# 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).

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.

### 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.