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[Michele Masseroni](https://orcid.org/0000-0003-1663-8239), Mario Gull, [Archisman Panigrahi](https://orcid.org/0000-0003-2619-431X), Nils Jacobsen, [Felix Fischer](https://orcid.org/0009-0007-7166-3222), [Chuyao Tong](https://orcid.org/0000-0003-4947-6002), [Jonas D. Gerber](https://orcid.org/0000-0002-4164-8765), Markus Niese, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Leonid Levitov](https://orcid.org/0000-0002-4268-731X), [Thomas Ihn](https://orcid.org/0000-0002-5587-6953), [Klaus Ensslin](https://orcid.org/0000-0001-7007-6949), [Hadrien Duprez](https://orcid.org/0000-0003-0506-126X)

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[Spin-orbit proximity in MoS2/bilayer graphene heterostructures](https://mdr.nims.go.jp/datasets/9e6d8539-2a39-42ed-8498-16ea7fc0cabf)

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Spin-orbit proximity in MoS2/bilayer graphene heterostructuresArticle https://doi.org/10.1038/s41467-024-53324-zSpin-orbit proximity in MoS2/bilayergraphene heterostructuresMichele Masseroni 1 , Mario Gull1, Archisman Panigrahi 2, Nils Jacobsen3,Felix Fischer 1, Chuyao Tong 1, Jonas D. Gerber 1, Markus Niese1,Takashi Taniguchi 4, Kenji Watanabe 5, Leonid Levitov 2, Thomas Ihn 1,Klaus Ensslin 1 & Hadrien Duprez 1Van der Waals heterostructures provide a versatile platform for tailoringelectronic properties through the integration of two-dimensional materials.Among these combinations, the interaction between bilayer graphene andtransition metal dichalcogenides (TMDs) stands out due to its potential forinducing spin–orbit coupling (SOC) in graphene. Future devices conceptsrequire the understanding of the precise nature of SOC in TMD/bilayer gra-phene heterostructures and its influence on electronic transport phenomena.Here, we experimentally confirm the presence of two distinct types of SOC –Ising (ΔI = 1.55meV) and Rashba (ΔR = 2.5meV) – in bilayer graphene wheninterfaced with molybdenum disulfide. Furthermore, we reveal a non-monotonic trend in conductivitywith respect to the electric displacementfieldat charge neutrality. This phenomenon is ascribed to the existence of single-particle gaps induced by the Ising SOC, which can be closed by a criticaldisplacement field. Our findings also unveil sharp peaks in the magneto-conductivity around the critical displacement field, challenging existing the-oretical models.Spin is emerging as a promising alternative or complement to chargefor information storage and processing1. Spin–orbit coupling (SOC)is crucial in spin-based devices, enabling manipulation of spin statesthrough time-dependent electric fields2,3. Bernal bilayer graphene(BLG) holds potential for spintronics4 and quantum computing5, withrecent studies indicating long spin relaxation times in BLG quantumdots6–8. However, intrinsic Kane–Mele (KM) SOC9 in graphene is weak(40–80 μeV)10,11. Various methods have been explored to enhanceSOC in BLG, including interfacing with high-SOC substrates. Transi-tion metal dichalcogenides (TMDs) have shown promise in thisregard, offering significant SOC enhancements (from 1 to 10meV)without compromising graphene’s electronic quality12–14. Addition-ally, the combination of BLG on WSe2 has recently been shownto host an unexpected superconducting phase, where the SOCseems to play a major role15,16, prompting further study of suchheterostructures.The extrinsic SOC induced in BLGby the TMDs is described by theHamiltonian17HSO =ΔI2ξszIσ +ΔR2ðξσxsy � σysxÞ, ð1Þwhere ξ = ±1 represents the valley index, sx,y,z denote spin Pauli matri-ces, σx,y and Iσ are Pauli and unit matrices operating on the sublatticedegree of freedom (A1,B1) within the layer in contactwith the TMD(seeschematic in Fig. 2e). The first term, known as the Ising SOC, actssimilarly to an effective out-of-plane magnetic field with a valley-dependent sign. It lifts the four-fold spin and valley degeneracy at theKReceived: 14 July 2024Accepted: 7 October 2024Check for updates1SolidState Physics Laboratory, ETHZürich, 8093Zürich, Switzerland. 2Department of Physics,Massachusetts Institute of Technology, Cambridge,MA02139,USA. 31st Physical Institute, Faculty of Physics, University of Göttingen, 37077 Göttingen, Germany. 4Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5Research Center for Electronic and Optical Materials, National Institute forMaterials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. e-mail: masmiche@phys.ethz.ch; ensslin@phys.ethz.chNature Communications |         (2024) 15:9251 11234567890():,;1234567890():,;http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-1663-8239http://orcid.org/0000-0003-2619-431Xhttp://orcid.org/0000-0003-2619-431Xhttp://orcid.org/0000-0003-2619-431Xhttp://orcid.org/0000-0003-2619-431Xhttp://orcid.org/0000-0003-2619-431Xhttp://orcid.org/0009-0007-7166-3222http://orcid.org/0009-0007-7166-3222http://orcid.org/0009-0007-7166-3222http://orcid.org/0009-0007-7166-3222http://orcid.org/0009-0007-7166-3222http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0003-4947-6002http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://orcid.org/0000-0002-4164-8765http://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/0000-0003-3701-8119http://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/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://orcid.org/0000-0003-0506-126Xhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-53324-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-53324-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-53324-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-53324-z&domain=pdfmailto:masmiche@phys.ethz.chmailto:ensslin@phys.ethz.chwww.nature.com/naturecommunications± points, forming spin-valley-locked Kramers doublets. The secondterm is a Rashba type of SOC18, favoring an in-plane spin polarizationperpendicular to the sublattice isospin vector.Intensive theoretical17,19–23 and experimental efforts12–14,24–28 inunderstanding and quantifying SOC proximity effects have led to arange of values for the relative strength of the two SOC termsdepending on the analysismethod. This is because the strength of SOCis often inferred indirectly, for example, through the extraction ofrelaxation times obtained from quantum interference effects such asweak antilocalization (WAL)29–31, or spin precessionmeasurements32. Incontrast, the fundamental frequency f =Δ(B−1) of the Shubnikov–deHaas oscillations (SdHOs) offers a direct measurement of the Fermisurface and is suitable for determining the band splitting induced bySOC33. However, the energy resolution of this technique is limited bythe broadening of the Landau levels, necessitating high electronmobilities and low disorder potentials.Here, we conductmagnetotransport experiments on a dual-gatedMoS2/BLG heterostructure. First, we analyze SdHOs to quantifyproximity-induced SOC.Our results confirm the presence of both Ising(ΔI = 1.55meV) and Rashba (ΔR = 2.5meV) SOC. Despite their compar-able strength, we show that the splitting of the low-energy bandsmainly arises from the Ising SOC. Additionally, we observe a non-monotonic conductivity response to an applied displacement fieldwhen BLG is charge-neutral. Our tight-binding calculations show howthe displacement fieldD opposes the Ising SOC, closing single-particlegaps in the spin-polarized bands at a critical value of Dc and causinglocalmaxima in the conductivity. In this critical field, the application ofan external magnetic field rapidly suppresses the conductivity, chal-lenging existing theoreticalmodels and suggesting the involvement ofmany-body interactions.ResultsProximity induced spin–orbit couplingDetermining the SOC gap in BLG viamagnetotransport experiments ischallenging due to disorder-induced density fluctuations δn. Shown inFig. 1a is the schematic of our sample, comprising BLG atop threelayers of MoS2, encapsulated within hexagonal boron nitride (hBN),and placed on a graphite bottomgate. The use of hBNdielectrics and agraphite layer minimizes density fluctuations34, evident from the low-density δn ~ 2 × 109 cm−2 at which conductivity saturates in our sample(Fig. 1b). High charge carrier mobilities [~5 × 105 cm2 (V s)−1 atn = 5 × 1010 cm−2, see Supplementary Note 2] indicate minimal impactof the MoS2 layer on BLG’s electronic properties compared to hBN-encapsulated Bernal BLG devices35.We analyze SdHOs at zero displacement field (D) and low mag-netic fields (B) to determine the band splitting induced by the SOC.Fig. 1 | Magnetotransport data at zero displacement field. a Schematic repre-sentation of the BLG/MoS2 heterostructure, illustrating a cross-section of the layers(top right panel) and highlighting the alignment of the BLG and MoS2 layers (bot-tom right panel). b The conductivity is plotted as a function of carrier density on alogarithmic scale. The measurement was conducted at a temperature of approxi-mately 30mK.Dotted (squared)markers represent data for electron (hole) doping.The solid red line depicts a linear fit, while the black dashed line indicates thesaturation of the conductivity. c Landau fan at zero displacement field (left panel)measured at a temperature T = 1.3 K. The right panel displays a vertical linecut atn = 2.7 × 1011 cm−2 (dashed line in the left panel). d Fast Fourier transform (FFT) ofthe Landau fan shown in (c). The FFT of Rxx(B−1) is calculated line-by-line for eachdensity. The vertical axis has been rescaled according to nSdH = 2ef/h, where f is thefrequency axis in Tesla, accounting for the valley degeneracy. Dashed linesrepresent densities obtained from theband structure in (e). The right panel shows avertical linecut at n = 2.7 × 1011 cm−2 (dashed line in the left panel). e Band structureof bilayer graphene with SOC (ΔI = 1.55meV and ΔR = 2.5meV) close to the K+ pointat zero displacement field. The bands are plotted along the relative momentum kx(or equivalently, along the line Γ−K−M of the first Brillouin zone). The horizontalaxis is scaled by the lattice constant a = 2.46Å. The color of the bands encodes thespin texture (violet for spin up and orange for spin down). f Density difference Δnobtained from the distance between the two peaks in the Fourier spectrum shownin (d). The red solid line represents a fit to the data, obtained from the bandstructure in (e) by determining the density of states and then the carrier densitiesn↓,↑ followed by calculating the difference ∣n↓−n↑∣. The fit yields the SOC para-meters: ΔI = 1.55 ± 0.10meV and ΔR = 2.5 ± 0.5meV. The shaded area indicates theuncertainty in the fitting parameters, reflected in the uncertainty in Δn.Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 2www.nature.com/naturecommunicationsFigure 1c shows the longitudinal resistance Rxx as a function of B andelectron density n at T = 1.3 K. Pronounced minima in resistance Rxxoccur at filling factors ν = ±4N (N an integer), characteristic of pristineBLG. In addition, small oscillation maxima appear in the SdH minima(highlighted by the arrow in the right panel), suggesting the presenceof a broken symmetry.Todetermine the oscillation frequencyof the SdHOs,we employ anumerical fast Fourier transform (FFT) of Rxx(1/B) calculated line-by-line for each density, as shown in Fig. 1d. The FFT reveals two clearfrequencies f1 and f2, resulting from the splitting of the Fermi surface,which is attributed to the influence of the MoS2 substrate through thespin–orbit proximity effect. The sum of the electron densities ni = 2efi/h (i = 1, 2) obtained from the SdHO matches the Hall density byaccounting for the twofold valley degeneracy, as expected.The two SOC terms in Eq. (1) yield distinct density dependenciesfor the spin–orbit splitting. The Ising SOC induces a constant splittingas a function of the Fermi energy (and hencedensity), while theRashbaterm leads to a splitting that increaseswith the Fermi energy. Althoughthe splitting in Fig. 1d initially appears constant with carrier density, acloser examination of Δn in Fig. 1f reveals a small but detectable slope.By aligning the density difference Δn obtained from the tight-bindingmodel (see the “Methods” subsection “Tight-binding model” andSupplementary Notes 3–5) with the data (illustrated by the red solidline), we find ΔI = (1.55 ± 0.10)meV and ΔR = (2.5 ± 0.5)meV. The theo-retically predicted densities with these parameters are overlaid againstthe total density in Fig. 1d as orange and violet dashed lines, demon-strating good agreement with the experimental data.We acknowledgethat the numerical outcome of the fit can be subtly influenced by thechoice of the tight-binding intralayer and interlayer coupling para-meters of BLG, referred to as the Slonczewski–Weiss–McClure para-meters. These parameters dictate the curvature of the bands, therebyaffecting the conversion between energy and density, as elaborated inSupplementary Note 4.We validate our findings at finite displacement fields, leveragingthe layer-dependent SOC induced by the asymmetric structure of oursample3. This layer selectivity is demonstrated in SupplementaryNote 6, where the electron wave function is polarized via the applieddisplacement field in one layer or the other, depending on its sign.Next, we continue the discussion by investigating the impact ofSOC on the electrical conductivity (σ) of BLG at charge neutrality (CN).Conductivity at charge neutralityMeasurements of σ reveal a non-monotonic dependence on the dis-placement field (Fig. 2a), which appears in a narrow density range(n ~ 1 × 1010 cm−2) around CN. A local minimum at D =0 is surroundedby conductivity maxima at D = ±Dc ≈ 12.5mV/nm, as highlighted in theline cut at n =0 presented in Fig. 2b.This dependence can be understood by taking into account theinfluence of SOC on the BLG band structure. From tight-binding cal-culations, we find that the Rashba SOC has little effect on the low-energy bands (see Supplementary Note 3 for more details). For thisreason, we consider only the Ising SOC in the following discussion. Theoutcome of the band structure calculations is presented in Fig. 2c,shown for the K+-valley and three characteristic interlayer potentialenergies (U). First, we consider the case U =0 in the left panel. Weobserve that theband structure comprises twopairs ofbands, one splitFig. 2 | Displacement field dependence of the conductivity at CN. aConductivityσ as a function of density and displacement fieldmeasured at T ≈ 30mK. b Verticalline cut of σ in (a) at CN. The conductivity shows local maxima at the criticaldisplacement Dc = ±12.5mV/nm. c Band structure of proximitized BLG at the Kpoint. The calculation includes an Ising SOC term with ΔI = 1.55meV. The bandstructure is shown for three characteristic interlayer potential energies: U =0 (leftpanel),U =0.775meV (central panel), andU = 2meV (right panel). The energy axis isadjusted such that E =0 corresponds to charge neutrality, which is marked by thehorizontal dashed lines. The color code represents the layer polarization: blueindicates polarization on layer 1, while red is on layer 2, as shown in the schematicsin the bottom right panel. The band structure shown in the left panel is the same asFig. 1c, where we color-coded the bands according to the spin texture.d Conductivity at CN for D =0 (violet) and D =Dc (orange) as a function of tem-perature in logarithmic scale. The crosses weremeasured in a dilution refrigerator,while the solid line was measured in a pumped He4 cryostat. The temperaturerange is divided into three regimes (A–C), according to relevant energy scales(Eth, Edis, ESO) defined in the main text. e Constant energy contours of the Fermienergy at charge neutrality for U =0 (left panel) andU =Uc (right panel). The Fermipockets are labeled according to their doping, electron e and holes h, and their spin(↑, ↓). Schematic: The side view of the BLG unit cell is schematically shown on aMoS2 substrate. The color bar underneath defines the polarization of the electronwave function on the low energy orbitals, which are localized on the lattice site A1and B2 (colored blue and red, respectively).Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 3www.nature.com/naturecommunicationsby the energy ΔI and partially layer-polarized on the bottom layer(blue), while the other pair is degenerate at two points along kx and ispartially polarized on the top layer (red). Due to their partial layerpolarization, the application of a displacement field shifts the twopairsof bands relative to each other based on their layer polarization.Notably, the calculations show that a band gap emerges only once theinterlayer potential energy exceeds the critical value Uc =ΔI/2 ≈0.8meV (right panel), i.e. once U counteracts the SOC, resulting inthe closure of the gap between bands with the same spin (see Fig. 2efor the spin texture). This elucidates why, in the experiment, theconductivity starts decreasingwith increasing displacement fields onlywhen D >Dc, and associates the maxima in the conductivity with theclosure of the SOC gaps. We verify this interpretation by convertingUcinto a displacement field, taking into account interlayer screening (seeSupplementary Note 7 for details). The conversion yields a displace-ment field of 11.4mV/nm, in good agreement with the experimentalvalue of Dc ≈ 12.5mV/nm. Furthermore, the local conductivity mini-mum at D =0 is observed only at low density and vanishes aroundn ~ 1 × 1010 cm−2, which corresponds to the density required to fill thespin–orbit splitting of the bands, as demonstrated in Fig. 1f.The local minimum in conductivity at D =0 prompts considera-tion of a potentially insulating phase arising from the presence of agap, as reported for BLG fully encapsulated in TMDs14. To verify this,we examine the temperature dependence of the conductivity inFig. 2d. Over the temperature range of 1–10 K, the conductivityincreases by one order ofmagnitude, indicative of insulating behavior.However, the data only conforms to the Arrhenius law within a verylimited temperature range (shown in Supplementary Note 8) andsaturates to rather large conductivity values at low temperatures.Similarly, the conductivity at the critical field Dc also increases withtemperature. Thus, although the insulating behavior is affected by theapplied displacement field, it is consistently observed across all dis-placement fields at CN.To further understand the temperature dependence, we comparethe thermal energy Eth = kBTwith the other characteristic energy scalesdeterminedbydisorder (Edis) and SOC (ESO). First, we take into accountthe disorder potential, which induces density fluctuations of the orderδn ≈ 2 × 109 cm−2. Thesefluctuations are converted into an energy scaleEdis ≈0.14meV using an effective mass approximation (m* ≈0.035m0,where m0 is the bare electron mass36) and taking into account thetwofold valley degeneracy. At low temperatures (Eth < Edis), the con-ductivity is governed by disorder-induced electron–hole puddles,causing the saturation of the conductivity in the temperaturerange labeled A in Fig. 2d. Second, the SOC introduces gapsESO =ΔI/2 ≈0.8meV between bands of the same spin at D =0, as illu-strated in Fig. 1c (see also Supplementary Note 9). The presence ofthese spin-resolved gaps, even without a real band gap, could explainthe insulating behavior of the conductivity. Effectively, if spin is con-served in thermal activation processes, carriers cannot be thermallyexcited from the highest occupied valence band into the lowestunoccupied conduction band, because these bands have oppositespin. Therefore, carriers thermally excited above the spin gap ESOshould result in an increase of conductivity with increasing tempera-ture, which is precisely happening in the temperature range labeled Bin Fig. 2d. In regime C (Edis < ESO < Eth), the thermal energy surpassesthe SOC gap, causing the conductivity to saturate again.Based on the results presented above, we attribute the depen-dence of conductivity on displacement field, density and temperatureto the presence of spin–orbit-induced gaps in the spin-polarized bandsin the absence of a global band gap.(B, D) phase diagramIn the final section of this work, we describe magnetotransport mea-surements at CN. Figure 3a illustrates the longitudinal conductivity σxxas a function of the out-of-plane magnetic field (B⊥) and displacementfield at a temperature of T ≈ 30mK. With the exception of the lowmagnetic field peaks atDc, which we discuss below, the phase diagramdepicted in Fig. 3a bears a resemblance to that of pristine BLG37,38.Drawing on previous studies37,39, we partition the parameter space intothree distinct regions.Phases (I) and (II), occurring at large displacement and magneticfields, respectively, are anticipated to mirror the behavior of the BLGsystem in the absence of SOC. This is attributed to the dominance ofenergy scales dictated by the externally applied parameters (B⊥ andD)over the SOC gap ΔI/2. Hence, we attribute phase (I) to the layer-polarized insulating state arising from the band gap induced by thedisplacement field, as illustrated in Fig. 2c. Phase (II) represents theinsulating state of the quantum Hall ν =0 state. In this phase, our biasspectroscopy measurements uncover the presence of a gap Eg∝B⊥(Fig. 3d), which qualitatively explains the observed B�1? suppression ofthe conductivity (see dotted and dashed lines in Fig. 3c). This behavioraligns with the canted antiferromagnetic phase observed in pristineBLG40. Moreover, the boundaries between Phase (I) and (II) (indicatedby white dotted lines and elaborated in detail in SupplementaryNote 10) exhibit common characteristics with those observed in pris-tine BLG: the insulator-insulator transition features enhancedconductance39,41, and the displacement field required to induce thetransition is D*(B)∝ 2e2B/h37,40,42,43.Phase (III), emerging at B =0 and D =0, is expected to differ fromBLG samples not in proximity with a TMD layer, due to the dominantSOC energy scale. Interestingly, a similar phase has been observed bySeiler et al.44 in BLG/WSe2 heterostructures and by Island et al.14 inWSe2/BLG/WSe2. In these cases, the insulating phase was attributed toFig. 3 | Magnetotransport data at CN. a Conductivity as a function of B⊥ and D atT = 30mK. This plot represents the (D, B) phase diagram of SOC proximitized BLG.Phase (I) is a gapped phase with a layer-polarized wave function. Phase (II) isanother gapped phase that has been attributed to the canted antiferromagneticphase in pristine BLG41,62. The phase boundary between phases (I) and (II) (straightdotted lines) is described in Supplementary Note 10. Phase (III) is a weakly insu-lating phase that arises from the presence of spin-gaps in the single-particle spec-trum. b σxx plotted against D for B⊥ =0 and two values of in-plane magnetic field:B∥ =0 (violet) and B∥ = 1.8 T (orange). c Linecuts of panel (a) at D =0 (violet) andD =Dc (orange). The black dashed and dotted lines highlight the 1/B dependence ofthe conductivity. d Differential conductance Gd = dI/dVSD as a function of the vol-tage bias VSD and magnetic field B⊥ at D =0. The dotted line highlights Eg∝ B⊥ (seediscussion in Supplementary Note 11).Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 4www.nature.com/naturecommunicationsa band gap induced either by electron correlations or SOC, respec-tively. Initially, the thermal activation in Phase (III) seems to supportthe presence of a gap. However, the temperature dependence can beexplained by spin–orbit gaps in spin-polarized bands without a globalband gap, as we have discussed above. In fact, while a clear gap isobserved at finite magnetic fields, no gap is present at B = 0 (Fig. 3d),ruling out the existence of a global gap.Now, we examine the sharpmagnetoconductivity peaks atDc (seean orange curve in Fig. 3c), a novel feature of spin–orbit proximitizedBLG not previously reported. With current theoretical models unableto fully explain these peaks, we explore various possibilities.At first glance, the sharp peak in the orange curve in Fig. 3cresembles the signature of WAL, expected in materials with strongSOC. This effect has been observed in numerous transport experi-ments in SOC-proximitized graphene13,25,29,30,45–47. However, with amean-free-path exceeding 1μm at finite density, the condition ℓϕ > ℓe(where ℓϕ and ℓe represent the phase-coherence length andmean-free-path, respectively) required to observe this effect would never befulfilled (ℓϕ ≤ 360nm if fitting the peak with a WAL model, as detailedin the Supplementary Note 12). Furthermore, the magnitude of thepeak (~3−4e2/h) exceeds what would be expected for WAL, whichtypically reaches up to 0.5e2/h per conducting channel. Additionally,quantum interference effects are typically suppressed with increasingtemperature due to the decrease in ℓϕ. In contrast, the magnitude ofthe peak in σxx remains robust against temperature changes (seeSupplementaryNote 12). For these reasons, we conclude that thepeakscannot arise from WAL.Typically, distinguishing between how a magnetic field affectsorbital or spin degrees of freedom involves tilting the fieldwith respectto the plane. Orbital effects couple exclusively to B⊥, while spin cou-ples to ∣B∣. In Fig. 3b, we compare the conductivity at B⊥ =0 for B∥ = 0and B∥ = 1.8 T (the maximum available in our system), where no sig-nificant effect is observed on σxx. The lack of an in-planemagnetic fielddependence is consistent with the presence of Ising SOC, which isexpected to align spins out-of-plane. Therefore, for an in-plane mag-netic field dependence in conductivity to occur, the Zeeman energyΔEZ = 2μB∣B∣ would need to become comparable to the spin–orbit gapΔI/2, estimated to occur at B > 6.7 T. In our experiments, the con-ductivity drops by nearly a factor of 2 at B⊥ ≈ 50mT. This magneticfield corresponds to aZeeman energyof only 6μeV,much smaller thanthe disorder. Therefore, it is unlikely that the Zeeman effect could beresponsible for the observed magnetoconductivity peaks.Since we could not find a suitable theoretical model relying solelyon free-electron physics, we speculate that the non-monotonic mag-netic field dependence of the conductivity at D = ±Dc (Fig. 3c) origi-nates from many-body effects at CN. Electron interactions, which arestrong near CN due to the lack of screening, were predicted to driveinstability towards an excitonic insulator phase, in which carriers invalleys K+ and K− display strong particle–hole correlations48–52. Pre-vious measurements, while reporting some promising results on gapopening at CN39,53, were not conclusive. This could be due to, amongother reasons, a reduction in exchange interactions in the valley sectorin the presence of spin degeneracy. In the present system, with spindegrees of freedom polarized by the SOC, carrier exchange respon-sible for themany-body physics at CN is expected to become stronger.Taking this as a starting point and assuming a correlated excitonicorder at CN similar to that occurring in quantum Hall bilayers (hererepresented by the K+ and K− valleys), we interpret the nonmonotonicbehavior seen in Fig. 3c as a transition between a valley-coherent orderand a valley polarized order. Formally this transition is analogous tothe transition between excitonic layer-coherent and layer-polarizedphases investigated in quantumHall bilayers54. Here it is driven by twocompeting mechanisms. On the one hand, themagnetic field interactswith the orbital magnetic moments arising from the Berry curvature55.The energy related to these orbital magnetic moments in a magneticfield is ΔEMðξÞ= �M � B= ξ g*vμBB, where g*v is an effective valleyg-factor, and ξ = ±1 is the valley index56,57. In nanostructures, the valleyg-factor has been reported to range from ~10 to above 100, dependingon the confinement potential58, thus significantly larger than the usualspin Zeeman energy. This valley-dependent energy shift leads to thelifting of the valley degeneracy, favoring a valley-polarized state. Onthe other hand, valley polarization is associatedwith layer polarizationonce Landau levels form, introducing an energy cost for the polariza-tion known as the capacitor-like Hartree energy37,38,43:EHðBÞ=14πε0εBLG2e2c0‘2B, ð2Þwhere εBLG is the dielectric constant of BLG and c0 is the distancebetween the two graphene layers. This energy counteracts the mag-netization energy, thus promoting valley-unpolarized states. Thesemechanisms compete with each other and might contribute to theobserved non-monotonic behavior in the conductivity as a function ofthe magnetic field.DiscussionIn this study, we demonstrated that two types of SOC are present inspin–orbit proximitized BLG. Despite the similar magnitudes of thetwo SOC terms, the band splitting at zero displacement field showslittle dependence on the total density, indicating that the Ising SOCpredominantly influences the splitting within the density range underinvestigation. Our results align with previous observations of Isingsuperconductivity in WSe2/BLG heterostructures15,16, suggesting thepotential for similar phenomena to occur in MoS2/BLG systems.Furthermore, we observed an insulating phase atD =0, leading toa non-monotonic electrical conductivity with respect to the displace-ment field. Insulating phases with a similar displacement field depen-dence have been also observed in charge neutral suspended BLG39,albeit with an intrinsic SOC two orders of magnitude weaker than inour sample9,59. While suspended BLG exhibits a gap at B = 0 andD =053,attributed to many-body correlations, our sample does not show thisbehavior (Fig. 3d), suggesting a different underlying mechanism. Theabsence of such correlated phases in hBN-encapsulated Bernal BLGsuggests that dielectric and gate screening effects may reduce therelevance of correlation phenomena. Thus, we conclude that SOCplays a crucial role in the emergence of the observed insulating phaseat D =0. This assertion aligns with findings by Island et al.14, whoreported a comparable insulating phase in BLG fully encapsulated inWSe2. While their explanation relied on SOC-driven band inversion,our observations suggest an alternative explanation, specificallysingle-particle SOC-induced gaps in spin-polarized bands in theabsenceof a global bandgap (Fig. 1e). Our conclusion is supportedby adetailed analysis of the SOC strength, a comparison between the bandstructure calculations and the displacement field dependence, as wellas temperature-dependent measurements.While the zero magnetic field data are understood in terms ofsingle-particle physics, we could not find a suitable theoretical modelto describe the data at the finite magnetic field. We speculate that thenon-monotonic magnetic field dependence of σxx atD = ±Dc originatesfrom many-body effects at CN. If this interpretation holds true, thesystem described here could serve as a platform to explore variousintriguing effects anticipated for excitonic phases, such as vortices,merons, and the Josephson effect for charge-neutral particles.Note from the authors. While preparing our manuscript, webecame aware of a related study by Seiler et al.44, who investigated theinterplay between SOC and Coulomb interaction in WSe2/BLG het-erostructures, drawing conclusions on the phase diagram of SOC-proximitized BLG. It is remarkable that very similar data was obtainedby two different groups, using a different TMD on bilayer graphene(MoS2 by our research group and WSe2 by Seiler et al.). While weArticle https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 5www.nature.com/naturecommunicationsattribute this observation to a single-particle effect (spin–orbit gaps inthe spin-polarized bands), Seiler et al. offer a different interpretationbased on electron-electron correlations.MethodsSample fabricationWe initiate the fabrication of our devices by assembling the hetero-structure using a polymer-based dry transfer technique. Each layer isobtained through mechanical exfoliation of bulk crystals onto silicon/silicon dioxide wafers. The heterostructure comprises, from top tobottom, hBN, bilayer graphene (BLG), three layers of MoS2, hBN, andgraphite.The relative alignment of BLG with the MoS2 layer is known toinfluence the strength of the SOC20,21. While themaximum induced SOCis anticipated around 15°–20°, the SOC is most stable against smalluncontrolled twist angle variations at 0°, ensuring better reproduci-bility. Therefore, during the fabrication process, we carefully align theedges of the MoS2 and BLG flakes, resulting in potential relative align-ments of 0° or 30°. At 30°, the SOC proximity is expected to vanish,leading us to conclude that the relative angle in our sample is 0°.Subsequently, the sample undergoes annealing in a hydro-argonatmosphere (H2/Ar:5%/95%) at 350 °C for 4 h to remove polymer resi-dues and enhance adhesion between the layers. The metallic top gateis defined using standard electron-beam lithography, followed byelectron-beam evaporation (chromium/gold) and lift-off processes.The mesa is dry-etched using a reactive ion etching process with aCHF3:O2 mixture (40:4).In the final fabrication step, metallic edge contacts are depositedusing electron-beam lithography, followed by electron-beam eva-poration (chromium/gold) and lift-off processes. After resistingdevelopment, we clean the contact area using an O2 reactive ionetching process before metal deposition. This ensures the resultingcontacts are ohmic and low resistive (<1 kΩ).Dual-gated deviceWe employ a dual gate structure that allows for independent tuning ofthe charge carrier densities n and displacement field D. The density isdefined asn=1eCBVBG +CTVTG� �+n0, ð3Þand the displacement field is defined asD=12CBVBG � CTVTG� �+D0, ð4Þwhere CB = 36.7 nF/cm2 and CT = 78.2 nF/cm2 are the capacitance perarea of the bottom and top gate, VBG and VTG are the voltages appliedto the bottom and top gate. Additionally, n0 = −6.3E10 cm−2 and D0/ε0 = −46mV/nm are offsets in the density and displacement field,respectively. These offsets are taken into account to compensate forthe asymmetries arising from factors such as the contact potentialdifference between hBN and MoS217.MeasurementsThe measurements were performed in a pumped Helium-4 cryostat(for temperatures above 1 K) or in a dilution refrigerator with a basetemperature < 10mK (estimated electronic temperature≈ 30mK).The four-terminal resistance was measured with constant inputcurrent, by using a series resistor of 10 or 100MΩ, depending on theresistance of the sample. The input voltage was generated at a fre-quency of roughly 31Hz with a Lock-in amplifier. The current ampli-tude ranged from 1 to 50nA.Thebias spectroscopymeasurementsweredone in a two-terminalsetup, where a DC voltage source was employed to generate thesource–drain bias and a home-made voltage-to-current converter wasused to detect the source–drain current.Tight-binding modelTo determine the band structure, we employ a four-band effectivetight-binding model for BLG in the basis (A1, B1, A2, B2), where A, B arethe twoatoms in the unit cell of a single graphene layer, and their indexrepresents the layer number36:H0 =�U=2 v0πy �v4πy v3πv0π �U=2+Δ γ1 �v4πy�v4π γ1 U=2 +Δ v0πyv3πy �v4π v0π U=20BBB@1CCCA, ð5Þwhere π = ℏ(ξkx + iky), π† = ℏ(ξkx−iky), U is the inter-layer potentialenergy difference, Δ is an energy difference between dimer andnon-dimer atoms, and vj =ffiffi3pa2_ γj . The parameters γj are theSlonczewski–Weiss–McClure (SWM) parameters given in the “Meth-ods” section (Table 1).We include the extrinsic SOC given by Eq. (1). The SOC lifts thespin degeneracy but does not mix states from different K-valleys.Therefore, the Hamiltonian becomes an 8 × 8 matrix with the basis(A1↑, A1↓, B1↑, B1↓, A2↑, A2↓, B2↑, B2↓). Since only layer 1 is in directcontact with the MoS2 layer, the SOC is taken into account only in thetop-left 4 × 4 block:HSO =HL1SO 00 0 !ð6ÞThe Ising and Rashba SOC components lead to the following HL1SO inmatrix form:ξ ΔI2 0 0 �i ΔRðξ�1Þ20 �ξ ΔI2 i ΔRðξ + 1Þ2 00 �i ΔRðξ + 1Þ2 ξ ΔI2 0i ΔRðξ�1Þ2 0 0 �ξ ΔI20BBBBB@1CCCCCA: ð7ÞIn the ordered basis (A1↑, A1↓, B1↑, B1↓, A2↑, A2↓, B2↑, B2↓), thefull Hamiltonian takes the form:H =H0 +HSO=ΔIξ2 � U2 0 v0πy �i ΔRðξ�1Þ2 �v4πy 0 v3π 00 � ΔIξ2 � U2 i ΔRðξ + 1Þ2 v0πy 0 �v4πy 0 v3πv0π �i ΔRðξ + 1Þ2 Δ+ ΔIξ2 � U2 0 γ1 0 �v4πy 0i ΔRðξ�1Þ2 v0π 0 Δ� ΔIξ2 � U2 0 γ1 0 �v4πy�v4π 0 γ1 0 Δ+ U2 0 v0πy 00 �v4π 0 γ1 0 Δ+ U2 0 v0πyv3πy 0 �v4π 0 v0π 0 U2 00 v3πy 0 �v4π 0 v0π 0 U20BBBBBBBBBBBBBBBB@1CCCCCCCCCCCCCCCCA:ð8ÞTable 1 | Values of the Slonczewski–Weiss–McClure (SWM)parameters in electron-Volt (eV)SWM parameters γ0 γ1 γ3 γ4 ΔExp.60 3.0 0.40 0.3 0.15 0.018Th.61 2.61 0.361 0.283 0.138 0.015Theexperimental values areobtained fromfits to infrareddata (ref. 60). Thesecond rowprovidesthe theoretical parameters obtained by ab initio calculations based on local density approx-imation (LDA)61. In this work, we use the experimental values to calculate the band structure.Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 6www.nature.com/naturecommunicationsBands and density of states. The bands are then obtained bynumerically diagonalizingH =H0 +HSO. Eachband is characterizedby aband index m, which labels the bands from the most negative (m =0)to the most positive (m = 7) energies.The density of states of band m is given byDmðEÞ=1AXξ ,kδðE � Emξ,kÞ ð9Þwhere ξ is the valley quantumnumber, andk is thewave vector.A = L2 isthe area in real space. Thedelta function is approximatedby aGaussianfunctionδðE � Em, ξ ,kÞ �1ffiffiffiffiffiffi2πpϵexp � ðE � Em, ξ ,kÞ22ϵ2 !, ð10Þwith an energy broadening of ϵ < 50μeV. The band structure Em,ξ,k iscalculated on a grid in k space with finite resolution Δk ~ 1 × 105m−1.Therefore the sum needs to be renormalized by the factorΔk2π=L� �2: ð11ÞEquations (9), (10) and (11) yield the density of states of band m:DmðEÞ=Δk2π� �2Xξ ,k1ffiffiffiffiffiffi2πpϵexp �ðE � Em, ξ ,kÞ22ϵ2 !: ð12ÞThe total density of states is obtained by summing over the bandindex m.The electron density is obtained by integrating over the conduc-tionband (m ≥ 4),while thehole density is obtainedby integratingoverthe valence band (m < 4)neðEFÞ =P7m=4R EF�1 DmðEÞdEnhðEFÞ =P3m=0R EF1 DmðEÞdE:ð13ÞOut of the 8 bands, we only consider the four low energy bands(2 ≤m ≤ 5), m = 4, 5 for the conduction band and m = 2, 3 for thevalence band. The total density is obtained by summation:nðEFÞ=neðEFÞ � nhðEFÞ: ð14ÞAccording to our definition, hole doping corresponds to a negativedensity.Data availabilitySource data and analysis scripts generated in this study have beendeposited in the ETH Research Collection under accession codehttps://doi.org/10.3929/ethz-b-000662935.References1. Burkard, G., Ladd, T. D., Pan, A., Nichol, J. M. & Petta, J. R. Semi-conductor spin qubits. Rev. Mod. Phys. 95, 025003 (2023).2. Rashba, E. I. & Efros, A. L. Orbital mechanisms of electron-spinmanipulation by an electric field. Phys. Rev. Lett. 91, 126405(2003).3. Khoo, J. Y., Morpurgo, A. F. & Levitov, L. On-demand spin-orbitinteraction from which-layer tunability in bilayer graphene. NanoLett. 17, 7003–7008 (2017).4. Avsar, A. et al. Colloquium: spintronics in graphene and other two-dimensional materials. Rev. Mod. Phys. 92, 021003 (2020).5. Trauzettel, B., Bulaev, D. V., Loss, D. & Burkard, G. Spin qubits ingraphene quantum dots. Nat. Phys. 3, 192–196 (2007).6. Gächter, L. M. et al. Single-shot spin readout in graphene quantumdots. PRX Quantum 3, 020343 (2022).7. Garreis, R. et al. Long-lived valley states in bilayer graphene quan-tum dots. Nat. Phys. 1–7 (2024). https://www.nature.com/articles/s41567-023-02334-7.8. Denisov, A. O. et al. Ultra-long relaxation of a Kramers qubit formedin a bilayer graphene quantum dot. Preprint at http://arxiv.org/abs/2403.08143 (2024).9. Kane, C. L. & Mele, E. J. Quantum spin hall effect in graphene. Phys.Rev. Lett. 95, 226801 (2005).10. Sichau, J. et al. Resonancemicrowavemeasurements of an intrinsicspin–orbit coupling gap in graphene: a possible indication of atopological state. Phys. Rev. Lett. 122, 046403 (2019).11. Duprez, H. et al. Spectroscopy of a single-carrier bilayer graphenequantum dot from time-resolved charge detection. Preprint athttp://arxiv.org/abs/2311.12949 (2023).12. Avsar, A. et al. Spin-orbit proximity effect in graphene. Nat. Com-mun. 5, 4875 (2014).13. Wang, Z. et al. Origin and magnitude of ‘designer’ spin–orbitinteraction in graphene on semiconducting transition metaldichalcogenides. Phys. Rev. X 6, 041020 (2016).14. Island, J. O. et al. Spin–orbit-driven band inversion in bilayer gra-phene by the van der Waals proximity effect. Nature 571,85–89 (2019).15. Zhang, Y. et al. Enhanced superconductivity in spin-orbit prox-imitized bilayer graphene. Nature 613, 268–273 (2023).16. Holleis, L. et al. Ising superconductivity and nematicity in Bernalbilayer graphenewith strong spin–orbit coupling. Preprint at http://arxiv.org/abs/2303.00742 (2023).17. Gmitra, M. & Fabian, J. Proximity effects in bilayer graphene onmonolayer WSe2: field-effect spin valley locking, spin–orbit valve,and spin transistor. Phys. Rev. Lett. 119, 146401 (2017).18. Rashba, E. I. Graphene with structure-induced spin–orbit coupling:spin-polarized states, spin zero modes, and quantum hall effect.Phys. Rev. B 79, 161409 (2009).19. Khoo, J. Y. & Levitov, L. Tunable quantum Hall edge conduction inbilayer graphene through spin–orbit interaction. Phys. Rev. B 98,115307 (2018).20. Li, Y. & Koshino, M. Twist-angle dependence of the proximity spin-orbit coupling in graphene on transition-metal dichalcogenides.Phys. Rev. B 99, 075438 (2019).21. David, A., Rakyta, P., Kormányos, A. & Burkard, G. Inducedspin–orbit coupling in twisted graphene–transition metal dichal-cogenide heterobilayers: Twistronics meets spintronics. Phys. Rev.B 100, 085412 (2019).22. Naimer, T., Zollner, K., Gmitra, M. & Fabian, J. Twist-angle depen-dent proximity induced spin–orbit coupling in graphene/transitionmetal dichalcogenide heterostructures. Phys. Rev. B 104,195156 (2021).23. Chou, Y.-Z., Wu, F. & Das Sarma, S. Enhanced superconductivitythrough virtual tunneling in Bernal bilayer graphene coupled toWSe2. Phys. Rev. B 106, L180502 (2022).24. Yang, B. et al. Strong electron-hole symmetric Rashba spin–orbitcoupling in graphene/monolayer transition metal dichalcogenideheterostructures. Phys. Rev. B 96, 041409 (2017).25. Zihlmann, S. et al. Large spin relaxation anisotropy and valley-Zeeman spin–orbit coupling in WSe2/graphene/h-BN hetero-structures. Phys. Rev. B 97, 075434 (2018).26. Wang, D. et al. Quantum Hall effect measurement of spin-orbitcoupling strengths in ultraclean bilayer graphene/WSe2 hetero-structures. Nano Lett. 19, 7028–7034 (2019).27. Ingla-Aynés, J., Herling, F., Fabian, J., Hueso, L. E. & Casanova, F.Electrical control of valley-zeeman spin–orbit-coupling–inducedspin precession at room temperature. Phys. Rev. Lett. 127,047202 (2021).28. Sun, L. et al. Determining spin–orbit coupling in graphene by qua-siparticle interference imaging. Nat. Commun. 14, 3771 (2023).Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 7https://doi.org/10.3929/ethz-b-000662935https://www.nature.com/articles/s41567-023-02334-7https://www.nature.com/articles/s41567-023-02334-7http://arxiv.org/abs/2403.08143http://arxiv.org/abs/2403.08143http://arxiv.org/abs/2311.12949http://arxiv.org/abs/2303.00742http://arxiv.org/abs/2303.00742www.nature.com/naturecommunications29. Wang, Z. et al. Strong interface-induced spin–orbit interaction ingraphene on WS2. Nat. Commun. 6, 8339 (2015).30. Wakamura, T. et al. Spin–orbit interaction induced in graphene bytransition metal dichalcogenides. Phys. Rev. B 99, 245402 (2019).31. Fülöp, B. et al. Boosting proximity spin–orbit coupling in graphene/WSe2heterostructures via hydrostatic pressure.npj 2DMater. Appl.5, (2021). https://www.nature.com/articles/s41699-021-00262-9.32. Benítez, L. A. et al. Tunable room-temperature spin galvanic andspin hall effects in van der Waals heterostructures. Nat. Mater. 19,170–175 (2020).33. Bergemann, C. Fermi surface measurements. In Encyclopedia ofCondensed Matter Physics (eds Bassani, F., Liedl, G. L. & Wyder, P.)185–192 (Elsevier, 2005).34. Rhodes,D.,Chae, S.H., Ribeiro-Palau, R. &Hone, J. Disorder in vanderWaalsheterostructuresof2dmaterials.Nat.Mater. 18, 541–549 (2019).35. Yankowitz,M.,Ma,Q., Jarillo-Herrero, P. & LeRoy, B. J. van derWaalsheterostructures combining graphene and hexagonal boronnitride. Nat. Rev. Phys. 1, 112–125 (2019).36. McCann, E. & Koshino, M. The electronic properties of bilayer gra-phene. Rep. Prog. Phys. 76, 056503 (2013).37. Nandkishore, R. & Levitov, L. Spontaneously ordered states inbilayer graphene. Phys. Scr. 2012, 014011 (2012)38. Knothe, A. & Jolicoeur, T. Phase diagramof a graphenebilayer in thezero-energy Landau level. Phys. Rev. B 94, 235149 (2016).39. Weitz, R. T., Allen, M. T., Feldman, B. E., Martin, J. & Yacoby, A.Broken-symmetry states in doubly gated suspended bilayer gra-phene. Science 330, 812–816 (2010).40. Kharitonov, M. Antiferromagnetic state in bilayer graphene. Phys.Rev. B 86, 195435 (2012).41. Maher, P. et al. Evidence for a spin phase transition at charge neu-trality in bilayer graphene. Nat. Phys. 9, 154–158 (2013).42. Gorbar, E. V., Gusynin, V. P. & Miransky, V. A. Energy gaps at neu-trality point in bilayer graphene in a magnetic field. JETP Lett. 91,314–318 (2010).43. Tőke, C. & Fal’ko, V. I. Intra-landau-level magnetoexcitons and thetransition between quantum hall states in undoped bilayer gra-phene. Phys. Rev. B 83, 115455 (2011).44. Seiler, A. M. et al. Layer-selective spin–orbit coupling and strongcorrelation in bilayer graphene. Preprint at http://arxiv.org/abs/2403.17140 (2024).45. Wakamura, T. et al. Strong anisotropic spin–orbit interaction inducedin graphene by monolayer ws2. Phys. Rev. Lett. 120, 106802 (2018).46. Tiwari, P., Srivastav, S. K. & Bid, A. Electric-field-tunable valleyZeeman effect in bilayer graphene heterostructures: realization ofthe spin–orbit valve effect. Phys. Rev. Lett. 126, 096801 (2021).47. Amann, J. et al. Counterintuitive gate dependence of weak antilo-calization in bilayer graphene/WSe2 heterostructures. Phys. Rev. B105, 115425 (2022).48. Kharitonov, M. Y. & Efetov, K. B. Electron screening and excitoniccondensation in double-layer graphene systems. Phys. Rev. B 78,241401 (2008).49. Nandkishore, R. & Levitov, L. Dynamical screening and excitonicinstability in bilayer graphene. Phys. Rev. Lett. 104, 156803 (2010).50. Zhang, F., Min, H., Polini, M. & MacDonald, A. H. Spontaneousinversion symmetry breaking in graphene bilayers. Phys. Rev. B 81,041402 (2010).51. Kharitonov, M. Y. & Efetov, K. B. Excitonic condensation in a double-layer graphene system. Semicond. Sci. Technol. 25, 034004 (2010).52. Throckmorton, R. E. & Das Sarma, S. Quantum multicriticality inbilayer graphene with a tunable energy gap. Phys. Rev. B 90,205407 (2014).53. Velasco, J. et al. Transport spectroscopy of symmetry-broken insu-lating states in bilayer graphene.Nat. Nanotechnol. 7, 156–160 (2012).54. Eisenstein, J. Exciton condensation in bilayer quantumhall systems.Annu. Rev. Condens. Matter Phys. 5, 159–181 (2014).55. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronicproperties. Rev. Mod. Phys. 82, 1959–2007 (2010).56. Overweg, H. et al. Topologically nontrivial valley states in bilayergraphene quantum point contacts. Phys. Rev. Lett. 121,257702 (2018).57. Banszerus, L. et al. Spin-valley coupling in single-electron bilayergraphene quantum dots. Nat. Commun. 12, 5250 (2021).58. Lee, Y. et al. Tunable valley splitting due to topological orbitalmagnetic moment in bilayer graphene quantum point contacts.Phys. Rev. Lett. 124, 126802 (2020).59. Konschuh, S., Gmitra, M., Kochan, D. & Fabian, J. Theory ofspin–orbit coupling in bilayer graphene. Phys. Rev. B 85,115423 (2012).60. Zhang, L. M. et al. Determination of the electronic structure ofbilayer graphene from infrared spectroscopy. Phys. Rev. B 78,235408 (2008).61. Jung, J. &MacDonald, A. H. Accurate tight-bindingmodels for the πbands of bilayer graphene. Phys. Rev. B 89, 035405 (2014).62. Kharitonov, M. Canted antiferromagnetic phase of the ν = 0 quan-tum hall state in bilayer graphene. Phys. Rev. Lett. 109,046803 (2012).AcknowledgementsWe thank Thomas Weitz, Anna Seiler, and Patrick Lee for fruitful dis-cussions.We thank PeterMärki, ThomasBähler, as well as the FIRST stafffor their technical support.We acknowledge support from the EuropeanGraphene Flagship Core 3 Project, the Swiss National Science Founda-tion via NCCR Quantum Science, and the H2020 European ResearchCouncil (ERC) Synergy Grant under Grant Agreement 95154. N.J.acknowledges funding from the International Center for AdvancedStudies of Energy Conversion (ICASEC). K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grant Numbers 20H00354 and23H02052) and World Premier International Research Center Initiative(WPI), MEXT, Japan.Author contributionsM.M., H.D., T.I., and K.E. conceived and designed the experiments. M.M.,M.G., and F.F. performed and analyzed the measurements with inputsfrom H.D., J.D.G., and M.N. M.M. designed the figures with inputs fromC.T. and H.D. M.M. andM.G. fabricated the device with inputs from J.G.,M.N., and H.D. A.P., N.J., and L.L. provided theoretical support. T.T. andK.W. supplied the hexagonal boron nitride. M.M. wrote the manuscriptwith inputs from H.D. All the coauthors mentioned above read andcommented on the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-53324-z.Correspondence and requests for materials should be addressed toMichele Masseroni or Klaus Ensslin.Peer review information Nature Communications thanks the anon-ymous, reviewer(s) for their contribution to the peer review of this work.A peer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 8https://www.nature.com/articles/s41699-021-00262-9http://arxiv.org/abs/2403.17140http://arxiv.org/abs/2403.17140https://doi.org/10.1038/s41467-024-53324-zhttp://www.nature.com/reprintswww.nature.com/naturecommunicationsOpen Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to theCreative Commons licence, and indicate if you modified the licensedmaterial. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-53324-zNature Communications |         (2024) 15:9251 9http://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/naturecommunications Spin-orbit proximity in MoS2/bilayer graphene heterostructures Results Proximity induced spin–orbit coupling Conductivity at charge neutrality (B, D) phase diagram Discussion Methods Sample fabrication Dual-gated device Measurements Tight-binding model Bands and density of states Data availability References Acknowledgements Author contributions Competing interests Additional information