Zero-energy modes and gate-tunable gap in graphene on hexagonal boron nitride

In this article, we derive an effective theory of graphene on a hexagonal boron nitride (h-BN) substrate. We show that the h-BN substrate generically opens a spectral gap in graphene despite the lattice mismatch. The origin of that gap is particularly intuitive in the regime of strong coupling between graphene and its substrate, when the low-energy physics is determined by the topology of a network of zero-energy modes. For twisted graphene bilayers, where inversion symmetry is present, this network percolates through the system and the spectrum is gapless. The breaking of that symmetry by h-BN causes the zero-energy modes to close into rings. The eigenstates of these rings hybridize into flat bands with gaps in between. The size of this band gap can be tuned by a gate voltage and it can reach the order of magnitude needed to confine electrons at room temperature.

Figure 1

Two-layer system made of graphene (top layer) and h-BN (bottom layer) with a lattice mismatch (exaggerated in the figure). (Red line) Moiré unit cell of the system.

Figure 2

Energy spectra in the moiré Brillouin zone for strong coupling $\varepsilon =L{\gamma }^2/\left[(V-m{v}^2)v\right]\gg 1$. (Left) $\eta =1$, as in twisted graphene bilayers [corresponding to the triangular network of zero-energy modes shown in Fig. 3a] with a metallic spectrum. (Right) $\zeta =1$ and $\eta =-0.5$ [corresponding to isolated rings of low-energy modes, as shown in Figs. 3b, 3c, 3d] with narrow bands separated by large gaps (red circle).

Figure 3

Spatial dependence of the mass term resulting from the local lattice misalignment in a bilayer as shown in Fig. 1. The red line indicates the moiré unit cell, and the $\pm$ signs specify the sign of the mass. (a) $\eta =1$: metallic state, with percolating zeros of $m^{\mathrm{eff}}$; (b) $\eta =0.9$; (c) $\eta =0.5$; (d) $\eta =-0.5$: the network of zeros breaks up into rings. In the strong coupling regime, those rings contain fully localized states (the states are localized by the periodic effective mass term in our low-energy theory, which opens a local gap in the blue regions of the above figure).

Figure 4

(Left panels) Scaling function $D(\varepsilon ,\eta ,\zeta )$ for realistic parameter values at $\theta =0$. The predicted crossover from linear scaling $D(\varepsilon ,\eta \ne 1,\zeta \ne 1)\propto \varepsilon$ at $\zeta \ne 1$ [(a) and (b)], Eq. (8), to cubic $D(\varepsilon ,\eta \ne 1,\zeta =1)\propto {\varepsilon }^3$ at $\zeta =1$ (c), Eq. (7), is confirmed. (Right) Predicted gap as a function of the interlayer bias $V$ (tunable by a perpendicular electric field) for parameter values from two fits to DFT: $\gamma =0.3\mathrm{eV}$, $\eta =-0.5$, $\zeta =1.07$ (d) and $\gamma =0.25\mathrm{eV}$, $\eta =-0.72$, $\zeta =1.19$ (e) (see text).