Interface Landau levels in graphene monolayer-bilayer junctions

Electronic structure of graphene monolayer-bilayer junction in a magnetic field is studied within an effective-mass approximation. The energy spectrum is characterized by the interface Landau levels, i.e., the locally flat bands appearing near the boundary region, resulting in a series of characteristic peaks in the local density of states. Their energies are independent of boundary types such as zigzag or armchair. In the atomic scale, the local density of states shows a Kekulé pattern due to the valley mixing in the armchair boundary while does not in the zigzag boundary.

Figure 1

Plots of $\nu \left(\epsilon \right)$, ${\nu }_{+}\left(\epsilon \right)$, and ${\nu }_{-}\left(\epsilon \right)$ with energy $\epsilon$ set to the vertical axis. Black and white circles represent the Landau levels of bulk monolayer and bilayer, respectively. ${\gamma }_1/\hslash {\omega }_B=3$ is taken for bilayer.

Figure 2

(Color online) Monolayer-bilayer graphene junctions with boundary types of (a) ZZ1, (b) ZZ2, and (c) AC.

Figure 3

Energy spectrum as a function of $X$ in boundary (a) ZZ1, (b) ZZ2, and (c) AC at magnetic field $\hslash {\omega }_B={\gamma }_1/3$ $\left(B\sim 13\text{T}\right)$. Black and white circles represent $P_K$ and $P_{K^{\prime }}$, respectively. Numbers between the levels indicate bulk filling factor in limits $X=\pm \infty$.

Figure 4

Energy spectrum as a function of $X$ in boundary ZZ2 at magnetic fields (a) $\hslash {\omega }_B={\gamma }_1/5$ $\left(B\sim 5\text{T}\right)$ and (b) ${\gamma }_1/10$ $\left(B\sim 1.2\text{T}\right)$.

Figure 5

(Color online) (Above) Monolayer-bilayer junction of type ZZ1, separated by infinite potential on white circles into independent monolayer and bilayer graphenes. (Below) Energy spectrum of junction ZZ1 (solid) and separated system (dashed) in magnetic field $\hslash {\omega }_B={\gamma }_1/3$. Left and right panels show the spectra of the $K$ and $K^{\prime }$ points, respectively, and the black and white circles are the interface Landau levels.

Figure 6

(Color online) Plots similar to Fig. 5 for ZZ2.

Figure 7

Wave functions near the interface Landau level of the $K^{\prime }$ point in ZZ1 at $\hslash {\omega }_B={\gamma }_1/3$. (a)–(f) correspond to the points in the energy spectrum in Fig. 5.

Figure 8

Local density of states of boundary ZZ2 with $\hslash {\omega }_B={\gamma }_1/3$ at the energy $\epsilon =0.631\hslash {\omega }_B$ near the lowest interface Landau level of $K$. The areas of circles in upper and lower panels represent the amplitude of LDOS at each atom on layers 1 and 2, respectively, while open and filled circles represent $A$ and $B$ sublattices, respectively. $l_B/a$ is about 30 at this magnetic field.

Figure 9

Plot similar to Fig. 8, showing the local density of states of the armchair boundary with $\hslash {\omega }_B={\gamma }_1/3$ at the energy $\epsilon =0.3\hslash {\omega }_B$.

Figure 10

(Color online) (Right) Averaged local density of states at different atomic sites and (left) corresponding energy spectrum in boundary ZZ2 at magnetic field $\hslash {\omega }_B={\gamma }_1/3$ $\left(B\sim 13\text{T}\right)$. The width of each bin in energy is taken as $0.02\hslash {\omega }_B$.