Flopping-mode spin qubit in a Si-MOS quantum dot

Spin qubits based on silicon metal-oxide semiconductor (Si-MOS) quantum dots (QDs) are promising platforms for large-scale quantum computers. To control spin qubits in QDs, electric dipole spin resonance (EDSR) has been most commonly used in recent years. By delocalizing an electron across a double quantum dots charge state, “flopping-mode” EDSR has been realized in Si/SiGe QDs. Here, we demonstrate a flopping-mode spin qubit in a Si-MOS QD via Elzerman single-shot readout. When changing the detuning with a fixed drive power, we achieve s-shape spin resonance frequencies, an order of magnitude improvement in the spin Rabi frequencies, and virtually constant spin dephasing times. Our results offer a route to large-scale spin qubit systems with higher control fidelity in Si-MOS QDs.

Spin qubits in silicon QDs are a leading candidate for building a quantum processor due to their long coherence time,
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Hwang et al., “ Operation of a silicon quantum processor unit cell above one kelvin,” Nature 580, 350–354 (2020). https://doi.org/10.1038/s41586-020-2171-6To implement EDSR in Si-MOS QDs, a rectangular micromagnet is deployed to generate an inhomogeneous magnetic field and an oscillating electric field resonant with the Larmor frequency is coupled to drive the spin states.20–2320. E. I. Rashba and A. L. Efros, “ Orbital mechanisms of electron-spin manipulation by an electric field,” Phys. Rev. Lett. 91, 126405 (2003). https://doi.org/10.1103/PhysRevLett.91.12640521. R. C. C. Leon, C. H. Yang, J. C. C. Hwang et al., “ Coherent spin control of s-, p-, d- and f-electrons in a silicon quantum dot,” Nat. Commun. 11, 797 (2020). https://doi.org/10.1038/s41467-019-14053-w22. X. Zhang, Y. Zhou, R.-Z. Hu et al., “ Controlling synthetic spin-orbit coupling in a silicon quantum dot with magnetic field,” Phys. Rev. Appl. 15, 044042 (2021). https://doi.org/10.1103/PhysRevApplied.15.04404223. R.-Z. Hu, R.-L. Ma, M. Ni et al., “ An operation guide of si-mos quantum dots for spin qubits,” Nanomaterials 11, 2486 (2021). https://doi.org/10.3390/nano11102486 During the conventional EDSR measurement, electrons in Si-MOS QD are confined in the quantum well, leading to a relatively small electric dipole.24,2524. M. Pioro-Ladrière, T. Obata, Y. Tokura et al., “ Electrically driven single-electron spin resonance in a slanting Zeeman field,” Nat. Phys. 4, 776–779 (2008). https://doi.org/10.1038/nphys105325. E. Kawakami, P. Scarlino, D. R. Ward et al., “ Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot,” Nat. Nanotechnol. 9, 666–670 (2014). https://doi.org/10.1038/nnano.2014.153 Driving single spin rotations in a DQD close to zero detuning where electron shuttles between two QDs, the flopping-mode EDSR increases the electric dipole in QDs.2626. M. Benito, X. Croot, C. Adelsberger et al., “ Electric-field control and noise protection of the flopping-mode spin qubit,” Phys. Rev. B 100, 125430 (2019). https://doi.org/10.1103/PhysRevB.100.125430 A longer spin coherence time (T2Rabi) with the same Rabi oscillation frequency (fRabi) has been achieved in Si/SiGe spin qubits by applying flopping-mode EDSR via dispersive readout.2727. X. Croot, X. Mi, S. Putz et al., “ Flopping-mode electric dipole spin resonance,” Phys. Rev. Res. 2, 012006 (2020). https://doi.org/10.1103/PhysRevResearch.2.012006 However, the small size and complicated distribution of Si-MOS QDs make cavity readout of a flopping-mode spin qubit in Si-MOS QDs difficult.1,9–11,13,19,211. M. Veldhorst, J. C. C. Hwang, C. H. Yang et al., “ An addressable quantum dot qubit with fault-tolerant control-fidelity,” Nat. Nanotechnol. 9, 981–985 (2014). https://doi.org/10.1038/nnano.2014.2169. W. Huang, C. H. Yang, T. Tanttu et al., “ Fidelity benchmarks for two-qubit gates in silicon,” Nature 569, 532 (2019). https://doi.org/10.1038/s41586-019-1197-010. L. Petit, H. G. J. Eenink, M. Russ et al., “ Universal quantum logic in hot silicon qubits,” Nature 580, 355–359 (2020). https://doi.org/10.1038/s41586-020-2170-711. K. W. Chan, H. Sahasrabudhe, W. Huang et al., “ Exchange coupling in a linear chain of three quantum-dot spin qubits in silicon,” Nano Lett. 21, 1517 (2021). https://doi.org/10.1021/acs.nanolett.0c0477113. W. Gilbert, T. Tanttu, W. H. Lim et al., “ On-demand electrical control of spin qubits,” arXiv:2201.06679 (2022).19. C. H. Yang, R. C. C. Leon, J. C. C. Hwang et al., “ Operation of a silicon quantum processor unit cell above one kelvin,” Nature 580, 350–354 (2020). https://doi.org/10.1038/s41586-020-2171-621. R. C. C. Leon, C. H. Yang, J. C. C. Hwang et al., “ Coherent spin control of s-, p-, d- and f-electrons in a silicon quantum dot,” Nat. Commun. 11, 797 (2020). https://doi.org/10.1038/s41467-019-14053-wHere, we demonstrate a flopping-mode single spin qubit in a Si-MOS QD via the Elzerman single-shot readout.2828. J. M. Elzerman, R. Hanson, L. H. Willems van Beveren et al., “ Single-shot read-out of an individual electron spin in a quantum dot,” Nat. 430, 431 (2004). https://doi.org/10.1038/nature02693 By setting gate voltages carefully, a DQD with appropriate tunneling rates of an electron from the QD to the reservoir is formed underneath adjacent electrodes. Then, we measure the EDSR spectra, Rabi oscillation, and Ramsey fringes. Due to the large interdot tunnel coupling 2tc, an s-shape spin resonance frequency (fq) as a function of the energy detuning (ε) is formed. We achieve an order of magnitude improvement in fRabi around ε=0 with the spin dephasing times (T2*) and spin coherence time of Rabi oscillation (T2Rabi) virtually constant.Figure 1(a) shows a scanning electron microscope (SEM) image of a typical Si-MOS DQD device, nominally identical to the one measured in Ref. 2323. R.-Z. Hu, R.-L. Ma, M. Ni et al., “ An operation guide of si-mos quantum dots for spin qubits,” Nanomaterials 11, 2486 (2021). https://doi.org/10.3390/nano11102486. The device was fabricated on a natural silicon substrate with a 70 nm thick isotopically enriched 28Si epi layer that has a residual 29Si concentration of 60 ppm. Overlapping aluminum gate electrodes were fabricated using multi-layer gate stack technology.2929. X. Zhang, R.-Z. Hu, H.-O. Li et al., “ Giant anisotropy of spin relaxation and spin-valley mixing in a silicon quantum dot,” Phys. Rev. Lett. 124, 257701 (2020). https://doi.org/10.1103/PhysRevLett.124.257701 The cobalt micromagnet integrated near QDs will be fully magnetized during the measurement, leading to a transverse magnetic field gradient for driving the spin qubits.2222. X. Zhang, Y. Zhou, R.-Z. Hu et al., “ Controlling synthetic spin-orbit coupling in a silicon quantum dot with magnetic field,” Phys. Rev. Appl. 15, 044042 (2021). https://doi.org/10.1103/PhysRevApplied.15.044042 The total magnetic field at the QDs (Btotal) is the sum of the external magnetic field Bext and the stray field from the micromagnet, as shown in the right bottom corner of Fig. 1(a). The device is in a dilution refrigerator at an electron temperature of Te=182.7±0.6 mK (see Sec. S2 in the supplementary material for details).The electrons are confined in the potential wells under gates LP and BC and form the DQD by selectively tuning gates LP, LB, and BC, as shown in Fig. 1(d). The corresponding charge stability diagram is shown in Fig. 1(b). (N1, N2) on the diagram labels the corresponding number of electrons. The black arrow illustrates the direction of ε between the DQDs. Gates MC and BC are designed to create a channel under the lead gate LG for electrons to tunnel between the electron reservoir and the DQD. Due to the small electrode size (∼30 nm) and the difference in the thermal expansion coefficient between the aluminum electrodes and SiO2 substrate surface, there is usually one quantum-well formed under each electrode gate in Si-MOS QDs, possibly forming complicated quadruple or more quantum dots in the device designed for the DQD.Then, we apply two-step pulse sequences to gate LB for the Elzerman single-shot readout, as shown by points R (Read) and C (Control) in Fig. 1(c). The gate LB is designed to modify the tunneling rate of electrons from the DQD to the electron reservoir, as shown in Fig. 1(e). Due to the capacitive coupling between gate LB, LP, and BC, we need to calibrate the gate voltages VLB, VLP, and VBC to maintain the tunneling rate for different ε during the measurements. We confirm that the transitions between points R and C are adiabatic, as discussed in Sec. S3 in the supplementary material.An external magnetic field is applied to the device for Elzerman single-shot readout. Bext is set to 605 mT to induce Zeeman splitting between spin states and fully magnetize the micromagnet magnetic field. As a result, ∼20 GHz microwave pulses are applied to the LP gate via a cryogenic bias-tee to manipulate the qubit. By using sequences of selective EDSR pulses with microwave burst of frequency (fs) at point C, we can perform single-qubit operations on the electron. The spin state is read out via state-to-charge conversion at point R, and a |↓⟩ electron is selectively loaded for initialization in the next pulse sequence.2323. R.-Z. Hu, R.-L. Ma, M. Ni et al., “ An operation guide of si-mos quantum dots for spin qubits,” Nanomaterials 11, 2486 (2021). https://doi.org/10.3390/nano11102486 The details of the measurement circuits are discussed in Sec. S1 in the supplementary material.To detect the EDSR spectra rapidly, we apply frequency-chirped microwave pulses (±2 MHz around the center frequencies (fs), 100 μs duration times) to gate LP before the end of the control phase.23,30–3223. R.-Z. Hu, R.-L. Ma, M. Ni et al., “ An operation guide of si-mos quantum dots for spin qubits,” Nanomaterials 11, 2486 (2021). https://doi.org/10.3390/nano1110248630. M. Shafiei, K. C. Nowack, C. Reichl et al., “ Resolving spin-orbit- and hyperfine-mediated electric dipole spin resonance in a quantum dot,” Phys. Rev. Lett. 110, 107601 (2013). https://doi.org/10.1103/PhysRevLett.110.10760131. A. Laucht, R. Kalra, J. T. Muhonen et al., “ High-fidelity adiabatic inversion of a 31P electron spin qubit in natural silicon,” Appl. Phys. Lett. 104, 092115 (2014). https://doi.org/10.1063/1.486790532. A. Sigillito, J. Loy, D. Zajac et al., “ Site-selective quantum control in an isotopically enriched 28Si/Si0.7Ge0.3 quadruple quantum dot,” Phys. Rev. Appl. 11, 061006 (2019). https://doi.org/10.1103/PhysRevApplied.11.061006 If the frequency sweeps through the spin resonance frequencies fq, the electron spin will end up in the excited state |↑⟩. By selectively setting VLP, VLB, and VBC, we perform the Elzerman readout with a fixed tunneling rate of approximately 150 Hz for the |↓⟩ electron to ensure consistency in the readout process, while ε increases from −4.5 to 4.5 meV along the exact transition line (0, 1) to (1, 0) (see Sec. S2 in the supplementary material for details). We measure the probability of electrons in the excited state (P↑) as a function of fs from 300 repeated single-shot readouts. For each ε, we repeat the measurement ten times, as mentioned in Ref. 2929. X. Zhang, R.-Z. Hu, H.-O. Li et al., “ Giant anisotropy of spin relaxation and spin-valley mixing in a silicon quantum dot,” Phys. Rev. Lett. 124, 257701 (2020). https://doi.org/10.1103/PhysRevLett.124.257701. The EDSR spectra over ε from −4.5 to 4.5 meV are shown in Fig. 2(b). There is an s-shape curve of increased P↑ with a width of 4 MHz of fs, where fq is located. We calibrate the peak of P↑ and extract fq as a function of ε in Fig. 2(c).To explain this s-shape feature, we focus on the Hamiltonian H of a single-electron occupied DQD system on the basis (|L↓⟩, |L↑⟩, |R↓⟩, |R↑⟩)26,3326. M. Benito, X. Croot, C. Adelsberger et al., “ Electric-field control and noise protection of the flopping-mode spin qubit,” Phys. Rev. B 100, 125430 (2019). https://doi.org/10.1103/PhysRevB.100.12543033. M. Benito, X. Mi, J. M. Taylor et al., “ Input-output theory for spin-photon coupling in Si double quantum dots,” Phys. Rev. B 96, 235434 (2017). https://doi.org/10.1103/PhysRevB.96.235434 H=12(−ε−Ez1−2tSO2tc0−2tSO−ε+Ez102tc2tc0ε−Ez22tSO02tc2tSOε+Ez2).(1)Here, Ez1 and Ez2 are Zeeman energies for the first and second QD, respectively, 2tc is the interdot tunnel coupling, and 2tSO=gμBb⊥ is the synthetic spin–orbit coupling induced by the transverse magnetic field difference (b⊥).The eigenenergies of this four-level system are shown in Fig. 2(a). The avoided crossings at ε=0 are generated by 2 tc. By diagonalizing the Hamiltonian in Eq. (1), we calculate the energy splitting (Es) between the lowest two energy levels as a function of ε. Therefore, the corresponding spin resonance frequency of the qubit is obtained through fq≡Es/h. For the situation of a small inhomogeneous field, i.e., b⊥,z≪|Ω−Ez|/gμB, where Ω=ε2+4tc2, Es is corrected by the transverse and longitude gradients to second and first orders, respectively,2626. M. Benito, X. Croot, C. Adelsberger et al., “ Electric-field control and noise protection of the flopping-mode spin qubit,” Phys. Rev. B 100, 125430 (2019). https://doi.org/10.1103/PhysRevB.100.125430 Es≃Ez−Ez2−ε22Ez(Ω2−Ez2)(gμBb⊥)2−εΩgμBbz.(2)Here, Ez=(Ez1+Ez2)/2 is the averaged Zeeman energy and δEz=(Ez1−Ez2)/2=gμBbz is the Zeeman energy difference generated by the longitudinal magnetic field difference (bz) of the micromagnet.We plot fq as a function of ε for Bext= 605 and 604 mT in Fig. 2(c). By fitting fq with Eq. (2), we obtain 2tc/h=914 ± 167 and 705 ± 40 GHz for Bext= 605 and 604 mT, respectively. The difference between the fitted splitting energy (ΔEz=19.790 ± 0.002−19.760±0.001 GHz) equals the difference between the external magnetic fields (gμBΔBext/h=28 MHz). δfq≡2δEz/h=56.8 ± 4.5 MHz for far detuned limits is shown in Fig. 2(c). In Ref. 2727. X. Croot, X. Mi, S. Putz et al., “ Flopping-mode electric dipole spin resonance,” Phys. Rev. Res. 2, 012006 (2020). https://doi.org/10.1103/PhysRevResearch.2.012006, the lowest Es occurs near ε=0, leading to a sweet spot for spin dephasing. However, in our device 2tc≫Ez, the second-order item Ez2−ε22Ez(Ω2−Ez2)(gμBb⊥)2 in Eq. (2) is suppressed, and there is no sweet spot approximately ε=0.After calibrating fq, we now use a microwave burst with a specific burst time (τB) to manipulate the spin qubit. First, we measure P↑ as a fs function with a fixed τB. Each point of P↑ in the curve is averaged from 300 repeated single-shot readouts. Then, we repeat the measurement ten times and sum P↑ with τB changing from 0 to 4 μs. The Rabi chevron is plotted in Fig. 3(a).Figure 3(b) illustrates fRabi as a function of ε with fixed microwave power PMW=0 dBm at the source. fRabi is symmetric about ε=0 and is an order of magnitude larger at ε=0 than the far detuned position. For every ε, the corresponding fRabi is extracted by fitting the Rabi oscillation with the function P↑(τB)=A· exp(−τB/T2Rabi)· sin(fRabiτB), as shown in Fig. 3(c).For a typical flopping-mode EDSR process, Ref. 2626. M. Benito, X. Croot, C. Adelsberger et al., “ Electric-field control and noise protection of the flopping-mode spin qubit,” Phys. Rev. B 100, 125430 (2019). https://doi.org/10.1103/PhysRevB.100.125430 gives fRabi as a function of ε for small b⊥, fRabi=4tc2gμBb⊥Ωc/Ω|Ω2−Ez2|.(3)Here, Ωc=edEac/ℏ is the Rabi frequency of charge qubits, proportional to the distance between the two QDs d, and the electric field with amplitude Eac. Ωc=15.8±0.8 GHz can be obtained from the relevant result of fRabi with a gμBb⊥ estimated as 0.232 μeV.2222. X. Zhang, Y. Zhou, R.-Z. Hu et al., “ Controlling synthetic spin-orbit coupling in a silicon quantum dot with magnetic field,” Phys. Rev. Appl. 15, 044042 (2021). https://doi.org/10.1103/PhysRevApplied.15.044042 We estimate d∼ 0.02 μm; thus, b⊥∼0.1 T/μm ·0.02 μm =2 mT.Figure 3(c) shows details of the Rabi oscillations for different ε. fRabi=1.262±0.002 MHz is achieved in the top panel. When ε increases to 1.5 and 3 meV, the Rabi frequencies decrease to fRabi=0.429 ± 0.003 and fRabi=0.135 ± 0.003 MHz, respectively. By fitting the Rabi oscillation to an exponentially decaying sinusoid, T2Rabi=6.46 ± 0.39 μs at ε=0, T2Rabi=5.53 ± 0.57 μs at ε=1.5meV, and

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