Edge Modes and Nonlocal Conductance in Graphene Superlattices

We study the existence of edge modes in gapped moiré superlattices of graphene monolayer ribbons on a hexagonal boron nitride substrate. We find that the superlattice bands acquire finite Chern numbers, which lead to a valley Hall effect. The presence of dispersive edge modes is confirmed by calculations of the band structure of realistic nanoribbons using tight binding methods. These edge states are only weakly sensitive to disorder, as short-range scattering processes lead to mean free paths of the order of microns. The results explain the existence of edge currents when the chemical potential lies within the bulk superlattice gap, and offer an explanation for existing nonlocal resistivity measurements in graphene ribbons on boron nitride.

Figure 1

Schematic representation of the even (left) and odd (right) parity scalar and vector potentials; each potential has an equivalent strength of 0.05 eV. The color scale gives the value of the local scalar potential. The vector potential describes changes in the local bond lengths due to the lattice mismatch with the substrate; lattice deformations are exaggerated by a factor of 10 for illustration purposes. Other sources of a gauge potential are also possible [42]. For details of the gauge potential, see Ref. [39].

Figure 2

(a) Subbands of a $50\times 50$ unit cell superlattice, with all perturbation potentials and an average unmodulated gap $\delta$, $(V_s^e,V_s^o,V_{\mathrm{\Delta }}^e,V_{\mathrm{\Delta }}^o,V_g^e,V_g^o,\delta )=(21,38,6,0,-42,-21,50)\mathrm{meV}$. The Chern densities for the conduction band (b) and first excited band (c) add up to Chern numbers 0 and 1, respectively.

Figure 3

(a) The band structure for nanoribbons 144 lattice sites in width, with a superlattice unit cell of $48\times 48$ graphene unit cells; we have labeled $k=k_{\parallel }d$. Flat bands localized at the nanoribbon edges lie within the average bulk gap, and produce a peak in the density of states. Most importantly, they provide conduction channels for low-energy edge transport. (b) The current distribution across nanoribbons of superlattice cell size $12\times 12$ and $48\times 48$ graphene unit cells. Assuming a small voltage and a midgap bias, only bands with energy $E_n$ in the ranges $-0.1\le E_n\le 0.1\mathrm{eV}$ and $-0.05\le E_n\le 0.05\mathrm{eV}$, respectively, are included. In both figures the superlattice parameters are the same as in Fig. 2.