Unusual Microwave Response of Dirac Quasiparticles in Graphene

Recent experiments have proven that the quasiparticles in graphene obey a Dirac equation. Here we show that microwaves are an excellent probe of their unusual dynamics. When the chemical potential is small, the intraband response can exhibit a cusp around zero frequency $\Omega$ and this unusual line shape changes to Drude-like by increasing the chemical potential $|\mu |$, with width linear in $\mu$. The interband contribution at $T=0$ is a constant independent of $\Omega$ with a lower cutoff at $2\mu$. Distinctly different behavior occurs if interaction-induced phenomena in graphene cause an opening of a gap $\Delta$. At a large magnetic field $B$, the diagonal and Hall conductivities at small $\Omega$ become independent of $B$ but remain nonzero and show a structure associated with the lowest Landau level. This occurs because in the Dirac theory the energy of this level, $E_0=\pm \Delta$, is field independent in sharp contrast to the conventional case.

Figure 1

Band structure of graphene. (a) The low-energy linear-dispersion $E(\mathbf{k})$ near the Dirac $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points for $B=0$. (b) A possible modification of the quasiparticle spectrum by the finite gap (Dirac mass) $\Delta$ due to interaction-induced phenomena [19, 20]. The chemical potential (indicated by the horizontal line) $\mu$ is shifted from zero by the gate voltage [2]. (c) Landau levels $E_n$ in the Dirac theory of graphene. For a given direction of the magnetic field $\mathbf{B}$ applied perpendicularly to graphene’s plane the lowest $(n=0)$ Landau level has the energy $E_0=-\Delta$ at $\mathbf{K}$ and $E_0=\Delta$ at ${\mathbf{K}}^{\prime }$. The presence of a field independent $n=0$ level causes an unconventional Hall effect and anomalies in the strong field microwave response.

Figure 2

The microwave conductivity ${\sigma }_{xx}(\Omega ,T)$ in units $e^2/h$ vs microwave frequency $\Omega$ in GHz. The long dashed line is for $T=1\mathrm{K}$ and the solid for $T=5\mathrm{K}$. In both cases $\mu =\Delta =0$ and $\Gamma =0.1\mathrm{K}$. The dash-dotted curve has a gap $\Delta =5\mathrm{K}$ and $T=1\mathrm{K}$.

Figure 3

Conductivity ${\sigma }_{xx}(\Omega ,T)$ in units $e^2/h$ vs microwave frequency $\Omega$ in GHz. The chemical potential is set at 6 K, the temperature at 0.5 K, and the impurity scattering rate is assumed constant set at 0.5 K. Long dashed curve $B=0.05\mathrm{T}$ and $\Delta =0$, solid curve $\Delta =0$, dash-dotted curve $\Delta =3\mathrm{K}$, and short dashed $\Delta =7\mathrm{K}$. The three last curves are all for $B=1\mathrm{T}$.

Figure 4

Conductivity ${\sigma }_{xy}(\Omega ,T)$ in units $e^2/h$ vs microwave frequency $\Omega$ in GHz. The chemical potential is set at $-10\mathrm{K}$, the temperature at 0.5 K and the impurity scattering rate assumed constant set at 4 K. The gap $\Delta$ is also indicated in the figure.