Time-resolved dynamics of the spin Hall effect

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 devices1,2,3,4. One realization of these spin–orbit phenomena, the spin Hall effect5,6, has been studied as a means of all-electrical spin-current generation and spin separation in both semiconductor and metallic systems. Previous measurements of the spin Hall effect7,8,9,10,11 have focused on steady-state generation and time-averaged detection, without directly addressing the accumulation dynamics on the timescale of the spin-coherence time. Here, we demonstrate time-resolved measurement of the dynamics of spin accumulation generated by the extrinsic spin Hall effect in a doped GaAs semiconductor channel. Using electrically pumped time-resolved 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 time-dependent diffusion analysis using both exact and numerical solution techniques and find that the underlying physical spin-coherence time differs from the dynamical rates of spin accumulation and decay observed near the sample edges.

Main

Theories have predicted5,6,12,13, and experiments confirmed7,9, that an electric current in a crystal with spin–orbit coupling gives rise to a transverse spin current through the spin Hall effect (SHE). Spin-dependent 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 spin-dependent 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 non-equilibrium 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 spin-coherence 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.

Steady-state observations of electrically generated spin accumulation7,15,16 are effective for inferring τ, but they cannot directly access the dynamical processes on the nanosecond timescale. In contrast, time-resolved spin dynamics with picosecond resolution are routinely measured using ultrafast optical pump–probe techniques17,18. Time resolution of bulk current-induced spin polarization was achieved using a photoconductive switch15, 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, current-induced spin polarization is a bulk phenomenon and consequently neither the steady-state4,15 nor the time-resolved15 measurements investigated spatial dynamics near the sample edge. Here, we combine the spatial resolution afforded by scanning Kerr microscopy19 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 n-doped GaAs channel. Details of the experimental technique are shown in Fig. 1 and are discussed in the Methods section.

Figure 1: Experimental design for time-resolved measurement of electrically generated spin polarization.
figure1

a, Schematic diagram of time-resolved measurement of the SHE (EOM, electro-optic modulator; BS, beam splitter; PD, photodiode; FG, function generator; RF, radiofrequency). b, Optical microscope image of the sample with coordinate system defined (units in micrometres) showing the origin (black circle) and the location for most of the measurements (red circle). The bright yellow regions are the gold contacts and the grey region is the GaAs channel. c, Illustration of the a.c. pulse scheme for lock-in detection. The pure a.c. components of the switch output (red) are added to a square wave at fV to create a triggered pulse train with amplitude modulation at fV≈1 kHz.

Figure 2b–d shows the Kerr rotation θK(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 tp=6 ns and amplitude V0=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<tp), spin polarization builds up owing to the SHE. After the pulse has passed (t>tp), the accumulated spin polarization undergoes decay and precession (Fig. 2b–d). We fit θK(t) for t>tp to an exponentially decaying cosine to extract an inhomogeneous depolarization time τ* and Larmor precession frequency νL=g μBB/h, where g is the Lande g-factor, μB is the Bohr magneton and h is Planck’s constant. Linear fits to νL(B) give |g|=0.346±0.002 (Fig. 2e), which is consistent with g expected for this doping level20. The depolarization time τ*=2.8±0.1 ns measured at a specific spatial location should differ from the physical spin-decoherence 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.

Figure 2: Time-resolved measurement of the spin Hall accumulation.
figure2

a, Voltage pulse profile V (t) measured by an oscilloscope (red line) and by reflectivity modulation ΔR/R (black diamonds). The arbitrary units of ΔR/R are scaled to match the scope voltage. The vertical dashed lines mark the time region 0<t<tp. bd, Representative scans of Kerr rotation θ(t) for tp=6 ns and y=126 μm. The blue lines are calculations from equation (1) with τ=4.2 ns, Ls=3.9 μm and y=126 μm. eνL extracted from cosinusoidal fits to θK(t). fh, Kerr rotation θK(B) as a function of magnetic field at three representative times. The blue lines are calculations from equation (1).

Typical optical studies of electrically generated spin accumulation measure the time-averaged projection of spin precession as a function of applied transverse magnetic field B. In analogy with the Hanle effect of luminescence depolarization, the z-axis spin polarization sz should depolarize for increasing B when 2πνLτ−1 (ref. 21). Therefore, coherence times in steady-state experiments are typically extracted from the linewidths of sz(B). Near the sample edge, sz(B) is a Lorentzian line shape analogous to the Hanle effect7, whereas it becomes more complicated away from the edges owing to the interplay of spin precession and diffusion11,21. In the current experiment, measurement of θK(B) does not represent a time-averaged steady-state 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 θK(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, θK(B) grows in a broad peak that narrows as t increases (Fig. 2f). Only for ttp>τ does θK(B) approach the Lorentzian line shape expected from a conventional Hanle analysis (Fig. 2g). For t>tp, θK(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 tp=15 ns to investigate accumulation dynamics with the current flowing for various V0. We fit θK(B=0) to an exponential saturation with a time constant τacc. For each V0, τacc is around 40% of the τ* measured from decay of the spin polarization (Fig. 3b). Both τ* and τacc decrease weakly with V0, which is expected owing to electron heating11,22.

Figure 3: Dynamics of spin Hall accumulation.
figure3

a, Kerr rotation θK(t) (upper panel), and inverse field width B1/2−1 (lower panel) with tp=15 ns at y=126 μm for V0=2.0 V (black), 1.5 V (red), 1.0 V (blue) and 0.5 V (green). The solid lines are fits from our model. b, The accumulation time τacc (green triangles), Hanle time τ1/2 (blue circles), decay time τ* (black squares) and coherence time τ (red diamonds) as a function of voltage. The error bars represent one standard error of the parameter obtained from a least-squares fitting routine. cθK(t) for y=126 μm (black), 124 μm (red), 122 μm (blue) and 120 μm (green). The solid lines are calculations with amplitude fixed by a fit at y=126 μm. d, Spatially resolved Kerr rotation θK(y) near the edge for t=5 ns (black), 7 ns (red), 9 ns (blue) and 11 ns (green). The inset shows the decay of jyz sz(y)/ y extracted at y=126 μm after the voltage pulse is complete (tp=6 ns).

We characterize the magnetic-field line shapes by their inverse half-width B1/2−1, which increases with t before quickly saturating (Fig. 3a). We can understand this evolution of B1/2 in a simple physical picture21. 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 tp) and have correspondingly more time for precession; hence, there is more depolarization for a given B and the Hanle curve narrows (Fig. 2g). For tτ, the average precession time is governed by τ rather than t and the Hanle width becomes constant as in a steady-state measurement. The coherence times τ1/2 calculated from the saturation of B1/2 near ttp agree with decay times τ* and are consequently also longer than the accumulation times τacc (Fig. 3b). Diffusion analysis of the SHE accumulation is necessary to reconcile the observed differences between the timescales τacc, τ* and τ1/2.

Spin accumulation from the SHE can be modelled using drift–diffusion equations when the spin-coherence time is much longer than the momentum scattering time21,23. The extrinsic SHE of the n-GaAs system is well suited to diffusive analysis because of its low spin–orbit coupling and relatively long spin-coherence time11,23. We treat our GaAs channel of width w as infinitely long because its length l is much larger than the spin diffusion length Ls=3.9 μm found from steady-state measurements at B=0. This assumption reduces the problem to only one spatial dimension and precludes the need for general two-dimensional modelling including spin drift11. The SHE generates a spin current transverse to the in-plane electric field and proportional to the spin Hall conductivity σSH, jji=σSHεijkEk. The total current of the i spin component along y from both diffusion and the SHE is jyi=−D ysiσSHE δi z, where D=Ls2/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 hard-wall boundary conditions normal to the edges at yw/2 (jyi=−D ysiσSHE δi 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 path21, but these effects can be ignored in the extrinsic case studied here.

We first develop an intuitive picture of the time-dependent 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 sz(y,t). The diffusion equation for sz 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, km=(2 m+1)π/w and λm=1/τ+km2Ls2/τ. For time-independent E, integrating equation (3) yields a series representation of the one-dimensional steady-state solution to the spin Hall diffusion equation7. To obtain a time-dependent solution, we assume an ideal square electric-field pulse of width tp and amplitude E0, E=E0[Θ(y+w/2)−Θ(yw/2)][Θ(t)−Θ(ttp)]. The corresponding source function is F(y,t)=σSHE0[δ(y+w/2)−δ(yw/2)][Θ(t)−Θ(ttp)] and the solution is found by integrating:

The three regimes of the term-by-term time-dependence function Tm(t) in equation (4) can each be observed in Fig. 2b. The lines in Fig. 3a represent fits of θK(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 Ls=3.9±0.2 μm was found independently from steady-state 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 τacc, τ* and τ1/2.

Figure 3c shows θK(t) and calculations from equation (4) convoluted with the laser profile for y=126, 124, 122 and 120 μm for V0=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, θK(t) does not grow exponentially in t and we cannot define the time constant τacc 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 τacc 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 spin-coherence 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 λm 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 τacc and τ* observed for spin accumulation and decay in Fig. 3b, but they contribute significantly only to sz 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 sz/ y at y=126 μm from spatial scans in Fig. 3d. The spatial derivative of sz(y) is proportional to the diffusive spin current and is non-zero at the sample edge in the presence of a compensating spin Hall current for 0<t<tp. 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=tp, the measured sz/ y relaxes with time constant τj=1±0.1 ns to satisfy the diffusive jyz=0 boundary condition (Fig. 3d, inset). Averaging of our diffusion solution over the finite laser spot size, we calculate the value τj=1.15 ns for the evolution time of sz/ y at y=126 μm, consistent with our observed value.

Introducing the magnetic field couples the spin components sx and sz. For this regime, we carry out numerical solutions to the system of coupled time-dependent linear differential equations represented by equation (1) using the best fit values for Ls and τ obtained from the earlier field-independent 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 θK(t) and θK(B). There are no free parameters except an overall scaling to match the amplitude of θK. The full experimental data set θ(t,B) at y=126 μm for tp=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.

Figure 4: Modelling boundary accumulation from the spin Hall effect.
figure4

a, Kerr rotation θK(B,t) at y=126 μm for tp=6 ns. b, Calculations of sz(B,t) from equation (1) with τ=4.2 and Ls=3.9 μm at y=126 μm.

The agreement between the experiments and calculations demonstrates that a single homogeneous decoherence time τ captures the various timescales observed in time-resolved 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 spin-coherence 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 high-frequency semiconductor devices.

Methods

Channels of width w and length l are processed from a 2- μm-thick silicon-doped GaAs epilayer on 200 nm of undoped Al0.4Ga0.6As grown on a semi-insulating (001) GaAs substrate by molecular beam epitaxy (Fig. 1b). The n-GaAs has doping density n=1×1017 cm−3 and mobility μ=3,800 cm2 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 in-plane 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 in-plane 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 mode-locked 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 electro-optic 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 tp, amplitude V0 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 θK of the reflected beam which is proportional to the spin polarization along the z axis sz. 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 fV=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 fV back onto the switched pulse train (Fig. 1c), resulting in a modulation of the pulse amplitude between +V0 and −V0 at frequency fV. θK is then measured with a lock-in 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 time-dependent reflectivity modulation ΔR/R of the optical beam during the pulse duration due to electron heating24,25 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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Correspondence to D. D. Awschalom.

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Stern, N., Steuerman, D., Mack, S. et al. Time-resolved dynamics of the spin Hall effect. Nature Phys 4, 843–846 (2008). https://doi.org/10.1038/nphys1076

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