3 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 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. 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 FIG. 2. (Color online) Electric field dependences of drift ve- locity v x (a), hot-electron temperature T e (c) and spin relax- ation time τ (e), calculated with the equilibrium LO phonons. The ratios of the quantities v x , T e and τ with the nonequilib- rium LO phonons to those with the equilibrium ones, η v$_{x}$ , η T$_{e}$ and η τ , are shown in (b), (d) and (f), respectively. In (e), the two curves with open squares and open circles are the elec- tric field dependences of τ with a magnetic field ( B = 2 T) along the − ˆ x -direction for T L = 50 K and 300 K, respec- tively [at T L = 50 K ( T L = 300 K), when E > 0 . 05 kV/cm ( E > 0 . 3 kV/cm) the spin relaxation time coincides with that without magnetic field and thus not shown]. FIG. 1. (Color online) Typical steady-state distributions of nonequilibrium electrons [(a) and (c)] and LO phonons [(b) and (d)] in momentum space. (a) and (b): T = 200 K and E = 0 . 3 kV/cm; (c) and (d): T = 50 K and E = 1 kV/cm. Note that in (b) and (d) the LO phonon distributions n$_{q}$ are rescaled by a factor 2 and 1 $_{7}$, respectively. 0.6 0.5 0.4 0.3 0.2 0.1 0 q y (1/nm) 0.4 0 -0.4 q$_{x}$ (1/nm) 0.4 0 -0.4 ( d ) 1 $_{7}$n$_{q}$ k y (1/nm) 0.4 0.2 0 -0.2 -0.4 k$_{x}$ (1/nm) 0.4 0.2 0 -0.2 -0.4 ( c ) f$_{k}$ 1 2 q y (1/nm)$^{0.4}$ 0 -0.4 q$_{x}$ (1/nm) 0.4 0 -0.4 ( b ) 2 n$_{q}$ k y (1/nm)$^{0.4}$ 0.2 0 -0.2 -0.4 k$_{x}$ (1/nm) 0.4 0.2 0 -0.2 -0.4 ( a ) f$_{k}$ 1 2  0  0.5  1  1.5  2  0  0.2  0.4  0.6  0.8  1 v x  (10 7  cm/s) E (kV/cm) (a) T$_{L}$ =50 K 100 K 150 K 200 K 300 K  0.7  1  1.3  1.6  1.9  2.2  0  0.2  0.4  0.6  0.8  1 v x E (kV/cm) (b) T$_{L}$ =50 K 100 K 150 K 200 K 300 K  1  1.4  1.8  2.2  0  0.2  0.4  0.6  0.8  1 T e /T L E (kV/cm) (c) T$_{L}$ =50 K 100 K 150 K 200 K 300 K  1  1.2  1.4  1.6  1.8  0  0.2  0.4  0.6  0.8  1 T e E (kV/cm) (d) T$_{L}$ =50 K 100 K 150 K 200 K 300 K  2  3  4  5  6  7  0  0.2  0.4  0.6  0.8  1  (ps) E (kV/cm) (e) T$_{L}$ =50 K 100 K 150 K 200 K 300 K B=2 T, T$_{L}$ =50 B=2 T, T$_{L}$ =300 K  1  1.1  1.2  1.3  1.4  0  0.2  0.4  0.6  0.8  1 E (kV/cm) (f) T$_{L}$ =50 K 100 K 150 K 200 K 300 K