  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 twisted trilayer graphene (MA-TTG). The absence of
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 inter-sublattice
electron-phonon matrix elements. We also obtain 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.
