Electronic structure theory of weakly interacting bilayers

We derive electronic structure models for weakly interacting bilayers such as graphene-graphene and graphene–hexagonal boron nitride, based on density functional theory calculations followed by Wannier transformation of electronic states. These transferable interlayer coupling models can be applied to investigate the physics of bilayers with arbitrary translations and twists. The functional form, in addition to the dependence on the distance, includes the angular dependence that results from higher angular momentum components in the Wannier $p_z$ orbitals. We demonstrate the capabilities of the method by applying it to a rotated graphene bilayer, which produces the analytically predicted renormalization of the Fermi velocity, Van Hove singularities in the density of states, and moiré pattern of the electronic localization at small twist angles. We further extend the theory to obtain the effective couplings by integrating out neighboring layers. This approach is instrumental for the design of van der Walls heterostructures with desirable electronic features and transport properties and for the derivation of low-energy theories for graphene stacks, including proximity effects from other layers.

Figure 1

(a) The generic graphene bilayer configuration with arbitrary translations and twists. The constituent monolayer crystal is described by the primitive vectors ${\mathit{a}}_1$ and ${\mathit{a}}_2$. For an interlayer pair, the coupling dependence is characterized by the projected distance $r$ and the angles ${\theta }_{12}$ and ${\theta }_{21}$ between $\mathit{r}$ and the nearest-neighbor bonds. (b) Decomposition of the Wannier function for monolayer graphene into the constant ($m=0$), $cos\left(3\theta \right)$ ($m=\pm 3$), and $cos\left(6\theta \right)$ ($m=\pm 6$) components.

Figure 2

Hoppings for the shifted graphene bilayer of (a) $AA$-type and (b) $AB$-type pairs as functions of the pair distance $r$; the spread of the hoppings at fixed $r$ indicates angular dependence. Decomposition into the constant $m=0$ (solid line), $cos\left(3\theta \right)$ (circle), and $cos\left(6\theta \right)$ (cross) components for the (c) $AA$-type pairs and (d) $AB$-type pairs. These curves are modeled by the $V_i\left(r\right)$ in Eq. (3).

Figure 3

Comparison between the tight-binding Hamiltonian (red lines) and ab initio DFT (blue circles) band structure calculations along $\mathrm{\Gamma }$-M-K-$\mathrm{\Gamma }$ in an $AB$-stacking (a) bilayer, (b) bulk; (c) $(M,N)=(6,5)$ twisted superstructure (${\theta }^{(6,5)}\approx 6.{01}^{\circ }$) and comparison to the folded monolayer band structure (green crosses).

Figure 4

(a) The renormalized Fermi velocity, ${\tilde{v}}_F/v_F$, as a function of the twist angle $\theta$ and VHS in the DOS (inset). (b) The band structure along ${\mathrm{K}\text{−}\mathrm{M}\text{−}\mathrm{K}}^{\prime }$ in an $(M,N)=(6,5)$ supercell ($\theta \approx 6.{01}^{\circ }$) compared to the folded monolayer bands (black dots). (c) The DOS for a $(M,N)=(6,5)$ supercell (green) compared with the DOS of the monolayer (black). The energy extrema in the bands in (b) correspond to the singular points of DOS. (d) Similar calculations for the $(M,N)=(31,30)$ supercell ($\theta \approx 1.{08}^{\circ }$). (e) The states around the Fermi level are localized at $AA$ sites forming a moiré pattern.