Effects of electrostatic fields and charge doping on the linear bands in twisted graphene bilayers

A twisted graphene bilayer consists of two graphene monolayers rotated by an angle $\ensuremath{\theta}$ with respect to each other. Theory predicts that charge-neutral twisted graphene bilayers display a drastic reduction of their Fermi velocity ${v}_{F}$ for $0\ensuremath{\lesssim}\ensuremath{\theta}\ensuremath{\lesssim}{20}^{\ensuremath{\circ}}$ and $40\ensuremath{\lesssim}\ensuremath{\theta}\ensuremath{\lesssim}{60}^{\ensuremath{\circ}}$. In this paper we present evidence for an additional anisotropic reduction of ${v}_{F}$ in the presence of external electrostatic fields. We also discuss in quantitative detail velocity renormalization for other relevant bands in the vicinity of the $K$ point. Except for a rigid energy shift, electrostatic fields and doping by metal atoms give rise to similar renormalization of the band structure of twisted graphene bilayers.