Abstract
The generation and manipulation of carrier spin polarization in semiconductors solely by electric fields has garnered significant attention as both an interesting manifestation of spin–orbit physics as well as a valuable capability for potential spintronics devices^{1,2,3,4}. One realization of these spin–orbit phenomena, the spin Hall effect^{5,6}, has been studied as a means of allelectrical spincurrent generation and spin separation in both semiconductor and metallic systems. Previous measurements of the spin Hall effect^{7,8,9,10,11} have focused on steadystate generation and timeaveraged detection, without directly addressing the accumulation dynamics on the timescale of the spincoherence time. Here, we demonstrate timeresolved measurement of the dynamics of spin accumulation generated by the extrinsic spin Hall effect in a doped GaAs semiconductor channel. Using electrically pumped timeresolved Kerr rotation, we image the accumulation, precession and decay dynamics near the channel boundary with spatial and temporal resolution and identify multiple evolution time constants. We model these processes with timedependent diffusion analysis using both exact and numerical solution techniques and find that the underlying physical spincoherence time differs from the dynamical rates of spin accumulation and decay observed near the sample edges.
Main
Theories have predicted^{5,6,12,13}, and experiments confirmed^{7,9}, that an electric current in a crystal with spin–orbit coupling gives rise to a transverse spin current through the spin Hall effect (SHE). Spindependent scattering of carriers by charged impurities (the extrinsic SHE)^{5,6,14} or the direct effect of spin–orbit coupling on the band structure (intrinsic SHE)^{12,13} causes spindependent splitting in momentum space and a resulting pure spin current. Although not locally observable, the presence of this bulk spin current can be inferred from the existence of nonequilibrium spin accumulation near sample boundaries. Whereas extrinsic spin Hall currents generated by impurity scattering evolve on momentum scattering timescales (<1 ps), spin Hall accumulation is expected to develop on the much slower spincoherence timescale τ (∼1 ns). As this timescale is of the same order as that desired for fast electrical manipulation of spin polarization in spintronics devices, understanding dynamics on this timescale is critical for both physical and practical insights into the extrinsic SHE processes.
Steadystate observations of electrically generated spin accumulation^{7,15,16} are effective for inferring τ, but they cannot directly access the dynamical processes on the nanosecond timescale. In contrast, timeresolved spin dynamics with picosecond resolution are routinely measured using ultrafast optical pump–probe techniques^{17,18}. Time resolution of bulk currentinduced spin polarization was achieved using a photoconductive switch^{15}, but only precessional dynamics were observed owing to the short duration of the ultrafast current pulse. Furthermore, in contrast to the boundary accumulation from the SHE, currentinduced spin polarization is a bulk phenomenon and consequently neither the steadystate^{4,15} nor the timeresolved^{15} measurements investigated spatial dynamics near the sample edge. Here, we combine the spatial resolution afforded by scanning Kerr microscopy^{19} with an optical probe pulse delayed relative to the electrical pump pulse to achieve both temporal and spatial resolution of spin polarization generated electrically by the extrinsic SHE in an ndoped GaAs channel. Details of the experimental technique are shown in Fig. 1 and are discussed in the Methods section.
Figure 2b–d shows the Kerr rotation θ_{K}(t) as a function of delay time t in a fixed magnetic field B applied along the y axis. The spin polarization is generated by the SHE from a voltage pulse V (t) with length t_{p}=6 ns and amplitude V_{0}=2 V (Fig. 2a). The laser is positioned at y=126 μm so as to be close to the boundary spin generation with minimal clipping of the spot by the edge. During the pulse (0<t<t_{p}), spin polarization builds up owing to the SHE. After the pulse has passed (t>t_{p}), the accumulated spin polarization undergoes decay and precession (Fig. 2b–d). We fit θ_{K}(t) for t>t_{p} to an exponentially decaying cosine to extract an inhomogeneous depolarization time τ^{*} and Larmor precession frequency ν_{L}=g μ_{B}B/h, where g is the Lande gfactor, μ_{B} is the Bohr magneton and h is Planck’s constant. Linear fits to ν_{L}(B) give g=0.346±0.002 (Fig. 2e), which is consistent with g expected for this doping level^{20}. The depolarization time τ^{*}=2.8±0.1 ns measured at a specific spatial location should differ from the physical spindecoherence time τ. Because spin polarization can diffuse owing to accumulation gradients, there is a second pathway for spin depolarization beyond decoherence that depends strongly on the measurement location. We reconcile the dynamically measured τ^{*} with the intrinsic decoherence time τ later in our discussion.
Typical optical studies of electrically generated spin accumulation measure the timeaveraged projection of spin precession as a function of applied transverse magnetic field B. In analogy with the Hanle effect of luminescence depolarization, the zaxis spin polarization s_{z} should depolarize for increasing B when 2πν_{L}∼τ^{−1} (ref. 21). Therefore, coherence times in steadystate experiments are typically extracted from the linewidths of s_{z}(B). Near the sample edge, s_{z}(B) is a Lorentzian line shape analogous to the Hanle effect^{7}, whereas it becomes more complicated away from the edges owing to the interplay of spin precession and diffusion^{11,21}. In the current experiment, measurement of θ_{K}(B) does not represent a timeaveraged steadystate accumulation, but rather a snapshot at a fixed time t of the dynamic behaviour of an electrically generated spin ensemble in a magnetic field.
Representative scans of θ_{K}(B) at y=126 μm are shown for t=2, 5 and 9 ns in Fig. 2f–h with the magnetic field applied along the y axis. For small t, θ_{K}(B) grows in a broad peak that narrows as t increases (Fig. 2f). Only for t∼t_{p}>τ does θ_{K}(B) approach the Lorentzian line shape expected from a conventional Hanle analysis (Fig. 2g). For t>t_{p}, θ_{K}(B) is primarily governed by spin precession, exhibiting characteristic periodic lobes of decreasing amplitude away from B=0 (Fig. 2h).
In Fig. 3a, we use a longer pulse t_{p}=15 ns to investigate accumulation dynamics with the current flowing for various V_{0}. We fit θ_{K}(B=0) to an exponential saturation with a time constant τ_{acc}. For each V_{0}, τ_{acc} is around 40% of the τ^{*} measured from decay of the spin polarization (Fig. 3b). Both τ^{*} and τ_{acc} decrease weakly with V_{0}, which is expected owing to electron heating^{11,22}.
We characterize the magneticfield line shapes by their inverse halfwidth B_{1/2}^{−1}, which increases with t before quickly saturating (Fig. 3a). We can understand this evolution of B_{1/2} in a simple physical picture^{21}. Soon after the pulse turns on at t=0, spins are all recently generated at the sample edge and have had little time for spin precession about B. For later times, spins have a larger spread in generation times (up to t_{p}) and have correspondingly more time for precession; hence, there is more depolarization for a given B and the Hanle curve narrows (Fig. 2g). For t≫τ, the average precession time is governed by τ rather than t and the Hanle width becomes constant as in a steadystate measurement. The coherence times τ_{1/2} calculated from the saturation of B_{1/2} near t∼t_{p} agree with decay times τ^{*} and are consequently also longer than the accumulation times τ_{acc} (Fig. 3b). Diffusion analysis of the SHE accumulation is necessary to reconcile the observed differences between the timescales τ_{acc}, τ^{*} and τ_{1/2}.
Spin accumulation from the SHE can be modelled using drift–diffusion equations when the spincoherence time is much longer than the momentum scattering time^{21,23}. The extrinsic SHE of the nGaAs system is well suited to diffusive analysis because of its low spin–orbit coupling and relatively long spincoherence time^{11,23}. We treat our GaAs channel of width w as infinitely long because its length l is much larger than the spin diffusion length L_{s}=3.9 μm found from steadystate measurements at B=0. This assumption reduces the problem to only one spatial dimension and precludes the need for general twodimensional modelling including spin drift^{11}. The SHE generates a spin current transverse to the inplane electric field and proportional to the spin Hall conductivity σ_{SH}, j_{j}^{i}=σ_{SH}ε^{ijk}E_{k}. The total current of the i spin component along y from both diffusion and the SHE is j_{y}^{i}=−D ∂_{y}s_{i}−σ_{SH}E δ_{i z}, where D=L_{s}^{2}/τ is the spin diffusion constant. For the extrinsic SHE (weak spin–orbit coupling), the spin decoherence at rate τ^{−1} is slow relative to momentum scattering, and s(y,t) will obey an approximate continuity equation including decay and precession terms:
The weak spin–orbit coupling of the GaAs system enables spin conserving hardwall boundary conditions normal to the edges at y=±w/2 (j_{y}^{i}=−D ∂_{y}s^{i}−σ_{SH}E δ_{i z}=0). Boundary conditions accounting for spin–orbit effects at the sample edge (such as in the intrinsic SHE) would require modification for effects on the scale of the mean free path^{21}, but these effects can be ignored in the extrinsic case studied here.
We first develop an intuitive picture of the timedependent spin Hall processes using the exact solution to equation (1) under the simplest conditions. For B=0, the components of s are uncoupled and equation (1) can be solved using a Green’s function to obtain an infinite series solution for s_{z}(y,t). The diffusion equation for s_{z} can be written as:
F(y,t) is a source function that contains all SHE terms. The homogeneous F=0 Green’s function for equation (2) is:
where m is an integer, Θ(t) is the Heaviside step function, k_{m}=(2 m+1)π/w and λ_{m}=1/τ+k_{m}^{2}L_{s}^{2}/τ. For timeindependent E, integrating equation (3) yields a series representation of the onedimensional steadystate solution to the spin Hall diffusion equation^{7}. To obtain a timedependent solution, we assume an ideal square electricfield pulse of width t_{p} and amplitude E_{0}, E=E_{0}[Θ(y+w/2)−Θ(y−w/2)][Θ(t)−Θ(t−t_{p})]. The corresponding source function is F(y,t)=σ_{SH}E_{0}[δ(y+w/2)−δ(y−w/2)][Θ(t)−Θ(t−t_{p})] and the solution is found by integrating:
The three regimes of the termbyterm timedependence function T_{m}(t) in equation (4) can each be observed in Fig. 2b. The lines in Fig. 3a represent fits of θ_{K}(t) to equation (4) keeping the first 200 terms in equation (4) and convoluting the solution with the Gaussian profile of the laser spot. As L_{s}=3.9±0.2 μm was found independently from steadystate spatial measurements, the only fit parameters are τ and an overall amplitude scaling. The best fit values for the parameter τ are plotted in Fig. 3b and are significantly longer than the experimentally measured timescales τ_{acc}, τ^{*} and τ_{1/2}.
Figure 3c shows θ_{K}(t) and calculations from equation (4) convoluted with the laser profile for y=126, 124, 122 and 120 μm for V_{0}=2 V using τ=4.2 ns obtained from the earlier fits. We fix the amplitude of the calculation from a fit to y=126 μm, and the remaining curves have no free parameters. For y away from the edge, θ_{K}(t) does not grow exponentially in t and we cannot define the time constant τ_{acc} as in Fig. 3a. Comparison of calculations from equation (4) with and without the spot size averaging reveals that the apparent asymmetry between the growth and decay times τ_{acc} and τ^{*} near the edge is primarily due to spatial averaging of these diffusion profiles over the Gaussian laser spot. The difference between τ^{*} and τ is real, however; dynamically measured spin polarization near the sample edge evolves with a faster time constant than the underlying spincoherence time.
We can understand the fast evolution of spin polarization from the interplay of diffusion and spin decoherence. As polarization gradients cause spins to diffuse away from the sample boundary, spin depolarization must occur faster than decoherence of the electrically generated spins. These dynamics are captured in the diffusion analysis of equation (4) by the fast decay rate λ_{m} of terms with large m in equation (4). Higher m terms are primarily responsible for the discrepancy between the best fit value for τ and the faster timescales τ_{acc} and τ^{*} observed for spin accumulation and decay in Fig. 3b, but they contribute significantly only to s_{z} near y=w/2 where all terms are in phase. In this boundary region, timescales should differ most from the coherence time τ.
We numerically calculate ∂ s_{z}/∂ y at y=126 μm from spatial scans in Fig. 3d. The spatial derivative of s_{z}(y) is proportional to the diffusive spin current and is nonzero at the sample edge in the presence of a compensating spin Hall current for 0<t<t_{p}. The spin Hall current itself tracks the pulse V (t) as fast as the momentum scattering timescales, but diffusive spin accumulation responds slower. After the spin Hall current disappears at t=t_{p}, the measured ∂ s_{z}/∂ y relaxes with time constant τ_{j}=1±0.1 ns to satisfy the diffusive j_{y}^{z}=0 boundary condition (Fig. 3d, inset). Averaging of our diffusion solution over the finite laser spot size, we calculate the value τ_{j}=1.15 ns for the evolution time of ∂ s_{z}/∂ y at y=126 μm, consistent with our observed value.
Introducing the magnetic field couples the spin components s_{x} and s_{z}. For this regime, we carry out numerical solutions to the system of coupled timedependent linear differential equations represented by equation (1) using the best fit values for L_{s} and τ obtained from the earlier fieldindependent analysis. For the numerical solutions, we use the exact pulse profile E(t)=V (t)/l measured from the oscilloscope (Fig. 2a) as a source. The curves in Fig. 2b–d and f–h are numerical calculations of θ_{K}(t) and θ_{K}(B). There are no free parameters except an overall scaling to match the amplitude of θ_{K}. The full experimental data set θ(t,B) at y=126 μm for t_{p}=6 ns is shown in Fig. 4a. Figure 4b shows the full calculation of spin accumulation from the numerical solution to equation (1) for y=126 μm.
The agreement between the experiments and calculations demonstrates that a single homogeneous decoherence time τ captures the various timescales observed in timeresolved measurements of the accumulation, decay and diffusive dynamics of boundary spin polarization due to the extrinsic spin Hall effect. Although diffusive timescales are set by the spincoherence time, evolution near sample boundaries can be limited by the faster response of the spin current. This spatial dependence of timescales could prove helpful for using electrically generated spin polarization in highfrequency semiconductor devices.
Methods
Channels of width w and length l are processed from a 2 μmthick silicondoped GaAs epilayer on 200 nm of undoped Al_{0.4}Ga_{0.6}As grown on a semiinsulating (001) GaAs substrate by molecular beam epitaxy (Fig. 1b). The nGaAs has doping density n=1×10^{17} cm^{−3} and mobility μ=3,800 cm^{2} V^{−1} s^{−1} at T=30 K. The sample is mounted in a helium flow cryostat so that the channel (x direction) is perpendicular to the externally applied inplane magnetic field B (y direction). All measurements are at temperature T=30 K. A voltage V (t) applied across annealed Ni/Ge/Au/Ni/Au ohmic contacts creates an inplane electric field E(t)=V (t)/l along x. We desire an impedance of ∼50 Ω to deliver the maximum broadband electrical power to the device; choosing w=256 μm and l=130 μm yields a device with d.c. resistance R=48 Ω at T=30 K.
Time resolution of SHE accumulation is achieved by electrically pumped Kerr rotation microscopy using a modelocked Ti:sapphire laser tuned to 1.51 eV that emits a 76 MHz train of ∼150 fs pulses. The pulse repetition rate is reduced to 38 MHz by pulse picking with an electrooptic modulator. Each laser pulse is divided into a trigger and a linearly polarized probe pulse. Spin polarization is generated at the sample edges by the SHE due to the current from a square electrical pulse of width t_{p}, amplitude V_{0} and 0.8 ns rise time applied to the sample from a pulse pattern generator triggered by the optical pump pulse. The linearly polarized probe beam is focused through a microscope objective to a 1 μm spot on the surface of the sample that can be scanned with submicrometre resolution. A balanced photodiode bridge measures the Kerr rotation of the linear polarization axis θ_{K} of the reflected beam which is proportional to the spin polarization along the z axis s_{z}. The leading edge of the electric pulse profile V (t) arrives at an electronically programmable delay time t before the arrival of the optical pulse. All reported measurements are taken at the centre of the length of the channel (x=0).
An absorptive radiofrequency switch alternates the centre conductor of the coaxial cable between the two complementary outputs of the pulse generator at frequency f_{V}=1.337 kHz. The radiofrequency switch passes only a.c. components of V (t), so the 0 V baseline is restored by adding a square wave at frequency f_{V} back onto the switched pulse train (Fig. 1c), resulting in a modulation of the pulse amplitude between +V_{0} and −V_{0} at frequency f_{V}. θ_{K} is then measured with a lockin amplifier analogous to the a.c. detection used in refs 7, 10, 11 but with a definite phase relationship between electrical and optical pulses.
The electrical pulse induces a timedependent reflectivity modulation ΔR/R of the optical beam during the pulse duration due to electron heating^{24,25} that tracks the profile V (t) measured by an oscilloscope (Fig. 2a). We use this effect to calibrate t=0 and confirm that the device acts as a proper 50 Ω termination owing to the minimal temporal pulse distortion at the sample.
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Acknowledgements
We thank NSF and ONR for financial support. N.P.S. acknowledges the support of the Fannie and John Hertz Foundation and S.M. acknowledges support through the NDSEG Fellowship Program.
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Stern, N., Steuerman, D., Mack, S. et al. Timeresolved dynamics of the spin Hall effect. Nature Phys 4, 843–846 (2008). https://doi.org/10.1038/nphys1076
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