Collision-dominated nonlinear hydrodynamics in graphene

Abstract

U. BRISKOT et al. PHYSICAL REVIEW B 92, 115426 (2015) and therefore is not a conserved quantity. However, it is conserved in the collinear scattering processes and hence the corresponding relaxation rate does not contain the logarithmic enhancement. Finally, the imbalance current j I is proportional to the sign of the quasiparticle energy and to the velocity. Similarly to the electric current, it does not experience logarithmically enhanced relaxation. The imbalance current is related to the quasiparticle number or imbalance density [10] n I = n + + n − , where n + and n − are the particle numbers in the upper (conduction) and lower (valence) bands. Neglecting the Auger processes, quasiparticle recombination due to, e.g., electron-phonon interaction, and three-particle collisions due to weak coupling, one finds that n + and n − are conserved independently. In this case, which will be considered in the rest of the paper, not only the total charge density n = n + − n − , but also the quasiparticle density n I is conserved.At times longer than τ g , physical observables can be described within the macroscopic (or hydrodynamic) approach. The existence of the three slow-relaxing modes in graphene implies a peculiar two-step thermalization.Short-time electron-electron scattering (at time scales up to τ g ) establishes the so-called "unidirectional thermalization" [24]: the collinear scattering singularity implies that the electron-electron interaction is more effective along the same direction. Within linear response [18], one can express the nonequilibrium distribution function in terms of the three macroscopic currents j , j E , and j I . The currents can then be found from the macroscopic equations. The currents j and j I are not conserved and can be relaxed by the electron-electron interaction. Close to charge neutrality, the corresponding relaxation rates can be estimated as [6,40] g . These rates enter the macroscopic equations as frictionlike terms. The macroscopic linear-response theory has the same form on time scales shorter or longer than τ ee .Beyond linear response, the scattering processes characterized by the time scale τ ee play an important role in thermalizing quasiparticles moving in different directions and thus lead to establishing the local equilibrium. This is the starting point for derivation of the nonlinear hydrodynamics, which is valid at time scales much longer than τ ee . In view of conservation of the particle number, energy, and momentum, as well as independent conservation of the number of particles in the two bands in graphene, we may write the local equilibrium distribution function as [12,14] where ε λ,k = λv g k denotes the energies of the electronic states with the momentum k in the band λ = ±, μ λ (r) the local chemical potential, the local temperature is encoded in β(r) = 1/T (r), and u(r) is the hydrodynamic velocity field which we define in the following (this field should not be confused with quasiparticle velocities v). The distribution function (1) follows from the sta...

Hydrodynamics in graphene: Linear-response transport

“…A generalization of our approach to nonlinear hydrodynamics in graphene will be reported in a subsequent publication [38].…”

“…In this paper, we are focusing on kinetic coefficients describing dissipative processes in the system within linear response. The nonlinear hydrodynamics of graphene will be discussed in a separate publication [38].…”

“…V A 2) the energy and imbalance currents in the two layers have the same direction leading to positive drag. Taking into account finiteness of the corresponding relaxation rates (Appendix D) refines the theory in analogy with including viscous terms into standard hydrodynamic theory [1,38]. The resulting theory contains four differential equations for the energy and imbalance currents [cf.…”

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“…It turns out, however, that the system possesses another (nearly) conserved density which is the 'imbalance' mode corresponding to the total number of electrons and holes (as opposed to their difference related to the charge density) [240][241][242][243][244][245][246][247]. Away from the neutral (particle-hole symmetric) regime its relaxation takes place through the Auger-type processes with the still slower rate of order Г imb ~ T 4 /μ 3 .…”