A single hole spin with enhanced coherence in natural silicon

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Device

The device is a four-gate silicon-on-insulator nanowire transistor fabricated in an industry-standard 300 mm CMOS platform11. The undoped [110]-oriented silicon nanowire channel is 17 nm thick and 100 nm wide. It is connected to wider boron-doped 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 (Si3N4) spacers. The gate stack consists of a 6-nm-thick SiO2 dielectric layer followed by a metallic bilayer with 6 nm of TiN and 50 nm of heavily doped polysilicon. The yield of the four-gate 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 silicon-on-insulator nanowire devices37,38, 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 surface-mount inductor (L = 240 nH) is wire bonded to the drain pad (see Extended Data Fig. 7 for the measurement set-up). This configuration involves a parasitic capacitance to ground Cp = 0.54 pF, leading to resonance frequency f = 449.81 MHz. The high value of the loaded quality factor Q ≈ 103 enables fast, high-fidelity charge sensing. We estimate a charge readout fidelity of 99.6% in 5 μs, which is close to the state-of-the-art for silicon MOS devices39. 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 ϕdrain of the reflected radiofrequency signal, which is measured through homodyne detection.

Energy-selective single-shot 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 VG2 and VG3 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 lever-arm parameter of gate G2 is αG2 ≈ 0.37 eV V−1, as inferred from temperature-dependence measurements. Comparatively, the lever-arm parameter of gate G1 with respect to the first hole under G2, αG1 ≈ 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 single-shot 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 EZ = gμBB where g is the g-factor, μB is the Bohr magneton and B is the amplitude of the magnetic field. This opens a narrow window for energy-selective readout using spin to charge conversion40. 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 spin-up hole can tunnel out of QD2 while only spin-down 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 single-shot 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 three-stage pulse sequence to optimize the readout. For that purpose, the tunnel rates between QD2 and the charge sensor were adjusted by fine tuning VG3 and VG4. For the spin-manipulation experiment discussed in the main text, we use a simplified two-stage 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 spin-up probability P after a given spin manipulation sequence, we repeat the single-shot readout a large number of times, typically 100–1,000 times.

Pulse sequences

For Ramsey, Hahn-echo, phase-gate and CPMG pulse sequences, we set a π/2 rotation time of 50 ns. Given the angular dependence of FRabi, 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 ttot = 10.26 h. For each realization, we varied the free evolution time τwait up to 7 μs, and averaged 200 single-shot spin measurements to obtain P (Extended Data Fig. 6a, top). The fringes oscillate at the detuning Δf = fMW1 − fL between the MW1 frequency fMW1 and the spin resonance frequency fL. To track low-frequency noise on fL, we make a Fourier transform of each fringe and extract its fundamental frequency Δf reported in Extended Data Fig. 6a (bottom). Throughout the experiment, fMW1 is set to 17 GHz. The low-frequency spectral noise on the Larmor frequency (in units of Hz2 Hz−1) is calculated (here we make use of two-sided power spectral densities, which are even with respect to the frequency) from Δf(t) as4:

$${S}_{\mathrm{L}}=\frac{{t}_{{{{\rm{tot}}}}}{\left|{{{\rm{FFT}}}}[{{\Delta }}f]\right|}^{2}}{{N}^{2}}\,,$$

(2)

where FFT[Δf] is the fast Fourier transform (FFT) of Δf(t) and N is the number of sampling points. We observe that the low-frequency noise, plotted in Extended Data Fig. 6b, behaves approximately as SL(f) = Slf(f0/f) with Slf = 109 Hz2 Hz−1, which is comparable to what has been measured for a hole spin in natural germanium41. To further characterize the noise spectrum, we add the CPMG measurements as coloured dots in Extended Data Fig. 6b4:

$${S}_{\mathrm{L}}\left({N}_{\uppi }/(2{\tau }_{{{{\rm{wait}}}}})\right)=-\frac{\ln ({A}_{{{{\rm{CPMG}}}}})}{2{\uppi }^{2}{\tau }_{{{{\rm{wait}}}}}},$$

(3)

where ACPMG is the normalized CPMG amplitude. As discussed in the main text, the resulting high-frequency noise scales as \({S}^{{{{\rm{hf}}}}}{({f}_{0}/f)}^{0.5}\), where Shf = 8 × 104 Hz2 Hz−1 is four orders of magnitude lower than Slf. This high-frequency noise appears to be dominated by electrical fluctuations, as supported by the correlations between the Hahn-echo/CPMG T2 and the LSESs. Additional quasi-static contributions thus emerge at low frequency, and may include hyperfine interactions (Supplementary Information, section 5).

Modelling

The hole wave functions and g-factors are calculated with a six-band kp model26. 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 (point-like positive charges with density σ = 5 × 1010 cm−2 at the Si/SiO2 interface and ρ = 5 × 1017 cm−3 in bulk Si3N4). 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 g-tensor 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 θzx by −25° and θzy by 10° in the calculations of Figs. 1, 2 and Extended Data Fig. 3.