{'fraction_non_alphanumeric': 0.025737059444924614, 'fraction_numerical': 0.00864304235090752, 'mean_word_length': 4.504228329809725, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Drift and Diffusion of Spins Generated by the Spin Hall Effect

Electrically generated spin accumulation due to the spin Hall effect is imaged in n-GaAs channels using Kerr rotation microscopy, focusing on its spatial distribution and time-averaged behavior in a magnetic field. Spatially-resolved imaging reveals that spin accumulation observed in transverse arms develops due to longitudinal drift of spin polarization produced at the sample boundaries. One- and two-dimensional drift-diffusion modeling is used to explain these features, providing a more complete understanding of observations of spin accumulation and the spin Hall effect.

The observation of current induced spin polarization 1 and the spin Hall effect 2 demonstrate all-electrical generation of spin polarization through the spin-orbit interaction. A detailed understanding of spin accumulation produced by these phenomena is a prerequisite to advancing spin-based electronics applications. Sample boundaries in spin-orbit systems are an active topic of research, affecting both the observed spin accumulation near and away from sample edges 3 and even the proper theoretical definition of a spin current 4 . For spin Hall systems in the extrinsic regime, many of these complications can potentially be avoided because the spin-orbit coupling is weak, enabling simple solutions to spin diffusion equations 5 which have been used to explain spin accumulation in a number of studies 2,3,6,7 .
Despite these expected simplifications in the analysis of the extrinsic spin Hall effect, spatially-resolved optical measurements have revealed spin accumulation which is not entirely explained by simple one-dimensional spin diffusion models 2,3 . In this Letter we measure spin accumulation with unambiguous contributions from both drift along electric fields and spin precession about an applied magnetic field. We develop a diffusion model in one-and two-dimensions which can reproduce the observations as well as allow reinterpretation of the details of recent experiments on the spin Hall effect. These results provide a more complete understanding of the behavior of electrically generated spin polarization in micron-scale devices.
The structures discussed in this manuscript are fabricated from a 2-µm thick epilayer of n-GaAs deposited on 200 nm of undoped Al 0.4 Ga 0.6 As grown by molecular beam epitaxy.
The n-GaAs is Si-doped with n = 3 × 10 16 cm −3 (at 30 K) for long spin coherence times as in Ref. 2. Using standard photolithography and wet etching, the n-GaAs is patterned into 300-µm long, 60-µm wide channels with a 60-µm long, 30-µm wide transverse arm (Fig.   1a). The n-GaAs is contacted by annealed Au/Ni/Au/Ge/Ni. The devices are mounted in a helium flow cryostat with the channel defined to be the x direction and a magnetic field B applied in the y direction. All measurements reported are at 30 K, and have been repeated in a variety of similar n-GaAs wafers and channels of various dimensions. An AC square voltage with frequency f V = 1.168 kHz is applied across the device, driving a current in the ±x direction and generating spin polarization s z in the ±z direction due to the spin Hall Spin accumulation is measured using a Kerr rotation microscope with a spatial resolution of ∼ 1 µm 8 . A linearly polarized beam from a pulsed Ti:sapphire laser tuned to λ = 821 nm is focused through a microscope objective normal to the sample surface and the polarization of the reflected beam is analyzed with a balanced photodiode bridge and lock-in amplifier operating at f V . The axis of linear polarization rotates by an angle θ K proportional to s z due to the Kerr effect. The microscope objective is scanned over the device to spatially map s z while using the reflectivity of the laser over the channel edges to calibrate the coordinate position.
Spatial profiles across the channel width (Fig. 1b) show a hyperbolic sine form for s z (y), as expected from solution of basic diffusion equations 2,5 . Spin accumulation in the channel is measured at least a few spin diffusion lengths (L s ∼ 5µm) away from the arm edge to minimize diffusive effects from the transverse arm. Sweeps of B produce a quasi-Lorentzian lineshape due to the time-averaged projection of s z known as the Hanle effect 2 (Fig. 1c).
The spin coherence time can be extracted from the half-width B 1/2 of a Lorentzian fit to the magnetic field scans, τ = /gµ B B 1/2 , where is the reduced Planck's constant, g = −0.44 is the g factor for GaAs, and µ B is the Bohr magneton. τ decreases with increasing applied E, which has been previously explained as increased D'yakonov-Perel spin scattering due to electron heating 9 . As a function of y, the central peak narrows away from the channel edge, causing an apparent increase in the extracted τ toward the center of the channel ( Fig. 1b inset). This feature is consistent with previous observations of the spin Hall effect 2,7 . Increased averaging in the current measurements also reveals that s z changes sign away from the central peak. Similarly, magnetic field scans in the transverse arm are more complicated than those in the main channel, exhibiting sign reversals and narrow central peaks ( Fig. 1d) that cannot be fit by a Lorentzian lineshape. The cause of the departures from simple Lorentzian behavior in both the channel and the arms is expected to be due to a combination of spin precession, drift, and diffusion 3 , but detailed calculations of both the spatial and B dependences of the experimental data have so far been lacking. The remainder of this Letter will present modeling necessary to understand the above observations of spin accumulation due to the spin Hall effect. In order to model spin accumulation, we assume that spin is generally conserved except for a decay with time constant τ . The sample boundaries are treated as hard walls; neither charge nor spin can penetrate sample edges, and the edge causes no spin-orbit induced spinflips. These approximations are both reasonable as GaAs is expected to be in the extrinsic limit of the SHE where spin-orbit coupling is weak. Under these assumptions, the vector spin density s i obeys a steady state continuity equation with additional spin precession and decay terms: where ω = gµ B B/ is the Larmor frequency vector and J i s is the spin current of the i-th component of spin, which is defined in the simplest fashion as a sum over all particles of the product of spin and velocity: J i s = s i v. A general treatment using a more complicated definition of spin current 4 , which is potentially necessary in the intrinsic regime, is not necessary in this case. For electrons with isotropic, spin-independent transport properties, the spin current tensor is: where µ is the charge mobility, E the electric field, D the diffusion constant, σ SH the spin Hall conductivity, and ijk the antisymmetric symbol. The first term represents drift of spin polarized carriers along the electric field, the second term governs diffusion of accumulated spin polarization, and the third is the spin current generated transverse to an electric field by the spin Hall effect.
Spin and charge coupling 5 is assumed to be negligible for simplicity, decoupling the electric field solution from the spin accumulation. This additional coupling does not appear to be necessary to capture the salient features of the experimental observations. Introducing the spin diffusion length L s = √ Dτ , the drift-diffusion equation in the channel is obtained from Eq. 1 and Eq. 2: In Eq. 3, we assume the system is in the linear transport regime, where σ SH is independent of electric field and the electron density is uniform. Because the electric field has no curl, the divergence of the spin Hall current is identically zero: For linear transport, the spin Hall current does not act as a source for spin accumulation in bulk; all spin accumulation arises from boundary accumulation at the channel edge. Recent modeling 11 has shown that bulk spin accumulation can indeed occur in a non-linear transport regime where the divergence of the spin Hall current is not identically zero; we do not study this effect in this manuscript. Eq. 3 represents a system of coupled linear differential equations for s i written entirely in terms of experimentally determined parameters.
These equations include drift and spin precession terms, both of which have demonstrable consequences in the experimental data presented in Fig. 1.
In the case of an infinitely long channel oriented along the x direction(which is applicable far from the contacts and transverse arms), s i will be independent of x and the local electric field will be a constant E = Ex, causing the drift term to vanish and validating the assumption σ SH (E) = σ SH of Eq. 3. Specializing to the case of B along y, Eq. 3 reduces to two non-trivial coupled differential equations for s x (y) and s z (y), which can be solved exactly by enforcing the boundary conditionn · J i s = 0. These diffusion equations including spin precession were solved for the case of spin accumulation in a one-dimensional system with polarized spins injected from ferromagnetic contacts 13 . The difference in this case is that the spin polarization is generated at the channel edge by the spin Hall effect. The We apply the model discussed above to two-dimensional geometries as well, focusing on the transverse arm originally investigated by Sih et al. 3 In this geometry, the drift term of the equations plays an important role. Two-dimensional finite element modeling is used to first solve for the electric field E(x, y) and then use Eqs. 1 and 2 to find the magnitude of spin polarization s z (x, y). As before, we fix L s = √ Dτ to be equal to that measured from magnetic field scans along y in Fig. 1b. Since the magnitude of the electric field, E, varies throughout the sample, we use in the model the τ (E) and µ(E) measured separately for this sample. We note that good agreement with the data is also observed by fixing τ and µ to their E = 0 values, with the reduced mobility being countered by the factor of five increase in τ . The results we present here use the measured τ (E) and µ(E), however, as this more likely describes the true dynamics of the mobile electron spins. The model accounts for both the B = 0 spatial profile and the field dependence observed in linecuts at x = 0 (Fig. 2d).