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

We devise a model to explain why twisted bi-layer graphene (TBLG) exhibits insulating behavior when $\nu=2,3$ charges occupy a unit moir\'e cell, a feature attributed to Mottness, but not for $\nu=1$, 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 $\nu$. We find, for $\nu=1$, $r_s$ fails to cross the threshold ($r_s = 37$) for the triangular lattice and metallic transport ensues. However, for $\nu=2$ and $\nu=3$, the thresholds, $r_s=22$, and $r_s=17$, respectively are satisfied for a transition to Wigner crystals (WCs) with a honeycomb ($\nu=2$) and kagome ($\nu=3$) structure. We believe, such crystalline states form the correct starting point for analyzing superconductivity.