  We investigate the many-body properties of graphene on top of a piezoelectric
substrate, focusing on the interaction between the graphene electrons and the
piezoelectric acoustic phonons. We calculate the electron and phonon
self-energies as well as the electron mobility limited by the substrate
phonons. We emphasize the importance of the proper screening of the
electron-phonon vertex and discuss the various limiting behaviors as a function
of electron energy, temperature, and doping level. The effect on the graphene
electrons of the piezoelectric acoustic phonons is compared with that of the
intrinsic deformation acoustic phonons of graphene. Substrate phonons tend to
dominate over intrinsic ones for low doping levels at high and low
temperatures.
