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Nat Commun
Nat Commun
Nature Communications
2041-1723
Nature Publishing Group UK London

39227367
51902
10.1038/s41467-024-51902-9
Article
A singlet-triplet hole-spin qubit in MOS silicon
http://orcid.org/0000-0001-8657-6681
Liles S. D. s.liles@unsw.edu.au

1
Halverson D. J. 1
Wang Z. 1
Shamim A. 1
http://orcid.org/0009-0007-3483-8986
Eggli R. S. 2
Jin I. K. 13
Hillier J. 1
Kumar K. 1
http://orcid.org/0000-0001-6634-3493
Vorreiter I. 1
http://orcid.org/0000-0003-1689-8356
Rendell M. J. 1
http://orcid.org/0000-0002-1729-3052
Huang J. Y. 45
Escott C. C. 45
http://orcid.org/0000-0003-0134-3657
Hudson F. E. 45
Lim W. H. 45
http://orcid.org/0000-0002-2342-0396
Culcer D. 1
http://orcid.org/0000-0003-1389-5096
Dzurak A. S. 45
http://orcid.org/0000-0001-7484-3738
Hamilton A. R. 1
1 https://ror.org/03r8z3t63 grid.1005.4 0000 0004 4902 0432 School of Physics, University of New South Wales, Sydney, NSW 2052 Australia
2 https://ror.org/02s6k3f65 grid.6612.3 0000 0004 1937 0642 Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
3 grid.7597.c 0000000094465255 Center for Emergent Matter Science, RIKEN, 2-1, Hirosawa, Wako-shi, 351-0198 Saitama Japan
4 https://ror.org/03r8z3t63 grid.1005.4 0000 0004 4902 0432 School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052 Australia
5 Diraq, Sydney, NSW Australia
3 9 2024
3 9 2024
2024
15 76902 1 2024
19 8 2024
© The Author(s) 2024
2024
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Holes in silicon quantum dots are promising for spin qubit applications due to the strong intrinsic spin-orbit coupling. The spin-orbit coupling produces complex hole-spin dynamics, providing opportunities to further optimise spin qubits. Here, we demonstrate a singlet-triplet qubit using hole states in a planar metal-oxide-semiconductor double quantum dot. We demonstrate rapid qubit control with singlet-triplet oscillations up to 400 MHz. The qubit exhibits promising coherence, with a maximum dephasing time of 600 ns, which is enhanced to 1.3 μs using refocusing techniques. We investigate the magnetic field anisotropy of the eigenstates, and determine a magnetic field orientation to improve the qubit initialisation fidelity. These results present a step forward for spin qubit technology, by implementing a high quality singlet-triplet hole-spin qubit in planar architecture suitable for scaling up to 2D arrays of coupled qubits.

Hole-spin qubits based on semiconductor quantum dots offer potential advantages over their electron-spin counterparts, such as fast qubit control and enhanced coherence times. Liles et al. report a hole-based singlet-triplet spin qubit in planar Si MOS device and develop a model to describe its dynamics.

Subject terms

Qubits
Quantum dots
Quantum information
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Silicon metal-oxide-semiconductor (MOS) quantum dots offer a compelling platform for implementing quantum circuits due to their compatibility with existing semiconductor foundry processes1,2. A leading approach for developing silicon qubits are silicon spin-qubits, due to their extended coherence times, minimal device footprint, and the potential for high integration densities3. The most straightforward implementation of a spin qubit uses single-spins4. This qubit encodes information using the ↑ and ↓ spin states of a localised electron. However, the adaptability of semiconductor quantum dot technologies permits a range of spin qubit implementations, each with unique advantages for quantum technology5.

The singlet-triplet qubit is an alternative form of spin qubit, which offers several advantages for implementation in scalable quantum circuits6–9. A singlet-triplet qubit uses the singlet (S=(↑↓−↓↑)/2) and unpolarised-triplet (T0=(↑↓+↓↑)/2) states of two exchange-coupled spins. While using two spins rather than one increases the fabrication footprint and the complexity of the eigenstates, singlet-triplet qubits offer specific advantages. Singlet-triplet qubits can be operated at very low magnetic fields (<5 mT), which enables compatibility with magnetic-sensitive components such as superconducting resonators10–13. Additionally, singlet-triplet qubits can be controlled using only baseband control pulses, with spectral components generally not exceeding 100 MHz. This reduces the cost and complexity of control hardware compared with single-spin qubits, which typically require GHz phase-controlled tones. Moreover, removing the need for GHz control tones is advantageous since high-frequency tones dissipate more power at the device than baseband pulses, and this extra power dissipation impacts qubit control fidelity14–17.

Hole spins in Group IV materials offer significant opportunities for use as fast scalable spin qubits18–22 because of the strong intrinsic spin–orbit coupling, which is vanishingly small for electron spins. The intrinsic spin–orbit coupling allows rapid electrical manipulation of hole-spin qubits with recent state-of-the-art demonstrations reaching control speeds in the sub-nanosecond range23. Importantly, this rapid control is possible without the need for additional bulky features such as micro-magnets or ESR strip-lines. Further, the g-factor24–30 and spin–orbit coupling31 for holes are both tunable, providing a wide range of in-situ control over hole-qubits. In addition, hole-spins have the potential for enhanced coherence times due to suppressed hyperfine coupling32, and the potential for configuring decoherence sweet-spots by tuning the spin–orbit interaction33–36.

Despite the opportunities holes offer, there currently are only limited studies of hole-based singlet-triplet qubits. Recently, a hole-spin singlet-triplet qubit was demonstrated in Ge37, where the strong spin–orbit coupling resulted in non-trivial qubit dynamics38. However, the spin–orbit effects in Si devices vary from Ge devices. Most notably, the splitting between the heavy hole and light hole bands is typically an order of magnitude larger in planar Ge devices compared with planar Si devices39,40. Since the spin–orbit interaction is influenced primarily by the heavy hole light hole splitting, the spin–orbit-driven phenomena can vary significantly between the two material systems. For example, the ratio of the in-plane to out-of-plane g-factors observed in Ge is significantly larger than in Si38. Given the difference in spin–orbit effects between Ge and Si, investigations into Si are needed to provide an understanding of Si holes.

Recent experiments in silicon FinFET’s have revealed an anisotropic exchange coupling for holes due to the spin–orbit interaction41, which may provide unique functionalities for hole-spin singlet-triplet qubits. Indeed, theoretical predictions have suggested that the non-trivial relationship between spin–orbit coupling and the site-dependent g-tensors may allow hole-spin singlet-triplet qubits to avoid leakage errors42. However, to-date, there are no demonstrations of a singlet-triplet qubit using holes in silicon.

In this work, we demonstrate a hole-spin singlet-triplet qubit formed in a planar MOS silicon double quantum dot. The planar structure provides a platform suitable for scaling up to the large arrays of coupled qubits needed for quantum circuits and error correction43–45. We identify the exact hole occupation of the double dot, which is critical for experimental reproducibility and detailed theoretical modelling of this system. In addition to characterising the key parameters of the qubit, we perform an investigation into the anisotropy of the two-hole eigenstates. By comparing the experimental results with a model that includes spin–orbit coupling and anisotropic site-dependent g-tensors, we identify key features in the eigenstates that allow the improvement of the initialisation fidelity and reduction in the readout errors.

Results

Device and operating regime

The hole-spin singlet-triplet qubit is formed using a planar-silicon double quantum dot device, fabricated using industrially compatible CMOS techniques. Figure 1a shows a model 3D cross-section of the double quantum dot region. Multi-layer palladium gates define the double quantum dot with P1 and P2 operating as plunger gates, while Jg provides in-situ control of the interdot tunnel coupling tc46. The device employs ambipolar charge sensing47, with an adjacent nMOS SET allowing the absolute charge occupation of each quantum dot to be determined.Fig. 1 Device operating point and energy spectrum.

a A 3D model of the device, showing a cross-section through the double quantum dot region. A full SEM image is shown in Supplementary Fig. 1. The tunnelling to the P1 dot was fast (<100 ns), while loading onto P2 dot was slow (>40 μs). This asymmetry in tunnelling allows latched readout. b Stability diagram of the (2,8)-(1,9) transition. The labels (N,M) indicate the number of holes in the P1 and P2 dot, respectively. The colour scale is the sensor current (Is) in pA. Zero detuning (ϵ = 0) is defined as the (2,8) and (1,9) charge degeneracy point, and positive detuning when the spins are separated into (1,9). Full details of the pulse sequence are discussed in Supplementary Note 2. c Eigenstates calculated using the singlet-triplet Hamiltonian HST defined in Supplementary Note 4. The coloured arrows indicate the energy transitions observed in the preceding experiments, and ΔST indicates the size of the avoided crossing between S and T±. Orange dashed lines show how the (1,9) states evolve in the absence of spin–orbit coupling. d The results of a spin funnel experiment with the pulse sequence are shown in the inset. The spin-funnel experiment was performed by initialising S2,8, followed by a rapid pulse to a point along the detuning axis (ϵ). At each Bx and ϵ, the state was allowed to evolve for a fixed separation time, τs = 100 ns, followed by a pulse to the readout point. The change in sensor signal (ΔI) due to this pulse indicates the likelihood of the returned state being singlet (low ΔI) or triplet (high ΔI) (described in more detail in the Methods). Red arrows indicate the Δg-driven oscillations. e The same pulse procedure as in d) except the magnetic field is fixed at Bx = 5 mT and we investigate the effect of varying the separation time (τS) at each detuning (ϵ). f The corresponding FFT at each detuning. Transparent lines indicated the best fit of the observed energy splittings to the eigenstates of HST, and the colours correspond to the transitions indicated in (c).

Figure 1b shows a stability diagram measured using the charge sensor. We perform all measurements in the (2,8)-(1,9) configuration, which is equivalent to a (2,0)-(1,1) spin system due to orbital shell filling48. We initialised singlet states by dwelling deep in (2,8) where S2,8 is the lowest energy eigenstate (point I). Manipulation of the state was performed by pulsing to a position along the detuning axis (ϵ) and dwelling there for a variable time τs. Readout of the state was performed by pulsing to point R (following the dashed trajectory), where latched Pauli–Spin–Blockade49,50 readout allowed identification of either the blocked triplet or the unblocked singlet states based on the average sensor current (see Methods and Supplementary Notes 1–3).

System Hamiltonian and eigenenergies

To model the two-spin system, we consider a 5 × 5 Hamiltonian, HST, which includes Zeeman, spin–orbit and orbital terms. The full details of the two-hole singlet-triplet Hamiltonian are provided in Supplementary Note 4. For the Zeeman Hamiltonian, we include independent 3 × 3 symmetric g-tensors for the left and right dot, g⃡L and g⃡R, respectively. Hole-spins in silicon are known to have strongly anisotropic g-tensors25–27,41, where variations in the g-tensor are produced by non-uniform strain27,29, spin–orbit coupling30 and differences in the confinement profile between the two dots24. Hence, we do not assume that g⃡L and g⃡R are correlated or share the same principle spin-axes. For the spin–orbit Hamiltonian we include a spin–orbit vector, tso = (tx, ty, tz), parameterising the effect of spin–orbit coupling in the laboratory reference frame indicated in Fig. 1a.

In Fig. 1c, we plot the eigenenergies of HST as a function of detuning. At negative detuning, the eigenstates are the (T+,T0,T−,S,S2,8) basis states. At large positive detuning, the eigenstates evolve into the S2,8 state and the four two-spin states (↑↑,↑↓,↓↑,↓↓), which are defined by the sum or difference of the Zeeman energy in the two dots. The ↑↓ and ↓↑ eigenstates have energy splitting given by1 EST0=J(ϵ,tc)2+ΔEZ2

where2 J(ϵ)=ϵ24+2tc2−ϵ2

3 ΔEZ=∣Δg*∣μB∣B∣

ϵ is the detuning energy, tc is the interdot tunnel coupling, Δg* is the difference in the effective g-factors for the applied magnetic field vector B (see Supplementary Note 4), and μB is the Bohr magneton. Since strong spin–orbit coupling results in an anisotropy in ∣Δg*∣ with respect to magnetic field orientation, we expect EST0 to exhibit a non-trivial anisotropy41. An avoided crossing occurs between the S2,8 and T±, with the amplitude of the avoided crossing (ΔST±) determined by the interplay between the spin–orbit vector and the difference in the projection of the g-tensors for the given field orientation38.

Figure 1d shows the charge sensor response of a spin-funnel experiment used to characterise the singlet-triplet system7. The spin-funnel experiment was performed by allowing a singlet state to time-evolve for τs = 100 ns at each Bx and ϵ. The change in sensor signal (ΔI) then measures the likelihood that the final state is either singlet (low ΔI) or triplet (high ΔI). A clear funnel edge is visible in ΔI when the detuning point coincides with the S and T± avoided crossing7. In addition, on the positive detuning side of the funnel edge, we see oscillations that result from Δg-driven S↔T0 oscillations (red arrows).

Figure 1e demonstrates the time evolution of the singlet at each detuning. The experimental procedure is the same as Fig. 1d, however here we varied the separation time (τS) at each detuning (ϵ) and held the magnetic field constant at Bx = 5 mT. Figure 1f shows the FFT of ΔI at each ϵ, revealing three clear oscillation frequencies, each with a distinct detuning dependence. Each oscillation frequency results from mixing between the three lowest eigenstates at the separation detuning (ϵ). The lowest frequency (red) results from oscillations between S↔T0 states, the middle frequency (blue) results from oscillations between S↔T±, and the highest frequency (green) results from oscillations between T0↔T±. The corresponding transitions are indicated by coloured arrows in Fig. 1c and a full description is provided in the methods.

We fit the observed frequencies in Fig. 1f to the eigenergies of the singlet-triplet Hamiltonian, HST and extract key parameters of the two-hole system. Transparent lines in Fig. 1f show the best fit, demonstrating good agreement between the observed and theoretical eigenenergies. Based on the best fit, we extract tc = 9 ± 1 μeV and two effective g-factors of 0.8 ± 0.1 and 1.2 ± 0.1 for Bx.

Anisotropic g-tensors and spin–orbit coupling

To characterise the key parameters of the two-hole system, we investigate the effect of magnetic field orientation on the two-hole eigenenergies. Figure 2a shows ΔI as a function of τS for a range of magnetic field orientations in the x-z plane, and Fig. 2b shows the resulting FFT of ΔI. Figure 2c, d repeat the same experiment for a rotation of the magnetic field through the x-y plane. Clear anisotropy with respect to magnetic field orientation can be observed, which results from the interplay between spin–orbit coupling and the orientation of the g-tensors. The visibility of the higher frequency (blue and green) oscillations also shows a strong dependence on the magnetic field orientation. In particular, the FFT amplitude of the higher frequency (blue and green) oscillations is suppressed for Bx and enhanced for By and Bz.Fig. 2 Eigenenergy anisotropy with respect to magnetic field orientation.

a, b Shows the sensor signal and resulting FFT when using the pulse sequence given in Fig. 1d) (initialise-separate-readout). Detuning is fixed at ϵ = 1.9 meV and a 10 mT magnetic field is rotated by 180∘ through the x-z plane. c, d Shows the same experiment for a rotation through the sample x-y plane. Transparent solid lines show the optimal fit of HST to the experimental data, and a zoom-in of the first 60 ns of (c) is reproduced later in the Methods, highlighting the multiple frequencies present. See Supplementary Note 5 for the full 360∘ data set in the x-y plane. e Shows the eigenenergies for ∣B∣ = 10 mT magnetic field applied at θ = 0∘ (purple), 110∘ (brown) and 180∘ (cyan) in the x-y plane, respectively. The y-axis ticks are in 0.5 μeV, and the x-axis ticks are separated by 1 meV. The energy splitting corresponding to the three FFT peaks in (d) are indicated by the red, blue and green vertical lines. The size and location of the S-T− avoided crossing varies with field orientation (black dashed circle), resulting in anisotropy in the Landau–Zener transition probability between S2,8→T− during the ramp-in/ramp-out. The black dashed lines indicated the splitting of the initial state when pulsing across the ΔST avoided crossing. f The solid black line is the calculated probability of the S2,8 loading into T− during the separation pulse (PT−, left axis). Blue markers indicate the amplitude of the S↔T− FFT peak in (d) (transparent blue). Both data were plotted as a function of the in-plane magnetic field angle. The trend in PT− correlates with the amplitude of the S↔T− oscillations in (d). The correlation between PT− and the S↔T− oscillation amplitude is expected since the S↔T− oscillations are enhanced as the probability of loading the T− state increases.

The 3 × 3 g-tensors for each dot and the spin–orbit orientation can be extracted by fitting the data in Fig. 2a–d to the eigenenergies of HST. The fitting procedure is discussed in the methods and Supplementary Notes 7, 8. The transparent lines in Fig. 2b, d indicate the frequency of the respective FFT peaks for the optimal fit parameters. For the optimal fit we find (tsox,tsoy,tsoz) = (−37 ± 2,107 ± 4,0 ± 20) neV, giving ∣tso∣ = 0.12 μeV. Notably, tso is oriented in-plane with the 2DHG, consistent with expectations for heavy holes in planar silicon51. Further, the in-plane spin–orbit vector has components in both tsox and tsoy, indicating that a combination of Rashba (oriented perpendicular to the double dot axis) and Dresselhaus (oriented parallel to the double dot axis) spin–orbit components are present52. The full g-tensors are presented in the methods. We find that the orientation of the g-tensor principle axes for left and right dots are slightly misaligned, which may result from differences in confinement profile or non-uniform strain between the left and right dots. The observation of misalignment in the g-tensor principle axes suggests accurate modelling of multiple quantum dot systems in silicon should incorporate site-dependent g-tensors with differing principle axes.

The anisotropy in the FFT amplitudes in Fig. 2a–d is caused by the probability of transitioning from S2,8 into T− during the pulse from (2,8) to (1,9). When pulsing from (2,8) to (1,9) the ΔST± avoided crossing causes the initial S2,8 state to be split between S and T− with a ratio determined by the Landau–Zener transition probability53 (see methods and Supplementary Note 6). Larger ΔST± favours T− states, while smaller ΔST± favours S. In Fig. 2e, we plot the energy spectrum of the two-hole system for various in-plane magnetic field orientations. The magnetic field orientation strongly influences the magnitude of ΔST± and the position in detuning (ϵΔ) at which the avoided-crossing occurs. As a result, the magnetic field orientation impacts the likelihood of populating the T− state during the separation ramp and thus impacts the amplitude of the S↔T− FFT peak.

We simulated the experimental pulse sequence using QuTIP54 and calculated the S2,8→T− transition probability (PT−), which yielded good agreement with the measured FFT amplitude. In Fig. 2f, the solid line shows the calculated PT− using the optimal fit parameters for a range of in-plane magnetic field orientations (See Supplementary Note 7 for details). The circles in Fig. 2f show the observed amplitudes of the S↔T− FFT peaks from Fig. 2d (blue). The trend in PT− matches the anisotropy in the measured amplitudes of the S↔T− FFT peaks from Fig. 2d, with both exhibiting a peak around θ = 120°, and an asymmetric reduction towards 0° and 180°. The correlation between FFT amplitude and the calculated PT− demonstrates that the model HST and optimal fit parameters capture the dynamics of the hole-spin qubit well (see Methods).

We now consider how hole-spin singlet-triplet qubits can be further optimised by using the anisotropic response of the system to a magnetic field. For the singlet-triplet qubit studied here, an optimal initialisation protocol would suppress the likelihood of the S2,8 state loading into the T− leakage state. In the presence of large spin–orbit coupling and/or large Δg, even rapid separation pulses may be unable to satisfy the non-adiabatic Landau–Zener requirement imposed by the large ΔST− avoided crossing. However, with knowledge of the orientation of spin–orbit vector (tso), and the g-tensors, we can identify optimum field orientations to minimise ΔST− and thus enhance the initialisation fidelity. Indeed, in Fig. 2f we have shown that the Bx magnetic field suppresses loading into the T− state, and therefore is an optimal field orientation for the singlet-triplet qubit in this system.

Finally, we comment on the implications of the fitting procedure used. Here we demonstrate a method that is able to characterise the two arbitrary g-tensors and the spin–orbit vector without the need for GHz EDSR tones that are typically used25,41. The fitting approach, which is described in detail in the methods and Supplementary Notes 7,8, leverages both the observed eigenenergies (FFT frequencies) and occupation probabilities (FFT amplitudes) in order to determine an optimal fit. While, we note that this method does not acquire a unique fit, we highlight that the optimal fits is sufficient to accurately reproduce the key qubit dynamics (see methods). Therefore this approach provides a method for extracting key qubit parameters for devices where EDSR is either not possible (such as in the very low magnetic field regime) or not available.

Coherent Δg-driven oscillations

We now turn to experiments to characterise the hole-spin qubit. Here, the qubit is defined using the S and T0 states of the double quantum dot. The simplified Hamiltonian for this system can be written as4 HST=J2σz+ΔEZ2σx

where J defines the exchange energy, ΔEZ = ∣Δg*∣μB∣B∣, ∣B∣ is the magnitude of the applied field vector, σx,z are the respective Pauli matrices and μB is the Bohr magneton. The Bloch sphere for this qubit system is shown in Fig. 3a. Rotations around the Bloch sphere can be driven by controlling J and ΔEZ at the separation point9,55, and Fig. 3b shows a schematic of the pulse sequence used.Fig. 3 Δg-driven coherent oscillations.

a Shows a schematic of the Bloch sphere for the singlet-triplet qubit. The Zeeman energy difference ΔEZ produces oscillations about the x-axis (S↔T0 oscillations), while exchange coupling J produces oscillations about the z-axis (ie ↑↓↔↓↑ oscillations). When both J and ΔEZ are non-zero, the angle of rotation with respect to the Bloch sphere z-axis is given by θ′ = arctan(ΔEZJ) as indicated by the red trajectory. b Shows the pulse sequence used to drive the Δg-driven oscillations. We initialised in S2,8, then applied a rapid separation pulse along the detuning axis. The system was then held at a fixed positive detuning (ϵ) for a separation time (τS), before performing readout. c Shows the observed S↔T0 oscillations as a function of separation time, for Bx = 20, 5 and 2 mT. The solid red line is the best fit for Eqn. (5) to the data. The singlet probability, PS, at each τS is extracted from the change in the normalised sensor signal ΔI (shown on the right axis, see Supplementary Note 3). d Shows the S↔T0 oscillations over a field range of Bx = ± 30 mT. e Shows the FFT of the oscillations observed in (d). A linear increase in the oscillation frequency is expected since hf = ΔEST0 (Eqn. (1)). The Bx magnetic field orientation was used for these experiments since it results in the least leakage into T− (Fig. 2). However, residual loading of the T− leakage state results in weak S↔T− (blue) and T0↔T− (green) oscillations for Bx. f Shows the T2* for different magnetic fields, where the solid line is the best fit of Eqn. (6) to both this data and the data in Fig. 4e. g Shows the effect of mixing chamber temperature on T2* for two different magnetic fields. h The qubit quality factor (Q) as a function of the Δg-driven oscillation frequency (fR), measured at TMC = 30 mK. All data in Fig. 3 was collected with Jg = 1.2 V. Error bars in (f–h) are the standard deviation in the measured value.

Figure 3c plots the measured singlet probability PS as a function of separation time τS for three different ∣Bx∣, demonstrating oscillations in PS. Solid lines in Fig. 3b show the best fit of the data to the equation5 PS=Acos(2πfRτs+ϕ)expτsT2*α+C

where fR is the Rabi frequency, τs is the separation time, ϕ is a phase offset, T2* is the qubit dephasing time, A is the oscillation amplitude, C is an offset, and α captures the noise colour (see Supplementary Note 5).

Analysis of the S↔T0 oscillations over a range of Bx was used to characterise the qubit control frequency (fR) and the coherence time (T2*). Figure 3d, e show a colour map of the S↔T0 oscillations and a corresponding FFT at each Bx. We resolve S↔T0 oscillations up to 150 MHz at 30 mT and have observed up to 400 MHz at 80 mT (Supplementary Note 5). To extract Δg and J we fit fR for each Bx to Eqn. (1). The fit yields J = 6 MHz at the separation point (ϵ = 1.9 meV), and ∣Δg∣ = 0.41 which is in agreement with the effective g-factors extracted from Fig. 1f. In Supplementary Note 5, we show electrical control over Δg* using the Jg gate, with a trend of dΔg*/dJg ≈ 0.9 V−1, demonstrating in-situ electrical control of the qubit control frequency.

We now show the decoherence in this qubit can be explained by fluctuations in both ϵ and ΔEZ and we quantify their magnitudes. Figure 3f shows T2* for a range of Bx, where each T2* has been extracted using a fit to Eqn (5). Dephasing is caused by fluctuations in the energy splitting between the S and T0 states56 (Eqn (1)). Hence, the variation in T2* can be modelled using6 1T2*=π2hJEST0dJdϵδϵ2+ΔEZESTδΔEZ2

where δϵ is the noise in detuning and δΔEZ is the effective magnetic noise. A common fit of the data in Figs. 3f, 4d (discussed later) was used to extract δϵ = 30 μeV and δΔEZ = 4 neV. These values are comparable to those previously reported for electrons in silicon devices8,57,58 and for holes in planar Ge devices37. The similarity in δϵ between holes in planar Si and planar Ge suggests that the highly disordered SiO2 oxide does not significantly enhance the effect of charge noise compared to planar Ge heterostructures, where the quantum dot is buried tens of nanometres below the surface. Further, the similarity in δϵ between this work in pMOS silicon and studies of electrons in nMOS silicon58 suggests that the level of charge noise is not impacted by the polarity of the gate bias. Finally, we note that the magnetic noise (δΔEZ) we measured appears to arise primarily from fluctuations in the Overhauser field, which is produced by the 5% residual 29Si nuclei (see Supplementary Note 5). This indicates that hole-spin qubits may similarly benefit from isotopic enrichment59.

In Fig. 3g, we plot T2* of the Δg-driven oscillations as a function of the fridge mixing chamber temperature (TMC). T2* is approximately independent of TMC up to 400 mK (kBT = 34 μeV), where T2* begins to drop. The T2* behaviour shows the same trend at Bx = 15 mT (ΔEST0=0.35μeV) and Bx = 10 mT (ΔEST0=0.23 μeV). Interestingly, we find that the noise colour (α) appears to whiten as the temperature of the fridge increases, consistent with recent experiments in other hole-spin qubits19 (see Supplementary Note 5).

Finally, in Fig. 3h, we show that the quality factor (Q=fR×T2*) of the Δg-driven singlet-triplet oscillations increases as the control speed increases. The quality factor quantifies the number of coherent oscillations that can be completed within the coherence time. A promising feature of this qubit is that Q increases with increasing fR, indicating that the qubit control speed can be increased without degrading the quality.

Coherent exchange-driven oscillations

In order to achieve full control of the singlet-triplet qubit it is necessary to produce control pulses that allow access to the full 3D Bloch sphere. We use exchange-driven oscillations to rotate the qubit around the z-axis of the Bloch sphere. By combining Δg-driven (x-axis) and exchange-driven (z-axis) rotations, it is possible to achieve full control over the qubit Bloch sphere.

Figure 4a, b show the experimental procedure for exchange-driven oscillations. A separation pulse with calibrated τS is first used to perform a Δg-driven π2 rotation to bring the state to the equator of the Bloch sphere. A rapid pulse to low detuning is then applied, which suddenly increases J, producing a change in the qubit rotation axis. The system is held at the exchange point for τE to drive oscillations around the z-axis, then a second Δg-driven π2 rotation is applied, followed by a pulse to the readout position.Fig. 4 Exchange-driven coherent oscillations.

a A schematic of the state evolution and b shows the pulse sequence used to achieve exchange-driven oscillations. A Δg-driven π/2 pulse brings the qubit to the equator (red dashed trajectory), then a rapid pulse to low detuning is used to suddenly increase J, changing the angle of rotation, resulting in exchange-driven oscillations around the equator (red trajectory). The orientation of the oscillation is tilted by θ′ from the z-axis. c Exchange-driven oscillations for three different detunings (ϵ = 0.5, 0.8 and 1.6 meV). Experiments were performed with a fixed magnetic field of Bx = 2 mT, such that a Δg-driven π/2 pulse takes 65 ns. The solid line shows a best fit to Eqn. (5), allowing extraction of J and T2* at each ϵ. d Exchange energy as a function of detuning, measured using the exchange pulse. The solid red lines show the fit to J(ϵ,tc)=ϵ24+2tc2−ϵ2, allowing the extraction of the tunnel coupling tc. e Tunnel coupling extracted for a range of different Jg gate voltages, where each tc is extracted from the ϵ dependence of the exchange-oscillation frequency. The trend in tc seems approximately linear in Jg (with the exception of an outlier at Jg = −1.35 V). While we would generally expect an exponential dependence of tc on the Jg voltage, the linear response is consistent with previous data in silicon MOSFET structures62. f Dephasing time T2* as a function of the detuning. The solid red line is a joint fit of Eqn. (6) to this data and the data in Fig. 3f. All data in (c, d, f) was collected for Jg = 1.2 V, and the inset in (f) shows the dephasing time as a function of the detuning for Jg = −1.4 V. Error bars in (d–f) are the standard deviation in the measured value.

Figure 4c demonstrates exchange-driven oscillations at three different detuning position (ϵ). Reducing ϵ at the exchange position increases the exchange energy J, so that the angle of rotation tends towards 0° with respect to the Bloch sphere z-axis (since J  ≫ ΔgμBB). In Fig. 4d, we plot J as a function of ϵ at the exchange point. The solid line shows the best fit of J(ϵ, tc), allowing the extraction of the tunnel coupling (see Figure caption). This experiment was repeated for a range of Jg gate voltages, and the resulting dependence of the tunnel coupling (tc) on the Jg gate voltage is shown in Fig. 4e, demonstrating smooth control of tc. Therefore, the exchange-driven oscillations are highly tunable, since J can be electrically tuned either by varying ϵ with the plunger gates, or by tuning tc using the Jg-gate. These results demonstrate coherent exchange-driven z-axis control of the singlet-triplet qubit.

Figure 4f presents T2* as a function of ϵ, which is used to characterise the coherence time of the exchange oscillations. The solid line shows the best fit to Eqn. (6), obtained by jointly fitting Figs. 4e, 3f. The trend in T2* is well explained by Eqn. (6), where charge noise (δϵ) dominates at low detuning due to enhanced dJ/dϵ, while Zeeman noise (δΔEZ) dominates at large detuning where dJ/dϵ→0.

Spin-echo measurement

Finally, we investigate the use of spin-refocusing to enhance the qubit coherence time. Given that J and ΔEZ are non-zero, rotations around the Bloch sphere occur at some angle offset from the pure x-axis or z-axis. Since the qubit trajectory is not solely around the x-axis or z-axis, the precise form of the refocusing pulse will vary as the qubit evolves. Therefore, complicated pulse engineering is required for perfect refocusing pulses60. Here, we implement a simplified procedure37, which employs a π rotation using an exchange pulse to enhance the observed coherence analogous to a Hahn echo. Figure 5a, b show the qubit evolution and pulse sequence for the refocusing procedure, respectively. The spin-echo experiment allowed the qubit to freely evolve for a time tf, during which N exchange-driven π rotations are carefully interlaced to provide the refocusing echo. Full details of the refocusing procedure are provided in the Methods.Fig. 5 Spin-echo measurement at Bx = 1.6 mT.

a Shows an example trajectory of the qubit-state for the refocusing pulse schematic of (b). Free evolution driven by Δg is allowed for a period of time τS(n) = (2n + 1/2)tπ (green trajectory), such that after any τS(n), the qubit will be at the equator. A refocusing pulse is incorporated as a π exchange pulse, followed by a second period of free evolution for τS(n). The total time to perform this sequence is the free evolution time, τf, which can be varied by increasing n, or repeating the cycle to include multiple refocusing pulses. The full pulse sequence results in the qubit refocusing to S after τf. c Residual S−T0 oscillations after a free evolution time (τf) of 1000 ns for no refocusing pulses (red), one refocusing pulse (blue) and two refocusing pulses (black). d Normalised peak-to-peak amplitude of the residual S−T0 oscillations as a function of free evolution time. For one (blue) and two (black) refocusing pulses, the oscillations are clearly extended compared with the data for no refocusing pulse. The amplitude is normalised against the shortest free evolution for each data set in order to account for the fidelity of the π exchange pulse. We extract T2Echo based on a fit to the decay envelope of Eqn. (6), using α = 2 (see Supplementary Note 5). The experiments were performed for Bx = 1.6 mT.

We demonstrate the enhancement of the qubit coherence by observing the residual Δg-driven singlet-triplet oscillations after the free evolution time, tf. Figure 5c shows the singlet-triplet oscillations after tf = 1000 ns for zero (red), one (blue) and two (black) refocusing pulses. When zero refocusing pulses are applied, the singlet-triplet oscillations are completely lost after 1000 ns of free evolution. However, with refocusing pulses, the singlet-triplet oscillations are visible even after tf > 1000 ns. Figure 5d shows the normalised amplitude of the residual Δg-driven singlet-triplet oscillations observed for a range of free evolution times. The application of one and two refocusing pulses clearly enhances the coherence of the qubit. We fit the decay of the peak amplitude to extract T2Echo = 1220 ± 150 ns for one refocusing pulse, and T2Echo = 1300 ± 200 ns for two refocusing pulses. When no refocusing pulses are applied, we find T2Echo = 550 ± 50 ns, consistent with the measurements in Fig. 3. Although we see an improvement of 120% by applying one pulse, we see no significant improvement when using two refocusing pulses. This suggests the maximum may have been reached for this simplified refocusing procedure.

In this work we have demonstrated a hole-spin singlet-triplet qubit in planar silicon. We demonstrate rapid S↔T0 oscillations exceeding 400 MHz, two-axis control via Δg-driven and exchange-driven oscillations, and enhancement of the qubit coherence time to  >1 μs using spin-echo procedures. Developing a complete model of the energy spectrum provided insight into spin qubit dynamics under rapid pulses across the S↔T− avoided crossing. The experimentally observed effects were well described by the model Hamiltonian. This demonstrates that additional modelling would be useful for simulating new methods to further optimise initialisation protocols in hole-spin qubits, such as identifying optimal magnetic field orientations or simulating more complex pulsing procedures.

We highlight this singlet-triplet qubit is implemented using planar MOS silicon, which is a platform that has future potential for scaling up to large qubit arrays, and implementing surface code error correction43–45. Therefore, the results presented here provide a robust foundation for up-scaling hole-spin qubits and offer insights for advancing hole qubit technology further. Furthermore, the analysis regarding the orientation of the spin–orbit vector, the Landau–Zener transitions, and the g-tensor effects are directly relevant to a range of spin qubit systems in silicon, including single spin, exchange only, and hopping qubits61. An open question that will impact the future direction of hole-spin qubit technology is the extent of variation of key qubit parameters (such as the g-tensor and spin orbit vector) across different devices, and how well these can be controlled in-situ. Therefore, a future direction for this research is investigating the effects of in-situ controlled operating parameters, such as occupation number and confinement shape, on the qubit operation.

Methods

Sample details

The device was fabricated from high-resistivity natural silicon. The multi-layer Pd gate stack is achieved using 2-nm ALD Al2O3 and a high-quality 5.9 nm SiO2 gate oxide. This device combines nMOS and pMOS capabilities in order to implement a single electron transistor (SET) as the charge sensor, while defining a hole double quantum dot46,47. Further details of the device operation are provided in Supplementary Note 1.

Experimental setup

All experiments were performed using a top-loading BlueFors XLD dilution fridge with a three-axis vector magnet. Unless otherwise stated, the experiments were performed with the fridge at base temperature where the mixing chamber thermometer was 40 mK. Previous measurements indicated that at base, this system achieves electron and hole temperatures of 120 mK. The device was fixed directly onto a brass sample enclosure that is thermally anchored to the probe cold-finger. GE varnish was used to mount the device directly onto a brass sample stage, and Al bond wires connected the sample to a homemade printed circuit board. All DC biasing was applied using a Delft IVVI digital-to-analogue converter using lines which each pass through individual 50-kHz low pass filters mounted to the cold-finger (40 mK). The current through the SET was amplified using a Basel SP983c I-V preamplifier. DC currents were then monitored using a Keithly 2000, and standard low-frequency lock-in techniques were implemented using an SR830. For all measurements, we used an integration time of 100 ms, since any longer integration times did not yield any increase in measured parameters (T2* or TEcho*).

Rapid pulses were applied to gates P1 and P2 in order to perform initialisation, control and readout out of the singlet-triplet qubit. The pulses were applied using a Tabor WX1284 with 1.25 GS/s. Fast pulses from the WX1284 were routed to gates P1 and P2 using homemade RC bias-tees (R = 330 kOhm, C = 1.2 nF) on the sample PCB at the mixing chamber. The pulses were transmitted from room temperature to the circuit board using coaxial cables with 15 dBm of cold attenuation for thermalisation. Further details of the pulse procedure and spin-to-charge conversion are discussed in Supplementary Notes 2, 3, respectively.

Landau–Zener transitions

Landau–Zener processes are discussed in relation to Fig. 2. The probability of maintaining the initial eigenstate when ramping across an avoided-crossing can be understood in terms of the Landau–Zener transition probability7 PLZ=exp−2πΔ2hv,

where v = dE/dt is the energy level velocity, and Δ is the size of the avoided crossing. For the two-hole-spin system, we initialised a S2,8, then pulsed (ramp time  ≈4 ns) to positive detuning. This pulse from negative to positive detuning traverses the ΔST± avoided crossing. The magnetic field orientation strongly influences both the magnitude of ΔST± and the position in detuning (ϵΔ) at which the avoided-crossing occurs. Therefore, the magnetic field orientation impacts the probability of transitioning from S2,8 into T− during the pulse from (2,8) to (1,9). This in turn influences the amplitude of the S↔T− FFT peak (blue), since this peak amplitude is proportional to the probability of occupying T−53. See Supplementary Note 6 for further discussion of Landau–Zener transitions.

Finally, we comment on the limitations of the Landau–Zener analysis for this experimental procedure. The derivation of the Landau–Zener formula assumes the energy separation term v = dE/dt is constant. However, due to the non-linear energy spectrum and the use of a fixed separation pulse (4ns rise time), the term v = dE/dt is not constant for our experimental protocol. Therefore, the Landau–Zener picture provides an approximate qualitative understanding of the underlying physics. In order to more accurately capture the dynamics induced by the pulse sequence, we have developed a QuTiP54 simulation procedure. This procedure allows input of the realistic experimental procedure and the calculation of realistic transition rates (see Supplementary Note 7).

Spin-echo measurement procedure

Here we provide full details of the experimental procedure used for the spin-echo measurement. This procedure was modelled on that developed for a planar Ge singlet-triplet hole-spin qubit37. The S2,8 state was pulsed to large detuning, producing Δg-driven oscillations. We allow a dwell time of tS(n) = (2n + 1/2)tπ, where n is an integer and tπ is the time for a Δg-driven π rotation (tπ ≈60 ns). Hence, after any tS(n) the state should be at the equator of the Bloch sphere. After time tS(n), an exchange-driven π rotation (tπJ≈7ns) is applied. Following this, we dwell at large separation for a second period tS = (2n + 1/2)tπ. Multiple refocusing pulses can be added by repeating the above cycle. The blue schematic in Fig. 5b) shows the sequence for one refocusing pulse, while the black schematic shows the sequence for two refocusing pulses. The free evolution time is the total time of the entire refocusing pulse sequence, tf=(2tS(n)+tπJ)N, where N is the number of exchange-driven π pulses used. For a fixed number of exchange pulses (N), we can vary the free evolution time by increasing the tS(n) dwell time. The entire pulse sequence is designed to refocus the qubit to a singlet state, regardless of the total free evolution time tf(n).

Given that the exchange pulses are shown to enhance coherence in Fig. 5c, it is counterintuitive that the amplitude of the N = 2 (black) oscillations are smaller than the N = 1 (blue) oscillations in Fig. 5c. However, this is due to the fidelity of the exchange pulse. The fidelity of the exchange pulse reduces the overall amplitude of the oscillations, which is the reason that the amplitude of the two-pulse data (black) is less than the one-pulse data (blue). The limited fidelity restricted the number of refocusing pulses we were able to apply, since, for more than three refocusing pulses, the amplitude of the oscillations were reduced to the noise floor of the singlet-triple visibility, even for short tf. As a result, we plot the normalised oscillation amplitude in Fig. 5c to allow comparison between the zero, one and two exchange pulse measurements.

Optimal fit for g-tensor and spin–orbit vector

We fit the frequencies of the three FFT peaks in Fig. 2a, c to predicted energy splitting in the singlet-triplet Hamiltonian HST (defined in Supplement Note 4). This allowed extraction of the g-tensors and spin–orbit vector for this hole-spin qubit. The best fit of the g-tensor for the two dots, is shown in Fig. 6. The spin–orbit vector (three components) and the g-tensors (six components each) result in a very large parameter space, with 16 parameters; tc, tx, ty, tz, six parameters for gL and six parameters for gR. This presents a challenge since the large number of free parameters prevents the identification of a unique fit to the FFT frequencies alone. In fact, we find more than 50 possible parameter combinations giving good fits to the observed frequencies.Fig. 6 Fitting parameters for the g-tensors and spin–orbit vector.

a–c The optimal fit g-factors gL (blue) and gR (red) plotted as a function of magnetic field orientation. There is a clear misalignment between the principle axes of the left and right g-tensors. The spin–orbit vector is plotted in green in (a). We note that while gL and gR are used to denote the two distinct g-tensors extracted from the fitting procedure, we are not able to assign either gL or gR to a dot under a specific plunger gate.

We are able to effectively constrain the fits by simultaneously considering the eigenstate anisotropy and the state occupation probability indicated by the FFT amplitude. This is described in more detail in Supplementary Notes 7, 8. However, the key aspect is that for each potential fit we are able to additionally simulate the probability that the S2,8 transitions into T− during the 4 ns ramp-in. By combining the two fitting approaches we are able to identify the optimal fit of the hole-spin system. In fact of the  ≈50 possible parameter combinations  <10 seemed to show reasonable state occupation probabilities, and one optimal fit was clearly identified (See Fig. 7).Fig. 7 Comparison of experimental results and QuTIP Simulation.

a The experimental data from Fig. 2c is reproduced over a shortened time scale. The colour scale is the normalised singlet probability. Black dashed lines serve as a guide to the eye for the Δg-oscillations. b Shows the full simulation of the experimental procedure using the QuTIP model described in Supplementary Notes 7,8. The black dashed lines are transposed from (a). We note that the experiment in (a) takes ~3 h to perform and is limited by the integration time of the SRS830 lock-in (0.3 ms). The simulation in (b), which uses QuTIP `sesolve' function, takes approximately 10 h to run, and is limited by the number of PC cores running simultaneously.

The parameters for the optimal fit aretc=13.7±0.1μeV

g⃡L=−0.78−1.13−1.45−1.130.85−0.27−1.45−0.271.91

g⃡R=−0.96−0.94−1.47−0.940.78−0.74−1.47−0.741.82

andtso=−37±2107±40±20neV

We use labels g⃡L and g⃡R to indicate a ‘left’ and ‘right’ g-tensor respectively, however this experiment is unable to assign g⃡L or g⃡R specifically to the P1 or P2 dots. Uncertainty in each of the g-factors was on the order of  ±0.02. The optimal fit here provides the parameters used for the solid lines in Fig. 2 of the main text. Figure 6 shows an illustration of the optimal fit g-tensors and spin–orbit vector t→so. Note that the principle axes of the left and right g-tensor are slightly misaligned with respect to each other.

Optimal fit validation using QuTIP modelling

A key method for validating the optimal fit was a comparison of the experimentally observed qubit dynamics with simulated data tht was generated using the best fit parameters. Figure 7a reproduces the experimental data from Fig. 2c (over a shortened time scale), allowing the non-trivial qubit dynamics to be more clearly identified. At θ = 0 the slower S↔T0 oscillations are the main component picked up in the FFT data of Fig. 2d. However, as θ increases, additional spectral components become more pronounced, with most significant effects observed at θ = 120°. These additional spectral components arise from both S↔T± oscillations and T0↔T±, which both become more prevalent as ΔST becomes both more open and moves to more positive detuning (See also Fig. 2e).

Figure 7b shows the results of a QuTIP54 simulation of the experimental procedure. The full details of the modelling procedure are provided in Supplementary Notes 7, 8. For Fig. 7b we use the optimal fit parameters and demonstrate excellent agreement between the simulated and observed qubit dynamics, thus validating the optimal fit parameters. See Supplementary Note 8 for further discussion.

Supplementary information

Supplementary Information

Peer Review File

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-024-51902-9.

Acknowledgements

All authors acknowledge funding from the Australian Research Council (Grants No. DP200100147 and No. FL190100167) and the US Army Research Office (Grant No. W911NF-23-1-0092). A.R.H. acknowledges an ARC industrial laureate fellowship (IL230100072). R.S.E. acknowledges the SNSF NCCR SPIN International Mobility Grant. Devices were made at the New South Wales node of the Australian National Fabrication Facility. All authors thank A. Saraiva, A. Sarkar, N. Dumoulin Stuyck and E. Vahapoglu for valuable discussions.

Author contributions

S.D.L. performed the experiments and analysis. F.E.H. and W.H.L. fabricated the device. Z.W., D.C. and S.D.L. developed the model for HST. D.J.H. and S.D.L. developed the QuTIP code used to simulate hole-spin dynamics. J.Y.H. and C.C.E. produced the 3D model Fig. 1a. R.S.E and S.D.L. performed fitting of HST. S.D.L. wrote the manuscript with input from all co-authors. All authors, including A.S., I.K.J., J.H., K.K., I.V., M.J.R., A.S.D. and A.R.H., contributed to the discussion and planning. A.R.H. supervised the project.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data related to this study have been deposited in the Zenodo database, accessible at https://zenodo.org/records/12803637 and https://github.com/ScottLiles/HoleSTQubit.git.

Code availability

The code generated in this study have been deposited in the Zenodo database accessible at https://zenodo.org/records/12803637 and https://github.com/ScottLiles/HoleSTQubit.git.

Competing interests

A.S.D. is CEO and director of Diraq Pty Ltd. C.C.E., F.E.H., W.H.L. and A.S.D. declare equity interest in Diraq Pty Ltd. The remaining authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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