Coulomb drag in graphene: Perturbation theory

Abstract

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 ${\ensuremath{\alpha}}^{2}T\ensuremath{\tau}\mathrm{min}(1,T/{\ensuremath{\mu}}_{1(2)})\ensuremath{\ll}1$ (here, $\ensuremath{\alpha}={e}^{2}/v$ is the strength of the Coulomb interaction, $T$ is temperature, ${\ensuremath{\mu}}_{1(2)}$ is the chemical potential of the two layers, and $\ensuremath{\tau}$ is the mean-free time). Our theory applies for arbitrary values of ${\ensuremath{\mu}}_{1(2)}$, $T$, 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 ${\ensuremath{\rho}}_{D}$ is a nonmonotonous function of ${\ensuremath{\mu}}_{1(2)}$ 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 ${\ensuremath{\mu}}_{1(2)}\ensuremath{\gg}v/(\ensuremath{\alpha}d)$, we recover the standard Fermi-liquid result ${\ensuremath{\rho}}_{D}^{\text{FL}}\ensuremath{\propto}{T}^{2}{({n}_{1}{n}_{2})}^{\ensuremath{-}3/2}{d}^{\ensuremath{-}4}$ (where ${n}_{i}$ 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 ${\ensuremath{\mu}}_{1(2)}$ (or the carrier densities) from its maximum value is characterized by a crossover from a logarithmic behavior ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{T}^{2}{({n}_{1}{n}_{2})}^{\ensuremath{-}1/2}\mathrm{ln}{n}_{1}$ 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 ${\ensuremath{\rho}}_{D}$ can not be described by a power law. On the contrary, the increase of the drag from the Dirac point (for ${\ensuremath{\mu}}_{1(2)}\ensuremath{\ll}v/d$) is described by the universal function of ${\ensuremath{\mu}}_{1(2)}$ measured in the units of $T$; if both layers are close to the Dirac point ${\ensuremath{\mu}}_{1(2)}\ensuremath{\ll}T$, then ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}/{T}^{2}$; in the opposite limit of low temperature, ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{T}^{2}/({\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2})$ and in the mixed case, ${\ensuremath{\mu}}_{1}\ensuremath{\ll}T\ensuremath{\ll}{\ensuremath{\mu}}_{2}$, we find ${\ensuremath{\rho}}_{D}\ensuremath{\propto}{\ensuremath{\mu}}_{1}/{\ensuremath{\mu}}_{2}$.