Hot-electron effect in spin relaxation of electrically injected electrons in intrinsic Germanium

The hot-electron effect in the spin relaxation of electrically injected electrons in intrinsic Germanium is investigated by the kinetic spin Bloch equations both analytically and numerically. It is shown that in the weak-electric-field regime with $E\lesssim 0.5$~kV/cm, our calculations has reasonable agreement with the recent transport experiment in the spin-injection configuration [Phys. Rev. Lett. {\bf 111}, 257204 (2013)]. We reveal that the spin relaxation is significantly enhanced at low temperature in the presence of weak electric field $E\lesssim 50$~V/cm, which originates from the obvious center-of-mass drift effect due to the weak electron-phonon interaction, whereas the hot-electron effect is demonstrated to be less important. This can explain the discrepancy between the experimental observation and the previous theoretical calculation [Phys. Rev. B {\bf 86}, 085202 (2012)], which deviates from the experimental results by about two orders of magnitude at low temperature. It is further shown that in the strong-electric-field regime with $0.5\lesssim E \lesssim 2$~kV/cm, the spin relaxation is enhanced due to the hot-electron effect, whereas the drift effect is demonstrated to be marginal. Finally, we find that when $1.4 \lesssim E\lesssim 2$~kV/cm which lies in the strong-electric-field regime, a small fraction of electrons ($\lesssim 5\%$) can be driven from the L to $\Gamma$ valley, and the spin relaxation rates are the same for the $\Gamma$ and L valleys in the intrinsic sample without impurity. With the negligible influence of the spin dynamics in the $\Gamma$ valley to the whole system, the spin dynamics in the L valley can be measured from the $\Gamma$ valley by the standard direct optical transition method.