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L. Banszerus, K. Hecker, S. Möller, E. Icking, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), C. Volk, C. Stampfer

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[Spin relaxation in a single-electron graphene quantum dot](https://mdr.nims.go.jp/datasets/e6efd00e-f817-46ff-96eb-fdb29873ba4a)

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Spin relaxation in a single-electron graphene quantum dotARTICLESpin relaxation in a single-electron graphenequantum dotL. Banszerus 1,2,5✉, K. Hecker1,2,5, S. Möller 1,2, E. Icking1,2, K. Watanabe 3, T. Taniguchi 4, C. Volk 1,2 &C. Stampfer 1,2The relaxation time of a single-electron spin is an important parameter for solid-state spinqubits, as it directly limits the lifetime of the encoded information. Thanks to the low spin-orbit interaction and low hyperfine coupling, graphene and bilayer graphene (BLG) have longbeen considered promising platforms for spin qubits. Only recently, it has become possible tocontrol single-electrons in BLG quantum dots (QDs) and to understand their spin-valleytexture, while the relaxation dynamics have remained mostly unexplored. Here, we reportspin relaxation times (T1) of single-electron states in BLG QDs. Using pulsed-gate spectro-scopy, we extract relaxation times exceeding 200 μs at a magnetic field of 1.9 T. The T1values show a strong dependence on the spin splitting, promising even longer T1 at lowermagnetic fields, where our measurements are limited by the signal-to-noise ratio. Therelaxation times are more than two orders of magnitude larger than those previously reportedfor carbon-based QDs, suggesting that graphene is a potentially promising host material forscalable spin qubits.https://doi.org/10.1038/s41467-022-31231-5 OPEN1 JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, Aachen, Germany. 2 Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, Jülich, Germany.3 Research Center for Functional Materials, National Institute for Materials Science, Tsukuba, Japan. 4 International Center for Materials Nanoarchitectonics, NationalInstitute for Materials Science, Tsukuba, Japan. 5These authors contributed equally: L. Banszerus, K. Hecker. ✉email: luca.banszerus@rwth-aachen.deNATURE COMMUNICATIONS |         (2022) 13:3637 | https://doi.org/10.1038/s41467-022-31231-5 | www.nature.com/naturecommunications 11234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31231-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31231-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31231-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31231-5&domain=pdfhttp://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-1855-1287http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0002-6237-6762http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-9527-317Xhttp://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362http://orcid.org/0000-0002-4958-7362mailto:luca.banszerus@rwth-aachen.dewww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsThe concept proposed by Loss and DiVincenzo to encodequantum information in spin states of QDs1 has laid thefoundation of spin-based solid-state quantum computa-tion. Spin qubits have been realized in III-V semiconductors2–4,as well as in silicon5–8 and germanium9. The lifetime of theinformation encoded in such qubits is ultimately limited by thespin relaxation time, T1. This relaxation time can be estimated viatransient current spectroscopy, where the excited spin state of theQD is occupied with the help of high-frequency voltage pulses,applied to one of the gates of the QD10–13. In single- and two-electron QDs in GaAs for example, T1 times up to 200 μs havebeen reported10,11. Group IV elements such as silicon, germa-nium and carbon are particularly interesting hosts for realizingspin qubits, thanks to their low nuclear spin densities and theabundance of nuclear spin free isotopes. While T1 times of up to1 s have been reported for silicon QDs with small spinsplittings14, T1 times of about 10 μs have been found in carbonnanotube QDs at low magnetic fields15,16. The latter is most likelylimited by the curvature-induced spin–orbit interaction innanotubes on the order of ΔSO ≈ 1 meV16. In contrast, flat gra-phene and BLG exhibit both low hyperfine coupling and smallKane-Mele type spin–orbit interaction on the order of40–80 μeV17–21, promising long spin lifetimes22. Early deviceswere based on etched QDs in single-layer graphene, where edgedisorder prevented control over the charge occupation of theQDs23–25, imposing currently a major roadblock for single-layergraphene based qubits. In contrast, BLG is particularly suitablefor realizing highly tunable QDs26,27, and important stepstowards the realization of spin qubits have already been achieved—such as the implementation of charge detection28,29, theinvestigation of the electron-hole crossover30 and the measure-ment of the spin–orbit gap in BLG20,21,31. However, electricalmeasurements of the spin relaxation time have remained elusivein both, single-layer graphene and BLG until now12,13. In thisletter, we report on the measurement of T1 times in a single-electron BLG QD. Our measurements confirm that the relaxationtime is sufficiently long to potentially operate a spin qubit,making graphene an interesting host material for bench-markingspin qubits.ResultsThe device consists of a BLG flake encapsulated in hexagonalboron nitride (hBN) placed on a global graphite back gate (BG),with two layers of metallic top gates. Figure 1a shows a scanningelectron microscopy image of the gate structure of the device (seemethods for details)30. To form a QD, we use the BG and splitgates (SG) to form a p-type channel connecting source (S) anddrain (D). The potential along the channel can be controlledusing a set of finger gates (FGs) and a QD is formed by locallyovercompensating the potential set by the BG using one of theFGs (see red FG in Fig. 1a, b), forming a p–n–p junction, wherean n-type QD is tunnel coupled to the p-type reservoirs. Theelectron occupation of the QD can be controlled down to the lastelectron using the FG potential VFG (see Supplementary Figs. 1, 2for details). The tunnel coupling between the QD and the channelcan be tuned using adjacent FGs (e.g., green FG in Fig. 1a, b),allowing also to realize configurations with strongly asymmetrictunnel barriers, as illustrated in the schematic of Fig. 1b. All otherFG potentials are kept on ground.Figure 1c shows the energy dispersion of the first orbital state ofthe QD as a function of an external out-of-plane magnetic field. InBLG, each single-particle orbital is composed of four states,because of the spin and valley degrees of freedom. In contrast tosilicon, the valley states in BLG are associated with topologicalout-of-plane magnetic moments, which originate from the finiteBerry curvature close to the K-points and has opposite sign for theK and K 0-valley32. At zero magnetic field, the Kane-Mele typespin–orbit interaction33 splits the four degenerate states into twoKramer’s pairs ð K "���; K 0 #���Þ and ð K #���; K 0 "���Þ with anenergy gap ΔSO20,21 (see inset of Fig. 1c). An out-of-plane mag-netic field, B⊥, lifts the degeneracy and each state shifts in energyaccording to the spin and valley Zeeman effect asEðB?Þ ¼ 12 ð± gs ± gνÞμBB?, with the Bohr magneton μB, the sping-factor gs= 2 and the valley g-factor gν. Considering typicalvalues of ΔSO ≈ 65 μeV and gν ≈ 3020, valley polarization of the twolowest energy states is achieved already at about 50 mT. In thisregime, the system can be treated as an effective two-level spinsystem with the ground state K 0 "���and excited state K 0 #���which are split by ΔE(B⊥)=ΔSO+ gsμBB⊥.The single-particle spectrum of the QD can be resolved byfinite bias spectroscopy measurements of the N ¼ 0 ! 1electron transition (Fig. 1d). At finite magnetic field, the twoenergetically lower K 0 valley polarized spin states as well as thenearly degenerate K-states can be well observed (see arrows anddashed lines in Fig. 1d). Figure 1e shows the extracted splittingΔE of the two spin states in the K 0-valley, from now on denoted as#���and "���, as a function of B⊥. From the slope we determineΔSO= 66 ± 8 μeV and gs= 1.93 ± 0.09, which is in good agree-ment with earlier experiments20,21.To gain insights on the relaxation of the #���excited state to the"���ground state, we now focus on transient current spectroscopymeasurements. First, we use a two-level pulse scheme11,12,34 toextract the combined tunneling and the overall blocking rate ofthe system. time in BLG QDs13. We therefore apply a finitemagnetic field of B⊥= 2.4 T to lift the spin and valley degeneracyand, furthermore, to reduce the tunneling rates to thereservoirs13,26, by altering the density of states in the reservoirs35and widening the tunneling barriers. Figure 2a shows the appliedsquare pulse scheme with amplitude VA and pulse widths τi andτm. During τi, the QD is emptied (initialized). If the ground state"���is in the bias window (eVSD) during τm, a steady current canbe observed. If the excited state #���is in the bias window duringτm, a transient current can be present, where electrons tunnelthrough the QD until one relaxes with a spin-flip or the groundstate "���gets occupied by direct tunneling from the reservoir.The current, I, through the device as a function of the pulseamplitude VA and VFG is shown in Fig. 2b. The two dominanttransitions originate from "���-transport during τi and τm (seewhite dashed lines in Fig. 2b and left schematic in Fig. 2a). If thepulse amplitude exceeds the energy splitting of "���and #���, atransient current can be observed during τm (see black dashedline in Fig. 2c and right schematic in Fig. 3a). Importantly, therise time of the pulses needs to be faster than the inverse tun-neling rates, such that the system cannot follow the pulse adia-batically (see Supplementary Fig. 3 for details).Studying the dependence of the transient current on the pulsewidth τm, we can extract quantitative information on the char-acteristic time scales of transient processes. Figure 2c shows theaverage number 〈n〉 of electrons tunneling per pulse cycle. Asexpected, in case of "���-transport, 〈n〉/pulse increases linearly withτm, where the slope is given by the combined tunneling rate of bothbarriers Γ= ΓSΓD/(ΓS+ ΓD) ≈ 6.6MHz. Transport via #���satu-rates, as the probability of blocking transport by relaxation ortunneling from the reservoir increases with τm. To enhance tran-sient currents, we establish an asymmetry between the source andthe drain tunneling rate ΓS≫ ΓD by tuning a FG adjacent to theQD. Assuming spin-independent tunnel rates, in this regime, thenumber of electrons tunneling via #���can be approximated byhni=pulse ¼ ΓDð1 � e�γτm Þ=2γ, with the blocking rate γ11. A fitof the data yields a blocking rate γ ≈ 7.9MHz and ΓD ≈ 6.6MHz.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31231-52 NATURE COMMUNICATIONS |         (2022) 13:3637 | https://doi.org/10.1038/s41467-022-31231-5 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsAs γ is on the order of ΓD, direct tunneling from the reservoir into"���dominates the blocking rate and, hence, the blocking rate onlyprovides a lower bound for the relaxation time T1.To extract T1, we then follow refs. 10,11 and include an addi-tional voltage step in the pulse scheme, which allows separatingthe relaxation from the measurement step. The correspondingthree-level pulse scheme is depicted in Fig. 3a, where the pulsesegments are described by pulse durations (τi, τh and τm) andcorresponding voltage values VAC= Vi, Vh, and Vm. During theinitialization step τi, the QD is emptied (see schematic i inFig. 3a). Next, both states "���and #���are pushed below the biaswindow in the loading and holding step (τh, Vh). If τh≫ γ−1, it isensured that an electron has tunneled into either one of the twostates (see schematic ii in Fig. 3a). Finally, to allow for spin-selective readout during the measurement step (τm, Vm), the QDlevels are aligned such that only an #���-electron (i.e., an electronthat has not relaxed) can tunnel out to the drain and contributeto the current (see schematic iii of Fig. 3a). Figure 3b shows〈n〉/pulse as a function of VFG and τh. The three transitionslabeled "���i;h;m originate from "���ground state transport duringc0 0.2 0.4 0.6 0.8 1.0010.5-1-0.50123.23 3.240 1.0 2.0 3.000.10.20.30.400.020.040.06deDSGFGSab FGchannelp-typequantum dotSG1.930.5Pulse sequenceBias teeFig. 1 Device schematics and single-particle spectrum. a False-color scanning electron microscopy image of the gate layout. The SGs define a narrowconducting channel connecting source and drain, while the FGs across the channel are used to form a QD. Bias tees connected to the FGs allow theapplication of AC pulses (VAC) and DC voltages (VFG) to the same gate. The scale bar corresponds to 1 μm. b Band schematic along the channel. One FG(red) is tuned to form a QD, while the tunnel coupling to the right lead, ΓD, is controlled using a neighboring FG (green). c Single-particle spectrum of a BLGQD as function of a out-of-plane magnetic field B⊥. Inset: At B⊥= 0 T, the spin–orbit interaction splits the four states of the first orbital into Kramer's pairswith spin–orbit gap ΔSO. d Finite bias spectroscopy measurement of the single-particle spectrum recorded at B⊥= 0.5T (see dashed line in c). Dashed lineshighlight the four single-particle states. e Measured energy splitting ΔE of the two K0-states, "���and #���, as a function of B⊥.3.238 3.242 3.246012340 200 400 6000.81.21.62.00.40.0a b0.40.2=2.4 00.511.52� �S D � �S Dc�timeFig. 2 Transient current spectroscopy. a The schematic depicts a square pulse with amplitude VA and pulse widths τi and τm. Bottom: possible processes ifthe GS (left) or ES (right) reside in the bias window, which depends on the DC gate voltage, VFG. b Current through the QD as a function of the VFG and thepulse amplitude VA at the transition from N ¼ 0 ! 1 electrons (VSD= 80 μV, f= 2.5 MHz, τm= τi, B⊥= 2.4 T). At low VA, only "���-transport is visibleduring τi and τm. At VA≈ 0.5 V, i.e., the pulse excitation exceeding the level splitting, a transient current via #���sets in. c Average number of electrons 〈n〉per pulse cycle (〈n〉/pulse= I(τi+ τm)/e) as a function of τm at τi= 0.2 μs and VA= 0.8 V (see black dashed line in b). As expected, the "���-transportshows a linear dependency on τm, corresponding to a steady tunnel current (see schematic in a), whereas the #���-transport saturates due to an occupationof the ground state (see schematic in a). The solid line represents a fit according to hni=pulse ¼ ΓDð1 � e�γτm Þ=2γ.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31231-5 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3637 | https://doi.org/10.1038/s41467-022-31231-5 | www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunications(τi,Vi), (τh,Vh) and (τm,Vm), respectively. As in Fig. 2c, the "���hamplitude increases linearly with the duration the ground state isin the bias window, while #���hsaturates with the characteristicblocking rate, γ, of the system. The peak labeled #���moriginatesfrom the electrons leaving the #���excited state to the drainduring the measurement step. The slight negative backgroundbetween #���mand "���i, stems from statistical backwards pump-ing of electrons during τi. The relaxation time, T1, can be deter-mined from the amplitude of the #���m-peak. In order tocontribute to #���m, electrons have to remain in the excited stateand not relax during τh. The amplitude of #���m as function of τhis directly proportional to the probability P↓(τh) of an electronremaining in the excited state during τh. Figure 3c–e show datasets for different B⊥ which have been normalized accordingto hnðτhÞi=hnð0Þi ¼ P#ðτhÞ=P#ð0Þ ¼ e�τh=T1 following ref. 11.Hence, the data is expected to follow an exponential decay,where T1 is the decay constant. The solid lines in Fig. 3c–e showthe exponential decay of P↓(τh)/P↓(0) for different values of T1.At B⊥= 1.7 T (Fig. 3c), no decay of P↓(τh)/P↓(0) as function ofτh can be observed within the noise level of the data and a lowerbound of T1 > 200 μs is estimated from the comparison of thedata with the calculated traces. At higher magnetic fields, i.e.,B⊥= 2.4 T (see Fig. 3d) a slight and almost linear decay ofP↓(τh)/P↓(0) can be observed, which is compatible withT1 ≈ 50 μs. When further increasing the magnetic field toB⊥= 2.9 T a clear exponential decay of P↓(τh)/P↓(0) withT1 ≈ 5 μs can be observed (see Fig. 3e).DiscussionFigure 4 shows T1 times extracted from exponential fits (rounddata points) to additional data sets as depicted in Fig. 3c–e as afunction of the energy splitting ΔE and, hence, B⊥ (see arrows).Decreasing the magnetic field from B⊥= 3 to 2 T, T1 increasesby almost two orders of magnitude from about 5 to 200 μs.For magnetic fields below B⊥= 2 T, no exponential decay ofP↓(τh)/P↓(0) can be fitted to the data anymore and only a lowerbound of T1 > 200 μs, can be stated (see triangular data points),limited by the signal-to-noise ratio of the measured data. Uponincreasing τh (during which no current tunnels through the QD),the average current and thus the measurement signal decreases,limiting τh to 10 μs, before the signal-to-noise ratio decreasesbelow one.Although our B⊥-field range is limited, the strong dependenceof the extracted T1 times as function of the magnetic field (bestdescribed by a power law of T1∝ B−8, see dashed line in Fig. 4)may provide important insights on the spin relaxation mechan-ism. From detailed (theoretical) studies of the B-field dependentT1 times in GaAs QDs36–38, Si QDs39–41 and single-layer gra-phene nanoribbon-based QDs42 it is known that the spin–orbitcoupling and the electron-phonon (e-ph) coupling, in particularthe coupling of piezoelectric or acoustic phonons to electrons37are playing a crucial role for relaxation. Indeed, it has been shownthat a power-law decrease of T1 as function of increasing spinsplitting ΔE∝ B originates in such systems from enhanced pho-non emission due to both, an increasing phonon density of stateand an increasing (accoustic) phonon momentum with increasingc edxaµs-2.92.41.720µs10µs200µs50µs5µs2 4 6 8 100 2 4 6 8 10 2 4 6 80µsµsµs0.20.40.60.81.01.20.03.234 3.238 3.24200.51.00246b20µs10µs200µs50µs5µs20µs10µs200µs50µs5µs,holding‘ ,measurement‘0 i    ii � �S D� �S Dtimeiii  Fig. 3 Measurement of the spin relaxation time. a Schematic of the applied three-level pulse train characterized by the voltages Vi, Vh, Vm and the times τi,τh, τm. During initialization (τi), the QD is emptied. Subsequently, both "���and #���are pushed below the bias window during τh allowing tunneling from thereservoirs into either of the states. Furthermore, relaxation from #���to "���is possible. In the readout step (τm), #���is aligned in the bias window, i.e., anelectron in #���can leave the QD contributing to the current. b Average number of electrons per pulse cycle 〈n〉/pulse= I(τi+ τh+ τm)/e as a function ofVFG and τh (τi= 0.4 μs, τm= 0.4 μs, Vi=−1 V, Vh= 0.6 V, Vm= 0 V and B⊥= 2.4 T). Individual line cuts of the data set are shown in Supplementary Fig. 4.c–e The probability P↓(τh)/P↓(0) of the electron to remain in the excited state during τh as a function of τh. Data has been acquired at B⊥= 1.7, 2.4, and2.9 T, respectively. Solid curves correspond to calculations considering different spin relaxation times T1. The error bars correspond to the current noiselevel in the measurement.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31231-54 NATURE COMMUNICATIONS |         (2022) 13:3637 | https://doi.org/10.1038/s41467-022-31231-5 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsΔE, which in turn leads to faster spin relaxation for larger B-fields37. The spin splitting ΔE is composed of the Zeeman split-ting, which increases linearly with B, as well as the constantZeeman-like Kane-Mele spin–orbit gap, ΔSO (c.f. Fig. 1e). Thusfor graphene and BLG ΔE∝ B is strictly speaking only valid if oneneglects ΔSO. The exact exponent of the power-law scalingdepends, however, sensitively on the system specific nature of (i)the e-ph coupling mechanisms, (ii) the phonons involved, (iii) thespin–orbit coupling, as well as (iv) the overall dimensionality ofthe system. For example, for GaAs QDs a T1∝ B−5 power law hasbeen reported for B-fields in the range of 2–6 T37,38, while forsmall B-fields and suppressed spin–orbit coupling also a T1∝ B−3dependence has been observed38. Interestingly, for Si QDs asignificantly stronger power law, T1∝ B−7, has been predictedand observed for B > 2 T39,41, which for multidonor QDs in Si isreduced to a T1∝ B−5 scaling, hlighting the sensitive dependenceon microscopic details. While for single-layer graphene armchairnanoribbon-based QDs an e-ph coupling dominated T1∝ B−5 istheoretically predicted for B < 3 T there is – to the best of ourknowledge—no theory yet for electrostatically confined QDs inBLG. As the e-ph coupling in single-layer graphene nanoribbonsand BLG are fundamentally different (just to mention the dif-ferent dimensionality and the dominant gauge-field coupling insingle-layer graphene43) it is very hard to make at the presentstage any prediction of what the theoretically expected power-lawdependence for BLG QDs should be. With almost certainty,electron-phonon coupling will also play an important role for BLGQDs and the observed strong B-field dependence of the T1 time,which gives hope for even longer times at smaller B-fields, may alsopoint to a modified BLG phonon bandstructure when encapsulatedin hBN. We expect that our experimental observation will triggerdedicated theoretical work on the spin relaxation in BLG QDs.It is important to mention, that our extracted T1 times can beconsidered as sufficiently long for single-electron spin manip-ulation and mark an important step towards the implementationof spin qubits in graphene. Interestingly, the reported T1 times aremore than two orders of magnitude larger than the valuesreported for carbon nanotubes in a similar magnetic field range15,most likely thanks to the smaller spin–orbit interaction in BLG.To investigate T1 times at smaller spin splittings, where spinqubits could be operated, the fabrication of devices with suffi-ciently opaque tunneling barriers is required, in order to achievelow tunneling rates at lower magnetic fields. Additionally, inte-grated charge sensors will be needed to allow for single-shotcharge and spin detection.MethodsThe device was fabricated from a BLG flake encapsulated between two hBN crystalsof ~25 nm thickness using conventional van-der-Waals stacking techniques. Agraphite flake is used as a BG. Cr/Au SGs with a lateral separation of 80 nm aredeposited on top of the heterostructure. Isolated from the SGs by 15 nm thickatomic layer deposited Al2O3, we fabricate 70 nm wide FGs with a pitch of 150 nm.In order to perform pulsed-gate experiments, the sample is mounted on acustom-made printed circuit board. The DC lines are low-pass-filtered (10 nFcapacitors to ground). All FGs are connected to on-board bias tees, allowing for ACand DC control on the same gate. The AC lines are equipped with cryogenicattenuators of −26 dB. VAC refers to the AC voltage applied prior to attenuation.All measurements are performed in a 3He/4He dilution refrigerator at a basetemperature of around 10 mK and at an electron temperature of around 60 mKusing standard DC measurement techniques. Throughout the experiment, a con-stant BG voltage of VBG=−3.5 V and a SG voltage of VSG= 1.85 V is applied todefine a p-type channel between source and drain.Data availabilityThe data supporting the findings are available in a Zenodo repository under accessioncode https://doi.org/10.5281/zenodo.6599004.Received: 4 November 2021; Accepted: 8 June 2022;References1. Loss, D. & DiVincenzo, D. P. Quantum computation with quantum dots.Phys. Rev. A 57, 120–126 (1998).2. Petta, J. R. et al. Coherent manipulation of coupled electron spins insemiconductor quantum dots. Science 309, 2180–2184 (2005).3. Nowack, K. C. et al. Single-shot correlations and two-qubit gate of solid-statespins. Science 333, 1269–1272 (2011).4. Shulman, M. D. et al. Demonstration of entanglement of electrostaticallycoupled singlet-triplet qubits. Science 336, 202–205 (2012).5. Veldhorst, M. et al. A two-qubit logic gate in silicon. Nature 526, 410–414 (2015).6. Zajac, D. M. et al. 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HNF—Helmholtz Nano Facility. J.Large Scale Res. Facil. 3, 112 (2017).AcknowledgementsThe authors thank G. Burkard, A. Hosseinkhani, and L. Schreiber for fruitful discussions, F.Lentz, S. Trellenkamp, and D. Neumeier for help with sample fabrication and J. Klos for helpwith the SEM micrographs. This project has received funding from the European Union’sHorizon 2020 research and innovation program under grant agreement No. 881603 (Gra-phene Flagship) and from the European Research Council (ERC) under grant agreement No.820254, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) underGermany’s Excellence Strategy—Cluster of Excellence Matter and Light for QuantumComputing (ML4Q) EXC 2004/1—390534769, through DFG (STA 1146/11-1), and by theHelmholtz Nano Facility44. K.W. and T.T. acknowledge support from the Elemental StrategyInitiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) and JSPSKAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).Author contributionsC.S. designed and directed the project; L.B., K.H., S.M., and E.I. fabricated the device,L.B., K.H., and C.V. performed the measurements and analyzed the data. K.W. and T.T.synthesized the hBN crystals. C.V. and C.S. supervised the project. L.B., K.H., C.V., andC.S. wrote the paper with contributions from all authors.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s41467-022-31231-5.Correspondence and requests for materials should be addressed to L. Banszerus.Peer review information Nature Communications thanks Dominik Zumbühl, and theother, anonymous, reviewer for their contribution to the peer review of this work.Reprints and permission information is available at http://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made. The images or other third partymaterial in this article are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. 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