Doped Twisted Bilayer Graphene near Magic Angles: Proximity to Wigner Crystallization not Mott Insulation

Abstract

We devise a model to explain why twisted bi-layer graphene (TBLG) exhibits insulating behavior when $ν=2,3$ charges occupy a unit moiré cell, a feature attributed to Mottness, but not for $ν=1$, clearly inconsistent with Mott insulation. We compute $r_s=E_U/E_K$, where $E_U$ and $E_K$ are the potential and kinetic energies, respectively, and show that (i) the Mott criterion lies at a density $10^4$ higher than in the experiments and (ii) a transition to a series of Wigner crystalline states exists as a function of $ν$. We find, for $ν=1$, $r_s$ fails to cross the threshold ($r_s = 37$) for the triangular lattice and metallic transport ensues. However, for $ν=2$ and $ν=3$, the thresholds, $r_s=22$, and $r_s=17$, respectively are satisfied for a transition to Wigner crystals (WCs) with a honeycomb ($ν=2$) and kagome ($ν=3$) structure. We believe, such crystalline states form the correct starting point for analyzing superconductivity.