Abstract
We identify the dominant source for lowfrequency spin qubit splitting noise in a highly isotopicallypurified silicon device with an embedded nanomagnet and a spin echo decay time \({T}_{2}^{\,\text{echo}}\) = 128 µs. The power spectral density (PSD) of the charge noise explains both, the clear transition from a 1/f^{2} to a 1/fdependence of the splitting noise PSD as well as the experimental observation of a decreasing timeensemble spin dephasing time, from \({T}_{2}^{* }\approx\) 20 µs, with increasing measurement time over several hours. Despite their strong hyperfine contact interaction, the few ^{73}Ge nuclei overlapping with the quantum dot in the barrier do not limit \({T}_{2}^{* }\), likely because their dynamics is frozen on a few hours measurement scale. We conclude that charge noise and the design of the gradient magnetic field are the key to further improve the qubit fidelity in isotopically purified ^{28}Si/SiGe.
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Introduction
Gatedefined quantum dots (QDs) are a promising platform to confine and control single spins, which can be exploited as quantum bits (qubits)^{1}. Unlike charge, a single spin does not couple directly to electric noise. Dephasing is dominated by magnetic noise, typically from the nuclear spin bath overlapping with the QD^{2}. The use of silicon as a qubit host material boosted the control of individual spins by minimizing this magnetic noise: in addition to the intrinsically low hyperfine interaction in natural silicon, the existence of nuclear spinfree silicon isotopes, e.g., ^{28}Si, allows isotopical enrichment in crystals^{3,4}. Controlling individual electrons and spins in highly enriched ^{28}Si quantum structures^{5,6,7,8} then opens the door to an attractive spin qubit platform realized in a crystalline nuclear spin vacuum. Indeed, twoqubit gates^{9,10,11,12} have recently been demonstrated in natural and enriched quantum films, while isotopical purification of ^{28}Si down to 800 ppm of residual nuclear spincarrying ^{29}Si allowed to push manipulation fidelities beyond 99.9% for a single qubit^{13,14} and towards 98% for two qubits^{15}. Qubit manipulation of individual spins is currently either realized with local ac magnetic fields generated by a stripline to drive Rabi transitions^{6,7,16} or via artificial spin–orbit coupling engineered by a micromagnet integrated into the device. This latter approach is advantageous by allowing the control of spin qubits solely by local ac electric fields^{10,11,13,17}, permitting excellent local control and faster Rabi frequencies. At the same time, it opens a new dephasing channel for electric noise, due to the static longitudinal gradient magnetic field of the micromagnet, competing with the magnetic noise. To fully exploit the potential of magnetic noise minimization through isotope enrichment in ^{28}Si/SiGe, two experimental questions thus become relevant for devices with integrated static magnetic field gradients: Firstly, to what extent electronic noise impacts the spin qubit dephasing compared to magnetic noise^{18} and, secondly, which role the natural SiGe potential wall barriers play for dephasing, since the hyperfine interaction of bulk Ge exceeds the one of bulk Si by a factor of approximately 100^{19,20}.
Here, we present an electron spin qubit implemented in a highly isotopically purified ^{28}Si/SiGe device, the quantum well of which is grown with 60 ppm residual ^{29}Si source material. It includes a magnetic field gradient generated by a nanomagnet which is integrated into the electronconfining device plane. We use Ramsey fringe experiments to investigate the splitting noise spectrum of the single electron spin down to 10^{−5} Hz. We find the frequency dependence of the fluctuation spectrum of the qubit splitting to be identical to the spectrum of the device’s electric charge noise over more than 8 decades. At low frequencies, below 5⋅10^{−3} Hz, both noise spectra decrease with 1/f^{2}. Above, they transit to a 1/f dependence as deduced from a Hahnecho sequence for the detuning, yielding \({T}_{2}^{\,\text{echo}}\) = 128 µs. From our observations, we conclude that electric noise dominates our qubit dephasing in a broad frequency range. It is also responsible for the observed decrease of \({T}_{2}^{* }\) with increasing measurement time^{21}. The highfrequency behavior is strikingly comparable to a device^{13} which differs in its layout, magnet design, and heterostructure, including isotope purification. Interestingly, although we show the ^{73}Ge in the quantum welldefining natural SiGe to represent a potential limitation for our device, our experiments suggest the nuclear spin bath to be frozen on a time scale of hours and to much less contribute to \({T}_{2}^{* }\) than expected at the ergodic limit.
Results
The device used for all measurements consists of an undoped ^{28}Si/SiGe heterostructure, confining a twodimensional electron gas in ^{28}Si. Metal gates allow to form a quantum dot (QD) containing a single electron (Fig. 1a). The charge state of the QD is detected via a single electron transistor (SET) located at the righthand side of the device. The large gate labeled M on the lefthand side is a single domain cobalt nanomagnet. Its straymagnetic field provides a magnetic field gradient^{22} for spin driving by electric dipole spin resonance (EDSR). For details of the device see the “Methods” section and ref. ^{23}. We apply an external magnetic field of 668 mT along the xdirection.
Power spectral density of the qubit energy splitting noise
First, we focus on the power spectral density (PSD) of the frequency detuning Δf of the qubit with respect to a reference frequency of f_{R} = 19.9 GHz. Δf is determined by a Ramsey fringe measurement, during which the microwave pulses are detuned from the reference by Δf_{MW} (Fig. 1b). We vary Δf_{MW} from −1 to 1 MHz in 100 steps. Each point of the spinup probability P_{↑} is an average of over 100 singleshot measurements. One Ramsey fringe, which is one measurement of Δf, takes 120 s. We fit Δf by applying the formula for the fringe pattern^{24}:
where \(\Phi =\sqrt{\Delta {f}^{2}+{f}_{R}^{2}}\), t_{e} is the evolution time between the two π/2 gates, and \({t}_{\frac{\pi }{2}}\) is the execution time of the π/2 gate. A and B are constants related to the qubit initialization and readout fidelity.
Figure 1c displays Ramsey fringes recorded during a measurement time t of 67 h. The white line tracks Δf during the fulltime period. We calculated the PSD S(f) of the qubit detuning with Welch’s method (Fig. 1d). For frequencies below ≈7⋅10^{−4} Hz, we find a S(f) ∝ 1/f^{1.97} dependence. It transitions into a region with a smaller exponent, here fitted with S(f) ∝ 1/f^{1.48} (blue line in Fig. 1d). Note that the slow drift towards increasing Δf in Fig. 1c cannot be induced by a discharge of the magnet in persistent mode, which would lead to a decreased energy splitting. It only appears when manipulating the qubit and is captured at most in the two lowest frequencies in the PSD in Fig. 1d.
Timeaveraged spin dephasing time \({T}_{2}^{* }\)
Having analyzed the qubit splitting noise S(f) in the lowfrequency regime, we now investigate its impact on the timeensemble spin dephasing time \({T}_{2}^{* }\). Starting at a lab time t, we record P_{↑} during a series of Ramsey sequences with varying t_{e} (Fig. 2a) in every line. We average as many consecutive P_{↑}(t_{e}) lines of this dataset as required to reach a total measurement time t_{m}. The averaged P_{↑}(t_{e}) is fitted with
An example of P_{↑}(t_{e}) averaged from a bundle of P_{↑}(t_{e}) lines measured over t_{m} = 10 min, starting at lab time t ≈ 10 h, is shown in Fig. 2b. We extract \({T}_{2}^{* }\)(t_{m} = 10 min) = 18 μs. To achieve better statistics, this procedure was executed consecutively for different bundles, i.e., different lab times t. We choose to offset the bundles by 25 lines giving overlap between them. This results in each \({T}_{2}^{* }({t}_{m})\) value being averaged from 900 \({T}_{2}^{* }({t}_{m})\) values, using different line bundles from the dataset displayed in Fig. 2a and a second dataset not shown here. Figure 2c shows these averaged \({T}_{2}^{* }({t}_{m})\) for t_{m} ranging between 38 s and 6.3 h. Remarkably, \({T}_{2}^{* }({t}_{m})\) drops monotonously with increasing measurement time without saturating for long t_{m}, qualitatively matching the qubit splitting noise PSD S(f), which keeps increasing towards low frequencies (Fig. 1d). In a rough approximation, considering splitting noise of the type S(f) = S_{0}/f^{α} with α ≳ 1, \({T}_{2}^{* }({t}_{m})\) induced by splitting noise is (see “Methods”):
Fitting \({T}_{2}^{* }({t}_{m})\) with only one α_{all} = 1.48 (green solid line) shows a clear deviation from the data points (Fig. 2c). Motivated by the variation of α in S(f) observed in Fig. 1d, we fit two separate ranges of t_{m} above and below \({t}_{m}=25\ \min\) (that is 6.7⋅10^{−4} Hz, which is very close to the transition point 7⋅10^{−4} Hz found in Fig. 1d). These ranges are characterized by α_{1} = 1.30 (red dashed curve) and α_{2} = 1.73 (blue dashed curve), and are thus in good qualitative agreement with the exponents found for the splitting noise in Fig. 1d. The quantitative deviation of both α_{i} (i = 1, 2) determined by the \({T}_{2}^{* }({t}_{m})\) compared to the exponents of the fits to the PSD results from the fact that the \({T}_{2}^{* }\) measurement integrates over the PSD from \({t}_{m}^{1}\) to \({t}_{e}^{1}\).
Hahnecho spin dephasing time \({T}_{2}^{\,\text{echo}\,}\)
Our spindetection bandwidth in the Ramsey fringe experiment sets a limit on the maximum frequency of S(f) in Fig. 1d. To gain information on S(f) at a higher frequency, we perform a Hahnecho experiment, that extends the Ramsey control sequence by a π_{X} gate between the two (π/2)_{X} gates, in order to filter out lowfrequency noise. The measured data (Fig. 2d) is fitted with
We find α = 1.003 ± 0.071 and \({T}_{2}^{\,\text{echo}\,}=128\pm 1.9\) μs. We can deduce that S(f) ∝ 1/f at a frequency of approximately \(f=1/{T}_{2}^{\,\text{echo}\,}=7.8\) kHz, in line with the observations in a device with an onchip micromagnet and 800 ppm residual ^{29}Si for f > 10^{−2} Hz^{13}. With the lowfrequency PSD (Fig. 1d) and these spin echo results, we conclude that the initial S(f) ∝ 1/f^{2} dependence observed at low frequencies transits to a S(f) ∝ 1/f dependence around 7⋅10^{−4}–1⋅10^{−3} Hz. With the detection bandwidth limit set by the Ramsey fringe experiment at approximately 3⋅10^{−3} Hz, we observe this gradual transition in our study, explaining α = 1.48 found in Fig. 1d. Remarkably, we find a 28% higher \({T}_{2}^{\,\text{echo}\,}\) compared to the device with an onchip micromagnet and 800 ppm residual ^{29}Si^{13}, indicating that overall the splitting noise is lower in our sample in this regime.
Power spectral density of charge noise
In order to investigate the impact of charge noise, we measure the charge noise in the qubit vicinity via the current noise of the SET sensor. This current noise is translated into gate equivalent voltagenoise by the variation dI_{SET}/dV_{QS} of the SET current by the voltage applied to the SET gate QS (see Fig. 1a). We measure the current noise ϵ_{SET} at a highly sensitive operation point of the SET and subtract from its PSD the noise spectrum measured when the SET is set to be insensitive to charge noise from the device, in order to remove noise originating from the measurement circuit^{25}. Figure 3a shows the measured PSD of the SET noise. As the data reveals two slopes, we fit with \({S}_{C}(f)={S}_{C1}/{f}^{{\alpha }_{1}}+{S}_{C2}/{f}^{{\alpha }_{2}}\). The fitted exponents of the S_{C}(f) spectrum are α_{1} = 1 ± 0.02 and α_{2} = 2 ± 0.05, respectively, the transition being at about 10^{−3} Hz. This frequency dependence is in very good agreement with the one observed for the qubit detuning PSD in Fig. 1d and with the qualitative trend extended to high frequencies with the Hahnecho experiment. Note that spin qubits in GaAs which dephase dominantly due to hyperfine interaction^{26,27,28} are also characterized by a 1/f^{2} dependence in their lowfrequency splitting noise PSD, which has been assigned to nuclear spin diffusion there^{29,30}. As discussed in the next section, nuclear spin diffusion seems implausible in our structure. It seems more likely that the appearance of the lowfrequency α_{2} = 2 ± 0.05 is due to a high density of twolevel charge fluctuators with very low switching rates.
To provide a quantitative estimation, we assume the charge noise at the SET to be similar to the one of the QD and the longitudinal gradient magnetic field to be isotropic for lateral QD displacements. Using the current noise trace ϵ_{SET}(t), the resulting frequency detuning is
where \(\frac{\mathrm{d}{V}_{\mathrm{QS}}}{\mathrm{d}{I}_{\mathrm{SET}}}=\frac{1}{35}\) mV/pA is the inverse of the current change through the SET induced by a change of the voltage on the gate. \(\frac{\mathrm{d}{x}_{\mathrm{QD}}}{\mathrm{d}{V}_{\mathrm{QD}}}=0.024\) nm/mV is the estimated displacement of the QD induced by voltage changes on the adjacent gates, represented by gateequivalent uncorrelated chargenoise applied to gate pL deduced from an electrostatic device simulation. \(\frac{\mathrm{d}{B}_{x}}{\mathrm{d}{x}_{\mathrm{QD}}}=0.08\) mT/nm is the simulated isotropic longitudinal gradient magnetic field at the QD position. The factor \(\frac{g{\mu }_{\rm{B}}}{\hslash }\), containing the electron gfactor (g ≈ 2), the Bohr magneton μ_{B}, and the reduced Planck constant ℏ, converts magnetic field to frequency. We convert the charge noise PSD into qubit splitting noise by Eq. (5) (right yaxis in Fig. 3a). We find this noisepower to vary by a factor of three, depending on the assumptions on the gatetodot distance and the direction of displacement triggered by the gate, an error which is small on the scale covered in Fig. 3a. Comparing the experiment and the quantitative estimation, the lowfrequency part of the PSD (blue dots in Fig. 3a) shows excellent agreement with the qubit splitting noise PSD S(f) in its frequency dependence and its magnitude.
In order to also include the highfrequency range (red dots in Fig. 3a) into the comparison, we simulated the spinup probability after a Ramsey gate sequence with evolution time t_{e} using
and included quasistatic noise during the free evolution time t_{e} from the full PSD in Fig. 3a. The simulated data points (black dots in Fig. 3b) yield \({T}_{2}^{* }=21\) μs for a measurement time \({t}_{m}=10\ \min\). Note that this estimated \({T}_{2}^{* }\) value is surprisingly close to the experimentally determined value \({T}_{2}^{* }=18\) μs found in Fig. 2b, given that the abovementioned accuracy of the qubit energysplitting noisepower estimation enters with \(\sqrt{3}\) into the estimation of \({T}_{2}^{* }\).
In summary, comparing data covering more than 8 frequency decades, the excellent agreement demonstrates that charge noise dominates the qubit splitting noise in our device and transits from a S(f) ∝ 1/f^{2} dependence to a S(f) ∝ 1/f dependence around 10^{−3} Hz.
Contact hyperfine interaction analysis
To complete our analysis of the splitting noise and the timeensemble spin dephasing time \({T}_{2}^{* }\), we estimate the magnetic noise impact due to the residual nonzero spin nuclei in our device. We can compute the resulting \({T}_{2}^{* }\) with (see “Methods”)
where N_{S} is the number of nuclei, p is the fraction of nuclei with finite nuclear spin, γ is the volume fraction of the wavefunction for which we want to calculate the influence on \({T}_{2}^{* }\), i.e., localized in the barrier or the quantum well. A is the hyperfine coupling constant per nucleus and I is the nonzero nuclear spin. In ref. ^{23}, we measured the orbital splitting of this QD to be 2.5 meV. Assuming a harmonic potential, we calculate the size of the QD, taken to be the fullwidthathalfmaximum of the ground state wavefunction. This yields a radius of ≈13 nm. By approximating the QD as a cylinder with height 6 nm we estimate the number of atoms in the QD volume to be N_{A} = 1.6⋅10^{5}. From Schrödinger–Poisson simulations, we estimate the overlap with the SiGe barriers to be γ_{B} ≈ 0.1%. We calculate the number of nonzero nuclear spins, which are relevant for the hyperfine coupling with the qubit (i.e., are within the cylindrical volume assigned to the QD), for the minimal residual concentration of 60 ppm ^{29}Si in the ^{28}Si strained QW layer, residual ^{29}Si and ^{73}Ge in the SiGe barriers with natural abundance of isotopes as, respectively:
The coupling constants are A_{Si} = 2.15 μeV and A_{Ge} ≈ 10⋅A_{Si}^{19,20}, respectively, with \({I}_{\text{}{}^{29}\text{Si}}=\frac{1}{2}\), \({I}_{\text{}{}^{73}\text{Ge}}=\frac{9}{2}\). Assuming the spin baths to be in the ergodic limit, each subset of nuclear spin results in the following dephasing times: \({T}_{2}^{* }\)(\({}_{{}^{29}Si}^{\,\text{QW}\,}\)) = 22 μs, \({T}_{2}^{* }\)(\({}_{{}^{29}Si}^{\,\text{barrier}\,}\)) = 30 μs, and \({T}_{2}^{* }\)(\({}_{{}^{73}Ge}^{\,\text{barrier}\,}\)) = 0.61 μs.
At first sight, it seems striking that \({T}_{2}^{* }\)(\({}_{{}^{29}Si}^{\,\text{QW}\,}\)) coincides with our experimental observation, but this only holds true for the shortest t_{m} in Fig. 2c. Attributing the dephasing solely to ^{29}Si would require the dephasing channel due to ^{73}Ge nuclei to be negligible and imply the implausible conclusion that the experimental matching of the PSDs of the magnetic and the charge noise is a simple coincidence. Additionally, ref. ^{13} reports the same \({T}_{2}^{* }\) for the same measurement time in a heterostructure with 800 ppm ^{29}Si. The observed S(f) ∝ 1/f^{2} in Fig. 1d by itself could be the highfrequency tail of nuclear PSD (as observed for GaAs) for nuclei having extremely long correlation time. But this is extremely unlikely considering the total magnitude of the PSD. If the S(f) ∝ 1/f^{2} region was dominated by ^{29}Si, it would imply the ergodic limit of \({T}_{2}^{* }\) to be reached for t_{m} even longer than the highest values reported here. But then, according to Fig. 2c, the nuclearinduced \({T}_{2}^{* }\) in the ergodic limit would be at most a few microseconds, in disagreement with >20 μs calculated for ^{29}Si. This leads us to rule out a dominant dephasing through ^{29}Si.
Notably, due to the strong hyperfine coupling of the ^{73}Ge in the barrier layers, the ^{73}Ge alone would dephase the qubit faster than observed in the experiment shown in Fig. 2c. This apparent contradiction would be resolved if the correlation time of the ^{73}Ge nuclear spin bath is larger than our measurement time of 6 h and thus the ergodic limit is not reached in our \({T}_{2}^{* }\)(t_{m}) measurement. The gradient magnetic field of the nanomagnet and the presence of the electron’s Knight shift^{31} may contribute to this effect of slowing down the nuclear spin diffusion.
Discussion
We have shown that in a highly purified ^{28}Si/SiGe qubit device grown from 60 ppm residual ^{29}Si source material, the natural abundance of ^{73}Ge nuclear spins in the potential barrier does not seem to dominantly contribute to the qubit dephasing time, despite their strong hyperfine coupling. One can thus conclude that, in this scenario, the improvement potential of qubit dephasing times that can be expected from an isotopical purification of the natural SiGe barrier of the heterostructure will be negligibly small.
In our device featuring a nanomagnet integrated into the gate layout for EDSR manipulation, we demonstrate charge noise to be the dominant qubit noise source in a frequency range of more than 8 decades by comparing the qubit energy splitting noise to the charge noise of the sensor SET. In the lowfrequency regime, the charge and the qubit splitting noise present an unexpected 1/f^{2} dependence below 1⋅10^{−3} Hz. Above, towards higher frequencies, both PSD transit to a 1/f dependence. This 1/f trend was recently also observed in a different type of device, featuring a micromagnet and 800 ppm ^{29}Si^{13}. From the Hahnecho experiment for our qubit detuning, we additionally deduce a remarkably high \({T}_{2}^{\,\text{echo}\,}=128\) μs. In accordance with the absence of a rolloff in the charge noise PSD down to at least 5⋅10^{−5} Hz, we also show \({T}_{2}^{* }\) to clearly and monotonously decrease for the measurement time which we increased from seconds to several hours. Our experimental \({T}_{2}^{* }\approx 18\) μs for a measurement time t_{m} = 600 s quantitatively results from the charge noise \({S}_{C}^{1/2}(1\,\text{Hz}\,)=0.47\) μeV/\(\sqrt{\,\text{Hz}\,}\), which falls within the range of 0.3–2 μeV/\(\sqrt{\,\text{Hz}\,}\) seen in literature^{32,33,34,35}. While the onchip integration of a micro or nanomagnet does not induce additional magnetic noise^{13,36}, minimizing the newly opened electric dephasing channel seems to be key for further significant improvement of spin qubit gate fidelities in highly purified ^{28}Si, compared to devices avoiding integrated static magnetic field gradients^{14,15,37,38}.
Methods
Device
The device studied in this work is fabricated with an undoped ^{28}Si/SiGe heterostructure^{23}. The layer structure is grown on a Siwafer by means of a solid source molecular beam epitaxy. A relaxed virtual substrate consisting of a graded buffer up to a composition Si_{0.7}Ge_{0.3} followed by a layer of constant composition Si_{0.7}Ge_{0.3} provides the basis for a 12 nm ^{28}Si quantum well (QW) grown using a source material of isotopically purified ^{28}Si with 60 ppm of remaining ^{29}Si. The QW is separated from the interface by a 45 nm Si_{0.7}Ge_{0.3} cap, which is protected from oxidation by a 1.5 nm Si cap.
A layer of 20 nm Al_{2}O_{3} grown by atomic layer deposition insulates the depletion gate layer depicted in Fig. 1a and the underlying heterostructure. The depletion gates are fabricated by means of electron beam lithography. A Co nanomagnet, colored green in Fig. 1a, is added to the depletion gate layer in order to provide a local magnetic field gradient for electric dipole spin resonance (EDSR). A second gate layer, insulated from the depletion gates by 80 nm of Al_{2}O_{3}, is used to induce a twodimensional electron gas in the QW via the field effect and provide reservoirs for the dotdefining and charge sensing parts of the device.
Setup
The device was measured in an Oxford Triton dilution refrigerator at a base temperature of 40 mK. All dc lines are heavily filtered using pifilters (f_{c} = 5 MHz) at room temperature followed by copperpowder filters and a secondorder RC lowpass filter at base temperature. The RC filter cutoff frequency is f_{c} = 10 kHz for the electron reservoirs and gates that are used for fast control. All other gates have a RC lowpass filter cutoff frequency of ~0.68 kHz. The copperpowder filter has an attenuation of 60 and 80 dB at 3 and 12 GHz, respectively. The electron temperature is 114 mK. Voltage pulses are applied via a Tabor Electronics WX2184C AWG. MW manipulation bursts for control are provided by a Rohde & Schwarz SMW200A signal source. MW signals can be added to the pL gate via a resistive biastee. The sensor signal for readout is amplified with a Basel high stability I–V converter SP983C at roomtemperature. The resulting voltage signal is digitized with an AlazarTech ATS9440 digitizer card.
Derivation of the measurement time dependence of \({T}_{2}^{* }\)
Using the qubit coherence function W(t_{e}) as defined in ref. ^{39}, we can approximate the dephasing due to splitting noise for noise models of the type S(f) = S_{0}/f^{α} with α ≳ 1 as
For low frequencies, \({F}_{\mathrm{FID}}/{f}^{2}\approx \frac{1}{2}{t}_{e}^{2}\) and thus for measurement times t_{m} ≫ t_{e}
When plugged into the expression for W(t_{e}), this gives us the relation
Thus, we get an expression for the dephasing time constant
Derivation of the nulcear bath induced \({T}_{2}^{* }\) formula
The wavefunction of the confined electron is given in the envelope function approximation by
where F(r) is the slowly varying envelope function and u(r) is the periodic part of the Bloch function. The envelope function F(r) is normalized to the volume of the primitive unit cell of silicon
where the volume of the primitive unit cell is
in which a is the lattice constant of Si. Here we have taken into account that in a cubic unit cell we have eight atoms while in the primitive unit cell of a diamondtype lattice we have two atoms. So the volume of the primitive cell is 1/4 of the volume of the cubic unit cell. Since the envelope in the nth primitive unit cell (PUC) is approximately constant, given by F_{n}, we have
Thus,
Since the whole wavefunction is normalized to unity we can write
from which we get
meaning ν_{0}∣u(r_{k})∣^{2} is a dimensionless quantity (denoted by η in ref. ^{2})
The hyperfine (hf) coupling Hamiltonian is
where k labels the spinful atoms and A_{k} is the hf coupling to the kth nucleus. Note that at magnetic fields along the zaxis being much larger than typical Overhauser fields from all the nuclei, we can neglect the transversal couplings in the above Hamiltonian and just consider S^{z}I^{z} interaction.
The hf coupling with a nuclear spin located at r_{k} is given by
where \({\mathcal{A}}\) is the hf coupling energy of one nucleus. It is related to gyromagnetic factors of the electron and nucleus and to the amplitude of the Bloch function at the nuclear site ∣u(r)∣^{2 }^{2}.
Let us note that due to the normalization of F discussed above the sum over all the A_{k} is
where p is the fraction of nuclei with a nuclear spin, e.g., p = 0.049 for natural Si. The factor of 2 comes from the fact that we have two atoms per PUC. For Silicon \({\mathcal{A}}\approx 2.15\) μeV
The rms value of the Overhauser field for unpolarized nuclei is given by^{28}
We now define the number of primitive unit cells encompassed by the wavefunction as^{27}
from which we get that
Here γ is the volume fraction of the electron wavefunction, in which the nuclei are positioned, for which we calculate the influence on \({T}_{2}^{* }\) of the electron spin. From this we get
We can rewrite this formula using the number of spins in a QD, N_{S} = 2pγN
Measurement cycle and magnetic field gradient
The measurement cycle (Fig. 4a) starts with a 1 ms long Ramsey pulse sequence, during which two \({(\frac{\pi }{2})}_{x}\) qubitgates separated by the evolution time t_{e} are generated by applying microwave pulses to gate pL (see Fig. 1a). The microwave pulses, which can be detuned from spin resonance by Δf_{MW}, displace the QD and thus drive EDSR in Coulomb blockade. The Ramsey sequence is followed by a measurement voltage pulse applied to gate T, during which the electron spin state projected on the external magnetic field direction is detected by spindependent tunneling to the reservoir and the QD is initialized in the spin ground state. This sequence containing initialization, control, and singleshot readout is repeated 100 times, with either the frequency Δf_{MW} or the evolution time t_{e} varied in each Ramsey sequence. This sequence is looped five times and followed by a pulse sequence designed to calibrate the chemical potential μ_{QD} of the QD, the tunnelcoupling to the reservoir t_{c}, and the operation point of the SET. The calibration sequence starts with 40 voltage ramps applied to gate T (20 up and 20 down) crossing the 0–1 charge transition and thus calibrating μ_{QD} of the spin groundstate as a function of V_{T}. During the second part of the calibration pulse, V_{T} is held constant for 20 ms and the time for loading an electron on the QD in its spinground state is observed, in order to update t_{c} and the sensitive operation point of the SET by averaging several calibration sequences. The whole initialization, Ramsey, spindetection, and calibration sequence is repeated N times.
Simulations of the longitudinal gradient magnetic field generated by the Co nanomagnet in the plane of the Si/SiGe quantum well are displayed in Fig. 4b.
Data availability
The data sets generated and/or analyzed during this study are available from the corresponding author upon reasonable request.
Change history
12 February 2021
A Correction to this paper has been published: https://doi.org/10.1038/s41534020002987
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Acknowledgements
The authors thank R. Huber and D. Weiss for technical support. This work has been funded by the German Research Foundation (DFG) within the project BO 3140/41 and under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1—390534769. This work has also been funded by the National Science Centre, Poland under QuantERA programme, Grant No. 2017/25/Z/ST3/03044. Project SiQuBus received funding from the QuantERA ERANET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme.
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L.R.S. and D.B. conceived and supervised the study. T.S. set up the experiment, performing the measurements and data analysis with A.H. and O.F., supported by L.R.S., A.S., F.S., and D.B. H.R. and N.V.A. fabricated the ^{28}Si source material. F.S., A.S., and D.B. grew the heterostructure, fabricated the DQD device with support from K.S., and conducted the device simulation. L.R.S. and L.C. analyzed the contact hyperfine interaction with support from T.S. and O.F. All authors discussed the results. D.B., L.R.S., T.S., F.S., O.F., A.H., A.S., and L.C. wrote the manuscript.
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Struck, T., Hollmann, A., Schauer, F. et al. Lowfrequency spin qubit energy splitting noise in highly purified ^{28}Si/SiGe. npj Quantum Inf 6, 40 (2020). https://doi.org/10.1038/s4153402002762
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DOI: https://doi.org/10.1038/s4153402002762
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