A single gap spin with enhanced coherence in pure silicon

System

The machine 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 huge. It’s linked 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 total 300 mm wafer reaches 90% and their room temperature traits exhibit wonderful uniformity (see Supplementary Info, part 6 for particulars).

Dispersive readout

Much like cost detection strategies just lately utilized to silicon-on-insulator nanowire units^37,38, we accumulate a big gap island underneath gates G3 and G4, as sketched in Fig. 1a. The island acts each as a cost reservoir and electrometer for the quantum dot QD2 positioned underneath G2. Nonetheless, in contrast to the aformentioned earlier implementations, the electrometer is sensed by radiofrequency dispersive reflectometry on a lumped aspect resonator linked to the drain quite than to a gate electrode. To this goal, a business surface-mount inductor (L = 240 nH) is wire bonded to the drain pad (see Prolonged Knowledge Fig. 7 for the measurement set-up). This configuration entails a parasitic capacitance to floor C_p = 0.54 pF, resulting in resonance frequency f = 449.81 MHz. The excessive worth of the loaded high quality issue Q ≈ 10³ permits 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 traces up with the drain Fermi vitality. This results in a dispersive shift within the section ϕ_drain of the mirrored radiofrequency sign, which is measured via homodyne detection.

Power-selective single-shot readout of the spin state of the primary gap in QD2

Prolonged Knowledge Fig. 1a shows the steadiness diagram of the machine as a operate of V_G2 and V_G3 when a big quantum dot (performing as a cost sensor) is amassed underneath gates G3 and G4. The dashed gray traces define the charging occasions within the quantum dot QD2 underneath G2, detected as discontinuities within the Coulomb peak stripes of the sensor dot. The lever-arm parameter of gate G2 is α_G2 ≈ 0.37 eV V⁻¹, as inferred from temperature-dependence measurements. Comparatively, the lever-arm parameter of gate G1 with respect to the primary gap underneath G2, α_G1 ≈ 0.03 eV V⁻¹, is way smaller. The charging vitality, measured because the splitting between the primary two prices, is U = 22 meV. Prolonged Knowledge Fig. 1b reveals a zoom on the steadiness diagram across the working level used for single-shot spin readout in the principle textual content. The three factors labelled Empty (E), Load (L) and Measure (M) are the successive phases of the readout sequence sketched in Prolonged Knowledge Fig. 1c. The quantum dot is initially emptied (E) earlier than loading (L) a gap with a random spin. Each spin states are separated by the Zeeman vitality E_Z = gμ_BB the place g is the g-factor, μ_B is the Bohr magneton and B is the amplitude of the magnetic discipline. This opens a slim window for energy-selective readout utilizing spin to cost conversion⁴⁰. Particularly, we align at stage M the centre of the Zeeman cut up vitality ranges in QD2 with the chemical potential of the sensor. On this configuration, solely the excited spin-up gap can tunnel out of QD2 whereas solely spin-down holes from the sensor can tunnel in. These tunnelling occasions are detected by thresholding the section of the reflectometry sign of the sensor to realize single-shot readout of the spin state. Typical time traces of the mirrored sign section at stage M, consultant of a spin up (spin down) in QD2, are proven in Prolonged Knowledge Fig. 1d. We used this three-stage pulse sequence to optimize the readout. For that goal, the tunnel charges between QD2 and the cost sensor have been adjusted by positive tuning V_G3 and V_G4. For the spin-manipulation experiment mentioned in the principle textual content, we use a simplified two-stage sequence for readout by eradicating the empty stage. The measure stage length is ready to 200 μs for all experiments, whereas the load stage length (seen as a manipulation stage length) ranges from 50 μs to 1 ms. To acquire the spin-up chance P_↑ after a given spin manipulation sequence, we repeat the single-shot readout a lot of instances, usually 100–1,000 instances.

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 F_Rabi, 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 length of every pulse sequence).

Noise spectrum

We measured 3,700 Ramsey fringes over t_tot = 10.26 h. For every realization, we various the free evolution time τ_wait as much as 7 μs, and averaged 200 single-shot spin measurements to acquire P_↑ (Prolonged Knowledge Fig. 6a, high). The fringes oscillate on the detuning Δf = ∣f_MW1 − f_L∣ between the MW1 frequency f_MW1 and the spin resonance frequency f_L. To trace low-frequency noise on f_L, we make a Fourier rework of every fringe and extract its elementary frequency Δf reported in Prolonged Knowledge Fig. 6a (backside). All through the experiment, f_MW1 is ready 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:

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 Knowledge Fig. 6b, behaves roughly as S_L(f) = S^lf(f₀/f) with S^lf = 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 Knowledge Fig. 6b⁴:

the place A_CPMG is the normalized CPMG amplitude. As mentioned in the principle textual content, the ensuing high-frequency noise scales as ({S}^{{{{rm{hf}}}}}{({f}_{0}/f)}^{0.5}), the place S^hf = 8 × 10⁴ Hz² Hz⁻¹ is 4 orders of magnitude decrease than S^lf. 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. Further quasi-static contributions thus emerge at low frequency, and should embrace hyperfine interactions (Supplementary Info, part 5).

Modelling

The opening wave features and g-factors are calculated with a six-band okay ⋅ p mannequin²⁶. The screening by the opening gases underneath gates G1, G3 and G4 is accounted for within the Thomas–Fermi approximation. As mentioned extensively in Supplementary Info, 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 Knowledge Fig. 3 correspond to a specific realization of this cost dysfunction (point-like optimistic prices with density σ = 5 × 10¹⁰ cm⁻² on the Si/SiO₂ interface and ρ = 5 × 10¹⁷cm⁻³ in bulk Si₃N₄). The ensuing variability, and the robustness of the operation candy spots with respect to dysfunction, are mentioned in Supplementary Info, part 1. The rotation of the principal axes of the g-tensor seen in Fig. 1d,e are likely because of small inhomogeneous strains (<0.1%); nevertheless, within the absence of quantitative pressure measurements, now we have merely shifted θ_zx by ∼−25° and θ_zy by ∼10° within the calculations of Figs. 1, 2 and Prolonged Knowledge Fig. 3.