Frictional drag between quantum wells mediated by phonon exchange

We use the Kubo formalism to evaluate the contribution of acoustic-phonon exchange to the frictional drag between nearby two-dimensional electron systems. In the case of free phonons, we find a divergent drag rate $\left({\tau }_D^{-1}\right).\mathrm{}$ However, ${\tau }_D^{-1}$ becomes finite when phonon scattering from either lattice imperfections or electronic excitations is accounted for. In the case of GaAs quantum wells, we find that for a phonon mean free path ${\mathcal{l}}_{\mathrm{ph}}$ smaller than a critical value, imperfection scattering dominates and the drag rate varies as $\ln ({\mathcal{l}}_{\mathrm{ph}}/d)$ over many orders of magnitude of the layer separation $d.\mathrm{}$ When ${\mathcal{l}}_{\mathrm{ph}}$ exceeds the critical value, the drag rate is dominated by coupling through an electron-phonon collective mode localized in the vicinity of the electron layers. We argue that the coupled electron-phonon mode may be observable for realistic parameters. Our theory is in good agreement with experimental results for the temperature, density, and $d$ dependence of the drag rate.