Quantum Hall studies of a semi-Dirac nanoribbon
Priyanka Sinha, Shuichi Murakami, Saurabh Basu
Abstract
Here we comprehensively investigate Landau levels, Hofstadter's butterfly, and transport properties of a semi-Dirac nanoribbon in a perpendicular magnetic field using a recently developed real-space implementation of the Kubo formula based on the kernel polynomial method. A Dirac ribbon is considered to compare and contrast our results for a semi-Dirac system. We find that with the Landau levels being nonequidistant from each other for the semi-Dirac case (which is true for a Dirac case as well), the flatness of the energy bands vanishes in the bulk and becomes dispersive for a semi-Dirac ribbon in contrast to a Dirac system. This feature is most discernible for intermediate values of the external field. We further compute the longitudinal $({\ensuremath{\sigma}}_{xx} \mathrm{and} {\ensuremath{\sigma}}_{yy})$ and the transverse or Hall $({\ensuremath{\sigma}}_{xy})$ conductivities, where the Hall conductivity shows a familiar quantization, namely, ${\ensuremath{\sigma}}_{xy}\ensuremath{\propto}2n$ (the factor of 2 includes the spin degeneracy), which is highly distinct from a Dirac system, such as graphene. We also observe anisotropic behavior in magnetotransport in a semi-Dirac ribbon owing to the dispersion anomalies in two different longitudinal directions. Our studies may have important ramifications for monolayer phosphorene.