Landau levels in biased graphene structures with monolayer-bilayer interfaces

The electron energy spectrum in monolayer-bilayer-monolayer and in bilayer-monolayer-bilayer graphene structures is investigated and the effects of a perpendicular magnetic field and electric bias are studied. Different types of monolayer-bilayer interfaces are considered as zigzag (ZZ) or armchair (AC) junctions which modify considerably the bulk Landau levels (LLs) when the spectra are plotted as a function of the center coordinate of the cyclotron orbit. Far away from the two interfaces, one obtains the well-known LLs for extended monolayer or bilayer graphene. The LL structure changes significantly at the two interfaces or junctions where the valley degeneracy is lifted for both types of junctions, especially when the distance between them is approximately equal to the magnetic length. Varying the nonuniform bias and the width of this junction-to-junction region in either structure strongly influence the resulting spectra. Significant differences exist between ZZ and AC junctions in both structures. The densities of states (DOSs) for unbiased structures are symmetric in energy whereas those for biased structures are asymmetric. An external bias creates interface LLs in the gaps between the LLs of the unbiased system in which the DOS can be quite small. Such a pattern of LLs can be probed by scanning tunneling microscopy.

Figure 1

Schematics of a MLG-BLG-MLG structure with (a) ZZ and (b) AC junctions. The BLG ribbon's width $W$ (yellow region) is $W=N\sqrt{3}{a}_0/4$ in the ZZ case, with $a_0=0.246$ nm the lattice constant and $N$ the number of atoms of the terminated layer in the black dashed box. In the AC case we have $W=n{a}_0$, with $n$ the number of lattice constants between the boundaries, i.e., the vertical green lines. The upper pictures show a side view of the systems.

Figure 2

Energy spectrum of an unbiased MLG-BLG-MLG structure $\left(U_1=U_2=0\right)$ versus $x_c/l_B=-k_y{l}_B$, with ZZ junctions in a magnetic field $B=10$ T, for three different widths of the BLG ribbon (yellow region): (a) $W_1=71.58$ nm, (b) $W_2=47.72$ nm, and (c) $W_3=11.93$ nm. Blue solid (red dashed) curves refer to the $K$ $\left(K^{\prime }\right)$ valley.

Figure 3

Electron velocity along the interfaces as a function of $x_c/l_B$, for the two lowest electron states of the spectrum shown in (a) Fig. 2 and (b) Fig. 2 at the $K$ (solid curves) and $K^{\prime }$ (dashed curves) valleys. The blue and red curves are for the first and second energy states, respectively.

Figure 4

Energy spectrum of a biased MLG-BLG-MLG structure $(U_1=-U_2=-50$ meV) as a function of $x_c/l_B$, with ZZ boundaries in the presence of a perpendicular magnetic field $B=10$ T, for BLG ribbon width (yellow region) (a) $W_2=47.72$ nm and (b) $W_3=11.93$ nm. Blue solid (red dashed) curves display the energy spectrum at the $K$ $\left(K^{\prime }\right)$ valley. Degenerate LLs of MLG are shown by green solid lines, and gray solid (dashed) lines indicate the LLs of biased BLG for the $K$ $\left(K^{\prime }\right)$ valley.

Figure 5

Probability densities for each sublattice corresponding to the points $\left(1\right),\left(2\right),\left(1^{\prime }\right)$, and $\left(2^{\prime }\right)$ shown in the energy spectrum of Fig. 4. The solid and dashed curves are for sites in layers 1 and 2, respectively, and the blue (red) color for sublattice $A$ $\left(B\right)$.

Figure 6

Energy spectrum of an unbiased MLG-BLG-MLG structure $\left(U_1=U_2=0\right)$ versus $x_c/l_B$ with AC junctions in a magnetic field $B=10$ T for three different widths of the BLG ribbon (yellow region): (a) $W_1=71.34$ nm, (b) $W_2=48.22$ nm, and (c) $W_3=12.30$ nm. The gray solid lines are the BLG LLs.

Figure 7

(Upper panels) Energy levels of a biased MLG-BLG-MLG structure $(U_1=-U_2=-50$ meV), as a function of $x_c/l_B$, with AC boundaries in a magnetic field $B=10$ T. The width of the BLG ribbon is (a) $W_2=48.22$ nm and (b) $W_3=12.30$ nm. The green lines show the degenerate MLG LLs, and the gray solid (dashed) ones are the $K$ $\left(K^{\prime }\right)$-valley LLs of BLG. The lower panels show the spinors of each sublattice corresponding to points (1)–(3) in panel (a).

Figure 8

Schematics of a BLG-MLG-BLG structure with (a) ZZ and (b) AC junctions. $W$ is the width of the MLG region. The upper pictures show a side view of the systems.

Figure 9

(Upper panels) Energy spectrum of an unbiased BLG-MLG-BLG structure $\left(U_1=U_2=0\right)$ as a function of $x_c/l_B$, with ZZ boundaries in a field $B=10$ T, for three different MLG ribbon widths (brown regions): (a) $W_1=71.58$ nm, (b) $W_2=47.72$ nm, and (c) $W_3=11.93$ nm. The blue solid (red dashed) curves refer to the $K$ $\left(K^{\prime }\right)$ valley. The LLs in MLG are shown by the green solid lines. The lower panels show the states for the energies marked by (1)–(6) in panels (b) and (c).

Figure 10

Energy spectrum of a biased BLG-MLG-BLG structure $(U_1=-U_2=-50$ meV) versus $x_c/l_B$, with ZZ boundaries in a field $B=10$ T. The MLG width (brown region) is (a) $W_2=47.72$ nm and (b) $W_3=11.93$ nm. The bulk MLG and BLG LLs are shown in green and gray solid lines, respectively.

Figure 11

Energy spectrum of an unbiased BLG-MLG-BLG structure $\left(U_1=U_2=0\right)$, as a function of $x_c/l_B$, with AC boundaries in a magnetic field $B=10$ T. The width of the MLG ribbon (brown region) is (a) $W_1=71.34$ nm, (b) $W_2=48.22$ nm, and (c) $W_3=12.30$ nm. The solid green lines are the LLs of bulk MLG.

Figure 12

Energy spectrum of a biased BLG-MLG-BLG structure $(U_1=-U_2=-50$ meV), versus $x_c/l_B$, with AC boundaries in a magnetic field $B=10$ T. The width of the MLG ribbon is (a) $W_2=48.22$ nm and (b) $W_3=12.30$ nm.

Figure 13

Density of states in MLG-BLG-MLG and BLG-MLG-BLG structures for ZZ (upper panels with $W_2=47.72$ nm) and AC (lower panels with $W_2=48.22$ nm) junctions. The blue curves pertain to an unbiased structure and the red ones to a biased one with $U_1=-U_2=-50$ meV.