  Stacked graphene multilayers with a small relative twist angle between each
of the layers have been found to host flat bands at a series of magic angles.
We consider the effect that Dirac point asymmetry between the layers, and in
particular different Fermi velocities in each layer, may have on this
phenomenon. Such asymmetry may be introduced by unequal Fermi velocity
renormalizations through Coulomb interactions with a dielectric substrate. It
also arises in an approximate way in tetralayer systems, in which the outer
twist angles are large enough that there is a dominant moire periodicity from
the stacking of the inner two layers. We find in such models that the flat band
phenomenon persists in spite of this asymmetry, and that the magic angles
acquire a degree of tunability through either controlling the screening in the
bilayer system or the twist angles of the outer layers in the tetralayer
system. Notably, we find in our models that the quantitative values of the
magic angles are increased.
