Eigenfrequencies of the randomly pinned drum and conductivity of graphene

Graphene is convenient material for nanomechanichal applications since high-frequency oscillations are easily accessible. In this Article, we consider graphene on a rough substrate attached to imperfections at random locations. We explore the statistics of low-lying phonon modes, which exert most influence on the conductivity of graphene. We find that the nearest neighbor spacings of low lying eigenfrequencies have the Wigner-Dyson probability distribution after averaging over the random configurations of disorder. Due to interaction of electrons with the oscillations of the membrane, an electron can be transfered to higher or lower energies, which is a manifestation of the phonon-assisted Tien-Gordon effect. The Tien-Gordon effect suppresses the conductivity of graphene. In the regime of low Fermi energies and small sizes of the sample an increase of conductivity is observed which we refer to Klein tunneling and electron-hole pair creation. Eventually, when the increase of the transmission becomes too prominent, the pair creation changes the ground state of the system, signalizing the limit of applicability of the single-particle Dirac equation used in this paper.