Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice

We theoretically investigate the electronic structures of moiré superlattices arising in monolayer/bilayer graphene stacked on hexagonal boron nitride (hBN) in the presence and absence of magnetic field. We develop an effective continuum model from a microscopic tight-binding lattice Hamiltonian and calculate the electronic structures of graphene-hBN systems with different rotation angles. Using the effective model, we explain the characteristic band properties such as the gap opening at the corners of the superlattice Brillouin zone (mini-Dirac point). We also investigate the energy spectrum and quantum Hall effect of graphene-hBN systems in uniform magnetic field and demonstrate the evolution of the fractal spectrum as a function of the magnetic field. The spectrum generally splits in the valley degrees of freedom ($K$ and $K^{\prime }$) due to the lack of the inversion symmetry, and the valley splitting is more significant in bilayer graphene on hBN than in monolayer graphene on hBN because of the stronger inversion-symmetry breaking in bilayer.

Figure 1

(a) Graphene-hBN moiré superlattice with $\theta =0^{\circ }$ and an exaggerated lattice constant ratio $a_{\mathrm{hBN}}/a=10/9$. Unit cell is indicated by a hexagon. (b) Superlattice Brillouin zone with the reciprocal lattice vectors spanned by ${\mathbf{G}}_i^{\mathrm{M}}$.

Figure 2

(a) Brillouin zone (BZ) folding in graphene-hBN moiré superlattice, where $a_{\mathrm{hBN}}/a$ is taken as 5/4 for the illustrative purpose. (b) Relative positions of the BZs centered at $K$ and $K^{\prime }$ valleys in the common BZ.

Figure 3

Band structures of monolayer graphene/hBN system with $\theta =0^{\circ }$ calculated by (a) the tight-binding model and (b) the effective continuum model, on the $k$-space path shown in Fig. 2. (c) Three-dimensional plot of the first and second electron and hole bands of $K$-valley, calculated by the continuum model.

Figure 4

Band structures of a monolayer graphene/hBN system with (a) $\theta =1^{\circ }$, (b) $2^{\circ }$, and (c) $5^{\circ }$, calculated by the effective continuum model.

Figure 5

Plots similar to Fig. 3, calculated for $AB$-bilayer graphene/hBN system with $\theta =0^{\circ }$.

Figure 6

Energy spectrum of monolayer graphene on a hBN system with $\theta =0^{\circ }$ as a function of magnetic field strength in (a) wide and (b) narrow ranges of energy. In each figure, the quantized values of Hall conductivity inside energy gaps are indicated by numbers in units $-e^2/h$ as well as shading filling the gaps. The Hall conductivity of the gray area cannot be determined by the present calculation. (c) Wannier diagram calculated for the energy spectrum in (a). Each gap is plotted as a line of which thickness is proportional to the gap width, and the color represents the quantized Hall conductivity. The color map for the Hall conductivity is the same as that in (a) and (b), except that the black circle represents the gap with Hall conductivity 0 in (c). (d) Energy spectrum originating from monolayer's $K$ region (black) and $K^{\prime }$ region (red).

Figure 7

Plots similar to Fig. 6 for $AB$-stacked bilayer graphene on a hBN system with $\theta =0^{\circ }$.

Figure 8

Landau level structure in bilayer graphene with the interlayer potential asymmetry $\pm V_0/2$.

Figure 9

Nonrotated, shifted bilayer of tight-binding honeycomb lattices with the same lattice constant.

Figure 10

(a) and (b): two different configurations of $AB$-stacked bilayer graphene on a hBN layer. (c) Electronic structures of the configurations (a) (dotted blue) and (b) (solid red).