Anomalous Absorption Line in the Magneto-Optical Response of Graphene

The intensity as well as position in energy of the absorption lines in the infrared conductivity of graphene, both exhibit features that are directly related to the Dirac nature of its quasiparticles. We show that the evolution of the pattern of absorption lines as the chemical potential is varied encodes the information about the presence of the anomalous lowest Landau level. The first absorption line related to this level always appears with full intensity or is entirely missing, while all other lines disappear in two steps. We demonstrate that if a gap develops, the main absorption line splits into two provided that the chemical potential is greater than or equal to the gap.

Figure 1

The absorption peaks and corresponding transitions. (a) Real part of the conductivity, $\mathrm{Re}{\sigma }_{xx}(\Omega )$ in units of $e^2/h$ vs frequency $\Omega$ in ${\mathrm{cm}}^{-1}$ for temperature $T=10\mathrm{K}$ and scattering rate $\Gamma =15\mathrm{K}$. Long dashed line (red), the chemical potential $\mu =50\mathrm{K}$ and the magnetic field $B={10}^{-4}\mathrm{T}$, dash-dotted line (black) $\mu =50\mathrm{K}$, solid line (blue) $\mu =510\mathrm{K}$, short-dashed line (green) $\mu =660\mathrm{K}$ all three for $B=1\mathrm{T}$. (b) A schematic of the allowed transitions between LLs $n=0,1,\dots 4$ shown as heavy (red) dots for both positive and negative Dirac cones. The pair of cones at points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ in the Brillouin zone [see Fig. 3b] are combined. Three values of chemical potential are shown as the (violet) heavy solid line. At the left, $\mu$ falls between $M_0$ and $M_1$; middle, between $M_1$ and $M_2$; and on the right, between $M_2$ and $M_3$. The first line on the left which corresponds to the first peak in dash-dotted line in the upper panel is unusual in that it arises from both interband and intraband transitions and ceases to occur for all other values of chemical potential, while all other interband lines first drop to half their intensity before disappearing entirely as $\mu$ crosses to higher energy levels.

Figure 2

Real part of the longitudinal conductivity, $\mathrm{Re}{\sigma }_{xx}(\Omega )$ in units of $e^2/h$ multiplied by $\Gamma /\sqrt{\hslash eB{v}_F^2/c}$ as a function of normalized photon energy $\Omega /\sqrt{2\hslash eB{v}_F^2/c}$. The temperature is $T=10\mathrm{K}$, the scattering rate $\Gamma =15\mathrm{K}$ and the chemical potential $\mu =680\mathrm{K}$. The long dashed (red) curve is for $B=0.75\mathrm{T}$, dash-dotted line (black) for $B=1\mathrm{T}$, solid line (blue) for $B=1.5\mathrm{T}$ and short-dashed line (green) for $B=3\mathrm{T}$.

Figure 3

The absorption peaks and the corresponding transitions when the lowest LL is split. (a) Real part of the longitudinal conductivity, $\mathrm{Re}{\sigma }_{xx}(\Omega )$ in units of $e^2/h$ vs frequency $\Omega$ in ${\mathrm{cm}}^{-1}$ for temperature $T=10\mathrm{K}$, $\Gamma =15\mathrm{K}$, $B=1\mathrm{T}$ and chemical potential $\mu =150\mathrm{K}$ for 4 values of the excitonic gap $\Delta$. Long dashed line (red) $\Delta =0\mathrm{K}$, dash-dotted line (black) $\Delta =100\mathrm{K}$, solid line (blue) $\Delta =150\mathrm{K}$, short-dashed line (green) $\Delta =200\mathrm{K}$. (b) Possible optical transitions between LLs with an excitonic gap $\Delta$. Two pairs of cones at $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points in the Brillouin zone are drawn. Only LLs with $n=0$ with energies $-\Delta$ (at $\mathbf{K}$ point) and $\Delta$ (at ${\mathbf{K}}^{\prime }$ point), and $n=1$ with energies $\pm M_1$, respectively, are shown. Three values of chemical potential are considered $|\mu |<\Delta$ (green), $\mu =\Delta$ (blue), and $|\mu |>\Delta$ (black). For the case $\mu =\Delta$ the state at energy $E_0=\Delta$ is both occupied and unoccupied with probability $1/2$.