Graphene on incommensurate substrates: Trigonal warping and emerging Dirac cone replicas with halved group velocity

The adhesion of graphene on slightly lattice-mismatched surfaces, for instance, of hexagonal boron nitride (hBN) or Ir(111), gives rise to a complex landscape of sublattice symmetry-breaking potentials for the Dirac fermions. Whereas a gap at the Dirac point opens for perfectly lattice-matched graphene on hBN, we show that the small lattice incommensurability prevents the opening of this gap and rather leads to a renormalized Dirac dispersion with a trigonal warping. This warping breaks the effective time-reversal symmetry in a single valley. On top of this an additional set of massless Dirac fermions is generated, which is characterized by a group velocity that is about half the one of pristine graphene.

Figure 1

(a) Renormalization of the Fermi velocity at the Dirac point as a function of $V_0$. The points are the results of the exact diagonalization of the low-energy Hamiltonian whereas the continuous line is the analytical result from second-order perturbation theory. (b) Energy of the Dirac point as a function of the external potential amplitude $V_0$. (c), (d) Fermi lines for different values of the potential amplitude $V_0$ close to the ${\mathbf{K}}_{+}$ (c) and the ${\mathbf{K}}_{-}$ (d) valleys. Energies and wave vectors are in units of $\hslash v_F\tilde{G}$ and $\tilde{G}$, respectively.

Figure 2

Energy dispersion relations for a graphene triangular moiré superlattice with $V_0=0.1\hslash v_F^0\tilde{G}$ close to the two graphene valleys ${\mathbf{K}}_{+}$ (a) and ${\mathbf{K}}_{-}$ (b). The arrows indicate the corners ${\tilde{\mathbf{K}}}_{\pm }$ of the SBZ where the Dirac cone replicas are generated. (c) Topology of the Fermi lines close to the emergent Dirac cones. Units are the same as in Fig. 1.

Figure 3

Behavior of the Dirac point energy ${\epsilon }_D({\mathbf{K}}_{\pm },{\tilde{\mathbf{K}}}_{\mp })$ (a) as a function of the external potential amplitude $V_0$. The continuous line is the result of the degenerate perturbation theory whereas the points are the result of the exact diagonalization. (b) Group velocity at the Dirac point replicas as a function of the strength of the potential $V_0$. (c), (d) Fermi lines close to the emergent Dirac points ${\epsilon }_D({\mathbf{K}}_{+},{\tilde{\mathbf{K}}}_{-})$ (c) and ${\epsilon }_D({\mathbf{K}}_{-},{\tilde{\mathbf{K}}}_{+})$ (d) for different values of the amplitude $V_0$. Units are the same as in Fig. 1.