Bending and flexural phonon scattering: Generalized Dirac equation for an electron moving in curved graphene

A generalized Dirac equation is derived in order to describe charge carriers moving in curved graphene, which is the case for temperatures above 10 K due to the presence of flexural phonons, or in bent graphene. Such interaction is taken into account by considering an induced metric, in the same spirit as the general relativity approach for the description of fermionic particle moving in a curved space-time. The resulting equation allows to include in a natural way the presence of other phonon branches as well as an external electromagnetic field. For a monochromatic sinusoidal bending of the graphene, the problem can be recasted as a Mathieu equation with a complex driven parameter, indicating the possibility of a resonance pattern.