Coupled electron-phonon transport and viscous thermoelectric equations

Non-diffusive, hydrodynamic-like transport of charge or heat has been observed in several materials, and recent pioneering experiments have proposed the possible emergence of electron-phonon bifluids. We introduce a first-principles computational framework to investigate these phenomena, showing that the macroscopic viscosity of electrons-phonon bifluids is microscopically determined by composite 'relaxon' electron-phonon excitations, and these excitations also describe electron-phonon drag effects on standard thermoelectric transport coefficients.

We demonstrate that these composite excitations emerge from the microscopic coupled electron-phonon Boltzmann transport equation, and show how the latter can be coarse-grained into a set of mesoscopic Viscous Thermoelectric Equations (VTE). The VTE unify the established hydrodynamic equation for electrons derived by Gurzhi [Sov. Phys. Usp. 11 (2) 1968], and the recently developed Viscous Heat Equations for phonons [Phys. Rev. X 10 (2022)], while also covering the mixed electron-phonon bifluid regime.

Using the Phoebe code (https://github.com/mir-group/phoebe), we apply this framework to predict phonon drag effects and to investigate the possibility of coupled electron-phonon hydrodynamics in several experimentally-reported materials.