Hydrodynamic Approach to Electronic Transport in Graphene: Energy Relaxation

Abstract

In nearly compensated graphene, disorder-assisted electron-phonon scattering or “supercollisions” are responsible for both quasiparticle recombination and energy relaxation. Within the hydrodynamic approach, these processes contribute weak decay terms to the continuity equations at local equilibrium, i.e., at the level of “ideal” hydrodynamics. Here we report the derivation of the decay term due to weak violation of energy conservation. Such terms have to be considered on equal footing with the well-known recombination terms due to nonconservation of the number of particles in each band. At high enough temperatures in the “hydrodynamic regime” supercollisions dominate both types of the decay terms (as compared to the leading-order electron-phonon interaction). We also discuss the contribution of supercollisions to the heat transfer equation (generalizing the continuity equation for the energy density in viscous hydrodynamics).

Anti-Poiseuille flow in neutral graphene

“…where n I,0 = π T 2 /(3v 2 g ) is the equilibrium value of the total quasiparticle density (i.e., at μ I = 0) and τ R is the recombination time [66,67]. The hydrodynamic velocity u satisfies the generalized Navier-Stokes equation [61]…”

“…where P and η are the thermodynamic pressure and shear viscosity. The full hydrodynamic equations [51,68] also includes the thermal transport equation [66] T ∂s ∂t…”

“…Here n E ,0 denotes the equilibrium value of the energy density similarly to n I,0 (i.e., at μ I = 0) and τ RE is the energy relaxation time (due to, e.g., supercollisions [66]). The last three terms in Eq.…”

“…Quasiparticle recombination refers to any scattering process that violates the "approximate" conservation of the number of particles in each individual band including the kinematically suppressed Auger processes, three-particle collisions, scattering by optical phonons [68,75], and the disorder-assisted electron-phonon coupling (or "supercollisions") [66,[76][77][78][79][80]. The resulting quasiparticle recombination is manifested by an additional term in the continuity equation (4b) for the total quasiparticle ("imbalance") density, first established in Ref.…”

“…In the reverse process, the phonon can be absorbed by an electron in the lower band scattering into the upper band (while the impurity compensates the momentum mismatch). Unlike the Auger or three-particle processes, supercollisions also lead to energy relaxation [66]. Taking into account recombination without energy relaxation leads to a problem: the continuity equations for energy and imbalance densities allow only homogeneous solutions, which are incompatible with the boundary conditions at the channel edges.…”