Abstract
Semiconductor spin qubits based on spin–orbit states are responsive to electric field excitations, allowing for practical, fast and potentially scalable qubit control. Spin electric susceptibility, however, renders these qubits generally vulnerable to electrical noise, which limits their coherence time. Here we report on a spin–orbit qubit consisting of a single hole electrostatically confined in a natural silicon metaloxidesemiconductor device. By varying the magnetic field orientation, we reveal the existence of operation sweet spots where the impact of charge noise is minimized while preserving an efficient electricdipole spin control. We correspondingly observe an extension of the Hahnecho coherence time up to 88 μs, exceeding by an order of magnitude existing values reported for hole spin qubits, and approaching the stateoftheart for electron spin qubits with synthetic spin–orbit coupling in isotopically purified silicon. Our finding enhances the prospects of siliconbased hole spin qubits for scalable quantum information processing.
Main
In the global effort to build scalable quantum processors, spin qubits in semiconductor quantum dots^{1} are progressively making their mark^{2}. We highlight, in particular, the achievement of single^{3,4} and twoqubit^{5,6,7,8} gate fidelities well above 99%, the first realizations of multiqubit arrays^{9,10} and a demonstrated compatibility with industrialgrade semiconductor manufacturing technologies^{11,12,13}.
Due to their long coherence time, electronspin qubits in silicon quantum dots have so far attracted the most attention^{2}. That said, their control requires addons such as metal microstrips^{3}, micromagnets^{4} or dielectric resonators^{14}, the largescale integration of which is technically challenging^{13}. Hole spin qubits, on the other hand, can circumvent this difficulty due to their intrinsically large spin–orbit coupling, which enables electricdipole spin manipulation. Over the last five years a variety of hole spin qubits have been reported in both silicon^{11,15} and germanium^{16,17,18,19} quantum dots. In all these qubits, quantum operations are performed using highfrequency gate voltage excitations.
The downside of allelectrical spin control is that the required spin–orbit coupling exposes the qubit to charge noise, leading to a reduced hole spin coherence. Recent theoretical works^{20,21,22}, however, have shown that, for properly chosen structural geometries and magnetic field orientations, careful tuning of the electrostatic confinement can bring the hole qubit to an optimal operation point where the effects of charge noise vanish to first order while enabling efficient electricdipole spin resonance. Here, using a single hole spin confined in natural silicon, we pinpoint the existence of operation sweet spots where the longitudinal spinelectric susceptibility is minimized, resulting in a large enhancement of the spin coherence time.
Numerical simulations are found in remarkable agreement with the experimental observations, and predict that such sweet spots are resilient to realistic amounts of disorder. This advocates the use of such sweet spots as a reliable way to decouple hole spin qubits from charge noise, thereby reinforcing the promises of emergent holebased quantum processors^{23}.
Device design and gfactor anisotropy
Our device consists of an undoped silicon nanowire of rectangular cross section in which the electrostatics is controlled by four gates (G1–G4) as shown in Fig. 1a,b. We define a large hole island below G3 and G4 to be used simultaneously as a reservoir and as a charge sensor for a single hole trapped in a quantum dot, QD2, under G2. Singleshot readout of this hole spin is performed by means of a spintocharge conversion technique based on the realtime detection of spinselective tunnelling to the reservoir, a widely used method often referred to as ‘Elzerman readout’^{24}. Tunnelling events are detected by dispersive radiofrequency reflectometry on the charge sensor (see Methods and Extended Data Fig. 1 for technical details).
In our device geometry, the first holes primarily accumulate in the upper corners of the silicon nanowire^{25}. Figure 1c displays the expected singlehole wave function in QD2, computed with a finitedifferences k ⋅ p model including the six topmost valence bands^{26} (see Methods and Supplementary Information, section 1). At low energy, that is, close to the valenceband edge, the hole wave function primarily contains heavyhole (HH) and lighthole (LH) components. The strong twoaxes confinement readily seen in Fig. 1c favours HH–LH mixing^{27,28}. This mixing is expected to manifest in the anisotropy of the hole gtensor, which carries information on the relative weight of the HH and LH components^{29,30,31}. To verify this, we measure the hole spin resonance frequency f_{L} while varying the orientation of the magnetic field B in the xz and yz planes. The effective gfactor g = hf_{L}/(μ_{B}∣B∣) (with μ_{B} the Bohr magneton and h the Planck constant) is plotted in Fig. 1d,e as a function of the magnetic field angles θ_{zx} and θ_{zy}, respectively. These maps highlight the strong anisotropy of the Zeeman splitting, with a maximal g = 2.7 close to the y axis (inplane, perpendicular to the wire) and a minimal g = 1.4 close the z axis (inplane, along the wire). The calculated gfactors are also plotted in the same figures as coloured solid lines. The agreement with the experimental data is remarkable. From the numerical simulation, we conclude that the measured gfactor anisotropy results from a strong electrical confinement against the side facet of the channel (along y), which prevails over the mostly structural vertical confinement (along x). The experimental gfactors and the small misalignment between the principal axes of the gtensor and the device symmetry axes are best reproduced by introducing a moderate amount of charge disorder in combination with small (∼0.1%) shear strains in the silicon channel (Extended Data Figs. 2 and 3, and Supplementary Information, section 1). The latter probably originate from device processing and thermal contraction at the measurement temperature^{32}.
Longitudinal spinelectric susceptibility
Given that the gfactor anisotropy is intimately related to the HH–LH mixing, which is controlled by the electrostatic confinement potential, the Larmor frequency is expected to be gatevoltage dependent. As a consequence, the hole spin coherence must be generally susceptible to charge noise. We thus measure the longitudinal spinelectric susceptibility (LSES) with respect to the voltages applied to the lateral gate G1 and to the accumulation gate G2, which we define as \({{{{\rm{LSES}}}}}_{{{{\rm{G1}}}}}=\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G1}}}}}}\) and \({{{{\rm{LSES}}}}}_{{{{\rm{G2}}}}}=\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G2}}}}}}\), respectively. In essence, LSES_{G1} and LSES_{G2} characterize the response of the Larmor frequency to the electricfield components parallel (z) and perpendicular (x,y) to the channel direction, respectively.
To probe the response to G2, we directly measure the spin resonance frequency f_{L} at different V_{G2} (Extended Data Fig. 4). The resulting LSES_{G2} is plotted as a function of the magnetic field angle θ_{zx} in Fig. 2a. The observed angular dependence is in good agreement with the theoretical expectation.
Noticeably, LSES_{G2} is positive along x and negative along z. Indeed, when increasing V_{G2}, the hole wave function extends proportionally more in the yz plane than in the vertical x direction, which increases g_{x} and decreases g_{y} and g_{z} (Extended Data Fig. 2b and Supplementary Information, section 1). As a result of the sign change, LSES_{G2} vanishes at two magnetic field orientations in the xz plane (marked by arrows in Fig. 2a), which are sweet spots for electricfield fluctuations perpendicular to the silicon channel.
To probe the response to G1, we introduce a pulse on V_{G1} in a Hahnecho sequence^{4} as outlined in Fig. 2b. This defines a phase gate, controlled by the amplitude δV_{G1} and duration τ_{z} of the pulse. Figure 2b displays the coherent oscillations recorded as a function of τ_{z} for three different pulse amplitudes. The frequency of these oscillations is expected to increase linearly with δV_{G1}, with a slope \({{{{\rm{LSES}}}}}_{{{{\rm{G1}}}}}=\frac{\partial {f}_{\mathrm{L}}}{\partial {V}_{{{{\rm{G1}}}}}}\). This is shown in Fig. 2c for different magnetic field orientations. LSES_{G1}, plotted in Fig. 2d as a function of θ_{zx}, ranges from −0.5 MHz mV^{−1} to −0.1 MHz mV^{−1}. Its magnitude is much smaller than that of LSES_{G2} because G1 is further from QD2 than G2 and its field effect is partly screened by the hole gas beneath. The numerically calculated LSES_{G1} (solid line) reproduces reasonably well the order of magnitude but not the angular dependence of the measured LSES_{G1}. This discrepancy may be due to inaccuracies in the description of the hole gases near QD2 and to unaccounted charge disorder and strains (see discussion in Supplementary Information, section 1). We also notice that LSES_{G1} never vanishes and that the minimum of ∣LSES_{G1}∣ happens to be almost at the same θ_{zx} as a zero of LSES_{G2}.
Coherence times and frequencydependent noise contributions
We now turn to the angular dependence of the hole spin coherence time and investigate its correlation with the longitudinal spinelectric susceptibility^{33}. To get rid of lowfrequency noise sources, we measure the coherence time using a conventional Hahnecho protocol^{2}. The control sequence, applied to G1 (see upper inset of Fig. 3a), consists of π_{x}/2, π_{y} and π_{ϕ}/2 pulses separated by a time delay τ_{wait}/2. For each τ_{wait}, we extract the averaged amplitude of the P_{↑} oscillation obtained by varying the phase ϕ of the last π/2 pulse, and normalize it to the P_{↑} oscillation amplitude in the zerodelay limit.
A representative Hahnecho plot is shown in Fig. 3a. We fit the echo amplitude to an exponential decay \(\exp ({({\tau }_{{{{\rm{wait}}}}}/{T}_{2}^{{{{\rm{E}}}}})}^{\beta })\), where the exponent β is left as a free parameter. The best fit is obtained for β = 1.5 ± 0.1, which implies a highfrequency noise with a characteristic spectrum \(S(f)={S}_{{{{\rm{hf}}}}}{({f}_{0}/f)}^{\alpha }\), where f_{0} = 1 Hz is a reference frequency and α = β − 1 ≈ 0.5 (we note that the same α value was reported for hole spin qubits in germanium^{10}).
To explore the angular dependence of \({T}_{2}^{{{{\rm{E}}}}}\) in the xz plane, we measure the decay of the Hahnecho amplitude for different values of θ_{zx}. The results, shown in Fig. 3b, reveal a strong anisotropy, with \({T}_{2}^{{{{\rm{E}}}}}\) ranging from 15 μs to 88 μs. Strikingly, the spin coherence time peaks at θ_{zx} = 99°, an angle between the minimum of ∣LSES_{G1}∣ and a zero of LSES_{G2}, highlighting a correlation with the correspondingly suppressed electrical noise. The extended coherence time is much longer than previously reported for hole spin qubits in both silicon (1.5 μs (ref. ^{15})) and germanium (3.8 μs (ref. ^{23}))^{34}. In addition, we notice that spin control remains efficient at all angles including θ_{zx} = 99^{∘}, where we could readily achieve Rabi frequencies F_{Rabi} as large as 5 MHz limited by the attenuation on the microwave line. The echo quality factor \({Q}^{{{{\rm{E}}}}}={F}_{{{{\rm{Rabi}}}}}\times {T}_{2}^{{{{\rm{E}}}}}\) also peaks at θ_{zx} = 99°, reaching Q^{E} ≈ 440 with further room for improvement (Supplementary Information, section 2 and Extended Data Fig. 5).
The observed angular dependence of \({T}_{2}^{{{{\rm{E}}}}}\) can be understood by assuming that the electrical noise is the sum of uncorrelated voltage fluctuations on the different gates Gi with respective spectral densities \({S}_{{{{\rm{G}}}}i}(f)={S}_{{{{\rm{G}}}}i}^{{{{\rm{hf}}}}}{({f}_{0}/f)}^{0.5}\). Given the Hahnecho noise filter function, the decoherence rate can then be expressed as (Supplementary Information, section 3):
Using the longitudinal spinelectric susceptibilities from Fig. 2a–d and leaving the weights \({S}_{{{{\rm{G}}}}i}^{{{{\rm{hf}}}}}\) as adjustable parameters, we achieve a remarkable agreement with the experimental \({T}_{2}^{{{{\rm{E}}}}}\) (coloured solid line in Fig. 3b). This strongly supports the hypothesis that the Hahnecho coherence time is limited by electrical noise. As already argued before, LSES_{G1} and LSES_{G2} indeed quantify the susceptibility of the hole spin to electric field fluctuations parallel and perpendicular to the channel, respectively.
The best fit in Fig. 3b is obtained with \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}={(1.7\,\upmu {{{\rm{V}}}}/\sqrt{{{{\rm{Hz}}}}})}^{2}\) and \({S}_{{{{\rm{G}}}}2}^{{{{\rm{hf}}}}}={(66\,{{{\rm{nV}}}}/\sqrt{{{{\rm{Hz}}}}})}^{2}\). We speculate that the large \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}/{S}_{{{{\rm{G}}}}2}^{{{{\rm{hf}}}}}\) ratio results from an artificial enhancement of \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}\) accounting for hidden sources of electric field fluctuations along the silicon nanowire. Certainly, equation (1) misses the contribution from the electrical noise on G3, the LSES of which could not be measured. For reasons of symmetry, we expect LSES_{G3} to be comparable to LSES_{G1}. A possible additional source of longitudinal electric field fluctuations are the randomly oscillating charges and dipoles in the silicon nitride spacers between the gates. Because these noise sources are closer to QD2 than is gate G1, and because they are much less screened by the hole gas beneath, they presumably make a large contribution to the apparent \({S}_{{{{\rm{G}}}}1}^{{{{\rm{hf}}}}}\) when lumped into ∝LSES_{G1} terms.
To further investigate the hole spin coherence, we implement Carr–Purcell–Meiboom–Gill (CPMG) sequences at the most favourable field orientation θ_{zx} = 99^{∘}. These consist in increasing the number of π pulses cancelling faster and faster dephasing mechanisms. Figure 3c displays the CPMG echo amplitudes as a function of the total waiting time τ_{wait} for series of N_{π} = 2^{n}π pulses, where n is an integer ranging from 1 to 8. The CPMG decay times \({T}_{2}^{{{{\rm{CPMG}}}}}\) extracted from Fig. 3c (see caption) are plotted against N_{π} in Fig. 3d. As expected, the data points follow a power law \({T}_{2}^{{{{\rm{CPMG}}}}}\propto {N}_{\uppi }^{\gamma }\), where \(\gamma =\frac{\alpha }{\alpha +1}\) for a ∝1/f^{α} noise spectrum^{4}. The bestfit value γ = 0.34 yields again α ≈ 0.5. For the largest sequence of 256 π pulses, we find \({T}_{2}^{{{{\rm{CPMG}}}}}=0.4\,{\mathrm{ms}}\), an exceptionally long coherence for a hole spin^{34}.
Finally, to gain insight into the lowfrequency noise acting on the hole spin, we perform systematic measurements of the inhomogeneous dephasing time \({T}_{2}^{* }\). To this aim, we apply Ramsey control sequences consisting of two π/2 pulses separated by a variable delay τ_{wait}. Contrary to Hahnecho, the dephasing induced by lowfrequency noise sources is not cancelled due to the absence of the refocusing π pulse. Figure 4a displays P_{↑} for a series of identical Ramsey sequences recorded on an overall time frame of 1 h, with each sequence lasting approximately 5.5 s. The next step is to average P_{↑}(τ_{wait}) on a subset of consecutive sequences measured within a total time t_{meas}. This way, an averaged Ramsey oscillation is obtained for each t_{meas}, the amplitude of which is fitted to a Gaussiandecay function yielding \({T}_{2}^{* }({t}_{{{{\rm{meas}}}}})\). Representative Ramsey data sets and corresponding fits are shown in Fig. 4b for three values of t_{meas}. The inhomogeneous dephasing time decreases with increasing t_{meas} due to the contribution of noise components with lower and lower frequency. To unveil the angular dependence of \({T}_{2}^{* }\), we repeat the same measurement for different magnetic field orientations. The results are plotted in Fig. 4c for the same three values of t_{meas}. The overall anisotropy of the Hahnecho decay time of Fig. 3b can still be identified, although it reduces at large t_{meas} starting from t_{meas} > 50 s.
However, if the 1/f^{0.5} charge noise prevailed over the whole mHz to MHz range, \({T}_{2}^{* }\) would be ∼50 μs when \({T}_{2}^{{{{\rm{E}}}}}\approx 88\,\upmu{\mathrm{s}}\) (Supplementary Information, section 3), well above the 7 μs seen in Fig. 4c. The power spectrum S(f) at low frequency can be extracted from the data of Fig. 4a (Extended Data Fig. 6). This reveals a 1/f^{α} noise with α closer to 1, and a power (at 1 Hz) four orders of magnitude larger than the one expected by extrapolating the highfrequency 1/f^{0.5} noise inferred from CPMG. The change of exponent α and amplitude of S(f) when going from the mHz to the MHz points to the presence of different mechanisms dominating the dephasing at low and high frequencies. We note that the \({T}_{2}^{* }\approx 1{\text{}}2\,\upmu{\mathrm{s}}\) measured at long t_{meas} is below but fairly close to the expected hole spin dephasing time due to hyperfine interactions with the naturally present ^{29}Si nuclear spins^{25} (see the dashed line in Fig. 4c, and Supplementary Information, section 5 for details). This suggests that lowfrequency dephasing may be partially due to such hyperfine interactions.
Conclusions
We report on a spin qubit with electrical control and singleshot readout based on a single hole in a silicon nanowire device issued from an industrialgrade fabrication line. The hole wave function and corresponding gfactors could be modelled with an excellent level of accuracy in these types of devices, denoting a relatively low level of structural and charge disorder. The holespin coherence was found to be limited by a 1/f^{0.5} charge noise at high frequencies (10^{4}−10^{6} Hz), with a strong dependence on the magneticfield orientation that could be faithfully accounted for by the spinelectric susceptibilities. A largely enhanced spin coherence was measured at the sweetspot angle, far beyond the current stateoftheart for holespin qubits and close to the best figures reported for ^{28}Si electronspin qubits electrically driven via a micromagnet. Our study of the inhomogeneous dephasing time revealed a much stronger noise at low frequencies (10^{−4}−10^{−2} Hz) that could be partially ascribed to the expected hyperfine interaction. In this scenario, the possible introduction of isotopically purified silicon devices would lead to a significant improvement of hole–spin coherence in the lowfrequency range. Finally, we would like to emphasize that such sweet spots should be ubiquitous in hole spin qubit devices^{21}, and that a careful design and choice of operation point can make them usefully robust to disorder (see example in Supplementary Information, section 1). The engineering of sweet spots should therefore open new opportunities for an efficient realization of multiqubit or coupled spinphoton systems^{35}.
Methods
Device
The device is a fourgate silicononinsulator nanowire transistor fabricated in an industrystandard 300 mm CMOS platform^{11}. The undoped [110]oriented silicon nanowire channel is 17 nm thick and 100 nm wide. It is connected to wider borondoped source and drain pads used as reservoirs of holes. The four wrapping gates (G1–G4) are 40 nm long and are spaced by 40 nm. The gaps between adjacent gates and between the outer gates and the doped contacts are filled with silicon nitride (Si_{3}N_{4}) spacers. The gate stack consists of a 6nmthick SiO_{2} dielectric layer followed by a metallic bilayer with 6 nm of TiN and 50 nm of heavily doped polysilicon. The yield of the fourgate devices across the full 300 mm wafer reaches 90% and their room temperature characteristics exhibit excellent uniformity (see Supplementary Information, section 6 for details).
Dispersive readout
Similar to charge detection methods recently applied to silicononinsulator nanowire devices^{37,38}, we accumulate a large hole island under gates G3 and G4, as sketched in Fig. 1a. The island acts both as a charge reservoir and electrometer for the quantum dot QD2 located under G2. However, unlike the aformentioned earlier implementations, the electrometer is sensed by radiofrequency dispersive reflectometry on a lumped element resonator connected to the drain rather than to a gate electrode. To this aim, a commercial surfacemount inductor (L = 240 nH) is wire bonded to the drain pad (see Extended Data Fig. 7 for the measurement setup). This configuration involves a parasitic capacitance to ground C_{p} = 0.54 pF, leading to resonance frequency f = 449.81 MHz. The high value of the loaded quality factor Q ≈ 10^{3} enables fast, highfidelity charge sensing. We estimate a charge readout fidelity of 99.6% in 5 μs, which is close to the stateoftheart for silicon MOS devices^{39}. The resonator characteristic frequency experiences a shift at each Coulomb resonance of the hole island, that is, when the electrochemical potential of the island lines up with the drain Fermi energy. This leads to a dispersive shift in the phase ϕ_{drain} of the reflected radiofrequency signal, which is measured through homodyne detection.
Energyselective singleshot readout of the spin state of the first hole in QD2
Extended Data Fig. 1a displays the stability diagram of the device as a function of V_{G2} and V_{G3} when a large quantum dot (acting as a charge sensor) is accumulated under gates G3 and G4. The dashed grey lines outline the charging events in the quantum dot QD2 under G2, detected as discontinuities in the Coulomb peak stripes of the sensor dot. The leverarm parameter of gate G2 is α_{G2} ≈ 0.37 eV V^{−1}, as inferred from temperaturedependence measurements. Comparatively, the leverarm parameter of gate G1 with respect to the first hole under G2, α_{G1} ≈ 0.03 eV V^{−1}, is much smaller. The charging energy, measured as the splitting between the first two charges, is U = 22 meV. Extended Data Fig. 1b shows a zoom on the stability diagram around the working point used for singleshot spin readout in the main text. The three points labelled Empty (E), Load (L) and Measure (M) are the successive stages of the readout sequence sketched in Extended Data Fig. 1c. The quantum dot is initially emptied (E) before loading (L) a hole with a random spin. Both spin states are separated by the Zeeman energy E_{Z} = gμ_{B}B where g is the gfactor, μ_{B} is the Bohr magneton and B is the amplitude of the magnetic field. This opens a narrow window for energyselective readout using spin to charge conversion^{40}. Namely, we align at stage M the centre of the Zeeman split energy levels in QD2 with the chemical potential of the sensor. In this configuration, only the excited spinup hole can tunnel out of QD2 while only spindown holes from the sensor can tunnel in. These tunnelling events are detected by thresholding the phase of the reflectometry signal of the sensor to achieve singleshot readout of the spin state. Typical time traces of the reflected signal phase at stage M, representative of a spin up (spin down) in QD2, are shown in Extended Data Fig. 1d. We used this threestage pulse sequence to optimize the readout. For that purpose, the tunnel rates between QD2 and the charge sensor were adjusted by fine tuning V_{G3} and V_{G4}. For the spinmanipulation experiment discussed in the main text, we use a simplified twostage sequence for readout by removing the empty stage. The measure stage duration is set to 200 μs for all experiments, while the load stage duration (seen as a manipulation stage duration) ranges from 50 μs to 1 ms. To obtain the spinup probability P_{↑} after a given spin manipulation sequence, we repeat the singleshot readout a large number of times, typically 100–1,000 times.
Pulse sequences
For Ramsey, Hahnecho, phasegate and CPMG pulse sequences, we set a π/2 rotation time of 50 ns. Given the angular dependence of F_{Rabi}, we calibrate the microwave power required for this operation time for each magnetic field orientation. We also calibrate the amplitude of the π pulses to achieve a π rotation in 150 ns. In extracting the noise exponent γ from CPMG measurements, we do not include the time spent in the π pulses (this time amounts to about 10% of the duration of each pulse sequence).
Noise spectrum
We measured 3,700 Ramsey fringes over t_{tot} = 10.26 h. For each realization, we varied the free evolution time τ_{wait} up to 7 μs, and averaged 200 singleshot spin measurements to obtain P_{↑} (Extended Data Fig. 6a, top). The fringes oscillate at the detuning Δf = ∣f_{MW1} − f_{L}∣ between the MW1 frequency f_{MW1} and the spin resonance frequency f_{L}. To track lowfrequency noise on f_{L}, we make a Fourier transform of each fringe and extract its fundamental frequency Δf reported in Extended Data Fig. 6a (bottom). Throughout the experiment, f_{MW1} is set to 17 GHz. The lowfrequency spectral noise on the Larmor frequency (in units of Hz^{2} Hz^{−1}) is calculated (here we make use of twosided power spectral densities, which are even with respect to the frequency) from Δf(t) as^{4}:
where FFT[Δf] is the fast Fourier transform (FFT) of Δf(t) and N is the number of sampling points. We observe that the lowfrequency noise, plotted in Extended Data Fig. 6b, behaves approximately as S_{L}(f) = S^{lf}(f_{0}/f) with S^{lf} = 10^{9} Hz^{2} Hz^{−1}, which is comparable to what has been measured for a hole spin in natural germanium^{41}. To further characterize the noise spectrum, we add the CPMG measurements as coloured dots in Extended Data Fig. 6b^{4}:
where A_{CPMG} is the normalized CPMG amplitude. As discussed in the main text, the resulting highfrequency noise scales as \({S}^{{{{\rm{hf}}}}}{({f}_{0}/f)}^{0.5}\), where S^{hf} = 8 × 10^{4} Hz^{2} Hz^{−1} is four orders of magnitude lower than S^{lf}. This highfrequency noise appears to be dominated by electrical fluctuations, as supported by the correlations between the Hahnecho/CPMG T_{2} and the LSESs. Additional quasistatic contributions thus emerge at low frequency, and may include hyperfine interactions (Supplementary Information, section 5).
Modelling
The hole wave functions and gfactors are calculated with a sixband k ⋅ p model^{26}. The screening by the hole gases under gates G1, G3 and G4 is accounted for in the Thomas–Fermi approximation. As discussed extensively in Supplementary Information, section 1, the best agreement with the experimental data is achieved by introducing a moderate amount of charge disorder. The theoretical data displayed in Figs. 1, 2 and Extended Data Fig. 3 correspond to a particular realization of this charge disorder (pointlike positive charges with density σ = 5 × 10^{10} cm^{−2} at the Si/SiO_{2} interface and ρ = 5 × 10^{17 }cm^{−3} in bulk Si_{3}N_{4}). The resulting variability, and the robustness of the operation sweet spots with respect to disorder, are discussed in Supplementary Information, section 1. The rotation of the principal axes of the gtensor visible in Fig. 1d,e are most probably due to small inhomogeneous strains (<0.1%); however, in the absence of quantitative strain measurements, we have simply shifted θ_{zx} by ∼−25° and θ_{zy} by ∼10° in the calculations of Figs. 1, 2 and Extended Data Fig. 3.
Data availability
All of the data used to produce the figures in this paper and to support our analysis and conclusions are available at https://zenodo.org/search?page=1&size=20&q=6638442. This repository includes the original data, jupyter notebooks for data analysis and figure plotting. Additional data are available upon reasonable request to the corresponding author.
Code availability
The code is part of the Commisariat à l’Energie Atomique et aux Energies Alternatives strategy and could not be made public. However, the authors are ready to collaborate with anyone interested in the modelling tools used in this work, as they already do with several international teams.
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Acknowledgements
This research has been supported by the European Union’s Horizon 2020 research and innovation programme under grant agreements number 951852 (QLSI project), number 810504 (ERC project QuCube) and number 759388 (ERC project LONGSPIN), and by the French National Research Agency (ANR) through the projects MAQSi and CMOSQSPIN.
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N.P. and B.Br. carried out the experiment with help from V.S., S.Z. and A.A. and under the supervision of X.J., R.M. and S.D.F. V.P.M., J.C.A.U. and Y.M.N. carried out the theoretical modeling. B.Be., H.N., L.H. and M.V. designed and supervised the fabrication of the device. M.U. and T.M. provided useful comments. N.P., B.Br., Y.M.N., R.M. and S.D.F. cowrote the papers with input from the other authors.
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Extended data
Extended Data Fig. 1 Single shot spin readout.
(a) Stability diagram of the device as a function of V_{G2} and V_{G3}. The dashed grey lines are guides to the eye highlighting charge transitions in QD2. The first hole tunnels into QD2 at V_{G2} ≈ − 650 mV. (b) Zoom on the stability diagram close to the working point used in the main text. The points labelled L (Load), M (Measure) and E (Empty) are the three stages of the pulse sequence applied to V_{G2} for spin readout. (c) (Top) Schematic of the three stages pulse sequence applied to V_{G2}. (Bottom) Schematic energy diagrams at the different stages of the pulse sequence. μ_{F} is the chemical potential of the charge sensor playing the role of reservoir. A random spin is charged during the load stage. At the measure stage, if the loaded spin is up, the hole is able to tunnel out and is replaced by a spin down. On the opposite, if the loaded spin is down, tunneling in or out is impossible. Finally, the dot is discharged during the empty stage. (d) Phase versus time during the measurement stage. The orange curve exhibits a ‘blip’ around t = 50 μs, which indicates that the dot experienced a discharge/charge cycle characteristic of a spin up loading (see c). On the contrary, the red curve shows no phase change, which can be interpreted as a spin down loading. The phase signal is integrated over 6 μs.
Extended Data Fig. 2 Modeled structure.
(a) The 17 nm thick and 100 nm wide silicon channel is connected to highly doped source and drain reservoirs and controlled by four gates G1...G4. (b) Dependence of the gfactors on the electric field in a simple setup with no hole gases below G1, G3 and G4. The gfactors g_{x}, g_{y}, and g_{z} are plotted as as a function of the difference of potential − V_{G2} between gates G2 and gates G1 and G3 (both grounded). When increasing − V_{G2}, the lateral electric field from the wrap gate squeezes the hole on the side facet of the channel [see panels (c) and (d)]. This strengthens heavyhole/lighthole mixing, which results in a decrease of g_{x}, and an increase of g_{y}. (c, d) Maps of the squared wave functions (red) in the cross section of the channel below gate G2, at the biases marked with an orange pentagon and a purple star in (b). The channel is colored in white, the gate G2 in gray and SiO_{2} in blue. The dashed gray lines are isopotential lines of the dot potential V_{QD}(r), spaced by 2 mV in (c) and by 10 mV in (d). The isodensity surface of the wave function in (d) that encloses 85% of the hole charge is represented in Fig. 1c of the main text. (e) LSES computed at the purple star in (b) as a function of θ_{zx} (for constant Larmor frequency f_{L} = 17 GHz). In all these calculations, a single positive charge is introduced on the left facet of the channel [pink dot in (c, d)] to lift the degeneracy between the left and right corner dots. Details about the modeling and dependence of the gfactors on bias conditions can be found in Supp. Info S1.
Extended Data Fig. 3 Comparison between the experimental and calculated gfactors.
(a)gfactors for a magnetic field in the xz (red) and yz (blue) planes, as a function of the angles θ_{zx} and θ_{zy}, respectively. The symbols are the experimental data, and the dotted lines are calculated in the pristine device at the experimental bias point (with the hole gases below G1, G3 and G4). The lateral electric field is however too weak at this bias point to match the experimental anisotropy g_{y} > g_{x}. The solid lines are calculated in a particular realization of a disordered device with roughness and positive charge traps at the Si/SiO_{2} interface and in Si_{3}N_{4} (see Methods and Supp. Info S1). These traps tend to strengthen confinement on the side facets (because they are much better screened near the corners of the wrap gate), which increases g_{y} and decreases g_{x}, as shown in Extended Data Fig. 2. Moreover, θ_{zx} is shifted by ≈ − 25^{∘} and θ_{zy} by ≈ 10^{∘} to account for the experimental rotations of the principal axes of the gtensor (resulting from residual strains, see Supp. Info S1). The polar plots of the gfactors and the LSES of this disordered device are shown in Figs. 1 and 2 of the main text, respectively. (b, c) Maps of the squared wave function (red) computed in the same disordered device, where (b) shows a transverse xy cross section at z = − 35 nm and (c) a planar yz crosssection at x = 0. The channel is colored in white, the gate G2 in gray, SiO_{2} in blue and Si_{3}N_{4} in yellow. The dashed gray lines are isopotential lines of V_{QD}(r), spaced by 20 mV. V_{QD}(r) is here measured with respect to the energy level of the hole. The robustness of the gfactors and operation sweet spots with respect to disorder is discussed in Supp. Info S1.
Extended Data Fig. 4 Measurement of LSESG2.
(a) Schematic representation of the pulse sequence used to monitor spin resonance. We burst on MW1 for 5 μs and average P_{↑} over 200 such sequences. (b) Average P_{↑} (blue dots) versus MW1 burst frequency at V_{plunge} = − 1 mV. This plot is in essence a line cut of a Rabi chevron at t_{burst} = 5 μs. The red dashed line is a fit used to extract the Larmor frequency. (c) Tracking of f_{L} as a function of V_{plunge}. The dashed blue line is a linear fit whose slope is equal to LSES_{G2}.
Extended Data Fig. 5 Rabi frequencies and quality factors.
(a) Rabi frequency as a function of magnetic field orientation θ_{zx}. The Larmor frequency f_{L} = 17 GHz is kept constant and the hole spin is manipulated by a microwave burst on gate G1 with power P_{MW1} = 15 dBm on top of the MW1 line. (b) Inhomogeneous quality factor \({Q}^{* }={F}_{{{{\rm{R}}}}abi}\times {T}_{2}^{* }\) as a function of the magnetic field orientation θ_{zx}. The data are calculated from the Rabi frequencies plotted in (a), and from the values of \({\overline{T}}_{2}^{* }\) measured for t_{meas} = 5.5 s (Fig. 4c). (c) Same as (b) for the echo quality factor \({Q}^{{{{\rm{E}}}}}={F}_{{{{\rm{R}}}}abi}\times {T}_{2}^{{{{\rm{E}}}}}\). In the present case, the Rabi frequency is minimal around the sweet spot (a). Nonetheless, the quality factors Q^{*} and Q^{E} do peak near the sweet spot owing to the much improved coherence times. They reach Q^{*} = 23 and Q^{E} = 276, with peaktovalley ratios of respectively ≈ 2.5 and ≈ 5.5. As discussed in section Supp. Info S2, we can achieve Rabi frequencies of at least 5 MHz at the sweet spot with a larger driving power P_{MW1} = 20 dBm, which results in Q^{*} ≈ 35 and Q^{E} ≈ 440. In principle, the quality factors may be further improved by driving with gate G2 and looking for the sweet spot in the xy plane (see Supp. Info. S1).
Extended Data Fig. 6 Noise spectrum.
(a) (top) Ramsey fringes as a function of τ_{wait} acquired during 10 hours, at θ_{zx} = 90^{∘}. Each fringe oscillates at the frequency Δf = f_{MW1} − f_{L}. A single fringe takes roughly 10 s to record. (bottom) Δf, obtained via Fourier transform of the Ramsey fringes, versus laboratory time. (b) Power spectral density of the noise on the Larmor frequency. The lowfrequency spectrum (RF) is calculated from (a) and is roughly proportional to 1/f, as outlined by the upper dashed line. The high frequency spectrum (colored dots) is extracted from CPMG measurements with N_{π} from 2 to 256, and is proportional to 1/f^{0.5} (lower dashed line).
Extended Data Fig. 7 Experimental setup.
Dilution fridge with all electrical connections to the sample. We operate in a dilution refrigerator system equipped with a threeaxis vector superconducting magnet. The main solenoid magnet produces a magnetic field of up to 6 T in the z direction, while both transverse Helmholtz coils ramp up to 1 T in the x and y directions. However, one of the axis was broken during the experiment. Therefore, after recording Fig. 1d of the main text, the sample was warmed up, physically rotated by 90^{∘}, and cooled down again to record Fig. 1e. 24 twisted pairs are filtered at the mixing chamber by 6 low pass filters. The DC gate voltages are generated by Itest high stability voltage sources (BE2141). To perform charge and spin manipulation, semirigid coaxial lines with 20 GHz bandwidth are routed to G1, G2 and G3 using onPCB bias tees. Microwave frequency signals are supplied by a vector signal generator (R&S SMW200A) with IQ modulating signals originating from two channels of an arbitrary waveform generator (AWG) Tektronix AWG5200. Other channels of the AWG are used to generate the pulse sequences. The homodyne readout of the resonator connected to the drain electrode is performed with a Zurich Instrument UHFLI lockin with an excitation power of − 105 dBm at the PCB stage. The reflected signal from the resonator is amplified at 4 K with an ultralow noise cryogenic amplifier LNFLNC0.23A.
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Supplementary sections 1–6.
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Piot, N., Brun, B., Schmitt, V. et al. A single hole spin with enhanced coherence in natural silicon. Nat. Nanotechnol. 17, 1072–1077 (2022). https://doi.org/10.1038/s4156502201196z
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DOI: https://doi.org/10.1038/s4156502201196z
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