# Valley-enhanced electrical manipulation of a single electron spin using
post-CMOS processed micromagnet

Bernhard Klemt  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38402
Grenoble, France  Victor El-Homsy  Univ. Grenoble Alpes, CNRS, Grenoble INP,
Institut Néel, 38402 Grenoble, France  Martin Nurizzo  Univ. Grenoble Alpes,
CNRS, Grenoble INP, Institut Néel, 38402 Grenoble, France  Pierre Hamonic
Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38402 Grenoble,
France  Biel Martinez-Diaz  Univ. Grenoble Alpes, CEA, IRIG, 38000 Grenoble,
France  Bruna Cardoso-Paz  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut
Néel, 38402 Grenoble, France  Baptiste Jadot  Univ. Grenoble Alpes, CEA, Leti,
F-38000 Grenoble, France  Emmanuel Chanrion  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38402 Grenoble, France  Vivien Thiney  Univ.
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38402 Grenoble, France
Renan Lethiecq  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38402
Grenoble, France  Benoit Bertrand  Univ. Grenoble Alpes, CEA, Leti, F-38000
Grenoble, France  Heimanu Niebojewski  Univ. Grenoble Alpes, CEA, Leti,
F-38000 Grenoble, France  Christopher Bäuerle  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38402 Grenoble, France  Maud Vinet  Univ.
Grenoble Alpes, CEA, Leti, F-38000 Grenoble, France  Yann-Michel Niquet  Univ.
Grenoble Alpes, CEA, IRIG, 38000 Grenoble, France  Tristan Meunier  Univ.
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38402 Grenoble, France
Matias Urdampilleta<EMAIL_ADDRESS>Univ. Grenoble Alpes,
CNRS, Grenoble INP, Institut Néel, 38402 Grenoble, France

###### Abstract

For semiconductor spin qubits, complementary-metal-oxide-semiconductor (CMOS)
technology is the ideal candidate for reliable and scalable fabrication.
Making the direct leap from academic fabrication to qubits fabricated fully by
industrial CMOS standards is difficult without intermediate solutions. With a
flexible back-end-of-line (BEOL) new functionalities such as micromagnets or
superconducting circuits can be added in a post-CMOS process to study the
physics of these devices or achieve proof of concepts. Once the process is
established it can be incorporated in the foundry-compatible process flow.
Here, we study a single electron spin qubit in a CMOS device with a
micromagnet integrated in the flexible BEOL. We exploit the synthetic spin
orbit coupling (SOC) to control the qubit via electric field and we
investigate the spin-valley physics in the presence of SOC where we show an
enhancement of the Rabi frequency at the spin-valley hotspot. Finally, we
probe the high frequency noise in the system using dynamical decoupling pulse
sequences and demonstrate that charge noise dominates the qubit decoherence in
this range.

††preprint: APS

##  I 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
computer [1, 2]. Many building blocks such as single-shot detection of
electron spin states [3, 4], electron manipulation in small arrays [5, 6, 7]
or single qubit operation using electron spin resonance [41] have been
successfully implemented. However, making the direct leap from academic
fabrication to qubits fabricated fully by industrial CMOS standards is
difficult without intermediate solutions. An important step is to process all
the parts of the qubits that are compatible with CMOS technology on an
industrial level. Only adding the components that are not yet compatible with
large scale integration by post-CMOS processing in an academic clean room.
This approach allows exploring the physics of these spin qubits and
characterize new 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 photon with spin
qubits [8] or for the readout of a quantum dot [9]. Following the same
principle, fabrication of magnetic materials for electric-dipole spin
resonance (EDSR) [10, 12, 13, 14, 15, 11] 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 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.

##  II Device fabrication and operation

The device, presented in Fig.1(a) is based on a $80\text{\,}\mathrm{nm}$ wide
silicon nanowire transistor from $28\text{\,}\mathrm{nm}$ fully depleted
silicon-on-insulator (FD-SOI) technology[16]. On top of a
$10\text{\,}\mathrm{nm}$ thick nanowire, $6\text{\,}\mathrm{nm}$
SiO${}_{\textrm{2}}$ gate oxide and $5\text{\,}\mathrm{nm}$ TiN /
$50\text{\,}\mathrm{nm}$ polysilicon gate metal are deposited. A single pair
of split gates with a gate length and gate separation of
$50\text{\,}\mathrm{nm}$ is patterned using optical and electron-beam
lithography. SiN spacers are created to protect the device during the in situ
doping of the reservoirs with phosphorus. Before the device is post-processed
in an academic clean room, it is buried in $200\text{\,}\mathrm{nm}$
SiO${}_{\textrm{2}}$ 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 gates
using EBL and standard lift-off techniques. The contact is ensured by an argon
ion beam milling step prior to Ti/Al gate deposition. The Ti/Al gates are
isolated from the $300\text{\,}\mathrm{nm}$ thick FeCo micromagnet, depicted
in Fig.1(b), by $12.5\text{\,}\mathrm{nm}$
Al${}_{\textrm{2}}$O${}_{\textrm{3}}$ grown with atomic layer deposition
(ALD). Finally, the device is encapsulated with $5\text{\,}\mathrm{nm}$ ALD-
grown Al${}_{\textrm{2}}$O${}_{\textrm{3}}$ and a post-fabrication annealing
is performed.

We use a FeCo micromagnet, 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.1(d), allows
for individual frequency detuning[17]. While useful for individual qubit
addressability, it also opens the door to dephasing [18, 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.1(d). It results in a finite dephasing gradient
in the range of few
$\text{\,}\mathrm{M}\mathrm{Hz}\mathrm{/}\mathrm{n}\mathrm{m}$ when the
electron is moved by charge noise along the quantization axis z. Simulations
of the micromagent (see Suppl. Mat. 2) yield a transverse gradient in the
range of $0.8\text{\,}\mathrm{m}\mathrm{T}\mathrm{/}\mathrm{n}\mathrm{m}$ and
it is limited by the encapsulation layer between the active region of the
device and the micromagnet ($250\text{\,}\mathrm{nm}$). The estimated stray
field induced by the micromagnet is in the range of $160\text{\,}\mathrm{mT}$,
as 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 other dot is operated as a single-electron transistor
(SET) in the many electron regime [20, 3]. 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 $G_{\text{QD}}$ and $G_{\text{SET}}$, see Fig.
1(c). 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
$30\text{\,}\mathrm{kHz}$ (see inset of Fig.1(c)).

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 (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 $B_{0}$. (c)
Stability diagram in the few electron regime for the quantum dot formed under
gate $G_{\text{QD}}$. The current flowing through the quantum dot formed under
gate $G_{\text{SET}}$ 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 from the micromagnet. The transverse
gradient $dB_{x}/dz$ (red) and the longitudinal gradient $dB_{z}/dz$ (blue)
are plotted against the position across the nanowire. The gray dashed line
represents the estimated position of the QD.

##  III Spin relaxation characterization

Spin detection is performed by Elzerman’s energy selective tunneling readout
at finite magnetic field [21, 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
given time. We then move to the measurement position previously described at
the $N=0$ to $N=1$ transition. We obtain the characteristic click signal at
$30\text{\,}\mathrm{kHz}$ bandwidth observed on Fig.2(a) where a current bump
is observed when the loaded electron was in the spin excited state. To further
characterize the readout configuration, we measure the probability to find the
system in the spin excited state as a function of the waiting time deep in the
$N=1$ charge state. We obtain a characteristic relaxation curve, depicted in
Fig.2(b) from which we extract a relaxation time $T_{1}$ around
$10\text{\,}\mathrm{ms}$. The measurement gives a poor visibility of $75\%$,
and is further reduced in the following of the paper, as the magnetic field is
decreased leading to a smaller $k_{\text{B}}T/E_{z}$ ratio, with $T$ the
temperature (estimated to $400\text{\,}\mathrm{mK}$),
$E_{z}=g\mu_{\text{B}}B_{z}$ the Zeeman energy, $k_{\text{B}}$ the Boltzmann
constant, $g=2$ the g-factor of electrons in silicon, and $\mu_{\text{B}}$ the
Bohr magneton [23].

Probing relaxation allows to extract more information about the QD spectrum
and its coupling to environment [24, 25, 26, 27]. In particular, we extract
the valley splitting $E_{\text{VS}}=$60\text{\,}\mathrm{\SIUnitSymbolMicro
eV}$$ by measuring $T_{1}$ as a function of magnetic field, see Fig2(c).
Fitting the relaxation dependence with external field shows that the main
contribution is Johnson-Nyquist noise for both contributions coming from the
presence of excited orbitals ($SO^{\text{JN}}$, in purple) and valleys
($SV^{\text{JN}}$, in pink), see Suppl. Mat. for the fitting parameters. The
phonon contribution $SO^{\text{Ph}}+SV^{\text{Ph}}$, plotted in blue in
Fig2(c), is negligible. This is in agreement with similar experiments where
hotspot has been obtained in the same range and with identical dependence with
magnetic field [28]. The strong contribution of the Johnson-Nyquist noise is
attributed to the enhanced spin-orbit interaction due to the magnetic gradient
combined with charge noise at the Larmor frequency [29].

Figure 2:  (a) Single shot traces representing typical scenarios of a spin-
down electron (purple) and a spin-up electron (yellow) being detected. (b)
Measurement of the spin relaxation time T${}_{\textrm{1}}$ for a total
magnetic field of $2.46\text{\,}\mathrm{T}$ including the contribution of the
micromagnet. An exponential decay is fitted to the data yielding a spin
relaxation time of T${}_{\textrm{1}}$ = $5.2\text{\,}\mathrm{m}\mathrm{s}$.
The initialization is performed by loading a random spin in the quantum dot.
(c) Spin relaxation rate 1/$\textrm{T}_{\textrm{1}}$ plotted for different
total magnetic fields. At $0.52\text{\,}\mathrm{T}$ the spin relaxation
hotspot is visible. The data is fitted to determine the contribution of spin-
orbit coupling (purple) spin-valley coupling (pink). Both phonon noise, as
well as Johnson–Nyquist noise are taken into account. The combination of spin-
orbit and spin-valley coupling (red) fits the data best. A fit using only the
phonon noise contribution is shown in blue.

##  IV Electric-dipole spin resonance

We probe the spectrum of the spin states by performing pulsed measurements
using electric-dipole spin resonance EDSR. The EDSR drive is applied to
$G_{\text{QD}}$ with a $1\text{\,}\mathrm{mV}$ amplitude. At fixed magnetic
field, we extract the Zeeman energy from a Lorentzian fit of the Larmor
resonance (Fig.3(a)). The Larmor frequency is then studied with respect to the
external magnetic field (Fig.3(b)). From such measurement, taking into account
that $g=2$, the micromagnet stray field is evaluated to be
$167\text{\,}\mathrm{mT}$. However, the magnetization of the micromagnet is
changing with the applied external magnetic field leading to different Larmor
frequencies depending on how the external magnetic field has been swept.
Tracking the Larmor frequency as a function of the external magnetic field
allows us to reconstruct its micromagnet minor hysteresis loop presented in
Suppl. Mat. in agreement with a partially saturated magnet.

We now move to the coherent manipulation of the qubit. We perform Rabi
oscillations by sweeping the duration of a $1\text{\,}\mathrm{mV}$ AC drive at
the Larmor frequency $f_{0}$. We obtain the data presented in Fig.3(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 scale [30]. The Rabi frequency over voltage drive ratio
corresponds to approximately $1\text{\,}\mathrm{MHz}$ per
$\text{\,}\mathrm{mV}$. It is worth noting that the drive amplitude is limited
to $1\text{\,}\mathrm{mV}$ in the present sample due to heating effect,
reducing drastically the visibility.

Using the above calibrated rotations we can perform a Hahn echo sequence to
extract the intrinsic dephasing time $T_{2}^{*}$ and the Hahn-Echo coherence
time $T_{2}^{\text{E}}$. $T_{2}^{*}$ is directly extracted from the envelope
of the echo, [43], as depicted in Fig.3(d) and falls in the
$500\text{\,}\mathrm{ns}$ range when measured over few tens of minutes. The
small difference with reported values in natural silicon, [31, 12] is
explained by a smaller size of quantum dot leading to a shorter hyperfine-
induced dephasing time. The Hahn echo time $T_{2}^{\text{E}}$ is longer by two
orders of magnitude, on the order of $36\text{\,}\mathrm{\SIUnitSymbolMicro
s}$, showing that the quasistatic noise due to hyperfine interaction has been
refocused.

Figure 3:  (a) Electric dipole spin resonance spectrum of the single electron
spin. Excited spin state probability is measured as a function of the
excitation frequency. The peak is centered around the Larmor frequency
$f_{0}$. The initialization is performed by waiting $10\text{\,}\mathrm{ms}$
at the measurement position where, in principle, only the ground state is
accessible. (b) $f_{0}$ is measured as in (a) and plotted as a function of the
external magnetic field. (c) Rabi oscillations for $1\text{\,}\mathrm{mV}$
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. (d) Echo envelope measured using a Hahn echo sequence. This
leads to an estimated T${}_{2}^{*}$ of $500\text{\,}\mathrm{ns}$.

##  V Valley enhanced EDSR

To investigate the interplay between valley mixing and SOC we perform EDSR as
a function of the magnetic field [32, 33]. Figure 4(a) presents Rabi
oscillations performed at different magnetic fields. It shows an increase of
the Rabi frequency as the field gets closer to the valley hotspot. This is
supported by figure 4(b) which plots both the Rabi frequency and the
relaxation rate as a function of the Larmor frequency. 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 SOC [29]. More precisely, the
microwave electric field allows a transition from two different valleys but
same spin ($\ket{\textrm{v}_{\textrm{-}},\downarrow}$ and
$\ket{\textrm{v}_{\textrm{+}},\downarrow}$) and the SOC couples the two
opposite spins in different valleys ($\ket{\textrm{v}_{\textrm{-}},\uparrow}$
and $\ket{\textrm{v}_{\textrm{+}},\downarrow}$) which eventually leads to an
opposite spins and same valley transition
($\ket{\textrm{v}_{\textrm{-}},\uparrow}$ to
$\ket{\textrm{v}_{\textrm{-}},\downarrow}$) [34].

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 $B_{z}$
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. Though, we do not observe a clear sweetspot in quality factor as
was proposed by Huang and Xu [29]. We therefore need to understand further the
noise spectrum of the qubit and in particular the influence of charge noise.

Figure 4:  (a) Rabi oscillations at different magnetic fields for constant
drive amplitude ($V_{AC}=$0.7\text{\,}\mathrm{mV}$$. (b) Rabi frequency and
relaxation rate as a function of magnetic field. The relaxation rate indicates
the presence of a hotspot close to $400\text{\,}\mathrm{mT}$. The Rabi
frequency follows the same trend as the relaxation rate by increasing in the
vicinity of this hotspot. Inset: colormap of Rabi oscillations as a function
of field.

##  VI Probing high frequency charge noise using dynamical decoupling

To probe noise at high frequency, we implement standard Carr-Purcell-Meiboom-
Gill (CPMG) pulse sequences, which consist in applying multiple $\pi$ pulses
instead of just one echo pulse in a given evolution time [35, 36, 37]. Such a
pulse sequence is characterized by the number of $\pi$ pulses $n_{\pi}$, the
total free evolution time $\tau$, and the duration of a $\pi$ pulse $t_{\pi}$.
The total time of one CPMG experiment is then given by
$t=\tau+n_{\pi}t_{\pi}$. The amplitudes obtained after a final refocusing
$\pi/2$ pulse are summarized in Fig.5(a). For $n_{\pi}$ $\pi$ pulses, the echo
amplitude $A(t)$ is fitted using the following equation [38]:

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

Where $T_{2}^{n_{\pi}}$ is the characteristic CPMG time for information loss,
for a given $n_{\pi}$. The parameter $\alpha$ is the noise color.

Another way to represent this dependence is to plot the CPMG-enhanced
$T_{2}^{n_{\pi}}$ with respect to $n_{\pi}$, see Fig5(b). 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 $\alpha=0.8$ over the frequency
range of CPMG pulse repetitions ($10^{4}-10^{6}$Hz). This is slightly
different from typical $\alpha=1$ in semiconductor structures [11, 39],
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 $n_{\pi}$ of $\pi$ pulses is limited due to the poor gate fidelity
induced by the large decoherence rate (see Suppl. Mat.).

Figure 5:  (a) Normalized echo amplitude as a function of total free evolution
time using CPMG sequences. These sequences are composed of a series of N
$\pi_{y}$ pulses, as depicted on the top of the figure for N=4, and finished
with a $\pi_{x}/2$ probe at different timings $\tau+\delta\tau$, where
$\delta\tau$ is swept to capture the whole echo. For each CPMG the echo is
renormalized using the reference echo amplitude for $\tau$=
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The different decay curves are
fitted using the following expression
$A(t)=exp{[-\left(t/T_{2}\right)^{\beta}]}$ with $\beta=1+\alpha$. (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${}_{2}(n_{\pi})\propto n_{\pi}^{\frac{\alpha}{1+\alpha}}$. (b) 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. (c) 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.

A complementary analysis of CPMG data consists in considering these pulse
sequences as tools for noise spectroscopy [11, 39]. 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 $\chi$ can be expressed as:

$\chi(t)=4\pi^{2}\int_{0}^{+\infty}dfS^{\prime}_{f_{L}}(f)\times
W(f,\tau,t_{\pi},n_{\pi})$ (2)

Suppl. Mat. gives details on this expression and on this section, following
closely Connors et al. [39]. In this expression, $S^{\prime}_{f_{L}}(f)$ is
the double-sided power spectral density (PSD) of the noise on the Larmor
frequency, and $W(f,\tau,t_{\pi},n_{\pi})$ is the spectral weighting function
corresponding to a specific pulse sequence. The normalized amplitude
$A^{n_{\pi}}(\tau)$ plotted in 5(a) 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
$S_{f_{L}}(f)=2S^{\prime}_{f_{L}}(f)=C/f^{\alpha}$, with $\alpha=0.8$ 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 $n_{\pi}$, 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 $f=n_{\pi}D/2\tau$, where
$D=\tau/(\tau+n_{\pi}t_{\pi})$ is the duty cycle of the pulse sequence. As a
set of CPMG experiments at fixed $n_{\pi}$ is characterized by the time
$T_{2}^{n_{\pi}}$, we chose to plot
$S_{f_{L}}(f^{n\pi}=n_{\pi}D/2T_{2}^{n_{\pi}})=C^{n_{\pi}}/(f^{n_{\pi}})^{\alpha}$
in Fig.5(c).

To confirm the origin of charge noise as the main source of decoherence at
high frequency, we now turn the PSD $S_{f_{L}}$ into a PSD $S_{v}$ 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 $\frac{dB_{z}}{dz}$. It summarizes with the following
expression:

$S_{f}=\frac{g\mu_{\text{B}}}{h}\times\frac{dB_{z}}{dz}\times\delta_{e}\times
S_{v}$ (4)

Where $h$ is the Planck constant and $\delta_{e}$ is the electron displacement
along $z$ due to gate voltage on $G_{\text{QD}}$. This displacement per
voltage can be extracted from the Rabi oscillations assuming:

$\delta_{e}=\frac{f_{\text{R}}}{V_{\text{AC}}\gamma\frac{dB_{x}}{dz}}$ (5)

with $V_{\text{AC}}$ the excitation voltage to drive Rabi oscillations at
frequency $f_{\text{R}}$ and $\frac{dB_{x}}{dz}$ the transverse gradient.
Using equation (5) and a Rabi frequency of
$f_{\text{R}}=$1\text{\,}\mathrm{MHz}$$ for
$V_{\text{AC}}=$1\text{\,}\mathrm{mV}$$, we obtain an electron displacement of
$$45\text{\,}\mathrm{pm}$/$\text{\,}\mathrm{mV}$$. Using equation (4) we plot
in Fig.5(c) 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.
5(d) at $1\text{\,}\mathrm{Hz}$ ($$1000\text{\,}\mathrm{\SIUnitSymbolMicro
V}$^{2}/$\text{\,}\mathrm{Hz}$$) to a PSD in terms of chemical potential using
the lever arm ($$0.25\text{\,}\mathrm{eV}$/$\text{\,}\mathrm{V}$$). We obtain
$$8\text{\,}\mathrm{\SIUnitSymbolMicro eV}$/$\text{\,}\mathrm{Hz}^{1/2}$$.
This value is one to two orders of magnitude higher compared to commonly
reported values in literature,[5, 39, 40] but similar to foundry fabricated
spin qubits [41]. 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
($\frac{dr}{dV}=$45\text{\,}\mathrm{pm}$/$\text{\,}\mathrm{mV}$)$ due to
confinement in the corner of the nanowire.

##  VII Conclusion

In conclusion, we have seen that a post-CMOS process can be used to implement
new 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 electric dipole magnetic resonance mechanisms: the one based on
bare synthetic SOC and the one induced by combining the spin-valley mixing
near the valley hotspot with the synthetic SOC. We further investigate the
coherent properties of the spin qubit which shows a decoherence time around
$500\text{\,}\mathrm{ns}$ due to hyperfine coupling to surrounding nuclear
spins. Refocusing this low frequency noise through dynamical decoupling pulse
sequences allowed to enhance the qubit coherence time by three orders of
magnitude, and to provide 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 charge noise by two orders of magnitude is at reach and
would contribute to CMOS-based spin qubits promising prospects.

##  VIII Acknowledgment

We acknowledge technical support from L. Hutin, D. Lepoittevin, I. Pheng, T.
Crozes, L. Del Rey, D. Dufeu, J. Jarreau, C. Hoarau and C. Guttin. We thank S.
De Franceschi and R. Maurand for fruitful discussions and I. De Moraes and N.
Dempsey for help with micromagnet fabrication. B.K., D.J.N. acknowledges the
GreQuE doctoral programs (grant agreement No.754303). The device fabrication
is funded through the Mosquito project (Grant agreement No.688539). This work
is supported by the Agence Nationale de la Recherche through the CRYMCO and
the PEPR PRESQUILE project. This project receives as well funding from the
project QuCube (Grant agreement No.810504) and the project QLSI (Grant
agreement No.951852).

##  Appendix A Appendix 1 - Micromagnet minor hysteresis loop

By taking measurements of the Larmor frequency at different magnetic fields,
the hysteresis of the micromagnet can be measured and its magnetisation can be
estimated. Fig. 6 a) shows measurements of the Larmor frequency for the
magnetic field being ramped up (green) and ramped down (blue). Taking into
account a g-factor of 2, the magnetisation of the micromagnet can be estimated
using eqn. (6). The magnetisation B${}_{\mu\text{-magent}}$ is the difference
between the external magnetic field B${}_{\text{ext,z}}$ and the expected
field from the Larmor frequency. $f_{\text{L}}$ is the Larmor frequency, h is
Planck’s constant and $\mu_{\text{B}}$ is the Bohr magneton.

$B_{\mu\text{-magnet}}=\frac{hf_{\text{L}}}{g\mu_{\text{B}}}-B_{\text{ext,z}}$
(6)

Due to the position of the coldfinger in our cryostat, the magnetic field felt
by the device is reduced compared to the field at the center of the magnet.
Therefore, the external magnetic field is adjusted such that a physical
hysteresis loop is formed. For an increasing external magnetic field the
magnetization needs to remain constant or has to increase. The configuration
where the magnetisation remains constant is plotted in Fig. 6 b). The field
generated by the micromagent is estimated to be $167\text{\,}\mathrm{mT}$.
This result represents a lower bound for the magnetisation of the micromagnet.
As a hard magnet behavior is expected for FeCo at cryogenic temperatures
[tbf], the choice of a constant magnetisation and a large hysteresis loop is
justified.

Figure 6: a) Larmor frequencies plotted for different magnetic fields. b)
Micromagnet magnetization depending on the external magnetic field. In both
panels, the magnetic field is lowered monotonously for the blue points and it
is increased monotonously for the green points.

##  Appendix B Appendix 2 - Micromagnetic simulations

In order to simulate the magnetic field created by the micromagnet, we follow
the development of ref.[42]. We consider perfect parallelepiped-shaped
magnets, as the impact of edge fluctuations on the magnetic field felt by the
qubit is negligible. We also consider that the magnets have uniform
magnetization along the $z$-axis.

The magnetic field produced by a single magnetic dipole reads:

$B_{\text{dip}}(r,\theta)=\frac{\mu_{0}}{4\pi}\times\frac{m}{r^{3}}\times\left(2\cos(\theta)\overrightarrow{r}+\sin(\theta)\overrightarrow{\theta}\right)$
(7)

with $\overrightarrow{r}$ and $\overrightarrow{\theta}$ the usual spherical
unit vectors. Integrating this formula over space yields the following
expression for the magnetic field created by a slab of magnetic material:

$\displaystyle B_{\alpha}(x,y,z)$
$\displaystyle=\frac{\mu_{0}m}{4\pi}\sum_{i,j,k=0}^{1}(-1)^{i+j+k}\times$ (8)
$\displaystyle\quad
G_{\alpha}\left(x,y,z,(-1)^{i}\frac{W}{2},(-1)^{j}\frac{D}{2},(-1)^{k}\frac{L}{2}\right)$
(9)

with $\alpha$ indicating the $B_{x},B_{y},B_{z}$ component of the field,
$(x,y,z)$ the Cartesian coordinates of some point outside the magnetic slab
expressed in the frame with origin at the center of the magnet, $m$ the
saturation magnetization, $(W,D,L)$ the dimensions of the slab along
$(x,y,z)$, and $G_{\alpha}$’s the functions defined as follows:

$\displaystyle G_{x}(x,y,z,x^{\prime},y^{\prime},z^{\prime})$
$\displaystyle=-\text{asinh}\left(\frac{y-y^{\prime}}{\sqrt{(x-x^{\prime})^{2}+(z-z^{\prime})^{2}}}\right)$
(10) $\displaystyle G_{y}(x,y,z,x^{\prime},y^{\prime},z^{\prime})$
$\displaystyle=-\text{asinh}\left(\frac{x-x^{\prime}}{\sqrt{(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}}\right)$
(11) $\displaystyle G_{z}(x,y,z,x^{\prime},y^{\prime},z^{\prime})$
$\displaystyle=$ $\displaystyle\frac{1}{2}\text{atan}$
$\displaystyle\left(\frac{2(z-z^{\prime})(y-y^{\prime})(x-x^{\prime})}{\sqrt{(z-z^{\prime})^{2}r^{2}-(x-x^{\prime})^{2}(z-z^{\prime})^{2}}}\right)$
(12)

with $r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}$. In
Fig. 7(a) and 7(b), we show 2D maps of the transverse gradient used for EDSR,
and the longitudinal dephasing gradient respectively. The red dashed line
shows the position of the cut where 1D plots of Fig. 1(c) are taken from.

The saturation magnetization of CoFe highly depends on the slab history and
stoichiometry. We adjust it thanks equation (6) from previous section, based
on Larmor spectroscopy.

Figure 7: EDSR gradient map

##  Appendix C Appendix 3 - Spin relaxation

The spin-valley relaxation hotspot, as well as the overall magnetic field
dependence of the spin relaxation rate is modeled by adding up the
contributions of spin-valley (SV) and spin-orbit (SO) couplings for both
Johnson–Nyquist (JN) noise and phonons (PH) following Ref. [27].

$\frac{1}{T_{1}}=\Gamma_{SV,JN}+\Gamma_{SV,Ph}+\Gamma_{SO,JN}+\Gamma_{SO,Ph}$
(13)

The relaxation rate induced by spin-valley coupling is described by eqns. 14
and 15 and corresponds to the pink curve in Fig. 2 (c). With $C_{i,j}$ (
$i\in\\{SV,SO\\}$, $j\in\\{JN,Ph\\}$) being the respective coupling strengths,
$E_{\textrm{Z}}$ the Zeeman energy, $E_{\textrm{0}}$ an arbitrary energy
reference, $k_{\textrm{b}}$ the Boltzmann constant and $T$ the temperature. F
is a function of the valley splitting E${}_{\textrm{VS}}$ and the spin-valley
gap $\Delta$. It is responsible for modeling the spin-valley hot spot. The
valley hotspot occurs due to the anticrossing of the
$\ket{\textrm{v}_{\textrm{-}},\uparrow}$,
$\ket{\textrm{v}_{\textrm{+}},\downarrow}$ states. The valley splitting
E${}_{\textrm{VS}}$ determines the B field at which it appears. $\Delta$ is
the spin-valley gap at the anti-crossing, and consequently defines the width
of the relaxation peak.
As $E_{\textrm{Z}}$ is proportional to the magnetic field, the global
dependence of each rate is given by the exponent of
$\left(\frac{E_{\textrm{Z}}}{E_{\textrm{0}}}\right)$. The Johnson–Nyquist
noise contribution is linear while the phonon noise contribution scales with
the power of five with $B$. However, far from the hotspot the contribution of
spin-valley coupling is negligible as the spin valley effects strongly
decrease.

$\displaystyle\Gamma_{SV,JN}$
$\displaystyle=C_{SV,JN}^{2}\left(\frac{E_{Z}}{E_{0}}\right)\frac{\exp(\frac{E_{Z}}{k_{b}T})+1}{\exp(\frac{E_{Z}}{k_{b}T})-1}F\left(E_{VS},\Delta\right)$
(14) $\displaystyle\Gamma_{SV,Ph}$
$\displaystyle=C_{SV,Ph}^{2}\left(\frac{E_{Z}}{E_{0}}\right)^{5}\frac{\exp(\frac{E_{Z}}{k_{b}T})+1}{\exp(\frac{E_{Z}}{k_{b}T})-1}F\left(E_{VS},\Delta\right)$
(15) $\displaystyle F\left(E_{VS},\Delta_{VS}\right)$
$\displaystyle=\frac{2\Delta_{SV}^{2}(E_{2}-E_{1})}{(E^{1}_{1}+\Delta_{SV}^{2})(E^{2}_{1}+\Delta_{SV}^{2})}\qquad\text{if
}E_{VS}\geq E_{Z}$ (16) $\displaystyle F\left(E_{VS},\Delta_{VS}\right)$
$\displaystyle=\frac{2(E_{1}E_{2}+\Delta_{SV}^{2})^{2}}{(E^{1}_{1}+\Delta_{SV}^{2})(E^{2}_{1}+\Delta_{SV}^{2})}\qquad\text{if
}E_{VS}<E_{Z}$ (17) $\displaystyle E_{1}$
$\displaystyle=E_{VS}-E_{Z}+\sqrt{(E_{VS}-E_{Z})^{2}+\Delta_{SV}^{2}}$ (18)
$\displaystyle E_{2}$
$\displaystyle=E_{VS}+E_{Z}+\sqrt{(E_{VS}+E_{Z})^{2}+\Delta_{SV}^{2}}$ (19)

The spin-orbit coupling is described by eqns. 20 and 21 and corresponds to the
purple curve. The Johnson–Nyquist noise contribution scales with the power of
three while the phonon noise contribution scales with the power of seven with
B.

$\displaystyle\Gamma_{SO,JN}$
$\displaystyle=C_{SO,JN}^{2}\left(\frac{E_{Z}}{E_{0}}\right)^{3}\frac{\exp(\frac{E_{Z}}{k_{b}T})+1}{\exp(\frac{E_{Z}}{k_{b}T})-1}$
(20) $\displaystyle\Gamma_{SO,Ph}$
$\displaystyle=C_{SO,Ph}^{2}\left(\frac{E_{Z}}{E_{0}}\right)^{7}\frac{\exp(\frac{E_{Z}}{k_{b}T})+1}{\exp(\frac{E_{Z}}{k_{b}T})-1}$
(21)

The fits show that Johnson–Nyquist noise dominates over phonons in almost the
entire range of B field, as can be seen in the red curve in Fig. 2(c). In
agreement with Ref. [20] phonons start to become important at
$2.5\text{\,}\mathrm{T}$, as can be seen in the blue curve, where only the
phonon contribution is plotted.

##  Appendix D Appendix 4 - $\pi$ pulse fidelity

The fidelity of pulses for the CPMG sequence is estimated by probing the
evolution of the echo amplitude with the number of pulses. The time delay
between each pulses is set to $2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to
allow the refocusing of quasistatic noise see fig. 5(a).

Figure 8:  Echo amplitude as a function of the number of pulses $n_{\pi}$.

##  Appendix E Appendix 5 - Hahn-Echo

The simplest dynamical decoupling pulse sequence implemented here to refocus
low frequency noise is the Hahn-Echo sequence. It consists in 1) Preparing the
electron spin in its down state 2) Applying a $\pi/2$ pulse along the $X$ axis
of the Bloch sphere to have a balanced superposition of up and down state with
known phase 3) Waiting a variable time $\tau_{\text{wait}}$: the qubit slowly
loses phase information 4) Applying a $\pi$ pulse along $X$ to refocus phase
noise that accumulated slower that $\tau_{\text{wait}}$ 5) Waiting
$\tau_{\text{wait}}+\Delta\tau_{\text{wait}}$ 6) Applying a $\pi/2$ pulse
along $X$ again 7) Measure the spin state.

The $\Delta\tau_{\text{wait}}$ added in the last waiting time enables to
measure an echo feature [43]. If we apply the last $\pi/2$ pulse after exactly
$\tau_{\text{wait}}$, we would measure the refocused signal, minus what has
been lost due to higher frequency noise. But measuring this echo amplitude for
variable $\Delta\tau_{\text{wait}}$ shows a broadening of the echo signal due
to intrinsic dephasing. We can then access $T_{2}^{*}$ by fitting the data to
the following expression :

$P_{\uparrow}(\Delta\tau_{\text{wait}})=B\times\exp\left[-\frac{1}{2}\left(\frac{\Delta\tau_{\text{wait}}}{C}\right)^{2}\right]+D$
(22)

$T_{2}^{*}$ is then given by the full width at half maximum of this Gaussian-
shaped curve: $T_{2}^{*}=2\sqrt{2\ln{2}}\times C$. In Fig. 11(a), we plotted
this echo for
$\tau_{\text{wait}}=3,10,35,$50\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. We
note that between $3\text{\,}\mathrm{\SIUnitSymbolMicro s}$ and
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$, the echo shows no visible
degradation. This is due to the SET signal level that varies between
experiment. This is why, for echo or CPMG experiments, before each shot of a
specific pulse sequence, we first measure a shot of a reference Hahn-Echo with
$\tau_{\text{wait}}=$3\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. The visibility
of this reference echo is used to renormalize the subsequent shot visibility.
In Fig.11(b), we plot the renormalized echo amplitude at
$\Delta\tau_{\text{wait}}=0$, for different waiting times
$\tau_{\text{wait}}$. The characteristic decay time of this curve is the Hahn-
Echo time, $T_{2}^{\text{E}}=$36\text{\,}\mathrm{\SIUnitSymbolMicro s}$$

Figure 9: Hahn-Echo final amplitude plotted for different waiting times
$\tau_{\text{wait}}$

##  Appendix F Appendix 6 - Analysis of CPMG data

Figure 10: (a) Spectral weight functions corresponding to CPMG sequences with
$t_{\pi}=$354\text{\,}\mathrm{ns}$$, and from left to right :
$\tau=30,96,$25.6\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and
$n_{\pi}=2,8,64$. (b) $C^{n_{\pi}}$ constants corresponding to assumed power
spectral density model $S_{f_{\text{L}}}(f)=C/f^{\alpha}$. Each point is an
average of these $C$’s computed from equation (F) at fixed $n_{\pi}$. (c)
Estimation of the error in numerical integration of relation (F), computed by
using the same numerical integration parameters for the integral of relation
(28) and comparing with expected result $\tau/2$.

In this section, we develop on how to get power spectral density (PSD) data
from CPMG experiments (main Fig.5), following closely [39]. The phase of the
qubit during an experiment containing a pulse sequence is given by:

$\phi(t)=2\pi\int_{-\infty}^{+\infty}dt^{\prime}f_{\text{L}}(t^{\prime})\times
y(t^{\prime},\tau,t_{\pi},n_{\pi})$ (23)

where $f_{\text{L}}$ is the Larmor frequency, and
$y(t^{\prime},\tau,t_{\pi},n_{\pi})$ describes the pulse sequence,
characterized by the total free evolution time $\tau$, the $\pi$ pulse time
$t_{\pi}$ and the number of pulses $n_{\pi}$.

We are interested in the time expectation value of the phase part of the qubit
state. If we assume that fluctuations in the Larmor frequency $f_{\text{L}}$
are Gaussian, fluctuations in the qubit phase $\phi(t)$ are Gaussian too, and
we can write:

$\langle\exp\left[i\phi(t)\right]\rangle\equiv\exp\left[-\chi(t)\right]=\exp\left[\frac{-\langle\phi(t)^{2}\rangle}{2}\right]$
(24)

The Wiener-Khinchin theorem states that the double-sided PSD is the Fourier
transform of the auto-correlation. Using this, and defining the spectral
weighting function from the Fourier transform of $y$:

$W(f,\tau,t_{\pi},n_{\pi})=|\hat{y}(f,\tau,t_{\pi},n_{\pi})|^{2},$ (25)

we finally get for the decay envelope:

$\chi(t)=4\pi^{2}\int_{0}^{+\infty}dfS^{\prime}_{f_{\text{L}}}(f)\times
W(f,\tau,t_{\pi},n_{\pi})$ (26)

The analytic expression of $W$ is given in ref.[44]. Supp. Fig.10(a) shows the
shape of $W$ for different pulse sequence parameters. We can see that for some
parameters, $W(f,\tau,t_{\pi},n_{\pi})$ is strongly peaked around
$f=n_{\pi}D/2\tau$ with $D=\tau/(\tau+n_{\pi}t_{\pi})$ is the duty cycle of
the pulse sequence. This explains why we can use CPMG experiments to probe the
qubit noise at specific frequencies. We also see that harmonics in $W$ can
become relevant, so that we can’t sum up the whole noise measured in a
specific CPMG experiment to the peak frequency. In order to analyze in the
same way all of our CPMG data, we proceed as follows. We assume the noise in
the system is described by a single-sided PSD of the form
$S_{f_{\text{L}}}(f)=C/f^{\alpha}$, where $\alpha$ has been fitted to 0.8 in
the main text. Plugging this PSD in equation (26), and remembering that
$\chi(t)$ is linked to measured CPMG amplitudes through
$\chi(t=\tau+n_{\pi}t_{\pi})=-\ln\left[A(\tau,t_{\pi},n_{\pi})\right]$, we
get:

$\displaystyle C=$
$\displaystyle-\ln\left(A(\tau,t_{\pi},n_{\pi})\right)\times$
$\displaystyle\left[4\pi^{2}\int_{0}^{+\infty}df\frac{W(f,\tau,t_{\pi},n_{\pi})}{f^{\alpha}}\right]^{-1}$
(27)

where we can compute the integral numerically. For this, we choose the upper
bound of integration so that it encompasses the first three relevant harmonics
of the spectral weight, given by odd multiples of the peak frequency. For each
$n_{\pi}$ we average the resulting $C$’s to get $C^{n\pi}$ values used in main
Fig.5. The $C^{n_{\pi}}$’s are plotted in Supp. Fig.10(b) and notably fall in
the same order of magnitude region. This suggests that the simple
$1/f^{\alpha}$ noise model is consistent on the relevant frequency range of
CPMG experiments.

The confidence in numerical integration is estimated thanks to the following
relation [39]:

$\int_{0}^{+\infty}dfW(f,\tau,t_{\pi},n_{\pi})=\frac{\tau}{2}$ (28)

For each computation of $C$, we computed this integral of $W$ using the same
integration parameters, with residual error coming mainly from the finite
upper bound for integration. We plot the relative error in Supp. Fig.10(c).

##  Appendix G Appendix 7 - Rabi time near the hotspot

We evaluate the evolution of the damping of Rabi oscillations as a function of
the magnetic field by fitting the Rabi oscillations presented in fig.3 (a).

Figure 11: Characteristic decay time of Rabi oscillations (black crosses) and
Rabi frequency (red dots) near the hotspot.

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