Optical Hall Conductivity in Ordinary and Graphene Quantum Hall Systems

We reveal from numerical study that the optical Hall conductivity ${\sigma }_{xy}(\omega )$ has a characteristic feature even in the ac ($\sim \mathrm{THz}$) regime in that the Hall plateaus are retained both in the ordinary two-dimensional electron gas and in graphene in the quantum Hall regime, although the plateau height is no longer quantized in ac. In graphene ${\sigma }_{xy}(\omega )$ reflects the unusual Landau level structure. The effect remains unexpectantly robust against the significant strength of disorder, which we attribute to an effect of localization. We predict the ac quantum Hall measurements are feasible through the Faraday rotation characterized by the fine-structure constant $\alpha$.

Figure 1

Exact diagonalization result for (a) the optical Hall conductivity ${\sigma }_{xy}({\epsilon }_F,\omega )$ with $\Gamma =0.2\hslash {\omega }_c$, (b) static [blue (dark gray)] and optical [red (light gray)] Hall conductivity ${\sigma }_{xy}({\epsilon }_F,\omega )$, and (c) ${\sigma }_{xy}({\epsilon }_F,\omega )$ with larger disorders $\Gamma =0.5\hslash {\omega }_c$ in usual QHE system.

Figure 2

Exact diagonalization result for (a) the optical Hall conductivity ${\sigma }_{xy}({\epsilon }_F,\omega )$ with $\Gamma =0.2\hslash {\omega }_c$, (b) static [blue (dark gray)] and optical [red (light gray)] Hall conductivity ${\sigma }_{xy}({\epsilon }_F,\omega )$, and (c) ${\sigma }_{xy}({\epsilon }_F,\omega )$ with larger disorders $\Gamma =0.5\hslash {\omega }_c$ for the graphene QHE system.

Figure 3

Exact diagonalization result for the optical Hall conductivity, ${\sigma }_{xy}({\epsilon }_F,\omega )$, plotted against Fermi energy and disorder strength $\Gamma$ for various values of frequency $\omega =0$, $0.4{\omega }_c$, $0.9{\omega }_c$, $1.5{\omega }_c$ in (a) the ordinary and (b) graphene QHE system.