Anisotropy-Driven Spin Relaxation in Germanium

A unique spin depolarization mechanism, induced by the presence of $g$-factor anisotropy and intervalley scattering, is revealed by spin-transport measurements on long-distance germanium devices in a magnetic field longitudinal to the initial spin orientation. The confluence of electron-phonon scattering (leading to Elliott-Yafet spin flips) and this previously unobserved physics enables the extraction of spin lifetime solely from spin-valve measurements, without spin precession, and in a regime of substantial electric-field-generated carrier heating. We find spin lifetimes in Ge up to several hundreds of nanoseconds at low temperature, far beyond any other available experimental results.

Figure 1

(a) The first Brillouin zone of germanium, showing isoenergetic surfaces of the four equivalent $L$-point conduction band valleys (green). In an external magnetic field $\overrightarrow{B}$, $g$-factor anisotropy causes electrons to experience an additional field oriented along the valley axis (red). (b) Equivalent magnetic field experienced by conduction electrons during intervalley scattering, where the four shorter (red) arrows represent a randomly fluctuating component. (c) Ordinary “spin-valve” effect in a spin-transport device is shown above the $B$ axis. The red solid and blue dashed curves are, respectively, the spin signals when the magnetic field is swept from negative to positive and vice versa to orient the injector to detector magnetization configuration parallel (P, higher signal level) or antiparallel (AP, lower signal level). Below it is the expected spin-valve effect in the presence of $g$-factor anisotropy and intervalley scattering, where signal decay with increasing field intensity is evident.

Figure 2

Experimental and simulated spin signal $\Delta I_{C2}$ vs applied magnetic field $B$, in an accelerating electric field caused by a voltage of $V_{C1}=0.6\mathrm{V}$ over the $325\mu \mathrm{m}$ transport distance in undoped Ge at a temperature of 41 K. Panel (a) shows the spin-valve effect for the $\overrightarrow{B}$ field along both $⟨110⟩$ (in blue) and $⟨100⟩$ (in red) due to switching injector or detector magnetizations in an in-plane $B$ field. The round (cross) markers are experimental results when $B$ is swept in the positive (negative) direction, with the solid (dashed) curve corresponding to theoretical simulation. Panel (b) shows coherent spin precession in an out-of-plane $B$ field. The diamond markers are experiment results. The solid (dashed) curve is the theoretical simulation with (without) the $g$-factor anisotropy-induced depolarization [Eq. (4)].

Figure 3

(a) Minor-loop-derived spin polarization at zero magnetic field as a function of accelerating voltage ($V_{C1}$) across the $325\mu \mathrm{m}$ undoped Ge spin-transport channel. A typical minor-loop measurement is shown in the inset. Comparison to the expected behavior from transit time depolarization with a static spin lifetime indicates electron heating and enhancement of intervalley scattering. (b) Spin valves with a $⟨110⟩$-oriented in-plane $B$ field for various electric fields showing the evolution of the anisotropy-driven depolarization due to shorter transit times and reduction in $g$-factor anisotropy-induced spin lifetime.

Figure 4

Temperature dependence of spin lifetime in undoped Ge. Low-field measurements at $V_{C1}=0.2\mathrm{V}$ yield a monotonically increasing spin lifetime with decreasing temperature, similar to the Elliott-Yafet theoretical prediction from Ref. 23. Electric-field-induced intervalley scattering causes enhanced suppression of lifetimes extracted from fitting the effects of Eq. (4) to the spin-valve measurement in a $⟨110⟩$-oriented longitudinal magnetic field at accelerating voltages $V_{C1}=0.9$ and 1.5 V. For comparison, lifetimes obtained by correlating transit time (Larmor clock from spin precession measurements) and zero-magnetic-field polarization (from minor-loop spin-valve measurements) show low-temperature depolarization inconsistent with Ref. 23.