Donor-Driven Spin Relaxation in Multivalley Semiconductors

The observed dependence of spin relaxation on the identity of the donor atom in $n$-type silicon has remained without explanation for decades and poses a long-standing open question with important consequences for modern spintronics. Taking into account the multivalley nature of the conduction band in silicon and germanium, we show that the spin-flip amplitude is dominated by short-range scattering off the central-cell potential of impurities after which the electron is transferred to a valley on a different axis in $k$ space. Through symmetry arguments, we show that this spin-flip process can strongly affect the spin relaxation in all multivalley materials in which time-reversal cannot connect distinct valleys. From the physical insights gained from the theory, we provide guidelines to significantly enhance the spin lifetime in semiconductor spintronics devices.

Figure 1

(a) Substitutional impurity atom in a Si crystal host. We consider typically employed group V donors such as $X=\{\mathrm{P},\mathrm{As},\mathrm{Sb}\}$. (b) Scheme of the dominant impurity-driven spin relaxation mechanism. The spin flip is governed by scattering off the central-cell potential after which the conduction electron is transferred to a valley on a different crystal axis in $k$ space. (c) Fine structure of the $1s$ state due to the central-cell potential. The scattering amplitude in (b) is governed by the impurity fine-structure parameters ${\mathrm{\Delta }}_{\text{so}}$ and ${\mathrm{\Delta }}_{\text{so}}^{\prime }$.

Figure 2

(a) Spin relaxation in heavily doped $n$-type Si for three common donor types, $\{\mathrm{P},\mathrm{As},\mathrm{Sb}\}$, at low temperature. Solid lines denote the theory results for the average spin lifetime (right axis), and symbols denote empirical values of the measured linewidth in EPR and spin injection experiments (left axis, open circle [15], open diamond [39], filled diamond [18], forward triangle [19], filled square [20], filled circle [21], star [22]). The scales on the left and right axes are related by $\delta H=1/{\gamma }_e{\tau }_s$ where ${\gamma }_e=1.7\times 10^7{\mathrm{s}}^{-1}·{\mathrm{Oe}}^{-1}$ is the electron gyromagnetic ratio. (b) Room-temperature mobility versus donor concentration showing a marginal dependence on donor identity (filled square [30], forward triangle [42], star [43], filled diamond [44], upward triangle [45], filled circle [46]). (c) Temperature dependence of ${\tau }_s$ in ${10}^{19}{\mathrm{cm}}^{-3}$ Si:P (solid line). Intervalley $f$ processes dominate the relaxation at all temperatures. (d) Doping concentration dependence of ${\tau }_s$ in Si:As at three temperatures. The decay of ${\tau }_s$ at the metallic regime ($N_d>2\times 10^{18}{\mathrm{cm}}^{-3}$) is due to transition from electron-phonon to electron-impurity dominated relaxation.