent developments in electronic-structure methods, we define and calculate the flexoelectric response of two-dimensional (2D) materials fully from first principles. In particular, we show that the open-circuit voltage response to a flexural deformation is a fundamental linear-response property of the crystal that can be calculated within the primitive unit cell of the flat configuration. Applications to graphene, silicene, phosphorene, boron nitride, and transition-metal dichalcogenide monolayers reveal that two distinct contributions exist, respectively of purely electronic and lattice-mediated nature. Within the former, we identify a key metric term, consisting in the quadrupolar moment of the unperturbed charge density. We propose a simple continuum model to connect our findings with the available experimental measurements of the converse flexoelectric effect.