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Qing Rao, Wun-Hao Kang, Hongxia Xue, Ziqing Ye, Xuemeng Feng, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Ning Wang, Ming-Hao Liu, Dong-Keun Ki

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[Ballistic transport spectroscopy of spin-orbit-coupled bands in monolayer graphene on WSe2](https://mdr.nims.go.jp/datasets/9f95f6d7-07cd-45e5-9b63-c2c87381cb4b)

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Ballistic transport spectroscopy of spin-orbit-coupled bands in monolayer graphene on WSe2Article https://doi.org/10.1038/s41467-023-41826-1Ballistic transport spectroscopy ofspin-orbit-coupled bands in monolayergraphene on WSe2Qing Rao1,6, Wun-Hao Kang2,6, Hongxia Xue 1, Ziqing Ye3, Xuemeng Feng3,Kenji Watanabe 4, Takashi Taniguchi 5, Ning Wang 3, Ming-Hao Liu 2,7 &Dong-Keun Ki 1,7Van der Waals interactions with transition metal dichalcogenides were shownto induce strong spin-orbit coupling (SOC) in graphene, offering great pro-mises to combine large experimental flexibility of graphene with unique tun-ing capabilities of the SOC. Here, we probe SOC-driven band splitting andelectron dynamics in graphene on WSe2 by measuring ballistic transversemagnetic focusing. We found a clear splitting in the first focusing peak whoseevolution in charge density and magnetic field is well reproduced by calcula-tions using the SOC strength of ~ 13meV, and no splitting in the second peakthat indicates stronger Rashba SOC. Possible suppression of electron-electronscatterings was found in temperature dependence measurement. Further, wefound that Shubnikov-de Haas oscillations exhibit a weaker band splitting,suggesting that it probes different electron dynamics, calling for a new theory.Our study demonstrates an interesting possibility to exploit ballistic electronmotion pronounced in graphene for emerging spin-orbitronics.The interfacial interactions with semiconducting transition metaldichalcogenides (TMDCs)have shown tobe highly efficient in inducingstrong spin-orbit coupling (SOC) in graphene1–3. For monolayer gra-phene, it was theoretically predicted to have twodistinctive terms, onethat couples out-of-plane spin and valley degrees of freedom (referredto as a spin-valley Zeeman term τzsz) and another that couples in-planespin and sublattice degrees of freedom, similar to the Rashba termðτzσxsy � σysxÞ, as follows:H =H0 +Δσz + λτzsz + λR τzσxsy � σysx� �, ð1Þwhere H0 is the graphene’s Dirac Hamiltonian, σ = ðσx ,σy,σz Þ is a Paulimatrix vector that acts on the sublattice degree of freedom in gra-phene, s = ðsx ,sy,sz Þ is a Pauli matrix vector that acts on spin, andτz = ± 1 identifies two different valleys in graphene (Δ≈0 due to a largelattice mismatch between the graphene and TMDCs)2,4–7. Both SOCterms induce band splitting in graphene, and their strengths (λ and λR)can be further tuned by an electric field perpendicular to the layers8–13,twisting14,15, or by pressure16.Combined with the high electronmobility and large experimentalflexibility of graphene, such a strong interface-induced SOCmakes thegraphene on TMDCs ideal for ballistic spin-orbitronics where ballisticelectronmotion canbe used to control or detect electron spin throughthe SOC17–22. It is particularly interesting as graphene has shown pro-nounced ballistic transport effects with large tunability, such astransverse magnetic focusing (TMF)23–25, Veselago lensing26,27, Fabry-Pérot interference28,29, and ballistic snake states30,31 among many oth-ers. They have also shown unique features originating from theReceived: 17 April 2023Accepted: 20 September 2023Check for updates1Department of Physics andHK Institute of QuantumScience & Technology, The University of Hong Kong, PokfulamRoad, Hong Kong, China. 2Department ofPhysics and Center for Quantum Frontiers of Research and Technology (QFort), National Cheng Kung University, Tainan 70101, Taiwan. 3Department ofPhysics and Center for Quantum Materials, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077 Hong Kong, China.4Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5Research Center forMaterials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 6These authors contributed equally: Qing Rao,Wun-Hao Kang. 7These authors jointly supervised this work: Ming-Hao Liu, Dong-Keun Ki. e-mail: minghao.liu@phys.ncku.edu.tw; dkki@hku.hkNature Communications |         (2023) 14:6124 11234567890():,;1234567890():,;http://orcid.org/0009-0004-4935-3074http://orcid.org/0009-0004-4935-3074http://orcid.org/0009-0004-4935-3074http://orcid.org/0009-0004-4935-3074http://orcid.org/0009-0004-4935-3074http://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/0000-0002-4902-5589http://orcid.org/0000-0002-4902-5589http://orcid.org/0000-0002-4902-5589http://orcid.org/0000-0002-4902-5589http://orcid.org/0000-0002-4902-5589http://orcid.org/0000-0001-5602-4181http://orcid.org/0000-0001-5602-4181http://orcid.org/0000-0001-5602-4181http://orcid.org/0000-0001-5602-4181http://orcid.org/0000-0001-5602-4181http://orcid.org/0000-0002-4638-2038http://orcid.org/0000-0002-4638-2038http://orcid.org/0000-0002-4638-2038http://orcid.org/0000-0002-4638-2038http://orcid.org/0000-0002-4638-2038http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-41826-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-41826-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-41826-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-41826-1&domain=pdfmailto:minghao.liu@phys.ncku.edu.twmailto:dkki@hku.hkrelativistic nature of Dirac electrons23–31. However, a vast number ofprevious studies on graphene-TMDC heterostructures have focusedon detecting spin relaxation due to electron scattering rather than theballistic motion1–3,7,16,32–37. Moreover, only a few studies have founddirect evidence for the SOC-induced band splitting by measuringbeatings in Shubnikov-de Hass (SdH) oscillations (for both mono- andbilayer graphene)3,10,38 or tracing changes in quantum capacitance (forbilayer graphene only)11. Not only to understand the effect of the SOCon the electronic properties of the system, such as the bandtopology39,40 but also to exploit its full potential on ballisticspintronics17–22, it is therefore essential to demonstrate the ballistictransport in graphene on TMDCs while simultaneously probing theirband structures and electron dynamics.To fill this missing link, we employ TMF technique in monolayergraphene on WSe2 as it can not only probe the SOC-induced bandsplitting but also investigate electron dynamics simultaneously (seeFig. 1a–d). TMF occurs when ballistic carriers injected from a narrowaperture (“injector”) at the edge of the sample are subject to a smallperpendicular magnetic field (B=Bz)41–43. Owing to the Lorentz force,the carriers follow skipping cyclotron orbits and focus on anothernarrow aperture (“collector”) at a distance (L) that equals an integermultiple of 2rc with a cyclotron radius rc = _kF=eB, where _ is thereduced Planck constant, kF is the Fermi momentum, and e is theelementary charge. Upon sweeping magnetic fields, the collector vol-tage will exhibit a set of resonance peaks at certain B-values deter-mined by kF ,Bj = ±2j_ej jL kF , ð2Þwhere j is an integer and ± represents electron and hole for the con-figuration shown in Fig. 1b. This enables the detection of Fermi surfaceconfigurations41–43. In the systems with multiple bands23, for instance,there will be multiple sets of resonance peaks at different B-valuesfrom which one can deduce their band structures. Moreover, TMFcan also be used to study or control electron dynamics as chargecarriers follow skipping cyclotron orbits during the process. In 2Delectron gas (2DEG) systems with SOC, the TMF was indeed used toprobe spin-orbit split bands and extract SOC strength by studying theseparation of the peaks17,19,21, deduce spin polarization by comparingtheir heights17, or focus spin-polarized current by controlling ballisticelectron motion17,19,20.Fig. 1 | Sample characteristics and TMFmeasurement scheme. a The schematicof monolayer graphene-multilayer WSe2 heterostructures. b Scanning electronmicroscope image of the device with a TMF measurement configuration (a scalebar: 2μm; a distance between the injector a and the collector c: L≈4:0 μm; a probewidth:w ≈0:3 μm). The two semicircles (S± ) illustrate trajectories of the carriers atdifferent spin-orbit-coupled bands S± shown in c under perpendicular magneticfield B. c The energy dispersion of graphene on WSe2 derived from the effectiveHamiltonian Eq. (1) in themain text using SOC strengths of λ= λR =8:9 meV ðλSOC �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 + λR2q= 12:5 meVÞ. The inset shows the simulated local current density map atthe focusing peak marked by a down triangle in d. d The corresponding TMFspectra, the conductance G between the injector and collector as a function ofmagnetic field, calculated for an effective three-terminal device at 6 different holedensities, n (in 1012cm−2) = −0.78,−0.93, −1.09, −1.24, −1.40, and −1.56 (frombottomto top) using the tight-binding model. See Supplementary Note 1 for more details.e The carrier density (n) dependence of the four-probe resistances Rxx of twodifferent samples 1 (blue solid line) and 2 (red broken line) measured at 1.5 K. Bothexhibit a sharp resistance peak at zero density, indicating high device quality. Theinset shows the non-local Hall resistance Rae,bf � Vbf =Iae� �, exhibiting a largenegative signal on the hole side originating from the ballistic transport. f TheLandau fan—the log Rxx� �as a function of n and B—plotted in a color scale (thedarker color corresponds to the lower resistance), showing a high-quality quantumHall effectmeasured at 1.5 K.On the hole side, the broken-symmetry states begin toappear at ~3 T (indicated by black down-triangles), which indicates higher qualityon the hole side, consistent with the large negative Rae,bf on the hole side shown inthe inset of e.Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 2Here, we demonstrate that all these studies are possible ingraphene-TMDCheterostructures.We first show a clear splitting of thefirst focusing peak in graphene on WSe2, whose evolution in chargedensity and magnetic field matches well with the theoretical calcula-tions shown in Fig. 1c, d using the SOC strength of ~13meV (see Sup-plementary Notes 1,2 for simulation details44,45) and no splitting in thesecond peak, which indicates a stronger Rashba SOC in the system.From the temperature dependence measurement, we also find a pos-sible suppression of electron-electron scatterings that may originatefrom the dielectric scattering of WSe2 layers and/or the induced SOC.Further, we show that Shubnikov-de Haas oscillations exhibit a weakerband splitting, suggesting that it probes different electron dynamics.Interestingly, a similar behavior was found in studies on 2DEG withSOC19,21, indicating that this is universal and a new theory is needed toexplain the phenomenon. Our study, therefore, places graphene-TMDC heterostructures as an interesting new material platform toexplore the effect of SOC on ballistic electron transport in low-dimensional systems.ResultsSample characterizationTo study the ballistic TMF effect, we used a dry pick-up and transfertechnique to assemble a stack of hexagonal boron-nitride (hBN),monolayer graphene, and multilayer WSe2 such that the graphene isprotected from the harsh chemical environments in the followingnanofabrication process46. Standard electron beam lithography andlift-off were carried out to make Hall bar devices on 285-nm-thick SiO2substrate with doped silicon underneath used as a gate to controlcharge density n (see Fig. 1a, b and Methods for fabrication details).Electron transport through the fabricated devices was measured in a1.5 K variable temperature insert with a 14-T superconducting magnetusing a standard low-frequency AC lock-in technique (see Methods).Classical and quantum Hall effect measurements were used to esti-mate gate capacitance and confirm that the graphene flakes in oursamples are monolayers (see Supplementary Fig. 1). Figure 1e showsthe carrier density dependence of the four-probe resistances of twodifferent samples 1 and2measured at 1.5 K, exhibitinghighqualitywithcarrier mobility of 200,000 ~ 400,000 cm2V-1s-1. Especially on the holeside, the symmetry broken quantum-Hall states were observed atmagnetic fields as low as ~3 T (marked by black down-triangles inFig. 1f). This indicates higher hole mobility in our sample than on theelectron side. It is also consistent with the observation of the largernegative non-local Hall resistance Rae,bf on the hole side (the inset ofFig. 1e) originating from the ballistic electron motion47(Rαβ,γδ � V γδ=Iαβ which refers to the resistancemeasured by sending acurrent from contact α to β and measuring the voltage between con-tacts γ and δ). Having such a high mobility—equivalently, a long meanfree path—is important to resolve the small splitting of the focusingpeak expected theoretically (Fig. 1d).Transverse magnetic focusing spectraThe TMF signal (Rnl =Vcj=Iai) is measured in a non-local configurationupon varying n and B, as depicted in Fig. 1b. Figure 2a, b show theresulting maps of Rnl n,Bð Þ from sample 1 and 2, respectively. Bothexhibit similar TMF spectra and their evolutions inn and B. Overall, thepositions of the j-th TMF peak in B follow Eq. (2) with kF =ffiffiffiffiffiffiffiffiffiπ nj jp, asexpected for monolayer graphene. However, on the hole side (wherewe found thehigher sample quality),wecan identify the splitting of thefirst focusing peak that evolves continuously inn andB andno splittingin the second (here, we only focus on the peaks that appear in alldensity range). Figure 2c, d furthermagnify the features by plotting 1Dcuts Rnl Bð Þ of the map at differentn, which qualitatively matches thesimulation result shown in Fig. 1d (see Supplementary Fig. 2a for the 1Dcuts of the map on the electron side for comparison). Moreover, dif-ferent from the TMFmeasurements on pristinemonolayer graphene23,we found a large TMF signal constantly exceeding 100 Ω around zerodensity (nearly two orders ofmagnitude larger than the values at finitedensities; see the dark red bands near zero density in the color mapsshown in Fig. 2a, b). All features found in the experiment (Fig. 2)matchwell with our expectations for the graphene with SOC and providevaluable insights about microscopic electron processes in the system,as discussed below.Analysis of the first focusing peakFirst, the first focusing peak splits due to the SOC-induced multiplebands S+ and S� in our sample (Fig. 1c). Note that such a prolongedsplitting in both n and B has not been observed in other graphenesystemswithout SOC23–25.Moreover, wewereable tofit thepositions ofthe first focusing peaks with calculations using the SOC strength ofλSOC �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 + λR2q= 13:9 meV and 12:1 meV for samples 1 and 2respectively asmarked by black dotted lines in Fig. 2a, b. Figure 3a andSupplementary Fig. 2b further emphasize the accuracy of the fitting byplotting the average of the normalized difference between the dataand the calculation hδB2i as a function of λ and λR in a color scale fordevices 1 and 2, respectively (a darker color indicates the smaller hδB2iso the better fitting; see the caption formore details). We note that thefitting works for any values of λ and λR as long as they satisfyλSOC = 13:0±4:7 meV for both samples, indicating that the λ and λRhave a similar effect on the splitting of the first TMF peak—equiva-lently, the band splitting—in the density range we studied. This isexpected as in the density range explored, the Fermi energy is largerthan both λ and λR, leading to an identical splitting in momentum,Δk =2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 + λ2Rq=_vF with Fermi velocity vF≈106 m � s�1 for both λ andλR. This is consistent with the previous SdH oscillationsmeasurementsthat probe electron bands3,38.In addition, we found that the amplitude of the first split peakcloser to zero B is always lower than that of the second (see, e.g.,Fig. 2c, d). Interestingly, in 2DEG with Rashba SOC, such unevenheights of the split peaks have been used as a signature of the spinpolarization in the system, even though more rigorous analysis andmore controlled experiments are needed to identify the origin of thespin polarization17,48–51. For instance, it was suggested that adiabatictransition between the quantized sub-bands formed at the injectorwith width w could polarize electron spin48,51. Slightly modifyingthe condition derived for 2DEG51, we get Δk2kF> 316λFw� �2! λR>3π8_vF λFw2 ≈0:25∼0:35 meV for our sample with Fermi wavelengthλF ≈ 30∼40 nm, and w≈ 300 nm. The SOC strengths estimated in ourstudy are well in the range (Fig. 3a), but due to the absence of thequantum point contacts in our device, we cannot confirm the forma-tion of the sub-bands at the injector. On the other hand, it was alsoshown that the relative heights of the split peaks could vary with thedistance between the injector and detector due to the difference inscattering lengths for carriers at different spin-orbit-split bands50 andthat the exact momentum dependence of the SOC should be con-sidered to understand the spin-polarization49. Interestingly, in gra-pheneonTMDCs, the presenceof both spin-valley Zeeman andRashbaSOC termswas predicted to induce a characteristic spinwinding of thespin-orbit-coupled bands that leads to a current-induced spinpolarization52, whichmay result in the uneven heights of the peaks. Forbetter understanding, we will need to conduct more sophisticatedexperiments, such as applying in-plane magnetic fields to controlZeeman energy17,49,53, using samples with various distances betweenthe injector and detector50, or using ferromagnetic contacts for spin-sensitive detection54. Nevertheless, we note that all these previousstudies have used the uneven heights of the split peaks as a signatureof the spin polarization in the system17,48–51. Thus, our study shows aArticle https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 3possibility of using TMF to detect spin polarization of the ballisticcarriers in graphene-TMDC heterostructures.Analysis of the second focusing peakThe second key finding of this study is the absence of the splitting inthe second focusing peak (Fig. 2), which provides more informationabout the nature of the SOC in the system. It first can be interpreted asthe scattering of charge carriers between the spin-orbit-coupled bandsS± at the sample edge, which leads to a single peak as illustrated in thebottom inset of Fig. 2a. To confirm this origin, we have further calcu-lated electron trajectories for the second peak with or without theinter-band transition in the sample in Supplementary Fig. 3. Asexpected, without the inter-band mixing, the second focusing peakalso exhibits splitting. This confirms that the absence of the splitting inthe second peak originates from the scattering between the bands S±at the sample edge.Interestingly, from the behavior of the second peak, we can learnmore about the relative strength of the spin-valley Zeeman and RashbaSOC terms, λ and λR, because the inter-band scattering at the edgedepends sensitively on the spin textures of the split bands S ± . Formore accessible discussion, let us first consider a single valley only anddiscuss the effect of the intervalley scattering later. As depicted inFig. 3b, c, when only spin-valley Zeeman term exists (in other words,when θSOC =0; see Fig. 3a), spins in S + (S�) band are aligned up (down)in z-direction. Thus, when backscattered at the edge, the electron atstateA in theband S+ will jump to state B in the samebandunless thereare enough magnetic impurities to flip the spin, which is unlikely inhigh-quality graphene samples like ours. This would lead to thesplitting of the second focusing peak as illustrated in Fig. 3b. On theother hand, when the Rashba term dominates (i.e., when θSOC =π=2),S± bands have an opposite spin winding such that the electron at thestate A in the band S+ will jump to the state C in the opposite band,leading to themerging of the peaks as depicted in Fig. 3d, e. Therefore,we can estimate that in our system, the Rashba term dominates.Similarly, studies on 2DEG systems with Rashba SOC have indeedshown no splitting in the second focusing peak19,21,55.To further confirm our analysis, we have simulated TMF spectrafor different values of θSOC in Fig. 3f. As shown in the figure, thesplitting in the second peak disappears rapidly as θSOC increases fromzero and becomes nearly invisible as θSOC≳π=4, consistent with ouranalysis above. It is, however, worthmentioning that in the simulation,we used an ideal edge, so wemay have underestimated the intervalleyscattering probabilities that occur in the real sample edge with atomicdefects56. Although this does not influence the Rashba-dominatingcase as the spinwindingdirection remains the same for the S ± bands indifferent valleys (see Fig. 3e), it can affect the result when the spin-valley Zeeman term dominates because the spin orientation for eachS± band becomes opposite in different valleys (see Fig. 3c). Thus, theintervalley scattering can lead to the scattering from state A at K valleyto state C’ at K’ valley, i.e. the backscattering between the S± bandsat the edge, when the spin-valley Zeeman termdominates, suppressingthe splitting in the second peak. Although more experimental andtheoretical works are required for a complete understanding of thisfeature, we can roughly assume that our sample has disordered edgeswith atomic scale defects with resonance energy near the chargeneutrality57, leading to the intervalley scattering rate close to or less-200    -100      0B (mT)-200  -100     0B (mT)n(1012cm-2)0454035302520151050-250 -200  -150 -100  -50  0      50B (mT)-200  -100    0B (mT)0.0-1.0-2.0n(1012cm-2)abd30201040n(1012cm-2)2.01.00.00 100     200B (mT)0.0-0.5-1.0-1.5c8-80-10   0   10−0.76n (in 1012 cm-2):−0.96−1.16−1.36−1.56−1.76−1.96−2.16−2.36−2.56n (in 1012 cm-2): (−1.56−1.09−0.62Fig. 2 | TMF spectra. aColor-scalemaps of TMF signal Rnl B,nð Þmeasured in sample1 at 1.5 K (top right: electron side; bottom left: hole side). The broken lines show thetheoretically calculated focusing peaks at λSOC = 13:9meV. Inset: carrier trajectoriesfor the first and second focusing peaks (top left and bottom right, respectively).b TMF map and c the corresponding 1D cuts measured in sample 2 at 1.5 K at n(in 1012cm-2) = −0.62, −0.78, −0.93, −1.09, −1.24, −1.40, and −1.56 (from bottom totop). The broken line shows the theoretically calculated focusing peaks atλSOC = 12:0 meV. d 1D cuts of the data from sample 1 shown in a. The black down-triangles in (c) and (d) mark some of the two split peaks for guidance.Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 4than that of the intravalley scattering in the density range explored. Inthis case, wewould still expect to see the splittingwhen θSOC =0. Thus,we believe that the absence of the second peak splitting in Fig. 2indicates the stronger Rashba SOC in our system.Large non-local resistance and temperature dependenceWecan also explain the observed largeRnl near zero density (Fig. 2a, b)as the presence of the spin Hall effect (SHE)1 in the system. From thevery weak temperature dependence of the conductance minimum atzero density (Fig. 4a), we first confirm that this is not from the gapopening at charge neutrality, whichmay have given a large TMF signalas found in the gapped trilayer graphene23. In contrast, we found alarge non-local Hall signal RHnl =Rae,cg near the charge neutrality thatexceeds the ohmic contribution by about 230Ω (Fig. 4b; here we usedHall probes that are further apart from those used in the inset of Fig. 1eto reduce the influence from the ballistic negative resistance). This isconsistent with the SHE found in a similar system1 previously, exceptthat the signal appears near zero density below 2:0∼ 3:0× 1011 cm−2 inour device. We attribute this to the crossover from the diffusiveregime, where the SHE occurs, to the ballistic regime, where the TMFeffect appears. In fact, we found that the density 2:0∼ 3:0× 1011 cm-2coincides well with the value above which the ballistic negative non-local resistance (the inset of Fig. 1e) and TMF signals appear (Fig. 2 andSupplementary Fig. 4) within the resolution of our experiment. Morein-depth studies on the crossover between the diffusive spin-Hall andballistic TMF effects or their coexistence may lead to a better under-standing of the charge transport in spin-orbit-coupled systems, andour study shows that it is possible in high-quality graphene-TMDCheterostructures.Additionally, the observation of both diffusive SHE at low densityand ballistic TMF peak splitting at higher density indicates that theSOC in our sample is induced by proximity with TMDC2,4–7 not bydefects1 as the defects would have strongly suppressed ballistictransport in the sample. It confirms that in graphene-TMDC hetero-structures—thanks to the atomically sharp interface—the atomicpotentials generated by the TMDC can influence graphene bandstrongly to create an effective Hamiltonian with distinctive SOC termsshown in Eq. (1)2,4–7. This is similar to how the atomic potentials ofFig. 3 | Analysis of TMF signal. a The color-scale map of the average differencehδB2i � ðP½ðΔB+ =B0Þ2 + ðΔB =B0Þ2�Þ=N as a function of λ and λR from sample 1 (N:number of data used). ΔB± is the difference between the predicted focusing peakpositions from the simulation for certain (λ, λR) and the real peak positions mea-sured in the experiment for the band S± , whereas B0 is the half of the maximumsplitting observed. Thus, the smaller hδB2i (darker in the map) indicates a betteragreement. A dashed white circle draws the best-fit value. We use a criterionhδB2i≤0:1 to extract the SOC strengths of λSOC = 13:9±4:0meV (and 12:0±3:5meVfor sample 2, see Supplementary Fig. 2b). θSOC in a is defined as cos�1 λ=λSOC� �.b–e Comparison of the electron trajectories for the second focusing peak (in theabsence of the intervalley scattering) and spin configurations for the two caseswhen there is only the spin-valley Zeeman term (b, c; θSOC =0) and when onlyRashba termexists (d, e; θSOC =π=2). The shapes of the resulting focusing peaks foreach case are shown in the inset of b and d. Without intervalley scattering, due tothe spin conservation, the electron at the edge (state A) will be backscattered to Bwhen θSOC =0 (b, c), leading to the splitting in the second peak, whereas whenθSOC =π=2, it will be transferred to state C (d, e). When the intervalley scattering ispresent, the backscattering from state A to C’ can occur even for the θSOC =0 case(c), leading to the suppression of the splitting in the second peak. See themain textfor details. f The calculated TMF spectra with varying θSOC at n= � 1:25 × 1012 cm-2when the overall SOC strength λSOC = 10meV.One can clearly see that the positionsof the first focusing peaks remain the same while the second peak shows multiplepeaks near θSOC =0.Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 5boron and nitrogen atoms in hBN creates the moiré minibands ingraphene-hBN moiré structure58–61. We note that such a proximityeffect does not require charge carriers in graphene to fill the energybands in TMDC (or hBN) and thus it can occur even when there are nodefect sites in TMDC that can sink charge carriers from graphene andsuppress the ballistic transport. Our observation also aligns well withother studies on similar graphene-TMDC heterostructures2,3,7,16,32–38,62.We now examine the temperature dependence of the TMFspectra in Fig. 4c, d to study the electron dynamics in the system.Upon increasing temperature, we found that the amplitude of theTMF spectra decreases (Fig. 4c). This suggests enhanced electronscattering at high temperatures. To identify the main scatteringmechanism in our system, we extracted the total area below thefirst focusing peaks A1 at varying temperatures from 1.5 K to 300 Kand calculated the relative scattering length fromLS=L0 = ðln½ðA1ð1:5KÞ=A1ðTÞ�Þ�1, which is proportional to the effec-tive scattering time (L0 is the length of the semi-circular electrontrajectory corresponding to the first focusing peak)24,25. Figure 4dshows the result exhibiting a clear T�1:8 dependence on bothelectron and hole side for different charge densities. This isbetween electron-phonon scattering (T�1) and electron-electron(e-e) scattering (T�2), indicating that although the e-e scattering isdominant, it is also slightly suppressed in our sample. In com-parison, similar TMF studies on graphene-hBNheterostructures24,25 have shown T�2 dependence in a widerange of temperatures and charge density. Thus, the T�1:8dependence found in our sample should be from the WSe2 itselfand/or the induced SOC. Although the exact origin is unclear, wenote that the WSe2 has a dielectric constant (ε0≈7:9) about twicelarger than hBN used in the previous studies (ε0≈3:8)63 that mayinduce more screening. In addition, recent studies64,65 have shownthat the SOC can affect electronic interaction phenomena such assuperconductivity in twisted or Bernal bilayer graphene. Thissuggests a possibility to measure ballistic transport effects ingraphene-TMDC heterostructures to study the effect of SOC one-e or electron-hole interaction phenomena, such as viscouscharge transport66–68, and electron-hole collisions69, orsuperconductivity64,65.Comparison with Shubnikov–de Haas (SdH) oscillationsTo further elucidate the band splitting in our system, we have mea-sured SdH oscillations at a higher magnetic field range. The results aresummarized in Fig. 5. In all the density ranges including the electronside, we found beatings in the oscillations originating from the spin-orbit-coupled split bands (Fig. 5b). For quantitative analysis, we per-formed fast Fourier transforms (FFT) and extracted the frequency f atwhich the spectra exhibit a peak (Fig. 5a)which is directly connected tothe area of the corresponding Fermi surface by f =nh=2e andn= kF2=2π assuming broken spin degeneracy due to SOC. It cantherefore be used to estimate the SOC strengths independently. Fig-ure 5c shows the result (we have selected the peaks that evolveFig. 4 | Temperature dependencemeasurements. a Temperature dependence ofthe local four-terminal conductance Gxx = 1=Rxx as a function of charge density,exhibiting a weak temperature dependence of the minimum conductance Gmin atzero density as magnified in the inset. bNon-local Hall resistance Rae,cg =Vcg=Iae asa function of carrier density n (solid blue line) compared with the calculated Ohmiccontribution (broken red line), consistent with the spin Hall effect. c Temperaturedependence of the TMF spectra at n= � 2:6× 1012 cm-2. The smooth backgroundsare extracted by aGaussianfilter with a full width at halfmaximumof0.2 T,which islarger than the oscillation period of TMF signals. d The relative scattering lengths(calculated from areas below the first focusing peaks) at n= � 2:6× 1012 cm-2(square), n= � 1:3 × 1012 cm−2 (triangle), and n= 2:6× 1012 cm-2 (circle) as a functionof temperature plotted in a log scale, which follows the T�1:8 dependence, indi-cated by the dashed red line. See the main text for more discussions.Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 6continuously in density only), which offers a good agreement with thecalculation using the SOC strengths along the circleffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ2 + λR2q= 3:4±0:7 meV (Fig. 5d). Interestingly, the fitting is slightlybetter (i.e., hδB2i is smaller) near λ=0 which indicates the largerRashba SOC in the system, consistent with our estimation from theanalysis on the second focusing peak (Fig. 3b–f). The absolute value is,however, about 4 times smaller than those extracted from the TMFdata (Fig. 3a and Supplementary Fig. 2b), indicating that the TMF andSdH oscillations probe different electron dynamics.To compare the results more directly, we have calculatedΔBSdH = 2_= ej jL� �kF1 � kF2�� ��, the expected size of the splitting of thefirst TMF peak in B using the kF -values extracted from the SdH oscil-lations (Fig. 5c), and plotted the splitting measured in TMF, ΔBTMF(extracted from Fig. 2), together in Fig. 5e after normalizing the valueswith the averaged peak positions of the two sub-peaks. As shown in thefigure, ΔBTMF=BTMF remains larger than ΔBSdH=BSdH in all densityrange accessed in the experiment. Interestingly, we found that similarbehavior was observed in 2DEG systems with SOC19,21, where it wassuggested21 that there might be a SOC term, such as a linear-k term,that does not affect the total area of the Fermi surface by shifting acircle in one momentum direction. Since the SdH oscillation requirescarriers to complete a full cyclotron motion while in TMF, they onlymake a half turn, it may be possible that one finds larger splitting inTMF than in SdH, as seen experimentally. However, it is also possiblethat the SdH oscillations require relatively large magnetic fields toform Landau levels which can induce non-negligible Zeeman energyand may affect the spin-valley Zeeman and Rashba terms in Eq. (1)differently (see Supplementary Note 3 formore discussions). Althoughmore studies are required to understand this discrepancy, ourmeasurement shows that the behavior occurs not only in semi-conductor heterostructures-based 2DEG systems19,21 but also in gra-phene when SOC exists. Therefore, there is likely a fundamental originbehind this phenomenon.Comparison with previous studiesIn Fig. 5d, we also include all SOC strengths extracted from the pre-vious measurements on monolayer graphene-TMDC heterostructuresfor comparison. Overall, the relaxation time analysis from weak anti-localization or spin-Hall effect measurements1,32,33,37 shows a con-siderable sample-to-sample variation (Fig. 5d). It can be from the factthat these measurements rely on the model to connect the spinrelaxation process in the system with the SOC strength, which is sen-sitive to the sample-specific electron scattering process7,37. On theother hand, TMF and SdH oscillations directly probe the size ofthe Fermi surface, which can be compared with theoretically calcu-lated band structures without considering details of the scatteringprocesses. The recent studyonSdHoscillation inmonolayer graphene-WSe2 heterostructures38 has indeed shown a SOC strength λSOC =2:51meV close toours (a dotted circle in Fig. 5d). Interestingly, the study onLandau level splitting62, which is closely related to the SdHoscillations,also showed a similar SOC strength (a square in Fig. 5d). Moreover, inour TMF study, we found similar SOC strengths in two different sam-ples (Fig. 3a and Supplementary Fig. 2b). This further elaborates thebenefits of carrying out the (ballistic) transport spectroscopy onunderstanding electronic properties of the system with SOC.DiscussionIn summary, we have successfully demonstrated the ballistic electronmotion in graphene on WSe2 by measuring TMF signals at differentn (in 1012 cm-2): Fig. 5 | SdHoscillations. a The FFT spectra derived from the SdHoscillation curvesas a function of carrier density n, for both electron and hole side. b RepresentativeSdH oscillation curves, which clearly show the beating patterns (marked by arrows;curves are vertically offset for clarity). c The frequency peak positions extractedfrom the FFT spectra at different carrier densities. The solid lines show the fittingwith calculated band structures using λSOC = 3:4 meV. d The color-scale map of theaverage difference hδB2i in λ and λR from the SdHoscillations with the best fit valuedrawn by the dashed white circle (see the caption of Fig. 3a for details). Using thesame criterion used in Fig. 3a hδB2i≤0:1, we get λSOC = 3:4±0:7 meV. Note thataround the circle near λ=0, the color becomes darker, indicating stronger RashbaSOC in the system consistent with discussions in Fig. 3b-f. The values from theprevious studies are included for comparison: a dotted purple circle from spin Halleffect1, a dashed blue circle from SdH oscillations38, a square from Landau levelsplitting62, and circle32, star33, and two filled green triangles37 (two different valuesare obtained from different spin relaxation mechanisms) from weak anti-localization measurements. Some studies1,32,33,62 used λR=2 and/or λ=2 in theHamiltonian Eq. (1), so we divided the values by half in the plot. e The normalizedsplitting ΔB=B of the first TMF peak (ΔBTMF=BTMF ) at different charge densitiesextracted from Fig. 2 compared with the splitting calculated from the SdH oscil-lations (ΔBSdH=BSdH).ΔBTMF andΔBSdH are the sizes of splitting inB, whileBTMF andBSdH are the averaged peak positions of two sub-peaks. Over the whole densityrange, the ΔBSdH=BSdH remains smaller than the ΔBTMF=BTMF .Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 7charge densities, magnetic fields, and temperatures. From the den-sity and magnetic field dependence of the first focusing peaks(Fig. 2), we confirmed that there exist two split bands in the system asexpected theoretically2,4–7 and estimated the SOC strength ofλSOC = 13:0±4:7 meV (Fig. 3a and Supplementary Fig. 2b). Moreinterestingly, by analysing the behavior of the second focusing peakthat shows no splitting and by carrying out quantum transportsimulations, we were able to learn that the Rashba SOC is likelydominant in our system (Fig. 3b-f). Both the presence of the bandsplitting and a stronger Rashba SOC are well reproduced in SdHoscillations measurements (Fig. 5) even though they showed asmaller SOC strength of λSOC =3:4±0:7 meV. A similar discrepancywas found in other 2DEG with Rashba SOC19,21, indicating that there isa fundamental reason behind it. This calls for a new theory.In addition to providing spectroscopic evidence of the spin-orbit-coupled bands, our work demonstrates that graphene on TMDCs cansupport ballistic transport that can be used not only to gain moreinsights into themicroscopic electronprocess in the systembut also toexploit various ballistic transport effects that are pronounced in gra-phene. It isparticularly interesting as, in contrast to the existing studieson graphene spintronics7,22,70,71, TMF separates spin-up and spin-downcarriers in real space. This enables the detection and measurement ofboth spins independently, instead of only the majority one injectedfrommagnetic contacts. This, therefore, offers an alternative venue forgraphene spintronic applications7,22,70,71. Similar strategies can also beused to study other 2Dmaterials or heterostructures with strong SOC,such as bilayer graphene-TMDC heterostructures10–13, blackphosphorus72, and more, which will offer new understandings aboutthe effect of SOC in these material systems.MethodsSample fabricationThe WSe2, hBN, and graphene flakes were exfoliated from corre-sponding crystals onto silicon wafers and examined under an opticalmicroscope. The flakes with suitable thicknesses and surfaces wereselected and assembled onto highly doped silicon substrates with285-nm-thick oxide, following the standard dry pick-up and transfertechnique46. The WSe2 flakes used in this study are around 20 nmand 40 nm for samples 1 and 2, respectively. After the assembly, thestacks were annealed at 250 °C for 2 h in a tube furnace in Ar/H2forming gas. 1D electrical contacts were fabricated on the annealedsample through a standard electron-beam lithography and reactive-ion etching (CF4/O2 mixture gas with flow rates of 5/25 sccm, RFpower: 60W), followed by electron beam evaporation of 5 nm Crand 50 nm Au films. The devices were finally shaped into Hall bars byanother electron-beam lithography and reactive-ion etchingprocess.Electrical measurementDevices were measured in a 1.5 K cryogen-free variable temperatureinsert (VTI) with a superconductingmagnet. The electrical signals weremeasured by applying a small low-frequency (17.777Hz) AC current of0.1–1μA between the source and drain terminals and measuring thevoltage drop between another two probes using a lock-in amplifier(Stanford Research SR830). The low-noise filters and amplifiers wereused to detect small TMF signals. The back gate was controlled byKeithley 2400 source-meter.Data availabilityThe data used in this study are freely available in the figshare databaseat https://doi.org/10.6084/m9.figshare.22644469.Code availabilityCodes to analyze the data and perform numerical calculations areavailable upon reasonable request.References1. Avsar, A. et al. Spin-orbit proximity effect in graphene. Nat. Com-mun. 5, 4875 (2014).2. Wang, Z. et al. Strong interface-induced spin–orbit interaction ingraphene on WS2. Nat. Commun. 6, 8339 (2015).3. Wang, Z. et al. Origin and magnitude of ‘designer’ spin-orbit inter-action in graphene on semiconducting transition metal dichalco-genides. Phys. 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Phys. 13, 888–893 (2017).AcknowledgementsThe work is financially supported by the National Key R&D Program ofChina (2020YFA0309600) and by the University Grants Committee/Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 9Research Grant Council of Hong Kong SAR under schemes of Area ofExcellence (AoE/P-701/20), ECS (27300819), and GRF (17300020,17300521, 17309722). K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 21H05233 and 23H02052) andWorld PremierInternational Research Center Initiative (WPI), MEXT, Japan. N.W.acknowledges support fromWilliamMong Institute of NanoScience andTechnology. W.-H.K. and M.-H.L. gratefully acknowledge National Sci-ence and Technology Council of Taiwan (grant numbers: MOST 109-2112-M-006-020-MY3 and NSTC 112-2112-M-006-019-MY3) for financialsupport and National Center for High-performance Computing (NCHC)for providing computational and storage resources.Author contributionsD.-K.K. conceived and supervised the project. M.-H.L. supervised thetheoretical part carried out by W.-H.K. Q.R. fabricated the samples andperformed the measurements, assisted by H.X. Some of the data wascollected in NW’s group with help from Z.Y. and X.F. Q.R. and W.-H.K.analyzed the data, and D.-K.K. and M.-H.L. interpreted them with inputfrom all authors. T.T. and K.W. synthesized the hBN crystals. Q.R., W.-H.K., M.-H.L., and D.-K.K. wrote the paper with input from all authors. Allauthors discussed the results.Competing interestsThe authors declare they have no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-41826-1.Correspondence and requests for materials should be addressed toMing-Hao Liu or Dong-Keun Ki.Peer review informationNature Communications thanks Jonghwa Eom,Shun-Tsung Lo and the other, anonymous, reviewer(s) for theircontribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2023Article https://doi.org/10.1038/s41467-023-41826-1Nature Communications |         (2023) 14:6124 10https://doi.org/10.1038/s41467-023-41826-1http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Ballistic transport spectroscopy of spin-�orbit-coupled bands in monolayer graphene�on WSe2 Results Sample characterization Transverse magnetic focusing spectra Analysis of the first focusing peak Analysis of the second focusing peak Large non-local resistance and temperature dependence Comparison with Shubnikov–de Haas (SdH) oscillations Comparison with previous studies Discussion Methods Sample fabrication Electrical measurement Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information