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# A single gap spin with enhanced coherence in pure silicon

### System

The system 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 large. It’s related to wider boron-doped supply and drain pads used as reservoirs of holes. The 4 wrapping gates (G1–G4) are 40 nm lengthy and are spaced by 40 nm. The gaps between adjoining gates and between the outer gates and the doped contacts are crammed with silicon nitride (Si3N4) spacers. The gate stack consists of a 6-nm-thick SiO2 dielectric layer adopted by a metallic bilayer with 6 nm of TiN and 50 nm of closely doped polysilicon. The yield of the four-gate units throughout the complete 300 mm wafer reaches 90% and their room temperature traits exhibit glorious uniformity (see Supplementary Data, part 6 for particulars).

Much like cost detection strategies just lately utilized to silicon-on-insulator nanowire units37,38, we accumulate a big gap island beneath gates G3 and G4, as sketched in Fig. 1a. The island acts each as a cost reservoir and electrometer for the quantum dot QD2 situated beneath G2. Nevertheless, not like the aformentioned earlier implementations, the electrometer is sensed by radiofrequency dispersive reflectometry on a lumped ingredient resonator related to the drain slightly than to a gate electrode. To this purpose, a industrial surface-mount inductor (L = 240 nH) is wire bonded to the drain pad (see Prolonged Information Fig. 7 for the measurement set-up). This configuration entails a parasitic capacitance to floor Cp = 0.54 pF, resulting in resonance frequency f = 449.81 MHz. The excessive worth of the loaded high quality issue Q ≈ 103 allows quick, high-fidelity cost sensing. We estimate a cost readout constancy of 99.6% in 5 μs, which is near the state-of-the-art for silicon MOS units39. The resonator attribute frequency experiences a shift at every Coulomb resonance of the opening island, that’s, when the electrochemical potential of the island strains up with the drain Fermi vitality. This results in a dispersive shift within the part ϕdrain of the mirrored radiofrequency sign, which is measured by means of homodyne detection.

### Pulse sequences

For Ramsey, Hahn-echo, phase-gate and CPMG pulse sequences, we set a π/2 rotation time of fifty ns. Given the angular dependence of FRabi, we calibrate the microwave energy required for this operation time for every magnetic discipline orientation. We additionally calibrate the amplitude of the π pulses to realize a π rotation in 150 ns. In extracting the noise exponent γ from CPMG measurements, we don’t embrace the time spent within the π pulses (this time quantities to about 10% of the period of every pulse sequence).

### Noise spectrum

We measured 3,700 Ramsey fringes over ttot = 10.26 h. For every realization, we various the free evolution time τwait as much as 7 μs, and averaged 200 single-shot spin measurements to acquire P (Prolonged Information Fig. 6a, high). The fringes oscillate on the detuning Δf = fMW1 − fL between the MW1 frequency fMW1 and the spin resonance frequency fL. To trace low-frequency noise on fL, we make a Fourier rework of every fringe and extract its basic frequency Δf reported in Prolonged Information Fig. 6a (backside). All through the experiment, fMW1 is about to 17 GHz. The low-frequency spectral noise on the Larmor frequency (in models of Hz2 Hz−1) is calculated (right here we make use of two-sided energy spectral densities, that are even with respect to the frequency) from Δf(t) as4:

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

(2)

the place FFT[Δf] is the quick Fourier rework (FFT) of Δf(t) and N is the variety of sampling factors. We observe that the low-frequency noise, plotted in Prolonged Information Fig. 6b, behaves roughly as SL(f) = Slf(f0/f) with Slf = 109 Hz2 Hz−1, which is corresponding to what has been measured for a gap spin in pure germanium41. To additional characterize the noise spectrum, we add the CPMG measurements as colored dots in Prolonged Information Fig. 6b4:

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

(3)

the place ACPMG is the normalized CPMG amplitude. As mentioned in the primary textual content, the ensuing high-frequency noise scales as ({S}^{{{{rm{hf}}}}}{({f}_{0}/f)}^{0.5}), the place Shf = 8 × 104 Hz2 Hz−1 is 4 orders of magnitude decrease than Slf. This high-frequency noise seems to be dominated by electrical fluctuations, as supported by the correlations between the Hahn-echo/CPMG T2 and the LSESs. Extra quasi-static contributions thus emerge at low frequency, and should embrace hyperfine interactions (Supplementary Data, part 5).

### Modelling

The opening wave features and g-factors are calculated with a six-band okp mannequin26. The screening by the opening gases beneath gates G1, G3 and G4 is accounted for within the Thomas–Fermi approximation. As mentioned extensively in Supplementary Data, part 1, one of the best settlement with the experimental information is achieved by introducing a average quantity of cost dysfunction. The theoretical information displayed in Figs. 1, 2 and Prolonged Information Fig. 3 correspond to a specific realization of this cost dysfunction (point-like optimistic fees with density σ = 5 × 1010 cm−2 on the Si/SiO2 interface and ρ = 5 × 1017 cm−3 in bulk Si3N4). The ensuing variability, and the robustness of the operation candy spots with respect to dysfunction, are mentioned in Supplementary Data, part 1. The rotation of the principal axes of the g-tensor seen in Fig. 1d,e are most likely attributable to small inhomogeneous strains (<0.1%); nonetheless, within the absence of quantitative pressure measurements, we have now merely shifted θzx by −25° and θzy by 10° within the calculations of Figs. 1, 2 and Prolonged Information Fig. 3.

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