Introduction

Dynamic nuclear polarization (DNP)1 occurs in many condensed matter systems, and is used for sensitivity enhancement in nuclear magnetic resonance2 and for detecting and initializing solid-state nuclear spin qubits3. DNP also occurs in two-dimensional electron systems4 via the contact hyperfine interaction. In both self-assembled5,6,7,8,9 and gate-defined quantum dots10,11,12,13, for example, DNP is exploited to prolong coherence times for quantum information processing. Closed-loop feedback12 based on DNP, in particular, is a key component in one- and two-qubit operations in singlet-triplet qubits11,14,15.

Despite the importance of DNP, it remains unclear what factors limit DNP efficiency in semiconductor spin qubits16. In particular, the relationship between the spin–orbit and hyperfine interactions17,18,19,20 has been overlooked in previous experimental studies of DNP in quantum dots, although several works have shown that the spin–orbit and hyperfine interactions contribute to spin relaxation21,22,23 under different conditions. It has been theoretically predicted, although not observed experimentally, that the spin–orbit interaction should limit DNP by providing a route for electron spin flips without corresponding nuclear spin flops18,20,24.

In this work, we show that spin–orbit coupling competes with the hyperfine interaction and ultimately quenches DNP in a GaAs double quantum dot14,25, even though the spin–orbit length is much larger than the interdot spacing. We use Landau–Zener (LZ) sweeps to characterize the static and dynamic properties of ΔST(t), the coupling between the singlet S and ms=1 triplet T+, and the observed suppression of DNP agrees quantitatively with a theoretical model. In addition to improving basic understanding of DNP in semiconductors, these results will enable enhanced coherence times in semiconductor spin qubits by elucidating the experimental conditions under which DNP is most efficient26.

Results

Hyperfine and spin–orbit contributions to the S−T+ splitting

Figure 1a shows the double quantum dot used in this work14,25. The detuning, ɛ, between the dots determines the ground-state charge configuration, which is either (1,1) (one electron in each dot) or (0,2) (both electrons in the right dot) as shown in Fig. 1b. To measure the S−T+ coupling, ΔST(t), the electrons are initialized in |(0,2)S〉, ɛ is swept through the S−T+ avoided crossing at ɛ=ɛST, and the resulting spin state is measured (Fig. 2a). When ɛ ≈ɛST, we may describe the double quantum dot by an effective two-state Hamiltonian

Figure 1: Experimental set-up.
figure 1

(a) Scanning electron micrograph of the double quantum dot. A voltage difference between the gates adjusts the detuning ɛ between the potential wells, and a nearby quantum dot on the left senses the charge state of the double dot. The gate on the right couples the double dot to an adjacent double dot, which is unused in this work. The angle between B and the z axis is ϕ. (b) Energy level diagram showing the two-electron spin states and zoom-in of the S−T+ avoided crossing. (c) The hyperfine interaction couples |(1,1)S〉 and |(1,1)T+〉 when the two dots are symmetric, regardless of the orientation of B, and the spin–orbit interaction couples |(0,2)S〉 and |(1,1)T+〉 when B has a component perpendicular to , the effective spin–orbit field experienced by the electrons during tunnelling.

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Figure 2: Measurements of σST.
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(a) Data for a series of LZ sweeps with varying rates, showing reduction in maximum probability due to charge noise. The horizontal axis is proportional to the sweep time. Upper inset: data and linear fit for fast sweeps such that 0<〈PLZ〉<0.1. Lower inset: in a LZ sweep, a |(0,2)S〉 state is prepared, and ɛ is swept through ɛST (dashed line) with varying rates. Here h=2πħ is Planck’s constant. (b) σST versus ϕ (dots) and simulation (solid line). (c) σST versus B for ϕ=0° and ϕ=90° (dots) and fits to equation (3) (solid lines). When ϕ=0°, ΔSO is fixed at zero, and the only fit parameter is σHF. When ϕ=90°, σHF is fixed at the fitted value, and ΔSO is the only fit parameter (see Methods section). Error bars are fit errors.

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in the {|T+〉,|S〉} basis (Fig. 2b), where tc=23.1 μeV is the double-dot tunnel coupling and B is the external magnetic field strength. In the absence of any noise, the probability for an S−T+ transition is given by the LZ formula27,28:

where β=d(ES−ET+)/dt is the sweep rate, with ES and ET+ the energies of the S and T+ levels. Following the LZ sweep, we interpret the experimentally measured triplet return probability as the LZ probability, PLZ.

Equation (2) predicts transitions with near-unity probability for slow sweeps. For large magnetic fields, however, we experimentally observe maximum transition probabilities of ∼0.5. As discussed in Supplementary Note 1 and shown in Supplementary Figs 1 and 2, this reduction is a result of rapid fluctuations in the sweep rate arising from charge noise. Even in the presence of noise, however, the average LZ probability 〈PLZ(t)〉 can be approximated for fast sweeps as , which is identical to the leading order behaviour in β−1 of the usual LZ formula29. Here 〈⋯〉 indicates an average over the hyperfine distribution and charge fluctuations. To accurately measure , we therefore fit 〈PLZ〉 versus β−1 to a straight line for values of β such that 0<〈PLZ〉<0.1 (Fig. 2a).

We first measure σST versus ϕ at B=0.5 T (Fig. 2b), where ϕ is the angle between the magnetic field B and the z axis (Fig. 1a). σST oscillates between its extreme values at 0° and 90° with a periodicity of 180°. Fixing ϕ=0° and varying B, we find that σST decreases weakly with B, but when ϕ=90°, σST increases steeply with B, reaching values >10 times that for ϕ=0°, as shown in Fig. 2c.

We interpret these results by assuming that both the hyperfine and spin–orbit interactions contribute to ΔST(t) and by considering the charge configuration of the singlet state at ɛST (Fig. 1b,c). The matrix element between S and T+ can be written as ΔST(t)=ΔHF(t)+ΔSO. ΔHF(t)=g*μBδB⊥(t) is the hyperfine contribution, which is a complex number, and it arises from the difference in perpendicular hyperfine field, , between the two dots30. Here and are coordinates perpendicular to B. (In the following, we set g*μB=1 and give the hyperfine field strength in units of energy.) ΔHF(t) couples |(1,1)S〉 to |(1,1)T+〉 when the two dots are symmetric. ΔSO is the spin–orbit contribution, which arises from an effective magnetic field (see Methods section) experienced by the electron during tunnelling17. Only the component of ΩSO⊥B causes an electron spin flip. ΔSO therefore couples |(0,2)S〉 to |(1,1)T+〉 when ϕ ≠0°, and ΩSO is proportional to the double-dot tunnel coupling tc (ref. 17), which is 23.1 μeV here. At ɛST, the singlet state |S〉 is a hybridized mixture: |S〉=cosθ|(1,1)S〉+sinθ|(0,2)S〉, where the singlet mixing angle approaches π/2 as B increases. Taking both θ and ϕ into account, we write17:

The data in Fig. 2b therefore reflect the dependence of ΔST(t) on ϕ in equation (3). The data in Fig. 2c reflect the dependence of ΔST(t) on θ. As B increases, θ also increases, and |S〉 becomes more |(0,2)S〉-like, causing ΔHF(t) to decrease. When ϕ=0°, ΔSO=0 for all B, but when ϕ=90°, ΔSO=ΩSO sinθ, and σST increases with B. Fitting the data in Fig. 2c allows a direct measurement of the spin–orbit and hyperfine couplings (see Methods section). We find and ΩSO=461±10 neV, corresponding to a spin–orbit length λSO≈3.5 μm (refs 24, 31) (see Methods section), in good agreement with previous estimates in GaAs32,33,34.

Spectral properties of the S−T+ splitting

We further verify that ΔST(t) contains a significant spin–orbit contribution by measuring the dynamical properties of PLZ(t). A key difference between the spin–orbit and hyperfine components is that ΔSO is static, whereas ΔHF(t) varies in time because it arises from the transverse Overhauser field, which can be considered a precessing nuclear polarization in the semiclassical limit30. To distinguish the components of ΔST(t) through their time dependence, we develop a high-bandwidth technique to measure the power spectrum of PLZ(t).

Instead of measuring the two-electron spin state after a single sweep, ɛ is swept twice through ɛST with a pause of length τ between sweeps (Fig. 3a) (see Methods section). Assuming that Stückelberg oscillations rapidly dephase during τ (refs 15, 32), and after subtracting a background and neglecting electron spin relaxation, the time-averaged triplet return probability is proportional to RPP(τ)≡〈PLZ(t)PLZ(t+τ)〉, the autocorrelation of the LZ probability (Fig. 3b). Taking a Fourier transform therefore gives SP(ω), the power spectrum of PLZ(t) (Fig. 3c,d). For PLZ(t)<<1, PLZ(t)∝|ΔST(t)|2, so , the power spectrum of |ΔST(t)|2. This two-sweep technique allows us to measure the high-frequency components of SP(ω), because the maximum bandwidth is not limited by the quantum dot readout time.

Figure 3: Correlations and power spectrum of PLZ(t).
figure 3

(a) Pulse sequence to measure RPP(τ) using two LZ sweeps. (b) RPP(τ) for ϕ=0° and B=0.1 T. The data extend to τ=200 μs, but for clarity are only shown to 75 μs here. (c) SP(ω) versus ϕ obtained by Fourier-transforming RPP(τ). At ϕ=0°, the differences between the nuclear Larmor frequencies are evident, but for |ϕ|>0°, the absolute Larmor frequencies appear, consistent with a spin–orbit contribution to σST. The reduction in frequency with ϕ is likely due to the placement of the device slightly off-centre in our magnet, and the reduction in amplitude of the difference frequencies occurs because the sweep rate β was increased with ϕ to maintain constant 〈PLZ〉 (see Methods section). (d) Line cuts of SP(ω) at ϕ=0°, 25° and 80°.

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Because it arises from the precessing transverse nuclear polarization, ΔHF(t) contains Fourier components at the Larmor frequencies of the 69Ga, 71Ga and 75As nuclei in the heterostructure, that is, , where α indicates the nuclear species, and the θα are the phases of the nuclear fields. Without spin–orbit interaction, contains only Fourier components at the differences of the nuclear Larmor frequencies. With a spin–orbit contribution, however, |ΔST(t)|2=|ΔSO+ΔHF(t)|2 contains cross-terms such as that give |ΔST(t)|2 Fourier components at the absolute Larmor frequencies. A signature of the spin–orbit interaction would therefore be the presence of the absolute Larmor frequencies in SP(ω) for ϕ ≠0° (ref. 35).

Figure 3b shows RPP(τ) measured with B=0.1 T and ϕ=0°. Fig. 3c shows SP(ω) for 0°≤ϕ≤90°. At ϕ=0°, only the differences between the Larmor frequencies are evident, but as ϕ increases, the absolute nuclear Larmor frequencies appear, as expected for a static spin–orbit contribution to ΔST(t). These results, including the peak heights, which reflect isotopic abundances and relative hyperfine couplings, agree well with simulations (Supplementary Fig. 3).

Dynamic nuclear polarization

Having established the importance of spin–orbit coupling at the S−T+ crossing, we next investigate how the spin–orbit interaction affects DNP. Previous research has shown that repeated LZ sweeps through ɛST increase both the average and differential nuclear longitudinal polarization in double quantum dots11. However, the reasons for left/right symmetry breaking, which is needed for differential DNP (dDNP), and the factors limiting DNP efficiency in general are only partially understood. Here we measure dDNP precisely by measuring δBz, the differential Overhauser field, using rapid Hamiltonian learning strategies36 before and after 100 LZ sweeps to pump the nuclei with rates chosen such that 〈PLZ〉=0.4 (see Methods section; Fig. 4a).

Figure 4: DNP quenching by spin–orbit coupling.
figure 4

(a) Protocol to measure DNP. δBz is measured before and after 100 LZ sweeps by evolving the electrons around δBz. (b) dDNP versus ϕ at fixed 〈PLZ〉=0.4 for B=0.8 and 0.2 T, and theoretical curves (solid lines). dDNP is suppressed for |ϕ|>0 because of spin–orbit coupling. (c) Data and theoretical curves for fixed 〈PLZ〉 collapse when normalized and plotted versus σHF/σST. Vertical error bars are statistical uncertainties and horizontal error bars are fit errors.

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Figure 4b plots the change in δBz per electron spin flip for B=0.2 and 0.8 T for varying ϕ. In each case, the dDNP decreases with |ϕ|. Because the spin–orbit interaction allows electron spin flips without corresponding nuclear spin flops, dDNP is suppressed as |ΔSO|=|ΩSO sinϕ sinθ| increases with |ϕ|. The reduction in dDNP occurs more rapidly at 0.8 T because θ, and hence ΔSO, are larger at 0.8 T than at 0.2 T. We gain further insight into this behaviour by plotting the data against σHF/σST, where (Fig. 4c). Plotted in this way, the two data sets show nearly identical behaviour, suggesting that the size of the hyperfine interaction relative to the total splitting primarily determines the DNP efficiency.

It is interesting to note that the peak DNP efficiency is less at B=0.2 T than at B=0.8 T. A possible explanation is that the electron–nuclear coupling becomes increasingly asymmetric with respect to the quantum dots at higher fields, because the singlet state becomes more |(0,2)S〉-like as θ increases37. The gradient build-up could also be due to an asymmetry in the size of the quantum dots38. As a result, we expect that the dDNP should be proportional to the total DNP, with a constant of proportionality that depends possibly on B, but not β or ϕ, in agreement with forthcoming theoretical and experimental work. We therefore explain our measurements of dDNP using a theoretical model in which we have computed the average angular momentum 〈δm〉 transfered to the ensemble of nuclear spins following a LZ sweep as:

where is the derivative of the LZ probability with respect to the magnitude of the splitting. (See Supplementary Note 2 for more details.) Neglecting charge noise, we have the usual LZ formula (equation (2)), and equation (4) reduces to

The data in Fig. 4b,c can therefore be understood in light of equation (5) because as the splitting σST increases with |ϕ|, the sweep rate β was also increased to maintain a constant 〈PLZ〉. Because the hyperfine contribution σHF is independent of ϕ, 〈δm〉 therefore decreases. The data collapse in Fig. 4c can also be understood from equation (5), assuming constant ΔST(t) and fixed PLZ. In this case, β∝|ΔST|2, as follows from equation (2), and hence . Measurements with fixed rate β also exhibit a similar suppression of dDNP (Supplementary Fig. 4 and Supplementary Note 2). In this case 〈PLZ〉 increases with |ϕ|, because of the increasing spin–orbit contribution to σST, and according to equation (5), 〈δm〉 therefore decreases.

The two theoretical curves in Fig. 4b,c are calculated using equation (5) multiplied by fitting constants C, which are different for the two fields, and agree well with the data. As discussed in Supplementary Note 2 and shown in Supplementary Fig. 5, we do not expect charge noise to modify the agreement between theory and data in Fig. 4b,c beyond the experimental accuracy. Finally, the peak dDNP value also approximately agrees with a simple calculation (Supplementary Note 3) based on measured properties of the double dot.

Discussion

In summary, we have used LZ sweeps to measure the S−T+ splitting in a GaAs double quantum dot. We find that the spin–orbit coupling dominates the hyperfine interaction and quenches DNP for a wide range of magnetic field strengths, unless the magnetic field is oriented such that B||ΩSO. A misalignment of B to ΩSO by only 5° at B=1 T can reduce the DNP rate by a factor of two, and DNP is completely suppressed for a misalignment of 15°. The techniques developed here are directly applicable to other quantum systems such as InAs or InSb nanowires and SiGe quantum wells, where the spin–orbit and hyperfine interactions compete. On a practical level, these results will improve coherence times in gate-defined quantum dot spin qubits by enabling more efficient DNP12, and the high-bandwidth correlation measurements demonstrated here offer a new tool to investigate nuclear dynamics in semiconductors. On a fundamental level, our findings suggest avenues of exploration for improved S−T+ qubit operation32 and underscore the importance of the spin–orbit interaction in the study of nuclear dark states37,38 and other mechanisms that limit DNP efficiency in central-spin systems.

Methods

Device details

The double dot is fabricated on a GaAs/AlGaAs hetereostructure with a two-dimensional electron gas located 90 nm below the surface. Au/Pd depletion gates are used to define the double-dot potential. The double dot is cooled in a dilution refrigerator to a base temperature of ∼50 mK. The double-dot axis is aligned within ≈5° of either the or [110] axes of the crystal, but we do not know which. In the latter case, both the Rashba and Dresselhaus spin–orbit fields are aligned with the z axis, and their magnitudes add17. In the former case, the Rashba and Dresselhaus contributions are also aligned with the z axis, but their magnitudes subtract. Because the spin–orbit field is aligned with the z axis in each case, we do not expect the orientation of the double dot to qualitatively change our results. When ϕ=90°, B lies in the plane of the crystal and parallel to the double-dot axis.

Measuring σST, ΔSO and σHF

In order to extract σST, we fit the measured 〈PLZ〉 versus β to a function of the form . We calibrate the sweep rate β using the spin-funnel technique25. We extract the spin–orbit and hyperfine strengths by fitting the data in Fig. 2c to a function of the form , with ΩSO and σHF as fit parameters. The singlet mixing angle is computed using the measured double-dot tunnel coupling tc=23.1 μeV.

ΔSO is held at 0 when fitting data for ϕ=0° to determine the hyperfine coupling. We also exclude data points for B<0.2 T in the fit, as the hyperfine contribution appears to decrease at very low fields. We determine the spin–orbit length as (refs 24, 31), where tc is the interdot tunnel coupling, and d≈100 nm is half of the interdot spacing. The curve in Fig. 2b is a simulation, not a fit, and is generated using the same equation with the fitted values of ΔSO and σHF.

Measuring RPP(τ)

Here we derive the triplet return probability after two consecutive LZ sweeps with a pause of length τ in between. In experiments, both sweeps were in the same direction, and ɛ was held in the (0,2) region between sweeps (Fig. 3a). If the first LZ sweep takes place at time t with probability PLZ(t), the probability for the two electrons to be in the T+ state is PLZ(t), whereas the probability to be in the S state is 1−PLZ(t). Then, the detuning is quickly swept into the (0,2) region, where electron spin dephasing occurs rapidly, and there is negligible T+ occupation in thermal equilibrium because the S and T+ states are widely separated in energy. After a wait of length τ, but before the second sweep, the triplet population is and the singlet population is , where T1 is the electron relaxation time. After the second sweep, the triplet occupation probability is

The second and third terms in equation (7) vary slowly with τ. Experimentally, these terms are found by fitting the measured triplet probability to an exponential with an offset and are subtracted. When T1>>τ, relaxation can be neglected, and the predicted time-averaged signal is 〈PT(t+τ)〉∝RPP(τ), where RPP(τ)≡〈PLZ(t)PLZ(t+τ)〉, the autocorrelation of the LZ probability. When ϕ=0°, T1>>τmax=200 μs, where τmax is the largest value of τ measured. The shortest relaxation time T1≈100 μs in these experiments time occurs when ϕ=90°, which is consistent with spin–orbit-induced relaxation23.

The effect of T1 relaxation is to multiply the measured correlation by an exponentially decaying window, which reduces the spectral resolution of the Fourier transform, but does not shift the frequency of the observed peaks. We expect statistical fluctuations in the amplitude of the hyperfine field to affect the spectrum in a similar way, although we expect this effect to be less than that of electron relaxation. The raw data, (Fig. 3b) consisting of 667 points (each a result of two sweeps with a 40% chance of a LZ transition), spaced by 300 ns, were zero padded to a size of 1,691 points to smooth the spectrum, and a Gaussian window with time constant 150 μs was applied to reduce the effects of noise and ringing from zero padding before Fourier transforming.

The magnetic resonance frequencies in Fig. 3c decrease with ϕ. The inhomogeneity of the x-coil in our vector magnet is 1.6% at 0.6 cm offset from the centre. Thus, the field could easily be reduced by >3% for a misplacement of the sample by 1 cm from the magnet centre. We have simulated the data in Fig. 3c in the main text based on the measured hyperfine and spin–orbit couplings and the known sweep rates. Assuming a 4.4% reduction in the field from the x-coil, we obtain good agreement between theory and experiment (Supplementary Fig. 3). The difference frequencies also decrease in strength with increasing ϕ, which happens because the sweep rate β was increased with ϕ to maintain constant 〈PLZ〉. This effect can be understood for fast sweeps, where the amplitudes of the difference frequencies should scale as Δ1Δ2/β, where the subscripts indicate different nuclear species.

We argued in the main text that only the difference frequencies should appear in the spectrum SP(ω) without spin–orbit coupling by considering the time dependence of |ΔST(t)|2 and because when PLZ(t)<<1. Since PLZ(t) contains only even powers of |ΔST(t)|, SP(ω) can generally be expressed in terms of differences of the resonance frequencies, but will not contain the absolute frequencies in the absence of spin–orbit coupling, regardless of the value of PLZ(t).

Measuring δBz

We measure δBz by first initializing the double dot in the |(0,2)S〉 state and then separating the electrons by rapidly changing ɛ to a large negative value25. When the electrons are separated, the exchange energy is negligible, and the magnetic field gradient δBz drives oscillations between |S〉 and |T0〉. In our experiments, we measure the two-electron spin state for 120 linearly increasing values of the separation time. The resulting single-shot measurement record is thresholded, zero padded and Fourier transformed. The frequency corresponding to the peak in the resulting Fourier transform is chosen as the value of δBz. This technique is related to a previously described rapid Hamiltonian estimation technique36.

Additional information

How to cite this article: Nichol, J. M. et al. Quenching of dynamic nuclear polarization by spin–orbit coupling in GaAs quantum dots. Nat. Commun. 6:7682 doi: 10.1038/ncomms8682 (2015).