Density of states and magneto-optical conductivity of graphene in a perpendicular magnetic field

The density of states (DOS) and the optical conductivity of graphene is calculated in the presence of a perpendicular magnetic field and where scattering on charged and short-range impurities is included. The standard Kubo formula is employed where the self-energy induced by impurity scattering and the Green’s function are calculated self-consistently including inter-Landau level (LL) coupling and screening effects. It is found that the scattering from those two types of impurities results in a symmetric LL broadening and asymmetric inter-LL coupling renormalizes the LL positions to lower energy. The peak position and intensity of the magneto-optical conductivity depends on the filling factor and the broadened DOS. Good agreement is found with recent cyclotron resonance measurements.

Figure 1

(a) The DOS as a function of energy at $B=5\text{T}$ and (b) the LL width at the Fermi energy $E_F$ as a function of magnetic field (or filling factor, upper axis) for a fixed electron density $n_e={10}^{12}{\text{cm}}^{-2}$ and charged impurity density $n_i^c={10}^{11}{\text{cm}}^{-2}$ limited. $D_0={\left(2\pi l_B^2\right)}^{-1}$ is the degeneracy of each LL. The vertical dotted line in (a) indicates the position of $E_F$.

Figure 2

(Color online) The DOS as a function of energy including inter-LL coupling for a larger charged impurity density $n_i^c=2\times {10}^{11}{\text{cm}}^{-2}$ in the presence of only charged impurity scattering for $B=5\text{T}$ and $n_e={10}^{12}{\text{cm}}^{-2}$. The solid black curve indicates the total DOS and the dashed color curves are the DOS of each separate LL. The vertical dotted lines indicate the positions of the nondisordered LLs. The Fermi level is located at $E_F/E_1=1.404$.

Figure 3

(Color online) The LL width as a function of magnetic field in the presence of only charged impurity scattering with magnetic field independent Thomas-Fermi screening wave vector $q_{\text{TF}}$. The solid red curve indicates the LL width at the Fermi energy. The parameters are the same as in Fig. 1.

Figure 4

The Fourier transform of the bare (dashed lines) and screened (solid lines) Coulomb potential in the presence of only charged impurity scattering and intra-LL screening $K_q$. The inset shows the potential in real space. The parameters are the same as in Fig. 1 and $B=5\text{T}$.

Figure 5

(Color online) The DOS as a function of energy (a) without and (b) with inter-LL coupling in the presence of only short-range scattering at $B=5\text{T}$. The solid black curves in (a) and (b) indicate the total DOS and the dashed color curves are the DOS of each LL. The vertical dotted lines in (a) and (b) indicate, respectively, the positions of the Fermi energy and the nondisordered LLs. The Fermi level in (b) is located at $E_F/E_1=1.436$ when $n_e={10}^{12}{\text{cm}}^{-2}$ and is shown by the dotted vertical line in (a).

Figure 6

(Color online) The DOS as a function of energy for different charged impurity density and single-particle level broadening due to short-range impurities when $n_e={10}^{12}{\text{cm}}^{-2}$ and $B=5\text{T}$ $\left(\nu =8.27\right)$. The solid black curves in (a) indicate the total DOS and the dashed color curves are the DOS of each LL. The vertical dotted line in (a) indicates the position of the Fermi energy $E_F$.

Figure 7

(Color online) The same as Fig. 6 but now for integer filling $\nu =10$.

Figure 8

(Color online) The LL width at $E_F$ as a function of magnetic field in the presence of both charged impurity and short-range impurity scattering including inter-LL coupling. Solid black, dashed olive, and dotted wine curves are for different strength of short-range broadening $\Gamma /E_1=0.05$, $0.08$ and 0.1, respectively. The dashed-dotted-dotted dark gray curves fit the width near half filling with $\sim B^{\alpha }$ where $\alpha =0.12$, 0.22, and 0.25, respectively. The other parameters $n_e$ and $n_i^c$ are the same as in Fig. 1a.

Figure 9

(Color online) The optical conductivity as a function of photon energy at $B=5\text{T}$ for two values of the filling factor (a) $\nu =8.27$ $\left(n_e={10}^{12}{\text{cm}}^{-2}\right)$ and (b) $\nu =10$ for different single-particle quantum level broadening induced by charged impurities and short-range impurities. The insets show a schematic LL ladder with allowed transitions indicated by arrows and the horizontal dotted line indicates the position of the Fermi energy.

Figure 10

(Color online) The optical conductivity as a function of photon energy for fixed filling factor $\nu =2$ for different magnetic fields at $n_i^c={10}^{11}{\text{cm}}^{-2}$ and $\Gamma =0.08E_1$.

Figure 11

(a) The peak position (full circle) and (b) the HWHM (-●-) of the $n=0\to 1$ transition vs $\sqrt{B}$. The solid line in (a) corresponds to the $n=1$ single-particle Landau level. The open square symbols in (b) are the experiment results from Ref. 11. The parameters are the same as in Fig. 10.

Figure 12

(Color online) (a) The optical conductivity as a function of photon energy at a fixed magnetic field $B=18\text{T}$ for different filling factor with $n_i^c={10}^{11}{\text{cm}}^{-2}$ and $\Gamma =0.1E_1$. Traces are offset for clarity. (b) The $n=1$ broadened DOS for the corresponding filling factor and (c) the peak position (-●-) of the $n=0\to 1$ transition as a function of the filling factor. The -○- symbols are the experiment results from Ref. 14.