Magneto-optical properties of multilayer graphene

The magneto-optical absorption properties of graphene multilayers are theoretically studied. It is shown that the spectrum can be decomposed into sub-components effectively identical to the monolayer or bilayer graphene, allowing us to understand the spectrum systematically as a function of the layer number. Odd-layered graphenes always exhibit absorption peaks which shifts in proportion to $\sqrt{B}$, with $B$ being the magnetic field, due to the existence of an effective monolayer-like subband. We propose a possibility of observing the monolayer-like spectrum even in a mixture of multilayer graphene films with various layers numbers.

Figure 1

(Top) Landau levels of the bilayer-type subband as a function of $\kappa$ with $\lambda =2\cos \kappa$. The magnetic field strength is taken as ${\Delta }_B∕{\gamma }_1=0.5$. (Bottom) Lists of $\kappa$ in $N$-layered graphene. Empty and filled circles represent even and odd $N$’s, respectively.

Figure 2

Real part of the dynamical conductivity of the monolayer (top) and bilayer (bottom) graphenes plotted against the frequency $\omega$ calculated for different magnetic fields (specified by ${\Delta }_B$) and $\Gamma ∕{\gamma }_1=0.01$. Dashed curves indicate the transition energies between several Landau levels in the ideal limit.

Figure 3

Real part of the dynamical conductivity of $N=1,2,\dots ,5$ layered graphenes as functions of the frequency, calculated for two different magnetic fields ${\Delta }_B∕{\gamma }_1=0.1$ and 0.2. We take $\Gamma ∕{\gamma }_1=0.01$.

Figure 4

Plots of Re ${\sigma }_{xx}\left(\omega \right)$ averaged over the layer numbers of $N=1–20$ for magnetic fields with ${\Delta }_B∕{\gamma }_1=0.1$, 0.2, and 0.3. Vertical solid and dashed lines represent the ideal transition energies for the monolayer-type $(\kappa =\pi ∕2)$ and bilayer-type $(\kappa =0)$ subbands, respectively.

Figure 5

(Top) Density plot of $\mathrm{Re}{\sigma }_{xx}(\omega ,{\Delta }_B)$ averaged over the layer numbers of $N=1–20$. $\Gamma$ is set to $0.01{\gamma }_1$. (Bottom) Corresponding plot for the transition energies between the several Landau levels.

Figure 6

(Top) Density of states averaged over the layer numbers of $N=1–20$ at ${\Delta }_B∕{\gamma }_1=0.3$. (Bottom) Corresponding plot for the local density of states on the top layer. Vertical solid and dashed lines indicate the ideal Landau level energies for the monolayer-type $(\kappa =\pi ∕2)$ and bilayer-type $(\kappa =0)$ subbands, respectively.

Figure 7

(Top) Density of states (measured from the zero-field value) averaged over the layer numbers of $N=1–20$, plotted in the $(\epsilon ,{\Delta }_B)$ plane. $\Gamma$ is set to $0.01{\gamma }_1$. (Middle) A similar plot for the local density of states on the top layer. (Bottom) Ideal Landau level energies corresponding to several dominant peaks in the above two panels.