Introduction

The length scale over which the travelling electron spin keeps the memory of its initial orientation is known as the spin-diffusion length1,2,3,4,5, which is a critical parameter for many spintronic devices6,7,8,9,10,11,12. In semiconductors, this distance is closely linked to spin relaxation and spin transport properties of conduction electrons, which in turn depend on the spin-orbit interaction. In quantum wells made of III–V semiconductors such as the model GaAs/AlGaAs system, the conduction-band (CB) spin splitting has two origins: the first one, called Bulk Inversion Asymmetry (BIA) or Dresselhaus term, is the consequence of the combined effect of spin-orbit interaction and the lack of crystal inversion symmetry13,14. The second contribution to the spin splitting, called Structural Inversion Asymmetry (SIA) or Rashba term, exists if the quantum well experiences an electric field15,16. The interplay of these two terms can yield longer relaxation times for electron spins oriented along some specific crystallographic directions17,18 or original effects, such as Spin helix19,20. After a few years of debates, it was recently shown that the Rashba-induced conduction-band spin splitting requires the application of a real electric field in the structure (through asymmetric doping or PIN structures), whereas CB potential gradients, which also break the inversion symmetry, yield negligible small Rashba splitting13,21,22.

Thanks to the unique relative orientation of the Dresselhaus and Rashba fields in (111)-oriented semiconductor quantum wells, we show here that the electron spin-diffusion length can be electrically controlled, a feature that was never demonstrated before. Moreover, the all-optical measurement of the electric field dependence of the electron spin-relaxation time and the spin-diffusion coefficient allows us to get a direct determination of both the Dresselhaus and Rashba coefficients independently.

Results

Electric field effect on the spin-diffusion coefficient

Spin-polarized electrons are photogenerated by exciting the non-intentionally doped quantum wells using a circularly polarized laser. The quantum wells are embedded into an NIP structure in which a reverse bias V can be applied, yielding the variation of the electric field E experienced by the spin-polarized electrons (Fig. 1). The spin-diffusion length Ls is simply linked to the spin lifetime Ts and the spin-diffusion coefficient Ds through (refs 23, 24). We performed an optical measurement of the spin-diffusion constant Ds by using the Transient Spin-Grating technique23, and we measured the electron spin lifetime Ts by the time-resolved photoluminescence spectroscopy25 (see Methods).

Figure 1: Gate control of the spin-diffusion length.
figure 1

(a) Variation of the spin-diffusion length as a function of the reverse bias V applied on the NIP device obtained from combined measurements of the electron-spin lifetime Ts and the electron spin-diffusion constant Ds. Insets: schematics of the gate control of the electron spin-diffusion length Ls in a (111)-oriented GaAs/AlGaAs quantum well. The red (V=5 volts) and blue (V=0) surfaces illustrate the spatial distribution of the spin density in the quantum well plane under laser excitation. (b) Schematic representation of the NIP junction with 20 GaAs/Al0.3Ga0.7As QWs embedded in its intrinsic region. The growth axis z corresponds to the <111> direction.

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A spin-density wave with wavelength Λ is created by the interferences of two cross-polarized pump laser beams (Fig. 2). The modulus of wave vector q of the spin grating is equal to q=2π/Λ. Figure 3a displays the decay of the spin-grating signal as a function of the delay between the pump and the probe laser pulses for a reverse bias V=1 volt. The decay rate Γ of this spin-density wave is simply given by23,26:

Figure 2: Transient spin grating.
figure 2

(a) Schematic of the transient spin-grating technique based on the interferences of the two cross-polarized pump beams yielding a spatially modulated spin density in the quantum well plane with a period Λ. (b) The spin-grating signal detected with a heterodyne technique based on a reference beam is recorded as a function of the time delay between the pump and the probe pulses (see Methods).

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Figure 3: Electron spin diffusion.
figure 3

(a) Time evolution of the spin-grating (SG) signal as a function of the delay between the pump and the probe laser pulses for V=1 volt. To separate the effects of spin diffusion and spin relaxation, the grating decay rate is measured for different grating wave vectors q by changing the phase mask period (see Methods). (b) Decay rate Γ of the SG signal for a reverse bias V=1 volt. (c) Spin-diffusion coefficient Ds as a function of the gate voltage V. The error bars correspond to the uncertainties of the slope determination (see Fig. 3b).

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To separate the effects of spin diffusion and spin relaxation, the grating decay rate Γ is measured for different grating wave vectors q by changing the phase mask period (see Methods). Performing the measurement of Γ for different grating periods (and hence q2 values) yields an accurate determination of Ds, whose dependence on the gate voltage has never been investigated. Figure 3b displays the variation of Γ as a function of q2 for an applied reverse bias V=1 volt. We measured a spin-diffusion coefficient Ds~20 cm2s−1. We performed the same experiments for different applied gate voltages. Figure 3c shows that the spin-diffusion coefficient Ds is almost constant whatever the applied vertical electric field is; this value is consistent with previously reported spin-diffusion coefficients in non-intentionally doped GaAs quantum wells27,28,29.

Gate voltage variation of the spin lifetime

In order to obtain the electric field dependence of the spin-diffusion length Ls, we need to determine the variation of the spin lifetime Ts measured independently as a function of the gate voltage. The striking feature of (111)-oriented quantum well is that the electron spin-relaxation time can be electrically controlled for any electron-spin orientation25,30. The accurate determination of the electric field dependence of the electron spin lifetime Ts has been obtained from spin-polarized time-resolved photoluminescence spectroscopy (Fig. 4a,b). Figure 4c displays the dependence of Ts as a function of the applied voltage, where 1/Ts=1/τs+1/τ; both the electron spin-relaxation time τs and the electron lifetime τ have been measured for each value of the reverse bias V. The electron-spin lifetime can be controlled with the applied electric field with an increase by a factor of about four when the reverse bias varies from 0–5 volts. We checked that the same gate voltage dependence of Ts was measured by time-resolved Kerr rotation experiments performed on the same set-up as the spin-grating measurements.

Figure 4: Spin Lifetime in the gate-control device.
figure 4

(a) Photoluminescence intensity (colour code in arbitrary units) as a function of both time and photon energy for a reverse bias V=5 volts. The white curve represents the PL intensity as a function of time when the emission is spectrally integrated. The decay time of this PL dynamics corresponds to the electron lifetime τ. (b) PL circular polarization degree Pc (colour code from Pc=0 to Pc=0.30) as a function of both time and photon energy for a reverse bias V=5 volts. Pc=(I+−I−)/(I++I−), where I+ and I− are the PL circular components, respectively, co- and counter-polarized with the incident laser beam. The white curve represents Pc as a function of time when the emission is spectrally integrated. The decay time of this Pc dynamics corresponds to the electron spin-relaxation time τs. (c) Gate control of the electron spin lifetime Ts. (d) Gate control of the spin-relaxation time τs.

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Figure 1a presents the resulting variation of the spin-diffusion length as a function of the applied voltage. It increases by more than a factor two when the amplitude of the electric field increases, reaching a value Ls~1.8 μm for |E|~55 kV cm−1 corresponding to V=5 volts. These results provide the proof of concept that the spin diffusion length can be electrically controlled in semiconductors. Here the increase of Ls is limited by the electron lifetime τ because for V>4 volts the spin relaxation time is much longer than τ in these non-intentionally doped quantum wells (Fig. 4d). In n-doped (111) quantum wells, which are of interest for spintronic applications11,12, we expect a much larger variation of Ls with the applied voltage that will be no more limited by the electron lifetime. In that case, the variation of LS with the voltage V will just depend on the square root of the electron spin-relaxation time τs, which can vary by several orders of magnitude with V25,30.

Discussion

The all-optical measurements of both the spin relaxation dynamics and the electron spin diffusion allow us to get a precise and independent determination of the Dresselhaus and Rashba coefficients, which are the key parameters to control and manipulate the carrier spins in semiconductors. Despite the numerous experimental and theoretical investigations, the reported values of the Dresselhaus and Rashba coefficients in GaAs vary strongly, ranging from |γ|=3 to 35 eV Å3 for Dresselhaus29,31,32,33,34,35 and r41 from 5–10 e Å2 for Rashba13,29,36. Most of the techniques used so far give only the ratio between the Rashba and Dresselhaus terms or allow the determination of only one type of spin splitting. They also require the investigation of n-doped quantum wells with electrical contacts31,37,38. The technique presented here, which yields the determination of both Dresselhaus and Rashba spin-orbit (SO) splitting in the same sample, could be applied for any doped or undoped III–V or II–VI cubic semiconductor quantum wells. The only requirement is to grow the structures on (111)-oriented substrates.

In a (111) quantum well, where the momentum component along the growth axis z is quantized, the spin precession vector Ω(k) due to the Dresselhaus (BIA) term for the lowest electron subband is written as39:

where is the averaged squared wave vector along the growth direction, k// is the in-plane wave vector () and γ is the Dresselhaus coefficient40. The weaker cubic terms in the Hamiltonian have been neglected here (at 50 K for a well width LW=15 nm, the average cubic Dresselhaus term is 10 times weaker than the linear one). For an electric field applied along the growth axis, the SIA (Rashba) component writes:

where α=2r41E is the Rashba coefficient (E is the electric field amplitude and r41 is a positive constant13). In contrast to (001) or (110)-oriented quantum wells, equations (2) and (3) show that the BIA and SIA vectors have exactly the same direction in the (111) QW plane and the same electron momentum dependence25. Thus the total spin precession vector has the form:

This spin precession vector will drive the well-known dominant D’yakonov–Perel electron spin-relaxation mechanism in the quantum well. The corresponding spin-relaxation time is given by14,41:

where is the mean square precession vector in the plane perpendicular to the electron spin direction (which is here along z) and is the electron momentum-relaxation time41. Substitution of equation (4) into equation (5) and taking the thermal average yields:

This means that the measurement of both the electron spin-relaxation time τs and the electron momentum-relaxation time as a function of the applied electric field gives a direct determination of both the Dresselhaus and Rashba coefficients.

The electron momentum-relaxation time can be obtained from the spin-grating experimental results presented above. On the basis of the Einstein relation, we have , where De is the charge diffusion constant, μ the electron mobility and m* the electron effective mass. For undoped quantum wells, the electron spin-diffusion coefficient is equal to the electron diffusion coefficient (De=Ds)29. As a consequence, the spin-diffusion coefficient Ds measured in Figure 3c allows us to measure ; we find ~200 fs with no variation with the applied electric field within the experimental uncertainties. Using this value and the measured variation of the electron spin-relaxation time τs as functions of the applied electric field, we have plotted in Fig. 5 the curve β+2r41E using equation (6). From the slope and the intercept of this curve, we get straightforwardly the Rashba and Dresselhaus coefficients (=2.5 × 1016 m−2 is obtained in the framework of a finite QW barrier calculation31). We find β=1.06±0.1 × 10−3 eV nm yielding a Dresselhaus coefficient γ=18±2 eV Å3, whereas the calculated value for GaAs is γ≅24 eV Å3 (refs 13, 35). From the slope of the curve in Fig. 5, we get the Rashba coefficient r41=7±1 e Å2, consistent with the calculated value r41=5.2 e Å2 (ref. 13).

Figure 5: Simultaneous measurement of the Dresselhaus and Rashba coefficients.
figure 5

Variation of β+2r41E as a function of the applied electric field E in the device. The red triangle schematically represents the slope of this function used to extract the Rashba coefficient r41. The intercept of β+2r41E with the ordinate axis gives β (β=4γ‹K z2›/), where γ is the Dresselhaus coefficient. The x axis error bars have been calculated from the uncertainties on the measured stark shift. The y axis error bars correspond to the uncertainties on the measurement of the spin-relaxation time τs and momentum-relaxation time .

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In conclusion, we have demonstrated for the first time the gate control of the electron spin-diffusion length in a semiconductor nanostructure. In addition to its interest for spintronic devices, this effect allows an accurate determination of both Dresselhaus and Rashba coefficients. It can be applied in any zinc-blende III–V and II–VI quantum wells and in wurtzite (0001) nanostructures as well42, as the principle relies only on symmetry considerations. We emphasize that this electrical control of the spin-diffusion length is effective whatever the injected carrier spin orientation is. This opens an avenue for spin manipulation and control in semiconductors.

Methods

Sample

The investigated sample is grown on (111)B-oriented semi-insulating GaAs substrate using molecular beam epitaxy. The (111)B substrate is misoriented 3° towards <111>A. The layer sequence of the NIP structure is: substrate/200 nm GaAs buffer/500 nm p-doped (p=1.2 × 1018 cm−3) GaAs/100 nm Al0.3Ga0.7As barrier/MQW/70 nm Al0.3Ga0.7As barrier/10 nm n-doped (n=2.8 × 1018 cm−3) GaAs (Fig. 1b). The Multiple Quantum Well (MQW) consists of 20 periods of GaAs/Al0.3Ga0.7As quantum wells with well width and barrier width LW=15 nm and LB=12 nm, respectively. The reverse bias voltage (V) is applied between the unetched top n-contact and the chemically etched p-contact.

Spin relaxation and spin transport measurements

In the Transient Spin-Grating experiment, the laser pulses are generated by a mode-locked Ti:sapphire laser with 120 fs pulse width and 76 MHz repetition frequency and are split into primary pump and probe beams with an average power of 5 and 0.5 mW, respectively (Fig. 2a). The wavelength is set to get the maximum signal of Kerr rotation for each reverse bias V through the standard time-resolved Kerr rotation technique18. Both pump and probe beams are focused on a phase mask (StockerYale. Inc.) with a period d. The phase mask splits each of the primary beams by diffraction into the m=±1 orders. The interference between the two pump beams with orthogonal linear polarization yields a spatially modulated circular polarization across the excitation spot. According to the optical interband selection rules, this interference pattern will generate a periodical spin density, called the spin grating. The period Λ of the transient spin grating is simply , where f1 and f2 are the focal lengths of two spherical mirrors. In our set-up, the focal lengths of the first and second spherical mirrors are fixed at f1=f2=30.4 cm. The spot sizes of both pump and probe beams are ~200 μm. The wave vector modulus q of the spin grating is equal to with a phase mask of d=5 μm. Phase masks with d=5, 6, 7 and 8 μm are used. A delayed probe beam with vertical linear polarization is incident on the spin grating and the corresponding diffracted beam is detected as a function of the time delay between the pump and the probe pulses (Fig. 2b). A heterodyne detection of the transient grating is implemented using a reference beam with horizontal linear polarization incident on the sample43. The four beams (two pump beams, probe and reference beams) are arranged in the boxcar geometry43. In this scheme based on two lock-in amplifiers, the pump beam is modulated by a photoelastic modulator with a frequency of 100 kHz, and the probe beam is mechanically chopped at 60 Hz. This heterodyne configuration yields an increase in the signal to noise ratio by more than two orders of magnitude. The spin-grating signal (proportional to the modulus the electric field of the diffracted probe beam) is simply given by ISG=A exp(−ΓΔt), where A is a constant, Γ is the decay rate of the spin grating and Δt is the time delay between the pump and probe pulses. The electron spin-relaxation time τs and electron lifetime τ are measured from time and polarization resolved photoluminescence experiments using circularly polarized picosecond Ti:Sapphire laser pulses and a synchroscan Streak camera with an overall time resolution of 8 ps44. All the measurements are performed at T=50 K.

Additional information

How to cite this article: Wang, G. et al. Gate control of the electron spin-diffusion length in semiconductor quantum wells. Nat. Commun. 4:2372 doi: 10.1038/ncomms3372 (2013).