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 the drag is described by a universal function of the chemical potentials of the layers measured in the units of temperature. When both layers are tuned close to the Dirac point (i.e. when both chemical potentials are much smaller than temperature), then the drag coefficient is proportional to the product of the chemical potentials. In the opposite limit of low temperature the drag is inversely proportional to the product of the chemical potentials. In the mixed case where the chemical potentials of the two layers belong to the opposite limits we find that the drag coefficient is proportional to the smaller chemical potential and inverse proportional to the larger. For stronger interaction and larger values of the inter-layer spacing 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 (if both chemical potentials are much larger than temperature, then these are small and may be neglected). In the case of strongly doped (or gated) graphene (i.e. when the chemical potential are much larger than the inverse inter-layer spacing) 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 the spacing.
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