Electron Interactions and Gap Opening in Graphene Superlattices

We develop a theory of interaction effects in graphene superlattices, where tunable superlattice periodicity can be used as a knob to control the gap at the Dirac point. Applied to graphene on hexa-boron-nitride ($G/h$-BN), our theory predicts substantial many-body enhancement of this gap. Tunable by the moiré superlattice periodicity, a few orders of magnitude enhancement is reachable under optimal conditions. The Dirac point gap enhancement can be much larger than that of the minigaps opened by Bragg scattering at principal superlattice harmonics. This naturally explains the conundrum of large Dirac point gaps recently observed in $G/h$-BN heterostructures and their tunability by the $G/h$-BN twist angle.

Figure 1

(a) Interaction-induced enhancement of couplings to an incommensurate (moiré) superlattice potential. The superlattice spatial wavelength $\lambda$, tunable by the twist angle between $G$ and $h$-BN (inset), controls the interaction effects via RG flow. Renormalization for the DP gap ${\Delta }_0$ and the gap ${\Delta }_1$ opened by Bragg scattering at the principal superlattice harmonics is shown by red and blue lines, respectively. Renormalization proceeds in two stages. Both gaps are enhanced from the noninteracting values ${\Delta }_0^{(0)}$ and ${\Delta }_1^{(0)}$; however, the gap ${\Delta }_0$ undergoes a much larger enhancement and, despite a small initial value, eventually overtakes ${\Delta }_1$. Both ${\Delta }_0$ and ${\Delta }_1$ grow faster than the carrier velocity $v$ (green line). (b) Density of states obtained using renormalized gaps and velocity. Parameters used: superlattice period ${\lambda }_0=14\mathrm{nm}$, scaling exponents $\beta =0.6$ and ${\beta }_v=0.3$, and initial values ${\Delta }_0^{(0)}\approx 0.02{\Delta }_1^{(0)}$ (see the text).

Figure 2

(a) Self-energy describing a gap opening at the Dirac point due to the bare sublattice-asymmetric superlattice potential [see Eq. (5)]. (b)–(d) Log-divergent diagrams contributing to gap renormalization at one loop. Here, solid black lines represent the electron Green’s function $G(ϵ,\mathbf{p})$, dashed lines represent coupling to the sublattice-asymmetric potential [the term $m_3{\sigma }_3$ in Eq. (4)], and the wavy lines represent the Coulomb interaction. (b) Vertex renormalization and (c) velocity renormalization arise from integration over $|\mathbf{b}|\lesssim |\mathbf{p}|\lesssim p_0$ (stage 1), giving Eq. (11). The contribution (d) arises from $|\mathbf{p}|<|\mathbf{b}|$ (stage 2), giving Eq. (12).