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 alpha T-2 tau min(1, T/mu(1(2))) << 1 (here, alpha = e(2)/v is the strength of the Coulomb interaction, T is temperature, mu(1(2)) is the chemical potential of the two layers, and tau is the mean-free time). Our theory applies for arbitrary values of mu(1(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 rho(D) is a nonmonotonous function of mu(1(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 mu(1(2)) >> v/(alpha d), we recover the standard Fermi-liquid result rho(FL)(D) alpha T-(n1n2)(2)-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 mu(1(2)) (or the carrier densities) from its maximum value is characterized by a crossover from a logarithmic behavior rho(D) alpha T-(n1n2)(2)-1/2 ln n(1) 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 rho(D) can not be described by a power law. On the contrary, the increase of the drag from the Dirac point (for mu(1(2)) << v/d) is described by the universal function of mu 1(2) measured in the units of T; if both layers are close to the Dirac point mu(1(2)) << T, then rho(D) alpha mu(1)mu(2)/T-2; in the opposite limit of low temperature, rho(D) alpha T-2/(mu(1)mu(2)) and in the mixed case, mu(1) << T << mu(2), we find rho(D) alpha mu(1)/mu(2).