Coulomb drag in graphene: perturbation theory

We study the effect of Coulomb drag between two closely positioned graphene monolayers. In the limit of weak electron-electron interaction and small inter-layer spacing ($\mu_{1(2)}, T\ll v/d$) the drag is described by a universal function of the chemical potentials of the layers $\mu_{1(2)}$ measured in the units of temperature $T$. When both layers are tuned close to the Dirac point, then the drag coefficient is proportional to the product of the chemical potentials $\rho_D\propto\mu_1\mu_2$. In the opposite limit of low temperature the drag is inversely proportional to both chemical potentials $\rho_D\propto T^2/(\mu_1\mu_2)$. In the mixed case where the chemical potentials of the two layers belong to the opposite limits $\mu_1\ll T\ll\mu_2$ we find $\rho_D\propto \mu_1/\mu_2$. For stronger interaction and larger values of $d$ the drag coefficient acquires logarithmic corrections and can no longer be described by a power law. Further logarithmic corrections are due to the energy dependence of the impurity scattering time in graphene (for $\mu_{1(2)}\gg T$ these are small and may be neglected). In the case of strongly doped (or gated) graphene $\mu_{1(2)}\gg v/d\gg T$ the drag coefficient acquires additional dependence on the inter-layer spacing and we recover the usual Fermi-liquid result if the screening length is smaller than $d$.