Eigenfrequencies of the randomly pinned drum and conductivity of graphene

TL;DR

This paper investigates the vibrational properties of graphene on a rough substrate, analyzing phonon mode statistics, their impact on conductivity, and the effects of electron-phonon interactions, revealing phenomena like the Wigner-Dyson distribution and phonon-assisted tunneling.

Contribution

It introduces a statistical analysis of low-lying phonon modes in disordered graphene and explores their influence on electronic conductivity and quantum effects.

Findings

Nearest neighbor spacings follow Wigner-Dyson distribution.

Phonon interactions induce a phonon-assisted Tien-Gordon effect reducing conductivity.

At low energies and small sizes, conductivity increases due to Klein tunneling and pair creation.

Abstract

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.