Direct and converse flexoelectricity in two-dimensional materials

Flexoelectricity, the generation of macroscopic polarization or voltage in response to a strain gradient, is expected to play a prominent role in two-dimensional (2D) crystals due to their extreme flexibility. Several attempts have been carried out to calculate the flexoelectric response of a monolayer (or few layers) due to a flexural deformation, but generally with remarkable disagreement in reported values. Here, building on recent developments in electronic-structure methods, we define and calculate the flexoelectric response of two-dimensional 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, BN 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 \emph{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.