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H. Dulisch, D. Emmerich, E. Icking, K. Hecker, S. Möller, L. Müller, [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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This document is the Accepted Manuscript version of a Published Article that appeared in final form in Nano Letters, copyright © 2025 American Chemical Society. To access the final published article, see https://doi.org/10.1021/acs.nanolett.5c02229.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Electric-Field-Tunable Spin–Orbit Gap in a Bilayer Graphene/WSe<sub>2</sub> Quantum Dot](https://mdr.nims.go.jp/datasets/a635b882-840e-47f6-b46e-a10d2b89159e)

## Fulltext

Electric field tunable spin-orbit gap in a bilayer graphene/WSe2 quantum dotH. Dulisch,1, 2 D. Emmerich,1, 2 E. Icking,1, 2 K. Hecker,1, 2 S. Möller,1, 2L. Müller,1 K. Watanabe,3 T. Taniguchi,4 C. Volk,1, 2 and C. Stampfer1, 2, ∗1JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany, EU2Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany, EU3Research Center for Electronic and Optical Materials,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan(Dated: June 11, 2026)We report on the investigation of proximity-induced spin-orbit coupling (SOC) in a heterostructureof bilayer graphene (BLG) and tungsten diselenide (WSe2). A BLG quantum dot (QD) in thefew-particle regime acts as a sensitive probe for induced SOC. Finite bias and magnetotransportspectroscopy measurements reveal a significantly enhanced SOC that decreases with the applieddisplacement field, distinguishing it from pristine BLG. Furthermore, our measurements demonstratea reduced valley g-factor at larger displacement fields, consistent with a weaker lateral confinementof the QD. Our findings show evidence of the influence of WSe2 across BLG layers, driven by reducedreal-space confinement and increased layer localization of the QD states on the BLG layer distant tothe WSe2 at higher displacement fields. This study demonstrates the electrostatic tunability of thespin-orbit gap in BLG/WSe2 heterostructures, which is especially relevant for the field of spintronicsand future spin qubit control in BLG QDs.I. INTRODUCTIONBernal stacked bilayer graphene (BLG) is character-ized by its low intrinsic spin-orbit coupling (SOC). The-ory predicts a Kane-Mele type SOC strength of around25-50µeV [1–3] whereas experiments have revealed val-ues in a range of 40-80µeV [4–8], mainly considered tobe enhanced due to proximity coupling to hBN [9]. Theintegration of transition metal dichalcogenides (TMDs)in graphene-based heterostructures has enabled an in-teresting approach for manipulating the SOC in two-dimensional materials. For example, coupling grapheneto a TMD, such as tungsten diselenide (WSe2), hasshown promising potential to enhance the intrinsicallylow SOC in graphene while preserving its exceptionallyhigh carrier mobility [10–17]. Of particular interest areBLG/TMDs heterostructures, thanks to the possibilityof tuning the band structure of BLG with an externalout-of-plane electric displacement field (D-field), whichintroduces a gate-tunable band-gap [18–20] and local-izes the charge carriers predominantly on one of thegraphene layers [21–23]. Since only the graphene layerin direct contact to the TMD is expected to exhibit sig-nificant proximity-induced SOC, this results in a split-ting of either the conduction or valence band depend-ing on the sign, i.e. direction of the D-field [22, 23].Furthermore, BLG/WSe2 heterostructures demonstratehigh electronic quality with charge carrier mobilities ex-ceeding 100.000 cm2(Vs)−1 allowing for ballistic trans-port [15, 24] and large spin diffusion lengths [25–27],making it suitable for the development of spin transis-∗ stampfer@physik.rwth-aachen.detors [28, 29]. This unique combination of high carrier mo-bility, gate-tunable SOC, and spin-dependent electronicband properties makes BLG/TMD heterostructures aversatile platform for spintronics applications, with im-plications for the design of next-generation spin-basedtransistors and quantum devices.Until now, proximity-induced SOC has been inves-tigated predominantly in bulk-like BLG/WSe2 devices,where the nearly perfect layer localization at low energieskeeps band mixing to a minimum. However, when lateralconfinement is introduced — as in gate-defined QDs inBLG— the dot wavefunction includes contributions fromhigher-momentum states. This leads to reduced layer lo-calization and a modified SOC strength. Here, we inves-tigate the proximity effect of WSe2 on the SOC in a BLGhole QD, where the QD wavefunction predominantly oc-cupies the graphene layer opposite to the WSe2 layer.By magnetotransport and finite-bias spectroscopy mea-surements in the single charge-carrier regime, we showthat the spin-orbit gap ∆SO in the BLG-QD exhibit apronounced dependence on the displacement field D, in-creasing at lower D-fields, while the spin g-factor remainslargely unchanged. Additionally, we find that the valleyg-factor decreases for larger displacement fields, indicat-ing a lateral size increase of the QD. We attribute thistrend to the interplay of enhanced layer localization inthe BLG conduction and valence bands and a reductionin lateral confinement at increased displacement fields.II. DEVICE FABRICATION AND OPERATIONThe device consists of a monolayer of WSe2 on topof a BLG flake, both encapsulated between two approx-imately 40 nm thick crystals of hexagonal boron nitridemailto:stampfer@physik.rwth-aachen.de2b)SD      FGSG hBNBGWSe2a)c)d) e)BLGFGV  (mV)G I  (nA)SDI  (nA)SDD/e (V/nm)0D/e (V/nm)0E-E (meV)FFIG. 1. (a) Schematic of the device showing the relevantmetal gate structure and the van der Waals heterostructure.The inset highlights the BLG/WSe2 stacking order. (b) Scan-ning force micrograph of the finished device showing the metalgate structure. (c) Tight-binding band structure calculationsclose to the K point at D-fields corresponding to D = 0,D = Dc and D > Dc, respectively. The color scale corre-sponds to the expectation value of the projection operatoron the graphene layer closer to the WSe2 (L1). The tight-binding band structure calculations are based on [3]. In ourcalculations we employ their model with the parameters listedin table 1 in [3], with the slight change of adding λ∆ to theparameters λI1 and λI2 to the block-matrix of HSOC whichcorresponds to layer 1 in the onsite orbital basis ΨAi , ΨBi ,where i denotes the layer index. (d) ISD through the con-ducting channel as a function of the effective gate voltage VGand D/ϵ0 with no voltage applied to the finger gates. Thefeature close to zero displacement field and zero doping is at-tributed to the band inverted phase appearing in TMD/BLGheterostructures due to increased Ising-type SOC, which leadsto a dip in conductance between the two critical field values(black arrows) ±Dc/ϵ0 [30, 31]. (e) Line cut at VG = 0 (seepanel (d)) showing the decrease in current close to D/ϵ0 = 0.(hBN). This van der Waals heterostructure is placed ontop of a graphite flake that functions as a back gate (BG).Three metal (Cr/Au) gate layers – split gates (SGs) andtwo layers of interdigitated finger gates (FGs), – are de-posited on top of the heterostructure, each separated bya 20 nm thick layer of Al2O3. Fig. 1(a) shows a schematicof the heterostructure, whereas Fig. 1(b) shows a scan-ning force microscope image of the gate structure. All theexperiments were performed in a dilution refrigerator ata base temperature of around 30mK.FGBGhQDa)b)c) d)D/e  = 371 mV/nm0 D/e  = 440 mV/nm0I (nA)SDV (mV)SDV (mV)SDDV  (mV)FG DV  (mV)FGV  (V)FGI  (nA)SDV  (V)FGdI /dV  (a.u.)SD SDFIG. 2. (a) ISD as a function of VFG showing the first fewCoulomb peaks of a hole QD measured at a displacementfield of D/ϵ0 = 440mV/nm and B⊥ = 200mT. The insetshows a schematic of the band-edge profile along the channel,with the QD being created by the induced n-p-n junction dueto the local inversion of the bands under the FG. (b) Finite-bias spectroscopy showing the differential conductance dI/dVat D/ϵ0 = 440mV/nm and B⊥ = 0T. (c) and (d) show aclose up on the first charge transition at D/ϵ0 = 371mV/nm(c) and D/ϵ0 = 440mV/nm (d) respectively at B⊥ = 0T.The white dashed line indicates the first excited state and itsintercept with the outline of the conductive region.The SGs are used to open a band gap in the BLGbeneath them, creating a narrow (∼ 200 nm wide) con-ducting channel connecting the source (S) and drain (D)contacts. The voltages applied to the BG and SGs allowto independently tune the effective gate voltage, VG =(VBG−V 0BG+β(VSG−V 0SG))/(1+β), and the displacementfield, D/ϵ0 = e/2 · (αBG(VBG −V 0BG)−αSG(VSG −V 0SG)),where β = αBG/αSG is the ratio between the geomet-ric lever arms of the BG and the SGs respectively, whileV 0BG and V 0SG are the offset voltages of the charge neu-trality point [20]. Figure 1(d) shows the source-draincurrent ISD as a function of VG and D/ϵ0. The datashows a non-monotonic dependency of the current nearthe charge neutrality point at low displacement fields. A3cut through the current map at VG = 0V reveals a localminimum in ISD at D/ϵ0 = 0mV/nm and two symmetri-cally centered local maxima at the critical field D = ±Dc(indicated by the black arrows in Figs. 1(d) and (e)).Similar features as in Fig. 1(d) have been observed inconductance measurements of dual gated suspended BLGdevices, which was attributed to a symmetry broken statedue to electron-electron interaction [32]. Recently, thisinverted gap phase was also observed in transport mea-surements of BLG fully encapsulated between WSe2 [33],as well as in samples with an one-sided contact to TMDs,such as WSe2 [30] and MoS2 [31]. In this case, the featureis attributed to band inversion caused by induced Ising-type SOC. Indeed, tight-binding band-structure calcula-tions indicate the existence of a critical field Dc for whichthe SOC-induced gap closes, see Fig. 1(c). At this criti-cal field, the band gap closes showing up as peaks in thecurrent (c.f. Fig. 1(e)), while the breaking of layer sym-metry leads to a dip in current at D/ϵ0 = 0V/nm. Thesignature of this inverted gap phase is less pronouncedin our data, due to our signal being dominated by theinduced conductive channel in the device, which is onlygated by the BG. Nevertheless, this is a clear evidenceof an enhanced proximity-induces spin-orbit coupling inthe BLG.The finger-gate structure on top of the split-gate al-lows us to modulate the potential profile along the con-ducting channel and to form a quantum dot, see insetin Fig. 2(a) [34–36]. We deplete the channel to forman electrostatically defined QD in BLG, by applying avoltage VFG to the central finger gate, while the remain-ing FGs are used to tune the tunnel barriers and iso-late the QD. Figure 2(a) shows the complete pinch-offof the current through the channel for increasing VFG,and the appearance of well-defined Coulomb peaks in-dicating the sequential filling of a QD with holes. Fig-ure 2(b) shows a finite-bias spectroscopy measurement,i.e. the normalized differential conductance as a func-tion of VFG and the bias voltage VSD. From the charac-teristic Coulomb diamond signature, we extract the gatelever arm α = ∆VSD/∆VFG, which allows us to translatechanges in gate voltage VFG into changes in electrochem-ical potential ∆µ.The Coulomb-blockade measurements show also signa-tures of the presence of a proximity-induced SOC gap inthe BLG. Indeed, we observe that the third Coulombdiamond is not properly closing (see dashed box inFig. 3(a)). This can be understood in terms of an imbal-ance of the four flavors (|K ↑⟩, |K ↓⟩, |K ′ ↑⟩ and |K ′ ↓⟩)in the lead regions, which for large displacement field,have a band-structure with spin split bands as shownas in the right panel of Fig. 1c. Due to this imbalance,the first two holes, |K ′ ↑⟩ and |K ↓⟩ can be easily filledand low-bias transport is possible (see upper schematicof Fig. 3(b)). However, the process of adding the 3rdhole, which corresponds to a |K ′ ↓⟩ or |K ↑⟩ state, issuppressed (see lower schematic). The blockade can beeither lifted by the bias voltage (see arrows in Fig. 3(a))−5.42 −5.40 −5.38 −5.36 −5.34 −5.32 −5.30ΔV  (V)FG-2.00.02.0V (mV)SD−0.25 0.00 0.25I  (nA)SD−3 0 3ΔV  (mV)FG60.00.4B (mT)⟂0.0 0.08 I  (nA)SDa)b) c)XFIG. 3. (a) Finite-bias spectroscopy of the first 4 holes in theQD measured at B = 0T. (b) Schematics depicting sequentialtunneling through a LG QD at the 1 to 2 transition (upperschematic) and the 2 to 3 transition (lower schematic) withspin-valley split bands in the source-drain lead regions (formore details see text). (c) Magneto transport spectroscopymeasurement of the 3rd hole at VSD = 60µV highlightingthat transport only becomes possible at finite magnetic field.making it possible to populate |K ′ ↓⟩ or |K ↑⟩ states inthe leads or by an out-of-plane B-field (see Fig. 3(c)).The latter will shift the states of the QD according totheir spin and valley g-factors, changing the state order-ing with increasing B⊥ (for details see Ref. [39]). In par-ticular, a state of the 2nd orbital, decreasing in energy,can become the new three-particle ground state, makingtransport possible.Finite-bias spectroscopy measurements also allow toinvestigate the excited states of the QD [37], as shownin Figs. 2(c) and (d) for two different displacement field.This type of measurement allows to determine the single-particle level splitting. We assume the first excited stateto coincide with the SO-gap. From finite-bias spec-troscopy measurements we estimate an orbital splittingof ∼ 500µeV. As discussed below, the values extractedfor the first excited state agrees very well with the SOCstrength estimated by magnetic field spectroscopy. Thedata of Fig. 2c present therefore a first indication thatthe SOC in the dot depends on the applied displacementfield.III. MAGNETIC FIELD SPECTROSCOPYThe size of the SOC gap in the dot, ∆SO, and the val-ues of the spin and valley g-factors, gs and gv can beexperimentally determined by measuring the shift of thefirst Coulomb resonance as function of an external mag-4a) b)BBBf)g)h)DV (mV)FGDV (mV)FGDV (mV)FGBI (nA)SDI (nA)SDI (nA)SDc)d)e)DV (mV)FGDV (mV)FGDV (mV)FGBI (nA)SDI (nA)SDI (nA)SDFIG. 4. (a) Schematic of the single particle energy levelsin dependence of the applied magnetic field. The black arrowindicates the difference in electrochemical potential of the dotlevel with respect to the leads. (b) Three exemplary datasetsextracted from the in-plane magneto-transport data shownin panel (f)-(h) together with the fits according to Eq. (2).The datasets are offset for better visibility. (c)-(e) Out-of-plane magneto-transport spectroscopy measurements for (c)D/ϵ0 = 302 mV/nm, (d) D/ϵ0 = 348 mV/nm, (e) D/ϵ0 = 418mV/nm. The fits according to Eq. (1) are overlaid as a dashedline for comparison. (f)-(h) In-plane magneto transport datafor the same displacement fields as panel (c)-(e).netic field, since this reflect directly the change in ground-state energy of the single particle states, see Fig. 4(a).This exhibits a linear dependence on the out-of-planemagnetic field B⊥ given by the relation∆µ =12(gs + gv)µBB⊥, (1)and the following dependence on the in-plane field B∥:∆µ =12√∆2SO + (gsµBB∥)2 (2)where µB is the Bohr magneton, gs the spin g-factor, gvis the valley g-factor, originating from the orbital val-ley magnetic moment caused by a non-vanishing Berrycurvature at the high symmetry points K and K ′ [38],and ∆SO is the spin-orbit gap. The shift of the firstCoulomb resonance as function of B∥ and B⊥ are shownin Figs. 4(c)-4(h) for different values of the applied dis-placement field D. Knowing the gate lever arm α fromfinite-bias spectroscopy measurements, it is possible totranslate the shift of the resonance in gate-voltage VFGinto a change of electrochemical potential ∆µ = α∆VFG,as shown in Fig. 4(b) for the measurements as functionof B∥.At large magnetic field, the peak position shifts ap-proximately linearly with B∥, ∆µ ≈ 12gsµBB∥, whichallows to extract the spin g-factor independently. Asexpected, the values of gs extracted in this way showno dependence on |D/ϵ0| within the margin of error, seeFig. 5(a). The main source of uncertainty arises from thedetermination of the lever arm, which depends on identi-fying the onset of the conducting region in the finite-biasspectroscopy measurements. Due to the inherent am-biguity in this onset, we estimate a systematic error ofapproximately∼ 10%. Taking the mean value of gs deter-mined in this way as fixed parameter, gs = 2.2±0.2, we fitEq. (1) and Eq. (2) to the data of Fig. 4, to determine gvand ∆SO respectively. The values determined in this wayare summarized in Fig. 5. In Fig. 5(c) we include alsovalues of ∆SO determined with finite-bias spectroscopy(down-pointing white triangles), which agree very wellwith those extracted with the procedure discussed above.Differently from the spin g-factor, both the valley g-factor, gv, and the spin-orbit coupling ∆SO show a cleardependence on the applied displacement field, both de-creasing with increasing |D/ε0|. The decrease of gv in-dicates a weaker lateral confinement for larger |D/ε0|,in agreement with earlier work [39]. This weaker lateralconfinement is also one of the effects that leads to theobserved reduction of ∆SO for increasing |D/ε0|, as dis-cussed in the following section.IV. TUNABLE PROXIMITY-INDUCEDSPIN-ORBIT GAPAt low displacement field, the extracted SOC strengthis significantly larger than what was observed in QDsformed in pure BLG-hBN heterostructures [6, 40], seealso gray data-points in Fig. 5(c)). Such an enhancementis a clear indication of proximity-induced SOC caused bythe WSe2 layer in our heterostructure. Interestingly, ourQD is expected to be located on the BLG layer oppositeto the WSe2. In fact, in gapped BLG, the valence andconduction bands near the K- and K ′-valleys are layer-polarized: one layer predominantly hosts the conductionband, while the other hosts the valence band. The po-larity of the displacement field determines which bandlocalizes on the top (WSe2-side) or bottom graphenelayer [22, 23]. In our device, the conduction band islocalized on the top-layer, which implies that for neg-ative doping, we fill states in the valence band localizedon the bottom graphene layer. To form a QD in thehole regime, the charge-carrier polarity must be locallyinverted, so the QD wavefunction predominantly derives5a) b)c)|D /    | (mV/nm)4gsgvsmall D large D|D /    | (mV/nm)|D /    | (mV/nm)D    (µeV) SOFIG. 5. (a) Spin g-factor gs determined from in-plane mag-netic field spectroscopy measurements for different displace-ment field strengths |D/ϵ0|. The data scatter around a meanvalue of 2.2 ± 0.2, indicated by the filled area, with no dis-cernible trend. (b) Same as (a), but for the out-of-plane fielddirection, showing the valley-g factor obtained by subtract-ing the spin g-factor from the slope of a linear fit to the data.The valley g-factor decreases with increasing electric field (D-field), suggesting reduced lateral confinement of the QD, con-sistent with [39]. (c) Comparison of ∆SO as a function of|D/ϵ0| for BLG/WSe2 (white) and pure BLG QD (grey) de-vices. The two methods used to extract the SOC parameter inthe case of the BLG/WSe2 device show good agreement and adecreasing trend with increasing displacement field strength,absent in the pure BLG data. The data for the pristine BLGQD devices stems from ref. [6].from bottom-layer states. Despite this nominal separa-tion, the measurements reveal a substantial proximity-induced SOC effect.One reason for this is that the layer localization ismost pronounced right at the K and K ′-points, whileaway from these high-symmetry points, the Bloch wave-functions experience increased mixing across both layers,providing overlap with the top graphene layer. In thequantum dot, the real-space confinement of the wave-function results in momentum-space mixing of Blochstates, enhancing the contribution from the proximitizedBLG layer and resulting in the observed enhanced ∆SO.However, for increasing the displacement field, the layer-polarization (i.e. layer localization) of the wavefunctionincreases (see schematics in Fig. 5(c)) and, at the sametime, the lateral confinement of the dot is reduced [39], asindicated by decreasing values of the valley g-factor, seeFig. 5(b), narrowing the distribution of the wave functionin momentum space around layer-polarized states. Thetwo effects contributes to the observed decrease of ∆SOwith increasing |D/ε0|.V. CONCLUSIONIn summary, we have experimentally investigated theproximity-induced SOC in a BLG QD formed adjacent toa WSe2 layer. Through magnetotransport and finite-biasspectroscopy measurements in the few-carrier regime, wefind a pronounced enhancement of the SOC gap ∆SOcompared to well studied pure BLG QDs, underscoringthe impact of the TMD on nearby graphene layers. Thisspin-orbit gap decreases with increasing |D/ϵ0|, which isin contrast to pristine BLG QDs, where no tuning of ∆SOwith D/ϵ0 was observed. This behavior can be under-stood as the interplay of two key effects: (i) a larger bandgap strengthens layer localization at the high-symmetrypoints K, K ′, and (ii) the lateral confinement of the QDdecreases at higher |D/ϵ0|, causing the wavefunction toinclude fewer mixed-layer states that would otherwise en-hance the SOC strength.Notably, we observe that the QD primarily resideson the graphene layer opposite to the WSe2, yet stillexhibits a sizable proximity-induced SOC. This high-lights the importance of momentum-space mixing ofBloch states and shows that a purely layer-localizedpicture is insufficient at experimentally relevant gatevoltages and QD sizes. Furthermore, our out-of-planemagnetotransport data reveal a systematic reductionof the valley g-factor with increasing |D/ϵ0|, consistentwith reduced lateral confinement and a larger real-spaceextent of the QD. Altogether, these findings demonstraterobust, tunable SOC in a BLG/WSe2 heterostructure,paving the way for future spintronic and QD-based spinqubit experiments where electrical control of spin-orbitcoupling may be used to realize spin-orbit driven qubitsand spin-based logic devices.While finalizing the manuscript we became aware ofanother recent work studying a BLG QD in proximityto a WSe2 layer [41]. Interestingly, their observeddependency of the proximity enhanced spin orbit gapon |D/ϵ0| is in very good qualitative and quantitativeagreement supporting our findings.6Acknowledgments The authors thank F. Lentz,S. Trellenkamp and M. Otto for help with samplefabrication and F. Haupt for help on the manuscript.This project has received funding from the EuropeanResearch Council (ERC) under grant agreement No.820254, the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) through SPP 2244(Project No. 535377524) and under Germany’s Ex-cellence Strategy - Cluster of Excellence Matter andLight for Quantum Computing (ML4Q) EXC 2004/1-390534769, and by the Helmholtz Nano Facility [42].K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 21H05233 and 23H02052) ,the CREST (JPMJCR24A5), JST and World PremierInternational Research Center Initiative (WPI), MEXT,Japan.Author contributions H.D, D.E. and E.I. fab-ricated the device. 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Journal of Large-Scale Research Facilities2017, 3, 112.http://arxiv.org/abs/2403.17140http://arxiv.org/abs/2403.17140 Electric field tunable spin-orbit gap in a bilayer graphene/WSe2 quantum dot Abstract Introduction Device fabrication and operation Magnetic field spectroscopy Tunable proximity-induced spin-orbit gap Conclusion References