Introduction

Foundry fabricated CMOS quantum dots offer a great opportunity to build high quality spin qubit devices with the prospect of creating a scalable quantum computer1,2. Many building blocks such as single-shot detection of electron spin states3,4, electron manipulation in small arrays5,6,7 or single qubit operation using electron spin resonance8 have been successfully implemented. However, scaling up from academic fabrication to an industrial processor requires intermediate developments. An important step is to process all the parts of the qubits that are compatible with CMOS technology on an industrial level, in order to only add the components that are not yet compatible with large scale integration by post-CMOS processing in an academic clean room. This approach allows to explore the physics of these spin qubits and characterize additional modules before their integration. Among the different modules, the development of superconducting resonators integrated in the post-CMOS process has proven to be successful in the coupling of photons with spin qubits9. Following the same principle, fabrication of magnetic materials for electric-dipole spin resonance (EDSR)10,11,12,13,14,15 could also be integrated in the BEOL.

In this paper, we study the physics related to the electrical control of a single electron spin qubit using EDSR. The device is fabricated using a hybrid optical/e-beam lithography process to achieve the proof-of-concept of micromagnet integrability. We exploit the resulting synthetic SOC to drive the qubit coherently and extract its coherence properties. Moreover, we show how the presence of valleys gives rise to a second driving mechanism thanks to spin-valley mixing combined with the synthetic SOC. Finally, we use dynamical decoupling to investigate the noise source at high frequency.

Results

Device fabrication and operation

The front-end manufacturing of the device depicted in Fig. 1a is performed in a 300 mm clean room using fully depleted silicon-on-insulator (FD-SOI) technology16, see Methods. Before the device is post-processed in an academic clean room, it is buried in 200 nm SiO2 with tungsten vias to contact the gates and reservoirs. In the post-CMOS process, the alignment with respect to the tungsten vias is made by depositing a set of markers and measuring the misalignment using scanning electron microscopy (SEM). Contact is established with Ti/Al metallic lines using Electron-beam lithography (EBL) and standard lift-off techniques. The contact is ensured by an argon ion beam milling step prior to Ti/Al deposition. The Ti/Al lines are isolated from the 300 nm thick FeCo micromagnet, depicted in Fig. 1b, by 12.5 nm Al2O3 grown with atomic layer deposition (ALD). Finally, the device is encapsulated with 5 nm ALD-grown Al2O3 and a post-fabrication annealing is performed.

Fig. 1: Device operation.
figure 1

a False-coloured SEM image of a CMOS quantum dot device similar to the one used in this work. A pair of split gates (purple) is deposited on top of the channel (yellow) and SiN spacers (dark pink) are created to protect the channel from in-situ doping of the reservoirs. b Cut through the device along the gates, showing the position of the SET and the QD. The micromagnet added on top is magnetized by the external magnetic field BZ,0. c Stability diagram in the few electron regime for the quantum dot formed under gate GQD. The current flowing through the quantum dot formed under gate GSET is used for charge sensing. An abrupt shift in the conductance peak corresponds to an electron entering or leaving the QD. In the inset a zoom on the transition used for the following measurements is shown. d Simulation of the magnetic gradients created by the micromagnet. The transverse gradient dBx/dz (red) and the longitudinal gradient dBz/dz (blue) are plotted against the position across the nanowire. The gray line represents the estimated position of the QD.

Full size image

We use FeCo micromagnets with a 450 nm gap, inducing a magnetic gradient of the field component along an externally applied magnetic field ("longitudinal gradient"), and a magnetic gradient of the field components orthogonal to this external field ("transverse gradient"). The longitudinal gradient, shown in Fig. 1d, allows for individual frequency detuning17. While useful for individual qubit addressability, it also opens the door to a new dephasing channel through charge noise18,19. In the current experiment, due to misalignment during the fabrication process observed by SEM, the spin qubit quantum dot is slightly shifted away from zero longitudinal gradient, see Fig. 1d. This results in a finite dephasing gradient in the range of few MHz/nm when the electron is moved by electrical noise along the quantization axis z. Micromagnetic simulations (see Supplementary Note 1) yield a transverse gradient in the range of 0.8 mTnm−1 and it is limited by the large distance between the active region of the device and the micromagnet (250 nm). The estimated stray field induced by the micromagnet is in the range of 160 mT, as it will be confirmed by Larmor spectroscopy.

Each of the two gates enables the accumulation of electrons in a so-called corner dot. The qubit dot (QD) contains a single electron whose spin is used as a qubit, and the second dot is operated as a single-electron transistor (SET) in the many electron regime3,20. To load a single electron in the QD, we start by characterizing the stability diagram of the QD-SET system as a function of the gate voltages GQD and GSET, see Fig. 1c. We can identify the different charge occupation regimes of the QD and perform real time measurements of tunneling events by sitting on a detector Coulomb peak at the N = 0 to N = 1 charge transition sampling data at 20 kHz (see inset of Fig. 1c).

Spin relaxation characterization

Spin detection is performed by energy selective tunneling readout at finite magnetic field21,22. We start by emptying the qubit dot in the N = 0 charge region, followed by a deep plunge in the N = 1 charge state for a waiting time τw. We then move to the measurement position: the N = 0 to N = 1 transition. We obtain a characteristic click observed in Fig. 2a. A current bump is observed when the loaded electron was in the spin-up state and the current remains at the base level when a spin-down electron was loaded. To further characterize the readout configuration, we measure the probability to find the system in the spin-up state deep in the N = 1 charge state, as a function of τw. We obtain a characteristic relaxation curve, depicted in Fig. 2b from which we extract a relaxation time T1 of 5.2 ms. The measurement gives a limited visibility of 75%, and is further reduced in the following of the paper, as the magnetic field is decreased leading to a smaller EZ/kBT ratio, with T the electron temperature (estimated to be 400 mK), EZ = gμBBz the Zeeman energy, kB the Boltzmann constant, g = 2 the g-factor of electrons in silicon, and μB the Bohr magneton23.

Fig. 2: Spin relaxation characterization.
figure 2

a Single shot traces representing typical scenarios of a spin-down electron (blue) and a spin-up electron (purple) being detected. b Measurement of the spin relaxation time T1. Spin-up probability as a function of loading time is fitted to an exponential decay, yielding a spin relaxation time of T1 = 5.2 ms. The initialization is performed by loading a random spin in the quantum dot. c Spin relaxation rate 1/T1 plotted for different externally applied magnetic fields. At an applied field of 0.36 T, an abrupt relaxation increase is visible (hotspot). It gives an estimation of the valley splitting energy EVS = 60 μeV taking into account a micromagnet stray field of 0.167 T (see Supplementary Note 2). The data is fitted to determine the contribution of spin-orbit coupling (purple) and spin-valley coupling (pink). The combination of spin-orbit and spin-valley coupling (red) fits the data best.

Full size image

Probing relaxation allows to extract more information about the QD spectrum and its coupling to environment24,25,26,27. In particular, the relaxation curve in Fig. 2c shows the presence of a hotspot where fast relaxation occurs (see Supplementary Note 3 for details about the fitting). This enhancement of relaxation rate is due to the presence of valley states which can mix with the spin degree of freedom. From this point we extract the valley splitting EVS = 60 μeV. It is worth noting that this value takes into account the micromagnet stray field (see Supplementary Note 2).

Electric-dipole spin resonance

We probe the spectrum of the spin states by performing pulsed measurements using EDSR. The EDSR drive is applied to GQD with an estimated 1 mV amplitude. At fixed magnetic field, we extract the Zeeman energy from a fit of the Larmor resonance (Fig. 3a). The Larmor frequency is then studied with respect to the external magnetic field, see Supplementary Note 2. From such measurements, taking into account that g = 2, the micromagnet stray field is evaluated to be in the range of 160 mT. However, the magnetization of the micromagnet is changing with the externally applied magnetic field leading to different Larmor frequencies depending on how the external magnetic field has been swept previously. Tracking the Larmor frequency as a function of the external magnetic field allows us to reconstruct its micromagnet minor hysteresis loop presented in Supplementary Note 2 in agreement with a partially saturated magnet.

Fig. 3: Spin coherent manipulation.
figure 3

a EDSR spectrum of the single electron spin. Spin-up state probability is measured as a function of the excitation frequency. The peak is centered around the Larmor frequency f0. The initialization is performed by waiting 1 ms at the measurement position where, in principle, only the ground state is accessible. b Power spectral density of qubit frequency. The Larmor frequency is measured over three days followed by a Fourier transform (black dots). The fit (red) corresponds to a model including three nuclear spins at different fluctuating frequencies. The 1/f trend is depicted with the pink line. c Rabi oscillations for 1 mV excitation amplitude. The data points (black) are fitted using the \(\cos (\omega t+\pi /4)/\sqrt{t}\) function to account for the slowly fluctuating nuclear field. Inset: Rabi frequency as a function of estimated driving amplitude on the gate. d Echo envelope measured using a Hahn echo with evolution time of 3 μs, leading to an estimated T\({}_{2}^{*}\) of 500 ns.

Full size image

We now use this spectroscopy to track the evolution of the qubit energy over time. For this purpose we measure the Larmor frequency over three days, and using Fourier transform we construct the noise power spectral density (PSD) curve in Fig. 3b. It shows fluctuations as strong as 10 MHz at low frequency, which is in good agreement with hyperfine interaction in natural silicon devices28. To fit the experimental data, we use a hyperfine model of nuclear spins with only three clusters in the frequency window. We obtain saturation of the PSD at very low frequency as expected for a finite number of fluctuators, but we do not reach the expected 1/f2 regime at higher frequency due the small measurement bandwidth in our case. In the following, we recalibrate the Larmor frequency every two minutes to minimize the influence of this quasi-static noise.

We now move to the coherent manipulation of the qubit. We perform Rabi oscillations by sweeping the duration of a 1 mV AC drive at the Larmor frequency f0. We obtain the data presented in Fig. 3b, c. We observe a non-exponential decay of the Rabi oscillations which is attributed to the non-Markovian nature and slow fluctuation of the nuclear spin bath over the spin rotation time scale29. The Rabi frequency over voltage drive ratio corresponds to approximately 1 MHz per mV. It is worth noting that the drive amplitude is limited to 1 mV in the present sample due to a heating effect, reducing drastically the visibility.

Using the above calibrated rotations we perform a Hahn echo sequence. We extract the intrinsic dephasing time \({T}_{2}^{* }\) from the width of the echo28, as depicted in Fig. 3d. It falls in the 500 ns range when measured over few minutes. The difference with reported values in natural silicon12,30 is explained by a smaller size of quantum dot leading to a shorter hyperfine-induced dephasing. Supplementary Fig. 2a in Supplementary Note 4 presents a calculation of the wavefunction size in the present device31. We approximate the dot volume to an ellipsoid of main axis (2 nm, 6 nm, 9 nm) giving a volume around 400 nm3 which is one order of magnitude smaller than standard SiGe quantum dots19.

Valley enhanced EDSR

To investigate the interplay between valley mixing and synthetic SOC, we perform EDSR as a function of the magnetic field32,33,34. Figure 4a presents Rabi oscillations performed at different magnetic fields. It shows an increase of the Rabi frequency as the field gets closer to the valley degeneracy (hotspot). This is supported by Fig. 4b which plots both the Rabi frequency and the relaxation rate as a function of the magnetic field. It is worth noting that measurement further on the hotspot is prevented by fast relaxation during readout. This increase of EDSR Rabi frequency at the hotspot is related to the presence of a second drive mechanism which involves the presence of valley mixing in the silicon QD combined with synthetic SOC35. More precisely, the microwave electric field allows a transition from two different valleys but same spin (\(\left\vert {{{{\rm{v}}}}}_{-},\downarrow \right\rangle\) and \(\left\vert {{{{\rm{v}}}}}_{{{{\rm{+}}}}},\downarrow \right\rangle\)) and the synthetic SOC couples the two opposite spins in different valleys (\(\left\vert {{{{\rm{v}}}}}_{-},\uparrow \right\rangle\) and \(\left\vert {{{{\rm{v}}}}}_{{{{\rm{+}}}}},\downarrow \right\rangle\)) which eventually leads to an opposite spin and same valley transition (\(\left\vert {{{{\rm{v}}}}}_{-},\uparrow \right\rangle\) to \(\left\vert {{{{\rm{v}}}}}_{-},\downarrow \right\rangle\))36.

Fig. 4: Valley-enhanced EDSR.
figure 4

a Rabi oscillations at different magnetic fields (364, 365, 368 and 385 mT) for a constant drive amplitude (VAC = 1 mV). b Experimental Rabi frequency and relaxation rate from spin-valley model as a function of the externally applied magnetic field. The Rabi frequency follows the same trend as the spin-valley contribution of the relaxation rate by increasing in the vicinity of the spin-valley degeneracy. Inset: colormap of Rabi oscillations as a function of field.

Full size image

The Rabi frequency monotonously increases with the valley mixing which indicates that the two mechanisms are adding up resulting in a larger transverse field in the rotating frame. A more quantitative estimation of the amplitude and phase of the two components of the driving fields in the rotating frame is impractical here. It would require to change both the direction of the synthetic SOC field and the external magnetic field Bz,0 to map and disentangle the evolution of the two mechanisms in space. While the presence of valleys offers an enhancement of EDSR driving speed, it also increases the susceptibility of the qubit to charge noise. We observe a similar trend between the Rabi frequency and the decoherence rate of Rabi oscillations, see Supplementary Note 5. Though, we do not observe a clear enhancement of the quality factor as was proposed in ref. 35. We therefore need to understand further the noise spectrum of the qubit and in particular the influence of charge noise.

Probing high frequency charge noise using dynamical decoupling

To probe noise at higher frequency, we implement dynamical decoupling sequences starting with a Hahn echo (see Supplementary Note 6). We obtain the echo amplitude as a function of evolution time fitted with a Gaussian decay plotted in Fig. 5a (black dots) which leads to \({T}_{2}^{{{{\rm{E}}}}}=36\,\mu {{{\rm{s}}}}\). This is two orders of magnitude longer than \({T}_{2}^{* }\) showing the efficiency of the refocusing pulse to filter out the quasi-static noise. To get rid of a larger noise spectrum, we implement Carr-Purcell-Meiboom-Gill (CPMG) pulse sequences, which consist in applying multiple π pulses in a given evolution time37,38,39. Such a pulse sequence is characterized by the number of π pulses nπ, the total free evolution time τ, and the duration of a π pulse tπ. The total time of one CPMG experiment is then given by t = τ + nπtπ. The amplitudes obtained after a final refocusing π/2 pulse are summarized in Fig. 5a. For nππ pulses, the echo amplitude A(t) is fitted using the following equation40:

$${A}^{{n}_{\pi }}(t)=\exp \left[-{(t/{T}_{2}^{{n}_{\pi }})}^{1+\alpha }\right]$$
(1)

where \({T}_{2}^{{n}_{\pi }}\) is the characteristic CPMG time for information loss, for a given nπ. The parameter α is the noise color.

Fig. 5: Dynamical decoupling and high frequency noise.
figure 5

a Normalized echo amplitude as a function of total free evolution time using CPMG sequences at Bz,0= 405 mT. These sequences are composed of a series of nππy pulses, as depicted on the top of the figure for nπ = 4 and finished with a πx/2 probe at different timings τ + δτ, where δτ is swept to capture the whole echo. For each CPMG the echo is renormalized using the reference echo amplitude for τ = 1 μs. The different decay curves are fitted using the following expression \(A(t)=\exp[-{\left(t/{T}_{2}\right)}^{\beta }]\) with β = 1 + α. b Evolution of CPMG-enhanced T2 with number of refocusing pulses. T2 values are extracted from the decay curves in (a) and fitted using \({T}_{2}^{{n}_{\pi }}\propto {n}_{\pi }^{\frac{\alpha }{1+\alpha }}\). c Noise power spectral density of the qubit energy fluctuation for the evolution time corresponding to the CPMG \({T}_{2}^{{n}_{\pi }}\) times. Treating the CPMG sequence as a band pass filter allows to extract the noise PSD around the pulse repetition frequency. The 1/f trend, characteristic of charge noise, is plotted along the data points. d Noise power spectral density in terms of effective gate voltage. The low frequency data points are extracted from time domain current measurements on the flank of a SET Coulomb peak followed by Fourier transform. The high frequency points correspond to data points from (a), turned into voltage fluctuations knowing the longitudinal magnetic gradient and the charge displacement due to gate voltage.

Full size image

Another way to represent this dependence is to plot the CPMG-enhanced \({T}_{2}^{{n}_{\pi }}\) with respect to nπ, see Fig. 5b. The pulse number dependence follows a power law \({n}_{\pi }^{\alpha /(\alpha +1)}\) which in our case is consistent with a noise color of around α = 0.8 over the frequency range of CPMG pulse repetitions (104 − 106Hz). This is slightly different from typical α = 1 in semiconductor structures11,41, indicating that a finite ensemble of two-level systems in the surroundings of the qubit dominates the noise at higher frequencies. It is worth noting that the number nπ of π pulses is limited due to the poor gate fidelity induced by the large decoherence rate (see Supplementary Note 7).

A complementary analysis of CPMG data consists in considering these pulse sequences as tools for noise spectroscopy11,41. We are interested in the phase information of the electron spin, at times t, which we can write as: \(\langle \exp \left[i\phi (t)\right]\rangle \equiv \exp \left[-\chi (t)\right]\), where the phase decay envelope χ can be expressed as:

$$\chi (t)=4{\pi }^{2}\int\nolimits_{0}^{+\infty }dfS^{\prime}_{{f}_{0}}(f)\times W\left(f,\tau ,{t}_{\pi },{n}_{\pi }\right)$$
(2)

Supplementary Note 8 gives details on this expression and on this section, following closely Connors et al.41. In this expression, \(S^{\prime}_{{f}_{0}}(f)\) is the double-sided power spectral density (PSD) of the noise on the Larmor frequency, and \(W\left(f,\tau ,{t}_{\pi },{n}_{\pi }\right)\) is the spectral weighting function corresponding to a specific pulse sequence. The normalized amplitude \({A}^{{n}_{\pi }}\) plotted in Fig. 5a embodies the loss of phase information in the corresponding CPMG experiment, giving:

$$\chi (t=\tau +{n}_{\pi }{t}_{\pi })=-\ln \left[{A}^{{n}_{\pi }}(\tau ,{t}_{\pi },{n}_{\pi })\right]$$
(3)

If we assume a single-sided PSD of the form \({S}_{{f}_{0}}(f)=2S^{{{\prime}}_{{f}_{0}}}(f)=C/{f}^{\alpha}\), with α = 0.8 found in the last section, we can compute C for all of our CPMG experiments using equation (2). We get \({C}^{{n}_{\pi }}\) values by averaging the results for C at a given nπ, taking into account the normalized CPMG amplitudes between 0.15 and 0.85, where the amplitude is less likely to come from instrumental errors. In order to visualize these results, we note that the spectral weighting function W peaks at f = nπD/2τ, where D = τ/(τ + nπtπ) is the duty cycle of the pulse sequence. As a set of CPMG experiments at fixed nπ is characterized by the time \({T}_{2}^{{n}_{\pi }}\), we chose to plot \({S}_{{f}_{0}}({f}^{n_\pi }={n}_{\pi }D/2{T}_{2}^{{n}_{\pi }})={C}^{{n}_{\pi }}/{({f}^{{n}_{\pi }})}^{\alpha }\) in Fig. 5c.

To confirm the origin of charge noise as the main source of decoherence at high frequency, we now turn the PSD \({S}_{{f}_{0}}\) into a PSD SV in terms of fluctuations of gate voltage: we mimic the effect of this charge noise by an effective noise on the gate voltage. This voltage noise translates into qubit frequency fluctuations by moving the electron along the z axis combined with a longitudinal gradient \(\frac{d{B}_{z}}{dz}\). It summarizes with the following expression:

$${S}_{{f}_{0}}=\frac{g{\mu }_{{{{\rm{B}}}}}}{h}\times \frac{d{B}_{z}}{dz}\times {\delta }_{e}\times {S}_{V}$$
(4)

where h is the Planck constant and δe is the electron displacement along z due to gate voltage on GQD. This displacement per voltage can be extracted from the Rabi oscillations assuming:

$${\delta }_{{{{\rm{e}}}}}=\frac{{f}_{{{{\rm{R}}}}}}{{V}_{{{{\rm{AC}}}}}\times \frac{g{\mu }_{{{{\rm{B}}}}}}{h}\times \frac{d{B}_{x}}{dz}}$$
(5)

with VAC the excitation voltage to drive Rabi oscillations at frequency fR, \(\frac{d{B}_{x}}{dz}\) the transverse gradient. Using equation (5) and a Rabi frequency of fR = 1 MHz for VAC = 1 mV, we obtain an electron displacement of 45 pm/mV. Using equation (4) we plot in Fig. 5c the PSD in terms of fluctuations in gate voltage. To compare with a standard charge noise measurement, we also add on the same plot the PSD extracted from SET fluctuations obtained by simply transforming SET current measurements recorded while sitting on the flank of a Coulomb peak. We obtain a good agreement between the high and low frequency data points falling on the 1/f trend. This pink noise behavior over 6 decades confirms the origin of the decoherence mechanism at high frequency to be charge noise.

To compare with other spin qubit platforms, we can convert the PSD from Fig. 5d at 1 Hz (1000 μV2Hz−1) to a PSD in terms of chemical potential using the lever arm (0.25eVV−1). We obtain 8 μeVHz−1/2. This value is one to two orders of magnitude higher compared to commonly reported values in literature5,41,42, but similar to foundry fabricated spin qubits8. In our case, this large charge noise amplitude points toward a degradation of device quality during the post-CMOS process which uses lower standard processes. However, the high frequency fluctuations of the qubit are similar to what is reported in other platforms with less nominal charge noise while the longitudinal gradient is comparable. Therefore, we attribute the lower influence of charge noise to a small charge displacement susceptibility (δe = 45 pmmV−1) due to confinement in the corner of the nanowire.

Discussion

In conclusion, we have seen that a post-CMOS process can be used to implement additional functionalities to foundry-fabricated devices. With this process, we patterned a micromagnet on top of a CMOS device, which enabled coherent manipulation of an electron spin through electric-dipole spin resonance. We identify two driving mechanisms: the one based on bare synthetic spin-orbit coupling and the one induced by combining the spin-valley mixing near the valley hotspot with the synthetic spin-orbit coupling. We further investigate the coherent properties of the spin qubit which shows a dephasing time around 500 ns due to hyperfine coupling to surrounding nuclear spins in the small volume of the quantum dot. Refocusing this low frequency noise through dynamical decoupling pulse sequences allowed to enhance the qubit coherence time by three orders of magnitude, and provided evidence that high-frequency noise has electrical origin, with an amplitude comparable to state-of-the-art spin qubit realizations. While the charge noise amplitude is relatively high, the induced decoherence at large frequency is relatively small, due to the strong confinement in the corner of the nanowire, at the cost of lower Rabi frequency. Lowering both the hyperfine quasistatic noise thanks to isotopic purification and the charge noise by two orders of magnitude would make these CMOS-based spin qubits a promising platform for quantum information.

Methods

Experimental setup

Experiments are performed in the CMOS device as shown in Fig. 1a using a dilution refrigerator with a base temperature of 120 mK. Charge sensing is performed by monitoring the current through the SET using a homemade IV converter followed by digitization using a NI BNC 2110 acquisition card. The voltage on gates is controlled using digital-to-analog converters controlled by a sbRIO-9208 FPGA board.

Front-end device fabrication

The device, presented in Fig. 1a, is based on a 80 nm wide silicon nanowire transistor from fully depleted silicon-on-insulator (FD-SOI) technology16. On top of a 10 nm thick nanowire, 6 nm SiO2 gate oxide and 5 nm TiN/50 nm polysilicon gate metal are deposited. A single pair of split gates with a gate length and gate separation of 50 nm is patterned using optical and electron-beam lithography (EBL). SiN spacers are created to protect the device during the in situ doping of the reservoirs with phosphorus.