A calculation of the energy exchange between phonons and electrons is done for a metal at very low temperatures. We consider the energy exchange due to two-phonon processes. Second-order processes are expected to be important at temperatures less than 1 K. We include two different second-order processes: (i) the Compton-like scattering of phonons, and (ii) the electron-dual-phonon scattering from the second-order electron-phonon interaction. It is found that the Compton-like process contains a singular energy denominator. The singularity is removed by introducing quasiparticle damping. For pure metals we find that the energy exchange depends upon the lifetime of the electrons and it is proportional to the temperature of the lattice as $T_L^8.$ The same calculation is performed for the electron-dual-phonon scattering and it is found that the temperature dependence is $T_L^9.$ The results can be applied to quantum dot refrigerators.

• Received 28 February 1997

DOI:https://doi.org/10.1103/PhysRevB.57.7679