Effective lattice model of graphene moiré superlattices on hexagonal boron nitride

We propose an effective lattice model of graphene moiré superlattices on hexagonal boron nitride (MLG/BN) based on ab initio calculations to study the evolution of electronic properties of relaxed MLG/BN superlattices with relative twist angle ($\theta$). The parameters for obtaining hopping terms and on-site energies in the lattice Hamiltonian are directly extracted from the ab initio electronic structure of shifted commensurate bilayers. The effect of the in-plane strain induced by the structural relaxation can be taken into account straightforwardly for the lattice model. We consider completely free MLG/BN heterostructures and those with constant interlayer distances. We find that the structural distortion due to the relaxation exhibits similar magnitude for $\theta$ smaller than about $0.5^{\circ }$. The gap at the primary Dirac points (${\mathrm{\Delta }}_P$) is maintained for $\theta <0.5^{\circ }$, while the gap at the secondary Dirac points (${\mathrm{\Delta }}_S$) decreases rapidly with small $\theta$, and such distinct behavior of ${\mathrm{\Delta }}_P$ and ${\mathrm{\Delta }}_S$ is roughly consistent with recent experimental observations. In addition, decreasing interlayer distance can significantly enhance ${\mathrm{\Delta }}_P$. This lattice model is sufficiently transparent and flexible to be employed to investigate various configurations of MLG/BN moiré superlattices.

Figure 1

The structure of a twisted MLG/BN moiré superlattice and the electronic structure of shifted commensurate bilayers of MLG/BN. (a) The rigid structure of the superlattice with a relative twist angle of $2.{0063}^{\circ }$, which is labeled by $\theta$. The top MLG layer is represented by hexagonal black lines and the B and N atoms in the bottom BN layer are represented by blue and red circles, respectively. ${\mathbf{a}}_{\mathbf{j}}^{\left(\mathbf{s}\right)}$ ($j=1,2$) are the basis vectors of the superlattice. The basis vectors of the MLG unit cell ${\mathbf{a}}_{\mathbf{j}}$ are shown in the enlarged inset, where ${\delta }_n$ ($n$ = 1–3) are the vectors from a sublattice-A site in the MLG layer to its three nearest neighbors and $\mathbf{d}$ is the in-plane local shift vector between the two layers. (b) The $H_0$ and $H_z$ maps as a function of $\mathbf{d}$ for the $h_{\mathrm{free}}$ case. The positions of high-symmetry stacking configurations (AA, AB, and BA) are labeled in the maps. (c) The band structures of a shifted bilayer with the $\mathbf{d}$ shown in (a) calculated by the ab initio method (solid lines) and the effective lattice model (dashed lines). (d), (e) The two-dimensional energy dispersion of the conduction band in (c) calculated using two different sets of k-points as described in the text. The smaller blue circle represents the corner of the BZ (K) and the larger red circle represents the shifted Dirac point (D).

Figure 2

(a) The maps of $t_1$ (left), $t_2$ (middle), and $t_3$ (right) as a function of $\mathbf{d}$. (b) The map of $\mathrm{\Delta }t={\sum }_{n=1}^3|t_n-t_0|/3$, where $t_0=(t_1+t_2+t_3)/3$.

Figure 3

(a) and (b) The total energy ($E_{\text{tot}}$), elastic energy ($E_{\text{el}}$), and interlayer interaction energy ($E_{\text{int}}$) of a supercell as a function of $\theta$ (a) for the $h_{\mathrm{free}}$ and $h_N$ cases and (b) for the $h_P$ case. In (a), the filled symbols denote the $h_{\mathrm{free}}$ case and the $h_N$ case is denoted by open symbols. (c), (d) The spatial distribution of the local interlayer interaction energy $V$ as a function of the position in the superlattice [$\mathbf{r}=(x,y)$ with the coordinates in the unit of the lattice constant of the supercell ($a^{\left(s\right)}$)] for the (c) $h_{\mathrm{free}}$ and (d) $h_P$ cases. The stackings around positions with the lowest $V$ are BA-like.

Figure 4

The band structures of the relaxed superlattices twisted by $0.{102}^{\circ }$ (left panels) and $1.{075}^{\circ }$ (right panels) (a) for the $h_{\mathrm{free}}$ and $h_N$ cases and (b) for the $h_P$ case. In (a), the solid lines denote the $h_{\mathrm{free}}$ case, the $h_N$ case is denoted by dashed lines, and the BZ of the superlattice is shown in the inset. The red and blue lines correspond to band states which are mainly contributed by the MLG states around the DPs at K and $-\mathrm{K}$, respectively. The Fermi levels are set to be zero and are represented by dashed green lines.

Figure 5

The gaps at PDPs (${\mathrm{\Delta }}_P$) and at SDPs (${\mathrm{\Delta }}_S$) of the relaxed superlattices as a function of $\theta$ (a) for the $h_{\mathrm{free}}$ and $h_N$ cases and (b) for the $h_P$ case. The gaps for the $h_P$ case without relaxation are also shown in (b).