Abstract

Spin–orbit coupling in semiconductors relates the spin of an electron to its momentum, and provides a pathway for electrically initializing and manipulating electron spins for applications in spintronics¹ and spin-based quantum information processing². This coupling can be regulated with quantum confinement in semiconductor heterostructures through band-structure engineering. Here we investigate the spin Hall effect^{3, 4} and current-induced spin polarization^{5, 6} in a two-dimensional electron gas confined in (110) AlGaAs quantum wells using Kerr rotation microscopy. In contrast to previous measurements^{7, 8, 9, 10}, the spin Hall profile shows complex structure and the current-induced spin polarization is out-of-plane. The experiments map the strong dependence of the current-induced spin polarization to the crystal axis along which the electric field is applied, reflecting the anisotropy of the spin–orbit interaction. These results reveal opportunities for tuning a spin source using quantum confinement and device engineering in non-magnetic materials.

Introduction

Previous measurements in bulk epilayers of n-GaAs and n-InGaAs (ref. 7) and in a two-dimensional hole gas⁸ provide experimental evidence for the spin Hall effect^{3, 4, 11, 12}, but it remains unclear whether the dominant mechanism is extrinsic or intrinsic. The extrinsic mechanism^{3, 4} is mediated by spin-dependent scattering, where spin–orbit coupling mixes the spin and momentum eigenstates. Alternatively, an intrinsic spin Hall mechanism has been proposed^{11, 12} that is an effect of the momentum-dependent internal magnetic field B_int. This internal field arises from spin–orbit coupling, which introduces a spin splitting for electrons with non-zero wavevector k in semiconductors lacking an inversion centre. For example, bulk inversion asymmetry exists owing to the zincblende crystal structure of GaAs and introduces the Dresselhaus spin splitting¹³, whereas structural inversion asymmetry is present in heterostructures that are not symmetric along the growth direction and leads to an in-plane spin splitting known as the Bychkov–Rashba effect¹⁴. The observation of the spin Hall effect in unstrained n-GaAs, in which the k-linear effective field is small¹⁵, suggests that the extrinsic effect is dominant in that system³. However, theoretical work argues that the cubic Dresselhaus term in GaAs could produce a non-negligible intrinsic spin Hall effect¹⁶.

Measurements in (110) quantum wells (QWs) may help distinguish between the two proposed mechanisms by allowing one to isolate the contributions of the Dresselhaus and Bychkov–Rashba fields. In two-dimensional systems, quantum confinement modifies the Dresselhaus spin splitting¹⁷. For the (110) QW, the Dresselhaus field is oriented along the growth direction, whereas this field is in-plane in conventional (001) heterostructures. As the Dresselhaus and Bychkov–Rashba fields are mutually perpendicular, one can tune the in-plane B_int with the Bychkov–Rashba effect and the out-of-plane B_int with the Dresselhaus field using engineered (110) heterostructures. In addition, two-dimensional systems provide a flexible architecture where carrier density, mobility and structural inversion asymmetry can be controlled using electric fields¹⁸.

Modulation-doped, digitally grown single QWs are grown by molecular beam epitaxy on (110) semi-insulating GaAs substrates. The QW structure behaves like a single 75 Å Al_0.1Ga_0.9As QW with Al_0.4Ga_0.6As barriers at a temperature T=30 K. For the optical measurements, a mesa is defined using a chemical etch (Fig. 1a) and contacts are made using annealed AuGe/Ni.

Figure 1: The spin Hall effect in a 2DEG.

a, The device schematic and measurement geometry. The light-blue region indicates the mesa and the yellow regions are the contacts. b, Kerr rotation (open circles) and fits (curves) as a function of applied in-plane magnetic field B_ext for x=-58.4 m (top, in red) and x=+58.4 m (bottom, in blue). The channel has width w=120 m, length l=310 m and mesa height h=0.1 m. A linear background is subtracted for clarity. c, B_ext scans as a function of position near the edges of the channel of a device fabricated along the [001] direction for V_p=2 V. Amplitude A₀, spin coherence time _s and reflectivity R are plotted for V_p=1.5 V (blue squares) and 2 V (red circles).

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The spin polarization in the two-dimensional electron gas (2DEG) is spatially resolved using low-temperature scanning Kerr rotation microscopy¹⁹ in the Voigt geometry. A linearly polarized beam is tuned to the absorption edge of the QW (wavelength =719 nm) and directed normal to the sample through an objective lens, providing 1.1 m lateral spatial resolution. The rotation of the polarization axis of the reflected beam provides a measure of the electron spin polarization along the beam direction. A square wave voltage with maximum amplitude V_p and frequency 511 Hz is applied to the device for lock-in detection. Measurements are performed in devices with electric fields applied along four different crystal directions in order to create a directional map of the internal fields. All of the data presented are measured at T=30 K and we take x=0 m to be the centre of the channel.

In Fig. 1b, we present Kerr rotation data as a function of the applied in-plane magnetic field B_ext for positions near the two opposite edges of a channel aligned along the [001] direction. These data correspond to a measurement of the Hanle effect using Kerr rotation²⁰ (_K) and indicate the presence of an out-of-plane spin polarization when the data can be fitted to a lorentzian A₀/[(_Ls)²+1], where A₀ is peak Kerr rotation, _L=g_BB_ext/ is the Larmor precession frequency, _s is the electron spin coherence time, g is the electron g factor, _B is the Bohr magneton and is the Planck constant. A₀ is of opposite sign for the two edges of the sample, which is a signature of the spin Hall effect.

In Fig. 1c, a one-dimensional spatial profile of the spin accumulation near the edges is mapped out by repeating B_ext scans as a function of position. There are two spin Hall peaks at each edge, one around x=58.6 m and one of smaller amplitude around x=56.4 m. This structure was not observed in measurements on bulk epilayers⁷ and could be due to an additional contribution from spin-polarized carriers undergoing spin precession about the in-plane Bychkov–Rashba field as they diffuse towards the centre of the channel. However, the asymmetry in |A₀| for the right and left edges and a spatial dependence of _s was also observed in previous measurements⁷. The reflectivity R shows the position of the edges of the channel, at x=59.4 m.

In the [001]-oriented device, electrically induced spin polarization is observed only at the edges of the channel. In contrast, devices fabricated along the , and directions also show spin polarization at the centre of the channel. Figure 2b shows data taken at x=0 m for electric field E along , and . As the polarization is along the growth direction and depends on the direction of E relative to the crystal axes, we attribute this effect to the Dresselhaus field. The application of an electric field results in a non-zero average drift velocity of the electrons, which produces a non-zero effective magnetic field that orients spins^{5, 6}. Although the opposite sign of A₀ for and may seem surprising as these directions are only separated by 35.3^° in the (110) plane (Fig. 2a), it is consistent with the calculated B_int due to the cubic Dresselhaus field in a (110) QW. This theory also predicts that B_int should be zero for E[001] (ref. 21), as observed.

Figure 2: Current-induced spin polarization in a 2DEG.

a, Relative orientations of crystal directions in the (110) plane. b, Kerr rotation (open circles) and fits (lines) as a function of B_ext for E (black), E (red) and E (green) at the centre of the channel. c, B_ext scans as a function of position near the edges of the channel of a device fabricated along with w=118 m and l=310 m for V_p=2 V. Amplitude A₀, spin-coherence time _s and reflectivity R are plotted for V_p=1.5 V (blue filled squares), 2 V (red filled circles) and 3 V (black open circles).

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Figure 2c shows a spatial profile of the spin polarization near the edges for a device aligned along . A₀ is negative across the entire channel, and |A₀| increases with increasing voltage. From -52 m<x<+52 m, |A₀| is nearly constant across the channel. However, |A₀| becomes smaller near the left edge of the channel, and a negative peak in A₀ is evident near the right edge, which is due to the spin Hall effect. The data for V_p=3 V suggest that there may be two spin Hall peaks, at x=55.5 and 57.5 m, similar to the two peaks with 2 m spacing observed in the [001] device. We also observe that A₀ increases more dramatically with voltage for the spin Hall peak near the right edge than for the current-induced spin polarization across the rest of the channel.

We continue examining the direction dependence of the current-induced spin polarization with spatial scans of a channel aligned along . Figure 3 shows the spatial profile of the spin polarization near the edges of the channel. A₀ is positive across the entire channel, and |A₀| is nearly constant from -26 m<x<+26 m. However, there is a small positive peak around x=-31 m and |A₀| diminishes near the right edge of the channel.

Figure 3: Spin polarization near the edges of a channel oriented along .

B_ext scans as a function of position near the edges of the channel of a device fabricated along with w=68 m and l=306 m for V_p=1.5 V. Amplitude A₀ and reflectivity R are also plotted.

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We also perform spatially resolved measurements of a device aligned along (Supplementary Information, Fig. S1). Again, we observe a uniform spin polarization in the centre of the channel and spin accumulation owing to the spin Hall effect at the edges of the channel. From our measurements on all four devices, we conclude that the spin Hall effect shows the same polarity for electric fields applied along all four crystal directions.

In Fig. 4, we present voltage dependences of A₀ and _s for the spin Hall peaks in the [001] device and the current-induced spin polarization in the , and devices. In Fig. 4a, we plot A₀ for the spin Hall peaks near the edges of the [001] channel and observe that |A₀| increases with increasing V_p. The nonlinearity of the increase in |A₀| could be due to changes in the spin Hall profile or in the electrical response of the device. In contrast, we observe in Fig. 4b that _s=545176 ps and it does not have a clear voltage dependence over this range.

Figure 4: Voltage dependence of the electrically induced spin polarization.

a,b, Amplitude A₀ (a) and spin-coherence time _s (b) of the spin Hall polarization as a function of voltage for x=-58 m (red) and x=+58 m (blue) for a device fabricated along [001]. c,d, A₀ (c) and _s (d) of the current-induced spin polarization as a function of voltage V_p for electric fields applied along (black), (red) and (green) measured at the centre of the channel (x=0 m).

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In order to explore the direction dependence of the current-induced spin polarization, we measure A₀ at x=0 m for devices aligned along , and as a function of V_p, which we plot in Fig. 4c. We observe that the amplitude of the current-induced spin polarization increases with increasing V_p, as expected. In addition, _s=1,344404 ps and does not show a clear dependence on voltage (Fig. 4d). The direction dependence of A₀ reflects the strong k dependence of the Dresselhaus field.

In order to determine the mechanism of the spin Hall effect, we quantify the Rashba coefficient by measuring the in-plane B_int for our sample. The Bychkov–Rashba field has magnitude |B_int|=|k|/g_B and is oriented perpendicular to k. B_int can be observed as a shift in a Hanle²² or field-dependent Kerr rotation curve¹⁵ when we apply a d.c. voltage V_d.c. along the [001] direction. Spins are injected optically into the QW and measured as a function of B_ext after a time delay of 6 ns. Figure 5a shows Kerr rotation as a function of B_ext for V_d.c.=-2 and +2 V. Lorentzian fits determine the centre of the peak, which is -B_int. In Fig. 5b, B_int as a function of V_d.c. can be fitted to a line with slope 1.77 mT V^-1 and we determine =1.810^-12 eV m. This small value for is reasonable because this QW was designed to be symmetric, as is a measure of the structural inversion asymmetry. This is also consistent with the observation that the current-induced spin polarization does not change significantly for the EB_ext geometry, where one would also measure spins that are oriented in-plane⁵. In addition, this value for B_int yields a spatial spin precession period²³ of 3.5 m, which is similar to the 2 m distance observed between the spin Hall peaks in the [001] and devices and suggests that the spacing between the spin Hall peaks could be due to spin precession. This relation could be confirmed by tuning with a gate voltage¹⁸. Calculations of the intrinsic spin Hall effect for Rashba spin–orbit coupling show that the spin Hall conductivity should be non-zero when the Rashba splitting is larger than the disorder broadening¹². The ratio _0p/10^-2, where ₀ is the spin-splitting energy and _p is the mean scattering time, and this ratio relates the strength of the spin–orbit coupling with impurity scattering²⁴. In addition, the Dresselhaus terms are oriented out-of-plane in our sample and should not contribute to the spin Hall conductivity. Therefore, our data suggest that the spin Hall effect that we observe is dominated by the extrinsic spin Hall mechanism.

Figure 5: Measurement of the Bychkov–Rashba spin splitting.

a, Kerr rotation as a function of B_ext for V_d.c.=-2 V (blue) and V_d.c.=+2 V (red). The data were taken with a laser spot size of 30 m. Lines are lorentzian fits. b, In-plane effective magnetic field B_int as a function of V_d.c..

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Spin–orbit engineering in two-dimensional systems allows for the manipulation of the magnitude and direction of the internal fields for sourcing spin polarization in non-magnetic semiconductors. Moreover, these interactions can be used to operate on electron spins by changing the direction of current, thereby enabling new degrees of control for quantum confined spintronic devices.

Acknowledgements

We acknowledge support from ARO, DARPA, NSF and ONR.

Competing interests statement:

The authors declare that they have no competing financial interests.

Received 1 May 2005; Accepted 24 June 2005; Published online 29 September 2005.

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1. Center for Spintronics and Quantum Computation, University of California, Santa Barbara, California 93106, USA

Correspondence to: D. D. Awschalom¹ e-mail: awsch@physics.ucsb.edu

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