Local density of states of electron-crystal phases in graphene in the quantum Hall regime

Abstract

We calculate, within a self-consistent Hartree-Fock approximation, the local density of states for different electron crystals in graphene subject to a strong magnetic field. We investigate both the Wigner crystal and bubble crystals with ${M}_{\text{e}}$ electrons per lattice site. The total density of states consists of several pronounced peaks, the number of which in the negative energy range coincides with the number of electrons ${M}_{\text{e}}$ per lattice site, as for the case of electron-solid phases in the conventional two-dimensional electron gas. Analyzing the local density of states at the peak energies, we find particular scaling properties of the density patterns if one fixes the ratio ${\ensuremath{\nu}}_{N}/{M}_{\text{e}}$ between the filling factor ${\ensuremath{\nu}}_{N}$ of the last partially filled Landau level and the number of electrons per bubble. Although the total density profile depends explicitly on ${M}_{\text{e}}$, the local density of states of the lowest peaks turns out to be identical regardless the number of electrons ${M}_{\text{e}}$. Whereas these electron-solid phases are reminiscent of those expected in the conventional two-dimensional electron gas in GaAs heterostructures in the quantum Hall regime, the local density of states and the scaling relations we highlight in this paper may be, in graphene, directly measured by spectroscopic means, such as, e.g., scanning tunneling microscopy.