Experimental extraction of donor-driven spin relaxation in n -type nondegenerate germanium

Using pure spin current transport measurements in lateral spin-valve devices, we study the spin relaxation in an $n$-type nondegenerate Ge layer, which is moderately doped Ge (P: $\ensuremath{\sim}{10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$). The obtained spin diffusion length $({\ensuremath{\lambda}}_{\mathrm{Ge}})$ of the nondegenerate Ge is two to three times greater than that of heavily doped degenerate Ge (P: $\ensuremath{\sim}{10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$) in the temperature range from 8 to 100 K. We find that the electron spin lifetime ($\ensuremath{\tau}$) for the nondegenerate Ge is monotonically increased with decreasing temperature $(T)$. The increase in $\ensuremath{\tau}$ at temperatures less than 50 K can be interpreted in terms of the donor-driven spin relaxation mechanism including the 1/$\sqrt{T}$ behavior in multivalley semiconductors, proposed by Song et al. [Y. Song, O. Chalaev, and H. Dery, Phys. Rev. Lett. 113, 167201 (2014)]. We note that it is important for the $\ensuremath{\tau}$ of the moderately doped nondegenerate Ge to partly consider the $T$-independent component of spin relaxation in addition to the 1/$\sqrt{T}$ component.