Coupled electron-phonon hydrodynamics and viscous thermoelectric equations

Non-diffusive, fluid-like transport of charge and heat has been observed in several materials, raising the question of whether they can emerge simultaneously and how they are related to bi-component electron-phonon fluids. Here we introduce a first-principles theory and computational framework to quantitatively describe these phenomena from atomistic to continuum scales in complex device geometries. Starting from the microscopic coupled electron-phonon Boltzmann transport equation, we formalize the emergence of composite "relaxon" electron-phonon excitations, show that they determine the viscosity tensors of the two fluids, and quantify the impact of electron-phonon drag on thermoelectric transport coefficients. We then demonstrate that the coupled Boltzmann equation can be coarse-grained into a set of mesoscopic Viscous Thermoelectric Equations, formally unifying Gurzhi's hydrodynamic equation for electrons [Sov. Phys. Usp., 1968] and the recently developed Viscous Heat Equations for phonons [PRX 10, 011019, 2020], while extending them to cover the intermediate regime of mixed electron and phonon fluids. We leverage this framework to elucidate how electron and phonon fluids can coexist or mix, rationalizing pioneering experiments on electron-phonon drag in graphite, and predicting smoking-gun signatures of non-diffusive behavior such as non-harmonic temperature and electric potential fields, and compressible thermoelectric backflow.