5 IV. COUCLUSION ACKNOWLEDGMENTS v$_{x}$ is affected largely by the nonequilibrium LO phonons, the spin precession frequency has a modification with the magnitude roughly proportional to the change of v$_{x}$ . In Fig. 3(a) we show the typical spin precession signals with the equilibrium and nonequilibrium LO phonons, respec- tively. In Fig. 3(b), the typical ratio of spin precession frequency obtained with the nonequilibrium LO phonons to that with the equilibrium ones is plotted in the region of the electric field where v$_{x}$ is large enough that spin precession signal with clear periods can be distinguished. Finally, further calculations show that the effect of nonequilibrium phonons on electron mobility, electron heating and electron spin relaxation decays with the decrease of electron density and the increase of impu- rity density. This can be easily understood as the LO phonons are driven to the nonequilibrium states by elec- trons and the increase of the electron-impurity scattering suppresses the effect caused by the electron-LO phonon scattering. In this work, we have studied the effect of nonequi- librium LO phonons on hot-electron spin relaxation in n -type (001) GaAs quantum wells. Under an in-plane electric field, the LO phonons can be driven away from their equilibrium states by electrons and then in turn af- fect the electron transport, electron heating as well as electron spin relaxation. In the presence of the nonequilibrium LO phonons, the electron drift velocity under electric field can be ei- ther increased (at low electric field and/or high lattice temperature) or decreased (at high electric field and/or low lattice temperature). This phenomenon is caused by the two competing effects: the momentum reabsorp- tion from phonons and the strengthened electron-phonon scattering. 9,11,13 The former tends to increase the elec- tron mobility whereas the latter tends to suppress it. The nonequilibrium LO phonons also impede the energy re- laxation of electrons and thus the electrons are further heated, especially when the lattice temperature is low. The nonequilibrium LO phonons effectively affect the hot-electron spin relaxation through the strengthening of the electron-phonon scattering, which tends to increase the spin relaxation time. This effect also dominates in the low temperature regime. Moreover, as the spin pre- cession frequency under the electric field is proportional to the electron drift velocity, it can be either increased or decreased when the nonequilibrium LO phonons are taken into account. Finally, it should be noticed that the effect of nonequilibrium phonons is more pronounced in systems with high electron density and low impurity density. FIG. 3. (Color online) (a) Typical spin precession signals cal- culated with the equilibrium and nonequilibrium LO phonons. Note that the absolute value of spin polarization | P | is in the logscale. T L = 100 K and E = 0 . 3 kV/cm. (b) The ratio of spin precession frequency obtained with the nonequilib- rium LO phonons to that with the equilibrium ones, shown at three typical temperatures in the electric field region where the spin precession signal can be clearly distinguished. This work was supported by the Natural Science Foundation of China under Grant No. 10725417, the National Basic Research Program of China under Grant No. 2006CB922005 and the Knowledge Innovation Project of Chinese Academy of Sciences. 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1  0  10  20  30  40  50 | P | t (ps) (a) T$_{L}$ =100 K, E=0.3 kV/cm Nonequilibrium LO phonons Equilibrium LO phonons  0.8  1  1.2  1.4  1.6  1.8  2  2.2  0  0.2  0.4  0.6  0.8  1 E (kV/cm) (b) T$_{L}$ =50 K 100 K 150 K ∗ Author to whom correspondence should be addressed; mwwu@ustc.edu.cn. 1 D. D. Awschalom, D. Loss, and N. Samarth, Semicon- ductor Spintronics and Quantum Computation (Springer, Berlin, 2002). 2 I. ˇ Zuti´ c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76 , 323 (2004); J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. ˇ Zuti´ c, Acta Phys. Slovaca 57 , 565 (2007); and references therein. 3 M. I. D’yakonov, Spin Physics in Semiconductors (Springer, Berlin, 2008). 4 M. W. Wu, J. H. Jiang, and M. Q. Weng, arXiv:1001.0606; and references therein. 5 M. I. D’yakonov and V. I. Perel’, Zh. ´ Eksp. Teor. Fiz. 60 , 1954 (1971) [Sov. Phys. JETP 33 , 1053 (1971)]. 6 G. L. Bir, A. G. Aronov, and G. E. Pikus, Zh. ´ Eksp. Teor. Fiz. 69 , 1382 (1975) [Sov. Phys. JETP 42 , 705 (1976)]. 7 R. J. Elliott, Phys. Rev. 96 , 266 (1954). 8 M. W. Wu, N. J. M. Horing, and H. L. Cui, Phys. Rev. B 54 , 5438 (1996). 9 R. Mickeviˇ cius, V. Mitin, G. Paulaviˇ cius, V. Kochelap, M. A. Stroscio, and G. J. Lafrate, J. Appl. Phys. 80 , 5145 (1996). 10 X. L. Lei and N. J. M. Horing, Phys. Rev. B 35 , 6281 (1987). 11 J. C. Vaissiere, J. P. Nougier, L. Varani, P. Houlet, L.