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state at time t = 0. After time t = 0, the spin-orbit coupling in the coherent term is switched on and elec- tron spins begin to relax with an initial spin polariza- tion P ( t = 0) = P$_{0}$ = 0 . 05. The spin relaxation time τ is obtained from the time evolution of spin polarization P ( t ) = 1 AN e ∑ $_{k}$Tr[ ρ$_{k}$ ( t ) σ$_{z}$ ], the electron drift velocity is the steady value of v ( t ) = 1 A ∑ $_{k}$Tr[ ρ$_{k}$ ( t )] ℏ k /m ∗ = v$_{x}$ ( t ) ˆ x and the hot-electron temperature T$_{e}$ is fitted out from the Boltzmann tail of the steady-state electron distribution. 17,19 

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In Fig. 1 the typical steady-state distributions of the nonequilibrium electrons and LO phonons in momentum space are plotted. It is shown by Fig. 1(a) and (c) that under the electric field along the − ˆ x -direction, the elec- trons gain a drift velocity along the ˆ x -direction. Fig- ure 1(b) and (d) show the corresponding distributions of nonequilibrium LO phonons. The LO phonons with ei- ther very large or small momenta [e.g., in the edge or center of the momentum space shown in Fig. 1(b) and (d)] are in equilibrium with the AC phonons. It is seen from the figure that the equilibrium distributions of the LO phonons are different due to the distinct lattice tem- peratures [200 K for Fig. 1(b) and 50 K for Fig. 1(d)]. It is interesting to see that the LO phonons with medi- ate momenta are driven far away from their equilibrium 

states by the hot-electrons. In the case with T$_{L}$ = 200 K and E = 0 . 3 kV/cm [Fig. 1(b)], the LO phonons with q$_{x}$ > 0 are emitted (and form a peak in the q$_{x}$ -positive momentum region) and those with q$_{x}$ < 0 are absorbed (and form a valley in the q$_{x}$ -negative momentum region). With the decrease of T$_{L}$ and/or the increase of E , the val- ley in the q$_{x}$ -negative momentum region is suppressed or even disappears, as shown in Fig. 1(d) for the case with T$_{L}$ = 50 K and E = 1 kV/cm. In any case, the total phonon density increases and a net positive momentum is gained by the LO phonons. 

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