2 III. RESULTS the electron KSBEs [Eq. (1)]: 4 In Eq. (1), ρ$_{k}$ represent the density matrices of elec- trons with in-plane momentum k , whose diagonal terms ρ$_{k}$$_{σσ}$ ≡ f$_{k}$$_{σ}$ ( σ = ± 1 / 2) represent the electron distribu- tion functions and the off-diagonal ones ρ$_{k}$ 1 $_{2}$− 1 2 = ρ ∗ k − 1 2 1 2 describe the inter-spin-band correlations for the spin co- herence. ∂ρ$_{k}$ ∂t ∣ dri = e E · ∇$_{k}$ ρ$_{k}$ are the driving terms from the external electric field. ∂ρ$_{k}$ ∂t ∣ coh are the coherent terms describing the coherent spin precessions due to the effec- tive magnetic fields from the Dresselhaus term 19,22 and the Hartree-Fock Coulomb interaction, as well as the op- tional external magnetic field in the Voigt configuration. ∂ρ$_{k}$ ∂t ∣ scat stand for the scattering terms of electrons, in- cluding the electron-LO/AC phonon, electron-impurity and electron-electron Coulomb scatterings. n$_{q}$ in Eq. (2) are the distributions of quasi-2D LO phonons with in- plane momentum q . ∂n$_{q}$ ∂t ∣ scat stand for the scattering terms of the LO phonons, including the phonon-phonon and phonon-electron scatterings. Expressions of the co- herent and scattering terms of electrons are given in detail in Refs. 17 and 20, except that the electron-LO phonon scattering term should be slightly modified here as the LO phonons are described as quasi-2D (this modi- fication makes no difference when the LO phonons are in equilibrium). 10,13,14 The electron-LO phonon scattering term in Eq. (1) reads with and S$_{k}$ ( <, > ) obtained by interchanging > and < from S$_{k}$ ( >, < ). The scattering term in Eq. (2) reads In the above equations ρ < k = ρ$_{k}$ , ρ > k = 1 − ρ$_{k}$ , n < q = n$_{q}$ and n > q = n$_{q}$ + 1. ε$_{k}$ = ℏ $^{2}$k$^{2}$/ 2 m ∗ rep- resents the energy of electron with momentum k and effective mass m ∗ = 0 . 067 m$_{0}$ , and ℏ ω$_{0}$ = 35 . 4 meV is the energy of the LO phonons. 23 A is the area of the well layer. M 2 q = 1 L ∑ q z g 2 qq $_{z}$| I ( iq$_{z}$ ) | 2 is the effec- tive electron-LO phonon scattering matrix element with g 2 qq z = e $^{2}$ω 0 2 ℏ $^{2}$ϵ 0 ( q $^{2}$+ q 2 $_{$_{z}$)}$( κ − 1 ∞ − κ − 1 0 ). 23 L is the size of the sam- ple along the ˆ z -direction. κ$_{0}$ = 12 . 9 and κ$_{∞}$ = 10 . 8 are the relative static and high-frequency dielectric constants respectively, 23 and ϵ$_{0}$ is the vacuum dielectric constant. | I ( iq$_{z}$ ) | 2 = π 4 sin 2 y y $^{2}$( y $^{2}$− π $^{2}$)2 with y = aq z 2 stands for the form factor under the infinite-depth well approximation. The first term on the right hand side of Eq. (5) represents the contribution from the phonon-phonon scattering in relax- ation time approximation. n 0 q = [exp( ℏ ω$_{0}$/k$_{B}$T$_{L}$ ) − 1] − 1 is the number of quasi-2D LO phonons in equilibrium with the AC phonons. The population relaxation time τ$_{pp}$ is contributed by anharmonic lattice vibrations (es- pecially the third-order anharmonicity), which in prin- ciple depends on the phonon momentum and lattice temperature. Moreover, distinctly heated nonequilib- rium LO phonons (depending on the heating and relax- ation of electrons) may have different relaxation times. In spite of these intricate factors involved in the LO phonon relaxation, we assume τ$_{pp}$ to be a constant only depending on the lattice temperature, by adopting τ$_{pp}$ fitted by Vall´ee and Bogani from the time-resolved co- herent anti-Stokes Raman scattering experiment. 24 It gives τ$_{pp}$ ( T$_{L}$ ) = τ 0 $_{pp}$/ { 1 + [exp(0 . 2 ℏ ω$_{0}$/k$_{B}$T$_{L}$ ) − 1] − 1 + [exp(0 . 8 ℏ ω$_{0}$/k$_{B}$T$_{L}$ ) − 1] − $^{1}$} with τ 0 pp ≡ τ$_{pp}$ (0) ≈ 9 ps. 24 This formula in fact depicts the dominant decay route of an LO phonon near the center of the Brillouin zone into a transverse AC phonon and a different LO phonon at the L critical point of the Brillouin zone. The re- laxation time approximation with a constant τ$_{pp}$ related only to the lattice temperature has been widely utilized in the study of hot-electron transport with the presence of nonequilibrium LO phonons. 10–14 We numerically solve the KSBEs following the scheme mainly laid out in Ref. 17, with the rate equation of the LO phonons discreted in the momentum space in a way similar to that for electrons. The impurity den- sity is set as zero and the electric field E = − E ˆ x with E ≥ 0. No magnetic field is applied except other- wise specified. The initial conditions at time t = 0 are chosen as the steady-state solution of Eqs. (1) and (2) in the absence of the spin-orbit coupling in the co- herent term ∂ρ$_{k}$ ∂t ∣ $_{coh}$. 17 Numerically, they are prepared from a state at t = − t$_{0}$ ( t$_{0}$ > 0) with f$_{k}$$_{σ}$ ( − t$_{0}$ ) = { exp[( ℏ $^{2}$k$^{2}$/ 2 m ∗ − µ$_{σ}$ ) /k$_{B}$T$_{L}$ ] + 1 } − $^{1}$, ρ$_{k}$$_{σ}$$_{−}$$_{σ}$ ( − t$_{0}$ ) = 0 and n$_{q}$ ( − t$_{0}$ ) = n 0 $_{q}$. Here µ$_{σ}$ are the electron chem- ical potentials determined by 1 A ∑ $_{k}$Tr[ ρ$_{k}$ ( − t$_{0}$ )] = N$_{e}$ and 1 A ∑ $_{k}$Tr[ ρ$_{k}$ ( − t$_{0}$ ) σ$_{z}$ ] = N$_{e}$P$_{0}$ , where P$_{0}$ = 0 . 05 is the spin polarization and N$_{e}$ = 4 × 10 11 cm − 2 is the electron area density. With the driving from the elec- tric field and the scattering, the system reaches a steady ∂ρ$_{k}$ ∂t = ∂ρ$_{k}$ ∂t ∣ ∣ dri + ∂ρ$_{k}$ ∂t ∣ ∣ coh + ∂ρ$_{k}$ ∂t ∣ ∣ scat , (1) ∂n$_{q}$ ∂t = ∂n$_{q}$ ∂t ∣ scat . (2) ∂ρ$_{k}$ ∂t ∣ ∣$^{e}$$^{-}$$^{LO}$ scat = −{ S$_{k}$ ( >, < ) − S$_{k}$ ( <, > ) + S$_{k}$ ( >, < ) † − S$_{k}$ ( <, > ) $^{†}$} , (3) S$_{k}$ ( >, < ) = π A ∑ k ′ M 2 k − k ′ ρ > k ′ ρ < $_{k}$[ n < k ′ − $_{k}$δ ( ε$_{k}$ − ε$_{k}$ ′ + ℏ ω$_{0}$ ) + n > k − k ′ δ ( ε$_{k}$ − ε$_{k}$ ′ − ℏ ω$_{0}$ )] (4) ∂n$_{q}$ ∂t ∣ ∣ scat = − n$_{q}$ − n 0 q τ$_{pp}$ − 2 π A M 2 q ∑ k δ ( ε$_{k}$ − ε$_{k}$$_{−}$$_{q}$ − ℏ ω$_{0}$ ) × [Tr( ρ > $_{k}$ρ < k − $_{q}$) n < q − Tr( ρ < $_{k}$ρ > k − $_{q}$) n > $_{q}$] . (5)