Piezoelectric surface acoustical phonon limited mobility of electrons in graphene on a GaAs substrate

We study the mobility of Dirac fermions in monolayer graphene on a GaAs substrate, restricted by the combined action of the extrinsic potential of piezoelectric surface acoustical phonons of GaAs (PA) and of the intrinsic deformation potential of acoustical eigen-phonons in graphene (DA). In the high temperature ($T$) regime the momentum relaxation rate exhibits the same linear dependence on $T$ but different dependences on the carrier density $n$, corresponding to the mobility $\mu\propto 1/\sqrt{n}$ and $1/n$, respectively for the PA and DA scattering mechanisms. In the low $T$ Bloch-Gr\"uneisen regime, the mobility shows the same square-root density dependence, $\mu\propto \sqrt{n}$, but different temperature dependences, $\mu\propto T^{-3}$ and $ T^{-4}$, respectively for PA and DA phonon scattering.