  Insight into why superconductivity in pristine and doped monolayer graphene
seems strongly suppressed has been central for the recent years' various
creative approaches to realize superconductivity in graphene and graphene-like
systems. We provide further insight by studying electron-phonon coupling and
superconductivity in doped monolayer graphene and hexagonal boron nitride based
on intrinsic phonon modes. Solving the graphene gap equation using a detailed
model for the effective attraction based on electron tight binding and phonon
force constant models, the various system parameters can be tuned at will.
Consistent with results in the literature, we find slight gap modulations along
the Fermi surface, and the high energy phonon modes are shown to be the most
significant for the superconductivity instability. The Coulomb interaction
plays a major role in suppressing superconductivity at realistic dopings.
Motivated by the direct onset of a large density of states at the Fermi surface
for small charge dopings in hexagonal boron nitride, we also calculate the
dimensionless electron-phonon coupling strength there, but the comparatively
large density of states cannot immediately be capitalized on, and the charge
doping necessary to obtain significant electron-phonon coupling is similar to
the value in graphene.
