Coulomb drag in graphene: Perturbation theory

We study the effect of Coulomb drag between two closely positioned graphene monolayers, assuming that transport properties of the sample are dominated by disorder. This assumption allows us to develop a perturbation theory in α²Tτmin(1,T/μ₁₍₂₎)1 (here, α=e²/v is the strength of the Coulomb interaction, T is temperature, μ₁₍₂₎ is the chemical potential of the two layers, and τ is the mean-free time). Our theory applies for arbitrary values of μ₁₍₂₎, T, and the interlayer separation d, although we focus on the experimentally relevant situation of low temperatures $T<v/d$. We find that the drag coefficient ρD is a nonmonotonous function of μ₁₍₂₎ and T in qualitative agreement with experiment [S. Kim et al., Phys. Rev. B 83, 161401(R) (2011)]. Precisely at the Dirac point, drag vanishes due to electron-hole symmetry. For very large values of chemical potential μ₁₍₂₎v/(αd), we recover the standard Fermi-liquid result ρDFL∝T²(n₁n₂)^-3/2d^-4 (where nᵢ is the carrier density in the two layers). At intermediate values of the chemical potential, the drag coefficient exhibits a maximum. The decrease of the drag coefficient as a function of μ₁₍₂₎ (or the carrier densities) from its maximum value is characterized by a crossover from a logarithmic behavior ρD∝T²(n₁n₂)^-1/2lnn₁ to the Fermi-liquid result. Our results do not depend on the microscopic model of impurity scattering. The crossover occurs in a wide range of densities, where ρD can not be described by a power law. On the contrary, the increase of the drag from the Dirac point (for μ₁₍₂₎v/d) is described by the universal function of μ₁₍₂₎ measured in the units of T; if both layers are close to the Dirac point μ₁₍₂₎T, then ρD∝μ₁μ₂/T²; in the opposite limit of low temperature, ρD∝T²/(μ₁μ₂) and in the mixed case, μ₁Tμ₂, we find ρD∝μ₁/μ₂.