Graphene, Dirac equation and analogue gravity

Abstract

We provide an updated study of some electronic properties of graphene nanoscrolls, exploiting a related curved space Dirac equation for the charge carriers. To this end, we consider an explicit parametrization in cylindrical coordinates, together with analytical solutions for the pseudoparticle modes living on the two-dimensional background. These results are then used to obtain a compact expression for the sample optical conductivity, deriving from a Kubo formula adapted to the 1+2 dimensional curved space. The latter formulation is then adopted to perform some simulations for a cylindrical nanoscroll geometry.

Dirac equation in curved spacetime: the role of local Fermi velocity

“…We start recovering the coordinates of a conventional three-dimensional spacetime in cylindrical coordinates as specified in [13,59]:…”

“…They consist in carbon-based structures obtained by rolling a graphene layer into a cylindrical geometry [72], and exhibit some exciting qualities due to their distinctively different conformation [80,81], making them conceptually interesting and experimentally relevant. Unusual electronic and optical properties of carbon nanoscrolls are due to their unique topology and peculiar structure [13,59,82]. From a geometrical point of view, nanoscrolls geometry can be parameterized in terms of Archimedeantype spirals in cylindrical coordinates [83][84][85].…”

“…From a geometrical point of view, nanoscrolls geometry can be parameterized in terms of Archimedeantype spirals in cylindrical coordinates [83][84][85]. Let us then consider the following parameterization for the cylindrical geometry of the layer surface [13,59]…”

“…The integer number N takes into account the windings of the wrapped graphene layer; from a physical point of view, it acts as a momentum cutoff, correctly restricting the analysis to the long wavelength continuum approximation. 2 The boundary conditions in the cylindrical symmetry give rise to a quantization condition on the transverse component of the charge carriers momentum k φ of the form [13,59]…”