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Spatial imaging of the spin Hall effect and current-induced polarization in two-dimensional electron gases

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 effect3,4 and current-induced spin polarization5,6 in a two-dimensional electron gas confined in (110) AlGaAs quantum wells using Kerr rotation microscopy. In contrast to previous measurements7,8,9,10, 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.

Main

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 effect3,4,11,12, but it remains unclear whether the dominant mechanism is extrinsic or intrinsic. The extrinsic mechanism3,4 is mediated by spin-dependent scattering, where spin–orbit coupling mixes the spin and momentum eigenstates. Alternatively, an intrinsic spin Hall mechanism has been proposed11,12 that is an effect of the momentum-dependent internal magnetic field Bint. 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 Bint with the Bychkov–Rashba effect and the out-of-plane Bint 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 Al0.1Ga0.9As QW with Al0.4Ga0.6As 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.
figure1

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 Bext for x=−58.4 μm (top, in red) and x=+58.4 μm (bottom, in blue). The channel has width w=120 μm, length l=310 μm and mesa height h=0.1 μm. A linear background is subtracted for clarity. c, Bext scans as a function of position near the edges of the channel of a device fabricated along the [001] direction for Vp=2 V. Amplitude A0, spin coherence time τs and reflectivity R are plotted for Vp=1.5 V (blue squares) and 2 V (red circles).

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 λ=719 nm) and directed normal to the sample through an objective lens, providing 1.1 μm 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 ±Vp 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 μm 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 Bext 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 rotation20K) and indicate the presence of an out-of-plane spin polarization when the data can be fitted to a lorentzian A0/[(ωLτs)2+1], where A0 is peak Kerr rotation, ωL=gμBBext/ħ is the Larmor precession frequency, τs is the electron spin coherence time, g is the electron g factor, μB is the Bohr magneton and ħ 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 Bext scans as a function of position. There are two spin Hall peaks at each edge, one around x=±58.6 μm and one of smaller amplitude around x=±56.4 μm. 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 τs was also observed in previous measurements7. The reflectivity R shows the position of the edges of the channel, at x=±59.4 μm.

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 μm 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 spins5,6. 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 Bint due to the cubic Dresselhaus field in a (110) QW. This theory also predicts that Bint should be zero for E[001] (ref. 21), as observed.

Figure 2: Current-induced spin polarization in a 2DEG.
figure2

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

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 μm<x<+52 μm, |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 Vp=3 V suggest that there may be two spin Hall peaks, at x=55.5 and 57.5 μm, similar to the two peaks with 2 μm 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 μm<x<+26 μm. However, there is a small positive peak around x=−31 μm and |A0| diminishes near the right edge of the channel.

Figure 3: Spin polarization near the edges of a channel oriented along .
figure3

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

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 τs 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 Vp. 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 τs=545±176 ps and it does not have a clear voltage dependence over this range.

Figure 4: Voltage dependence of the electrically induced spin polarization.
figure4

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

In order to explore the direction dependence of the current-induced spin polarization, we measure A0 at x=0 μm for devices aligned along , and as a function of Vp, which we plot in Fig. 4c. We observe that the amplitude of the current-induced spin polarization increases with increasing Vp, as expected. In addition, τs=1,344±404 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 α by measuring the in-plane Bint for our sample. The Bychkov–Rashba field has magnitude |Bint|=α|k|/gμB and is oriented perpendicular to k. Bint can be observed as a shift in a Hanle22 or field-dependent Kerr rotation curve15 when we apply a d.c. voltage Vd.c. along the [001] direction. Spins are injected optically into the QW and measured as a function of Bext after a time delay of 6 ns. Figure 5a shows Kerr rotation as a function of Bext for Vd.c.=−2 and +2 V. Lorentzian fits determine the centre of the peak, which is −Bint. In Fig. 5b, Bint as a function of Vd.c. can be fitted to a line with slope 1.77 mT V−1 and we determine α=1.8×10−12 eV m. This small value for α is reasonable because this QW was designed to be symmetric, as α 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 EBext geometry, where one would also measure spins that are oriented in-plane5. In addition, this value for Bint yields a spatial spin precession period23 of 3.5 μm, which is similar to the 2 μm 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 α 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 Δ0τp/ħ10−2, where Δ0 is the spin-splitting energy and τp 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.
figure5

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

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.

Methods

Sample growth and device preparation

Conditions for the (110) growth are similar to those described in ref. 25; the substrate temperature is 490 C, the As4 beam equivalent pressure is 1.6×10−5 torr and the growth rate of GaAs is 0.5 monolayers s−1. The samples consist of four 14 Å GaAs layers with Al0.4Ga0.6As barriers separated by 6 Å Al0.4Ga0.6As spacers. The barriers are delta-doped with silicon at 200 Å from the QW structure on both the surface and the substrate side, with doping densities of 1.4×1012 and 5.6×1011 cm−2, respectively. In addition, silicon doping at 1×1018 cm−3 is present within the QW region. Conventional Hall measurements at T=5 K determine the sheet density ns=1.9×1012 cm−2 and mobility μ=940 cm2 V−1s−1. Devices are aligned to the natural cleaves along [001] and such that an electric field E can be applied along the in-plane directions [001], , and . Using time-resolved Kerr rotation26, we determine |g|=0.33 for this sample and τs=766 ps at Bext=0.2 T. The longitudinal spin coherence time is 3,250 ps at Bext=0 T. The relatively long spin-coherence times observed in (110) QWs27 compared with (001) 2DEGs28 is due to the suppression of the D’yakonov–Perel’ spin relaxation mechanism29. The data presented in this paper are from devices processed from one sample, but measurements performed on devices fabricated from a second sample verify the reproducibility of our results.

Measurement of Bychkov–Rashba field

The shift in field-dependent Kerr rotation is used to measure the in-plane Bint as a function of applied voltage in order to determine α. As the contact resistance is large compared with the resistance of the channel, we consider the voltage drop across the channel Vc=(Rc/RT)Vd.c., where Rc=980 Ω is the resistance of the channel and RT=10.3 kΩ is the total resistance of the device. As 〈k〉=μVcme*/ħ l, where the in-plane effective mass me*=0.074 me from a 14-band K·p calculation, and the spin-splitting energy Δ0=gμBBint, we determine α=1.8×10−12 eV m.

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Acknowledgements

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

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Correspondence to D. D. Awschalom.

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Sih, V., Myers, R., Kato, Y. et al. Spatial imaging of the spin Hall effect and current-induced polarization in two-dimensional electron gases. Nature Phys 1, 31–35 (2005). https://doi.org/10.1038/nphys009

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