  Graphene moire superlattices are outstanding platforms to study correlated
electron physics and superconductivity with exceptional tunability. However,
robust superconductivity has been measured only in magic-angle twisted bilayer
graphene (MA-TBG) and magic-angle twisted trilayer graphene (MA-TTG). The
absence of a superconducting phase in certain moire flat bands raises a
question on the superconducting mechanism. In this work, we investigate
electronic structure and electron-phonon coupling in graphene moire
superlattices based on atomistic calculations. We show that electron-phonon
coupling strength lambda is dramatically different among graphene moire flat
bands. The total strength lambda is very large (lambda>1) for MA-TBG and
MA-TTG, both of which display robust superconductivity in experiments. However,
lambda is an order of magnitude smaller in twisted double bilayer graphene
(TDBG) and twisted monolayer-bilayer graphene (TMBG) where superconductivity is
reportedly rather weak or absent. We find that the Bernal-stacked layers in
TDBG and TMBG induce sublattice polarization in the flat-band states,
suppressing intersublattice electron-phonon matrix elements. We also obtain the
nonadiabatic superconducting Tc that matches well with the experimental
results. Our results clearly show a correlation between strong electron-phonon
coupling and experimental observations of robust superconductivity.
