Linear-response theory of Coulomb drag in coupled electron systems

We report a fully microscopic theory for transconductivity, or, equivalently, momentum transfer rate, of Coulomb coupled electron systems. We use the Kubo linear-response formalism, and our main formal result expresses the transconductivity in terms of two fluctuation diagrams, which are topologically related, but not equivalent to, the Aslamazov-Larkin diagrams known from superconductivity. Results reported elsewhere are shown to be special cases of our general expression; specifically, we recover the Boltzmann equation result in the semiclassical clean limit, and the memory function results for dirty systems with constant impurity scattering rates. Furthermore, we show that for energy-dependent relaxation times, the final result is not expressible in terms of standard density-response functions. Other results include (i) at T=0, the frequency dependence of the transfer rate is found to be proportional to Ω and ${\mathrm{\Omega }}^2$ for frequencies below and above the impurity scattering rate, respectively, and (ii) the weak localization correction to the transconductivity is given by δ${\mathrm{\sigma }}_{21}^{\mathrm{WL}}$∝δ${\mathrm{\sigma }}_{11}^{\mathrm{WL}}$+δ${\mathrm{\sigma }}_{22}^{\mathrm{WL}}$.