Coupled electron-phonon hydrodynamics and viscous thermoelectric equations

Abstract

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 electron-phonon bifluids. 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 bifluid viscosity tensor, 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 mixed electron-phonon bifluid regime. We employ this framework to elucidate the conditions under which electron and phonon fluids can coexist and mix, rationalizing pioneering experiments on electron-phonon drag in graphite. Finally, we rely on these findings to predict smoking-gun signatures of non-diffusive behavior such as non-harmonic temperature and electric potential fields, and compressible thermoelectric backflow.

Figures (8)

• Figure 1: Influence of electron-phonon drag and doping on bulk transport coefficients in graphite. We show first-principles predictions for the in-plane Seebeck coefficient (a), electrical conductivity (b), and total thermal conductivity (c), accounting for electron-phonon drag (solid) or not (dashed)---the shaded area quantifies the contribution of electron-phonon drag. (d) and (e) show the independent components of the electron and phonon viscosity tensors ($\eta^{jl,\rm ee}_{ik}$, $\eta^{jl,\rm pp}_{ik}$, respectively) obtained from the coupled epBTE. Colors distinguish doping concentrations: $-10^{20} {\rm cm}^{-3}$ (yellow), $-2.5{\times} 10^{18} {\rm cm}^{-3}$ (blue), and $+2.5{\times} 10^{18} {\rm cm}^{-3}$ (red). Blue circles are experiments from Ref. ayache_observation_1980, and red circles from Ref. de_combarieu_thermoelectric_1973 (annealed 520K sample).
• Figure 2: Signatures of electron-phonon drag on the structure of relaxon excitations. Scatter points are contributions to the Peltier coefficient from individual relaxons, shown in (a) without drag and (b) with drag contributions to the scattering matrix, for $n=1\times10^{20} cm^{-3}$ carriers at a temperature of 70 K. The $x$ axis shows the product of velocities for relaxon $\xi$ (\ref{['eq:v2eff']}), while the $y$ axis shows its lifetime. The area of the scatter points shows the contribution of each relaxon to the Peltier coefficient, with positive (negative) contributions shown as closed (open) circles. The coupling ratio shown by the colorbar quantifies whether a relaxon is electron-dominated (blue), phonon-dominated (orange), or mixed (purple).
• Figure 3: Influence of temperature on electron-phonon drag in weakly hole-doped graphite. Contributions to the Peltier coefficient from individual relaxons as a function of increasing temperature, from (a) 50 K, (b) 80 K, (c) 110 K, for a carrier concentration of $n=2.5\times10^{18} cm^{-3}$. As in \ref{['fig:eigenvectors_drag_vs_nodrag']}, we plot the relaxation time of each relaxon against a product of relaxon velocities, with scatter point area indicating the contribution of relaxon $\xi$ to the Peltier coefficient. Positive and negative contributions are represented as closed and open circles, respectively. The colorbar shows the coupling ratio (CR), which indicates if a relaxon is primarily electron-like (blue) or phonon-like (orange). As we increase temperature across a-c, the velocity of phonon-like relaxons approaches that of the electron-like ones; this corresponds to a stronger drag effect and a larger phonon contribution to the thermoelectric coefficients.
• Figure 4: Doping-dependent heat and charge backflow, and inversion of temperature or voltage driven by electron-phonon hydrodynamics in a tunnel-chamber device made of graphite. The in-plane device directions coincide with the basal-plane directions of graphite, and the device is very long in the off-plane direction. Upper row: in-plane heat flux $\bm{Q}$ (streamlines) and temperature field $T$ (colormap) obtained from the VTE solution for high electron doping (carrier concentration $-10^{20}\; {\rm cm}^{-3}$, a) or weak hole doping ($+2.5{\cdot}10^{18}\; {\rm cm}^{-3}$, b), and from the standard DTE (c shows the strongly electron-doped case, $-10^{20}\; {\rm cm}^{-3}$, which is practically indistinguishable from the hole-doped case, see \ref{['fig:DTE_doping_effect']}). Bottom row: corresponding in-plane charge current $\bm{J}$ (streamlines) and voltage $V$ (colormap). In all cases, the boundary conditions are: a temperature difference and a potential difference are applied to the tunnel ($T = 70\pm 0.1 \mathrm{\ K}$ and $V = \mp 1\mathrm{\ mV}$ at $y =\mp 1.25 \mathrm{\ \mu m}$); the other boundaries are adiabatic, electrically insulating, and, in the VTE, "frictionless" (see text for details). In strongly electron-doped samples, the VTE predict the emergence of viscous backflow and vortices for both heat (a) and charge (d), accompanied by temperature and voltage gradients that are inverted in the chamber compared to the tunnel. In weakly hole-doped samples, the VTE predict heat vortices and temperature inversion (b), while no charge backflow and no potential inversion are found (e). The DTE always predict irrotational flow for heat and charge, and correspondingly no inversion of temperature or potential (c and f).
• Figure 5: Non-diffusive transport & non-harmonic potential. The difference between the effective potential averaged over a circular probe and the effective potential at the centre of the probe, as a function of the probe radius. Inset: outline of the tunnel-chamber geometry (adapted from dragasevic2023) and the position of the probe within the device. By the mean-value property, the average over the probe should be exactly the same as the value at the centre of the probe for a harmonic function, showing that $V_{\rm eff}$ is harmonic in the diffusive regime, but non-harmonic in the hydrodynamic regime.