4 In Fig. 2(a) the electric field dependence of electron drift velocity v$_{x}$ at various lattice temperatures is plot- ted, with the LO phonons treated as the equilibrium ones in the calculation. The ratio of v$_{x}$ obtained with the nonequilibrium LO phonons to that with the equilibrium ones is shown in Fig. 2(b). From Fig. 2(b), one notices that when the nonequilibrium phonon effect is taken into account, v$_{x}$ can be either increased or decreased. In fact, the influence of nonequilibrium phonons on hot-electron transport consists of two competing effects: the reab- sorption of momentum from the nonequilibrium phonons tends to increase the electron mobility, while the en- hanced electron heating strengthens the electron-phonon scattering (including both the electron-AC phonon and electron-LO phonon scatterings) and tends to decrease the electron mobility. 9,11,13 Generally the former (latter) effect dominates in the regime with low (high) electric filed and/or high (low) temperature, 9,11,13 as indicated in Fig. 2(b). As mentioned previously, when the electric field is low and/or the lattice temperature is high, the val- ley of the LO phonon distribution in q$_{x}$ -negative momen- tum space is pronounced, thus it substantially suppresses the back scattering of electrons in momentum space by absorbing the q$_{x}$ -negative phonons and hence makes the electron distribution more forward-peaked. Finally, when the lattice temperature is high enough (e.g., T$_{L}$ = 300 K), the effect of nonequilibrium phonons becomes weak and less sensitive to the electric field, mainly due to the shorter LO phonon relaxation time ( τ$_{pp}$ ≈ 1 . 9 ps when T$_{L}$ = 300 K). 24 From Fig. 2(c), where the electric field dependence of the hot-electron temperature T$_{e}$ under various lattice temperatures is shown, one indeed finds that the heating of electrons by the electric field is quite obvious when the electric field is high and the lattice temperature is low. 17,19,21 From Fig. 2(d), where the ratio of T$_{e}$ ob- tained with the nonequilibrium LO phonons to that with the equilibrium ones is plotted, one finds that with the nonequilibrium phonon effect considered, electrons are further heated as expected. 11–13 Moreover, when T$_{L}$ is low, η$_{T}$ e shows a nonmonotonic behavior. That is caused by the decay of the heating efficiency with the increase of electric field in the presence of nonequilibrium phonons: With the increase of electric field, the number of the LO phonons increases and the rate of electron energy relaxation through the electron-LO phonon scattering in- creases as well. This effect is more pronounced when the lattice temperature is low, where the nonequilibrium LO phonons can be considerably accumulated with the in- crease of electric field, due to the long phonon relaxation time ( τ$_{pp}$ = 7 . 3 ps when T$_{L}$ = 50 K) 24 as well as the small equilibrium phonon distribution. The electric field dependence of the spin relaxation time τ under various temperatures is plotted in Fig. 2(e). From the figure, one notices that τ generally increases with E in the regime under investigation. Two rea- sons lead to this phenomenon: (I) Under the electric field along the − ˆ x -direction, a net effective magnetic field along the − ˆ x -direction is induced via the Dressel- haus spin-orbit coupling. 17,19 With this effective mag- netic field, spins begin to precess around it and thus the in-plane spin relaxation is mixed with the out-of-plane one. 25,26 The two-dimensional electron system in (001) GaAs quantum well has an in-plane spin relaxation rate smaller than the out-of-plane one in the framework of the D’yakonov-Perel’ relaxation mechanism. Thus when the electric field is applied, τ is increased due to the ef- fective magnetic field. 25,26 The effective magnetic field decreases with the increase of T$_{L}$ , because it is propor- tional to v$_{x}$ 17,19 and v$_{x}$ decreases with increasing T$_{L}$ [as shown in Fig. 2(a)]. Therefore the mixing of the in-plane and out-of-plane spin relaxations is obvious in the low temperature regime. In fact, when E = 0 . 05 kV/cm, the effective magnetic field is ∼ 1 T when T$_{L}$ ∼ 50-100 K and ∼ 0 . 1 T when T$_{L}$ = 300 K. As a result, the effective magnetic field causes an abrupt increase of τ [Ref. 25] with the increase of E from 0 to 0.05 kV/cm for the cases with T$_{L}$ ∼ 50-100 K but a slow increase of τ with the increase of E from 0 to 0.3 kV/cm for the case with T$_{L}$ = 300 K. (II) The heating of electrons by the electric field enhances the electron-phonon scattering and thus increases the spin relaxation time τ in the strong scat- tering limit. 4,17–21 This effect, only important in the low temperature regime where the heating effect is strong [as shown in Fig. 2(c)], is responsible for the continuing increase of τ with E when E > 0 . 05 kV/cm and T$_{L}$ ∼ 50- 100 K. To make the underlying physics depicted above more pronounced, a magnetic field B = 2 T is applied along the − ˆ x -direction for the cases with T$_{L}$ = 50 K and 300 K. The corresponding electric field dependences of τ are plotted as curves with open squares ( T$_{L}$ = 50 K) and open circles ( T$_{L}$ = 300 K) in Fig. 2(e). With this large ex- ternal magnetic field, the in-plane and out-of-plane spin relaxations are efficiently mixed even when E = 0. One finds that for the case with T$_{L}$ = 300 K, τ almost keeps unchanged with the increase of electric field, while for the case with T$_{L}$ = 50 K τ keeps on increasing with E due to the strong heating effect. Finally, it is noted that when T$_{L}$ = 300 K, there is a marginally decreasing tendency of τ with E when E is near 1 kV/cm. This is caused by the enhanced inhomogeneous broadening of the effective magnetic field from the Dresselhaus spin-orbit coupling due to the drifting and heating of the electric field. 4,17–21 This effect is easier to take place when the lattice tem- perature is high. 17,19 In Fig. 2(f), the ratio of the spin relaxation time ob- tained with the nonequilibrium LO phonons to that with the equilibrium ones is shown. With the nonequilib- rium LO phonons taken into account, τ is generally in- creased due to the strengthened electron-phonon scat- tering. Therefore the increase of τ corresponds to the increase of T$_{e}$ , as shown in Fig. 2(d). However, when the lattice temperature is high enough (e.g., T$_{L}$ = 300 K), the modification on τ induced by the nonequilibrium phonons can not be seen. Moreover, as the spin preces- sion frequency is proportional to v$_{x}$ (Refs. 17 and 19) and