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The hot-electron effect in the spin relaxation of electrically injectedelectrons in intrinsic Germanium is investigated by the kinetic spin Blochequations both analytically and numerically. It is shown that in theweak-electric-field regime with $E\lesssim 0.5$~kV/cm, our calculations hasreasonable agreement with the recent transport experiment in the spin-injectionconfiguration [Phys. Rev. Lett. {\bf 111}, 257204 (2013)]. We reveal that thespin relaxation is significantly enhanced at low temperature in the presence ofweak electric field $E\lesssim 50$~V/cm, which originates from the obviouscenter-of-mass drift effect due to the weak electron-phonon interaction,whereas the hot-electron effect is demonstrated to be less important. This canexplain the discrepancy between the experimental observation and the previoustheoretical calculation [Phys. Rev. B {\bf 86}, 085202 (2012)], which deviatesfrom the experimental results by about two orders of magnitude at lowtemperature. 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 thehot-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 thestrong-electric-field regime, a small fraction of electrons ($\lesssim 5\%$)can be driven from the L to $\Gamma$ valley, and the spin relaxation rates arethe same for the $\Gamma$ and L valleys in the intrinsic sample withoutimpurity. 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 measuredfrom the $\Gamma$ valley by the standard direct optical transition method.

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