Coulomb drag in monolayer and bilayer graphene

We theoretically calculate the interaction-induced frictional Coulomb drag resistivity between two graphene monolayers as well as between two graphene bilayers, which are spatially separated by a distance "$d$". We show that the drag resistivity between graphene monolayers can be significantly affected by the intralayer momentum-relaxation mechanism. For energy independent intralayer scattering, the frictional drag induced by inter-layer electron-electron interaction goes asymptotically as $\rho_D \sim T^2/n^4d^6$ and $\rho_D \sim T^2/n^2d^2$ in the high-density ($k_F d \gg 1$) and low-density ($k_F d \ll 1$) limits, respectively. When long-range charge impurity scattering dominates within the layer, the monolayer drag resistivity behaves as $\rho_D \sim T^2/n^3d^4$ and $T^2 \ln (\sqrt{n} d) /n$ for $k_F d \gg 1$ and $k_F d \ll 1$, respectively. The density dependence of the bilayer drag is calculated to be $\rho_D \propto T^2/n^{3}$ both in the large and small layer separation limit. In the large layer separation limit, the bilayer drag has a strong $1/d^4$ dependence on layer separation, whereas this goes to a weak logarithmic dependence in the strong inter-layer correlation limit of small layer separation. In addition to obtaining the asymptotic analytical formula for Coulomb drag in graphene, we provide numerical results for arbitrary values of density and layer separation interpolating smoothly between our asymptotic theoretical results.