Renormalized Magic Angles in Twisted Asymmetric Graphene Bilayers*

The process of stacking graphene multilayers with a small relative twist angle between each of the layers has been found to produce flat bands at a series of “magic angles.” A number of recent studies have investigated  the  effects  that  varying  the  number  of  layers or changing the stacking arrangement (AA/AB/ABC etc.)  has  on  the  values  of  the  magic  angles. Here, by contrast, we consider how magic angles and flat bands are affected when the Fermi velocity at the Dirac points at the corners of a moire Brillouin zone are different.  This can occur through differing Fermi velocity renormalizations due to differing dielectrics near the two layers.  Such models are also approximately realized in 4-layer structures where the twist angles are not all the same. We model each system using a low-energy theory based on a renormalized Bistritzer-MacDonald model that involves two different renormalized Fermi velocities.  We find that our renormalized BM model supports significantly larger magic angles than the magic angles that are present in TBG.