Landau Levels and Quantum Hall Effect in Graphene Superlattices

We show that, when graphene is subjected to an appropriate one-dimensional external periodic potential, additional branches of massless fermions are generated with nearly the same electron-hole crossing energy as that at the original Dirac point of graphene. Because of these new zero-energy branches, the Landau levels at charge neutral filling become $4(2N+1)$-fold degenerate (with $N=0,1,2,\dots$, tunable by the potential strength and periodicity) with the corresponding Hall conductivity ${\sigma }_{xy}$ showing a step of size $4(2N+1)e^2/h$. These theoretical findings are robust against variations in the details of the external potential and provide measurable signatures of the unusual electronic structure of graphene superlattices.

Figure 1

(a) Schematic diagram of a Kronig-Penney type of potential applied to graphene with strength $U_0/2$ inside the gray regions and $-U_0/2$ outside with lattice period $L$ and barrier width $L/2$. (b) Electron energy in units of ${\epsilon }_L$ ($\equiv \hslash v_0/L$; for example, if $L=20\mathrm{nm}$, ${\epsilon }_L=33\mathrm{meV}$) versus wave vector near the Dirac point in pristine graphene. (c) The same quantity as in (b) for a GS with $U_0=6\pi {\epsilon }_L$. (d) Number of Dirac points (not including spin and valley degeneracies) in a GS versus $U_0$.

Figure 2

Electron energy (in units of ${\epsilon }_L=\hslash v_0/L$) versus $k_y$ with $k_x=0$ in GSs shown in Fig. 1 for several different values of barrier height $U_0$ (specified in each panel in units of ${\epsilon }_L$).

Figure 3

Landau level energy $E_i$ (in units of ${\epsilon }_B\equiv \hslash v_0/l_B$ with $l_B=\sqrt{\hslash c/eB}$) versus the Landau level index $i$ ($i=0,\pm 1,\pm 2,\dots$) in GSs formed with a 1D Kronig-Penney potential for several different values of barrier height $U_0$, with lattice period $L=0.5l_B$. The LLs now have a finite width $\Delta E$ (shown not to scale and exaggerated in the figure) arising from the $k_y$ dependence of the energy of the electronic states in a perpendicular magnetic field [38]. Note the 3-fold and the 5-fold degeneracies around $E_i=0$ in (d) and (f), respectively. (If the spin and valley degeneracies are considered, those become 12-fold and 20-fold, respectively.)

Figure 4

Hall conductivity ${\sigma }_{xy}$ versus carrier density (with an artificial broadening for illustration) for a 1D Kronig-Penney GS with $U_0$ near $6\pi \hslash v_0/L$ (solid red line) is compared to that of pristine graphene (dashed blue line).