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[Emma C. Regan](https://orcid.org/0000-0002-9100-6031), Zheyu Lu, Danqing Wang, [Yang Zhang](https://orcid.org/0000-0003-4630-5056), [Trithep Devakul](https://orcid.org/0000-0002-4129-897X), Jacob H. Nie, [Zuocheng Zhang](https://orcid.org/0000-0001-7851-6101), Wenyu Zhao, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Sefaattin Tongay](https://orcid.org/0000-0001-8294-984X), Alex Zettl, [Liang Fu](https://orcid.org/0000-0002-8803-1017), [Feng Wang](https://orcid.org/0000-0001-8369-6194)

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[Spin transport of a doped Mott insulator in moiré heterostructures](https://mdr.nims.go.jp/datasets/d1767b8a-f706-4062-b111-683da1ba03fe)

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Spin transport of a doped Mott insulator in moiré heterostructuresArticle https://doi.org/10.1038/s41467-024-54633-zSpin transport of a doped Mott insulator inmoiré heterostructuresEmma C. Regan 1,2,3,12, Zheyu Lu1,2,3,12, Danqing Wang1,2,3,12, Yang Zhang 4,5,6,Trithep Devakul 4, Jacob H. Nie7, Zuocheng Zhang 1, Wenyu Zhao1,Kenji Watanabe 8, Takashi Taniguchi 9, Sefaattin Tongay 10, Alex Zettl1,3,11,Liang Fu 4 & Feng Wang 1,3,11Moiré superlattices of semiconducting transition metal dichalcogenide het-erobilayers are model systems for investigating strongly correlated electronicphenomena. Specifically, WSe2/WS2 moiré superlattices have emerged as aquantum simulator for the two-dimensional extended Hubbard model.Experimental studies of charge transport have revealed correlated Mottinsulator and generalizedWigner crystal states, but spin transport of themoiréheterostructure has not yet been sufficiently explored. Here, we use spatiallyand temporally resolved circular dichroism spectroscopy to directly image thespin transport as a function of carrier doping and temperature in WSe2/WS2moiré heterostructures. We observe diffusive spin transport at all hole con-centrations at 11 Kelvin — including the Mott insulator at one hole per moiréunit cell — where charge transport is strongly suppressed. At elevated tem-peratures the spin diffusion constant remains unchanged in theMott insulatorstate, but it increases significantly at finite doping away from the Mott state.The doping- and temperature-dependent spin transport can be qualitativelyunderstood using a t–Jmodel, where spins can move via the hopping of spin-carrying charges and via the exchange interaction.Despite the complexity of strongly interacting quantum systems, themacroscopic dynamics of conserved quantities like spin and chargecan often be described by emergent classical hydrodynamics. At lowtemperatures, transport properties are highly sensitive to the groundstate and the nature of low-lying quasiparticles, long wavelengthorder parameter fluctuations, and the presence of superfluidity. Athigher temperatures, transport can be understood in terms of theNernst–Einstein relation arising from the diffusive spreading ofconserved quantities. While both spin and charge are carried byelectrons, the hydrodynamic coefficients governing spin andcharge diffusion are generally distinct. Understanding how spintransport emerges from a microscopic model is an important ques-tion in studying quantum dynamics in strongly interacting quantummatter.Received: 28 February 2024Accepted: 15 November 2024Check for updates1Department ofPhysics, University ofCalifornia at Berkeley, Berkeley,CA,USA. 2GraduateGroup inAppliedScience andTechnology, University ofCaliforniaatBerkeley, Berkeley, CA, USA. 3Material Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA. 4Department of Physics, MassachusettsInstituteof Technology,Cambridge,MA,USA. 5Department of Physics andAstronomy,University of Tennessee, Knoxville, TN,USA. 6MinH. KaoDepartment ofElectrical Engineering and Computer Science, University of Tennessee, Knoxville, TN, USA. 7Department of Physics, University of California at Santa Barbara,Santa Barbara, CA, USA. 8Research Center for Electronic andOptical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan. 9ResearchCenter for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan. 10Department of Physics, Ma. School forEngineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, USA. 11Kavli Energy NanoSciences Institute at University of CaliforniaBerkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA. 12These authors contributed equally: Emma C. Regan, Zheyu Lu, Danqing Wang.e-mail: liangfu@mit.edu; fengwang76@berkeley.eduNature Communications |        (2024) 15:10252 11234567890():,;1234567890():,;http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0002-9100-6031http://orcid.org/0000-0003-4630-5056http://orcid.org/0000-0003-4630-5056http://orcid.org/0000-0003-4630-5056http://orcid.org/0000-0003-4630-5056http://orcid.org/0000-0003-4630-5056http://orcid.org/0000-0002-4129-897Xhttp://orcid.org/0000-0002-4129-897Xhttp://orcid.org/0000-0002-4129-897Xhttp://orcid.org/0000-0002-4129-897Xhttp://orcid.org/0000-0002-4129-897Xhttp://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://orcid.org/0000-0001-7851-6101http://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/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0001-8294-984Xhttp://orcid.org/0000-0002-8803-1017http://orcid.org/0000-0002-8803-1017http://orcid.org/0000-0002-8803-1017http://orcid.org/0000-0002-8803-1017http://orcid.org/0000-0002-8803-1017http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://orcid.org/0000-0001-8369-6194http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54633-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54633-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54633-z&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-54633-z&domain=pdfmailto:liangfu@mit.edumailto:fengwang76@berkeley.eduwww.nature.com/naturecommunicationsThe moiré superlattice, formed by stacking two-dimensional (2D)materials with a relative twist angle or lattice mismatch, offers a uniqueplatform for studying strongly correlated physics in a highly tunablesetting1–17. Semiconductor transitionmetal dichalcogenide (TMD)moirésuperlattices, such as WSe2/WS2, has emerged as a quantum simulatorof the extended triangular lattice Hubbard model1,2. Crucially, the nar-row moiré bandwidth allows access to a new parameter regime ofdoped Mott insulators, characterized by the separation of scalest≪kBT≪U, typically inaccessible in traditional solid-state materialswhere kBT is usually much lower than t. Here t is the hopping ampli-tude, kBT is the thermal energy, andU is the onsite Coulomb repulsion.Measurements of charge transport in the WSe2/WS2 moiré superlatticehave revealed correlated insulators, including Mott1,2 and generalizedWigner crystal states1, at specific filling of the superlattice unit cell.Yanhao et al., provided direct transport measurements showing resis-tance peak atMott state2. Emma et al., provided an alternative approachnamed optically detected resistance and capacitance (ODRC) andshowed a Mott insulating state as well1. Theories predict that spinphysics, such as quantum spin liquids and superfluid spin transport, canalso emerge in correlated states of moiré heterostructures18–20. Whilethe presence of spin moments with antiferromagnetic interaction hasbeen observed, experimental characterization of spin transport phe-nomena in moiré heterostructures has been challenging so far.Here we use spatial- and temporal-resolved circular dichroismspectroscopy to directly image non-equilibrium spin transport inWSe2/WS2 moiré heterostructures, giving insight into the behavior ofspin in a doped 2D Hubbard model.Results & DiscussionIn the TMD moiré superlattice, the optical selection rules provide anopportunity to conveniently create and probe spin excitations usingcircularly polarized light21–23. A circularly polarized beam generates anexciton population in one valley, which is also spin-polarized due tothe spin–valley locking in TMDs. Although this spin polarization istypically short-lived in TMD monolayers due to the electron-holeexchange interaction, long-lived spin lifetimes can be realized in type IIheterostructures, like WSe2/WS2, where the electron and hole occupyseparate layers24. Importantly, previous studies have shown that onecan optically create a pure spin excitation, with no associated chargeexcitation, that has a lifetime of many microseconds in a hole-dopedWSe2/WS2 heterostructure1,24.Using this approach, we optically generate a local spin populationin the hole-doped WSe2/WS2 moiré superlattice and image their spatialand temporal evolution (Fig. 1b). We observe diffusive spin transportthat is reduced compared to the spin transport in heterostructureswithout a large wavelength moiré superlattice. The spin transportremains significant at the Mott state with one hole per moiré unit cell(p=p0= 1) at 11 K, where charge transport is strongly suppressed1,2. Thisobservation is consistent with spin-charge separation in the Mott state.Furthermore, the spin transport exhibit unusual temperature and dop-ing dependence in themoiré heterostructure. This spin transport can becaptured by a t–J model on a triangular lattice, where spin conductionoccurs via nearest neighbor hopping t of spin-polarized free carriersand via exchange coupling J. At the Mott state, holes are immobile, soantiferromagnetic exchange between charges on neighboring super-lattice sites is responsible for the spin diffusion. When additional elec-trons or holes are doped into the Mott state, the resulting empty anddoubly occupied sites allow for conduction of spin-polarized holes.Here we study the spin transport in a gated WSe2/WS2 moirésuperlattice (device D1, see Supplementary Fig. 1). The WSe2 and WS2monolayers are stacked with near-zero-degree twist angle and encap-sulated in hBN. Few-layer graphite flakes form the electrostatic gateand contact, allowing for the injection of electrons and holes into thedevice. The ~4% lattice mismatch between the WSe2 and WS2 layersresults in a moiré superlattice with a period of aM~8 nm. Figure 1cshows the doping-dependent reflection contrast spectrum of deviceD1 in the energy range around the WSe2 intralayer excitons. The threeabsorption peaks are signatures of the moiré superlattice25, where thestrong moiré potential results in three WSe2 intralayer excitons stateswith different spatial characters. The lowest-energy peak is stronglyenhanced when the heterostructure is doped to one electron or holeper moiré unit cell (p0 = 1.8× 1012 cm−2, see Methods) due to the for-mation of Mott insulator states.We use a spatial- and temporal-resolved pump–probe technique24to image spin transport in a hole-doped WSe2/WS2 moiré superlattice(Fig. 1b, Supplementary Fig. 2). We exploit the unusual ultrafastdynamic processes in the TMD heterostructures to achieve pure spinexcitation with circularly polarized light in hole-doped WSe2/WS2heterostructures24. Briefly, a circularly polarized pump at 1.807 eVselectively excites K valley excitons in the WSe2 layer. Due to thespin–valley locking in TMDs, the K-valley excitons are composed ofspin-up electrons and holes. The electrons transfer to the WS2 layerwithin 100 fs, and their spins depolarize at the nanosecond time scale.The spin-depolarized electrons in the WS2 layer can then recombinewith holes of either spin in the WSe2 layer over ~100 ns. After therecombination, net spin excitation of holes (i.e., more spin-up holesthan spin-down holes) remains in the WSe2/WS2 moiré superlattice.Thenet spin polarization is equivalent to a net valley polarization in theTMD heterostructure due to the spin–valley locking. Therefore, theobserved spin transport can also be viewed as valley transport. WeFig. 1 | Optical generation and detection of spin transport in aWSe2/WS2moirésuperlattice. a An optically-generated 1D spin–valley polarization (up arrows)drives a spin current in the doped moiré superlattice (red circles are holes) at theMott insulator state (one hole permoiré unit cell, p=p0 = 1).b A circularly polarizedpump beam (yellow) creates a local spin polarization in the hole-doped moirésuperlattice. The time-resolved circular dichroism of a probe beam (orange)measures the evolution of the 1D spin polarization in space and time as thepump–probe spatial separation (Δx) is scanned. A voltage V on the capacitorformed by the graphite gate and the moiré superlattice is used to tune the equili-brium hole concentration, p. c Doping-dependent reflection contrast spectrum ofdevice D1, a WSe2/WS2 moiré superlattice. The three moiré exciton states arelabeled as I, II, and III. The probe beam is near-resonant with exciton I.Article https://doi.org/10.1038/s41467-024-54633-zNature Communications |        (2024) 15:10252 2www.nature.com/naturecommunicationsfocus the pump beam into a line to create a one-dimensional (1D) spinexcitation with an FWHM of approximately 1 μm that propagatesthrough the sample. We probe the evolution of the spin polarizationvia the pump-induced circular dichroism of a probe beam at 1.703 eV,near-resonant with the lowest-energy WSe2 exciton. By tuning thespatial separation (Δx) and the temporal delay between the pump andprobe beams, wemonitor the spin population in both space and time.Using the spatial-temporal pump-probe technique, we generateandmonitor a pure spin population in device D1 as we tune the chargeconcentration from near charge neutral to past the Mott state(p=p0 = 1) at 11 K,well below theMott transition temperature of ~150K2.Figure 2a shows the evolution of the spin-polarized holepopulation fora subset of the hole concentrations measured (see Supplementary Fig.3 for full dataset). The horizontal and vertical axes represent thetemporal and spatial separation between the pump and probe pulses,respectively. At zero-time delay, the optically-generated spin polar-ization is localized at the pump position (Δx < 1 μm), and the circulardichroism is negligible away from the pump. After a finite time delay,the spin-polarized holes move out of the excitation region, and thecircular dichroism increases away from the pump. Notably, spintransport is observed at all doping levels, including at p=p0 = 1 wherethe system is electrically insulating. Therefore, the spin transport isdecoupled from the charge transport in the Mott insulator at p=p0 = 1.Like in large-twist angle heterostructures24, the spin transport inthe moiré superlattice is diffusive for all measured hole concentra-tions. The spin-polarized hole density Δps x, tð Þ can be described by adiffusion-decay model:Δps x, tð Þ= Δp0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ σ20 + 4Dst� �q e� x2σ20+4Ds te�tτ ð1Þwhere Δp0 is the total number of pump-induced spin-polarized holes,and σ0 is the FWHM of the pump beam, Ds is the spin diffusionconstant, and τ is the spin lifetime. x and t are the space and timecoordinates respectively. We fit the experiment data according to thediffusion-decay model based on the nonlinear least squares method.From the fitting results, we get the fitted parameters along with theirconfidence intervals. We achieve > 0.9 R2 and <10−4 root-mean-squareerror (RMSE) levels across all fittings which convinces the quality ofour fittings (Supplementary Fig. 5 and Supplementary Table 1).Figure 2b and c show the measured circular dichroic (CD) signal(circles) and corresponding fits to the diffusion-decaymodel (lines) forp=p0 = 0.5 and p=p0 = 1, respectively. The doping-dependent spinlifetime has been reported1, and our result is shown in SupplementaryFig. 6. At low hole concentration, the spin lifetime increases withdoping because the spin lifetime becomes decoupled from the chargelifetime24. The spin lifetime is very long at the Mott insulator states1,followed by a rapid decrease at higher doping. The detailed spinlifetime behavior shows some variation between different samples1,possibly due to variations in defect density, residue strain, orinhomogeneities. Currently, there is no microscopic understandingof the spin lifetime behavior in moiré heterostructures. In this paper,we focus on the spin-diffusive transport.Figure 2d shows the extracted spin diffusion constant at varioushole concentrations in the superlattice at 11 K, which shows a smoothdependence on the hole concentration in the WSe2/WS2 moiré het-erostructure. Notably, the spin diffusion constant does not exhibit anoticeable decrease at the Mott insulator state. This is in strikingcontrast to the electrical transport behavior of the moiré hetero-structure, where pronounced suppression of conductance is observedat the Mott insulator state in previous studies1,2. It provides signaturesof the decoupling of the spin and charge transport in the MottFig. 2 | Diffusive spin transport in a dopedWSe2/WS2 moiré superlattice. a Thespatial-temporal evolution of a spin excitation in the WSe2/WS2 heterostructure atvarious hole concentrations, labeled in units of p=p0. The horizontal and verticalaxes represent the temporal and spatial separation between the pump and probepulses, respectively. The color represents the circular dichroic signal on a log scale.b, c Horizontal cuts from a (dots) and fits the diffusion-decay model (lines) forp=p0 = 0.5 (b) and p=p0 = 1 (c). The spin transport is well described by a diffusion-decay model for all hole concentrations. d Extracted spin diffusion constant atvarious initial hole concentrations in the moiré superlattice. Error bars represent95% confidence intervals obtainedwhen fitting the data to a diffusion-decaymodel.Article https://doi.org/10.1038/s41467-024-54633-zNature Communications |        (2024) 15:10252 3www.nature.com/naturecommunicationsinsulator state. The lack of suppression of spin diffusion constant atthe Mott state indicates that the spin transport is dominated by spin-spin couplings rather than diffusion, and the effect from disorder-induced itinerant holes is small. Otherwise, one expects to observe astrong increase of the spin diffusion constant from the higher itineranthole concentration away from theMott density, similar to the increaseof the charge conductance from the higher itinerant hole concentra-tion away from the Mott density.Figure 3 compares the doping-dependent spin diffusion constantfor device D1 to a second near-zero twist angle WSe2/WS2 hetero-structure D2, and a large twist angle WSe2/WS2 heterostructure. Thetwo near-zero twist anglemoiré superlattice samples show similar spindiffusion constants for a given hole concentration, which are morethan an order of magnitude smaller than that of the large twist angleheterostructure. The reduction of spin diffusion in the moiré super-lattice is a manifestation of strongly correlated electron states in themoiré superlattice flat bands.Wemeasure the doping-dependent spin transport at two elevatedtemperatures of 22 K and 45 K. The extracted spin diffusion constantsare shown in Fig. 4. At the Mott state, highlighted in grey, the spindiffusion dependsweakly on temperature.When temperature changesfrom 11 K to 22 K (100% change) and 45 K (309% change), the diffusionconstant increases only by 11% and 53% respectively for the Mott state(p=p0 = 1). When additional electrons or holes are doped to the Mottinsulator, the spin diffusion becomes notably faster at elevated tem-peratures. For example, the spin diffusion constant increases by 130%and 317% respectively for p=p0 = 1.3 and 296% and 1078% respectivelyfor p=p0 = 0.2 when the temperature increases from 11 K to 22 Kand 45K.We can understand this doping and temperature-dependent spintransport in the moiré heterostructures through an effective Hubbardmodel on the triangular lattice26. For WSe2/WS2, the hopping ampli-tude t is much smaller than the on-site Coulomb repulsion U27. In thisregime, for ν ≤ 1, the Hubbard model reduces to the t–Jmodel,HtJ = �tPhiji,σðcyiσcjσ +h.c.Þ+ JPhijið~Si �~Sj �ninj4 Þ which captures the essentialphysics. Here ν is the filling factor of the moiré superlattice,~Si is thespin-12 operator given by ~Si =Pσσ0ciσσσσ02 ciσ0 and ni is the particlenumber operator given by ni =Pσcyiσciσ, where σ is the Pauli matrix,and ciσ are electron annihilation/creation operators on the site i. Athigher density ν > 1, the essential physics can again be captured by aneffective t–J model at filling 2� ν, but with modified parameters.Theoretical calculation of the spin diffusion constant is highlychallenging in such systems but is possible under certain assumptionsand limits28–37. We consider the spin diffusion constant from the t–Jmodel via the Nernst–Einstein relation Ds = σs=χs, where σs and χs arethe d.c. spin conductivity and compressibility in equilibrium. Weobtain analytic expressions for Ds in the high-temperature limit of thet–J model (corresponding to the realistic separation of scalest≪kBT≪U of the Hubbard model) in two limits, given byD2s =9πδ2t2a424�20δ�4δ2 J =09πJ2a4 1�2δ + δ2� �16 7�5δð Þ t =08><>:ð2Þwhere δ = ν � 1j j is the doping away from 1. Some physical insight canbe gained from these expressions. For small δ, we find Ds=t �ffiffiffiffiffi3π8qδincreases with δ in the J =0 limit, which can be understood as anincrease in the number of itinerant spin-current carriers. In the t =0limit, Ds=J �ffiffiffiffiffi9π112q1� 914 δ� �instead decreases with δ, which can beunderstood as a decrease in the number of local spinmoments. At hightemperatures, the observed increase of Ds away from the Mottinsulator indicates that the dominant contribution to spin transport isvia the increase of itinerant spin carriers, in agreement with thetheoretical expectation that J=t =4t=U≪1.We further address the general temperature and doping depen-dence through exact diagonalization, and indeed find that theobserved trends can be described by realistic model parameters. Weextract an exchange coupling J � 0:1 meV from themeasuredDs at theMott insulator. Here, the weak temperature dependence and Einsteinrelation implies a T-linear spin resistivity ρs = Dsχs� ��1 � T , since thespin compressibility χs � 1=ð4TÞ for T≫J. The asymmetric increase ofDs upon doping away from theMott insulator implies that the effectivehopping amplitude t is much larger for ν>1 than for ν<1. A possibleexplanation is that holes for ν>1 fill a charge transfer band27, which canagain be described by a t–Jmodel with an enhanced hopping throughthe physics of Zhang and Rice38. An alternative possible explanation iselectron-assisted hopping due to Coulomb repulsion within a moirésite that effectively increases the size of the localized orbital andtherefore the hopping integral39.The complex interplay of spin and charge physics in dopedquantum magnets has received recent theoretical and experimentalattention in the platform of ultracold atomic Hubbard modelsimulators40. Spin transport has been observed in 1D spin chains in thelow41,42 and high43 temperature regimes, and in the 2D square latticeHubbard model near half filling28,44. Here we have directly visualizedspin in a highly non-equilibrium setting, approaching the level ofcontrol possible in ultracold atomic simulators but in a solid-stateFig. 3 | Spin diffusion in aligned and misaligned WSe2/WS2 heterobilayers.Extracted spin diffusion constant from two near-zero-twist angle moiré super-lattices (black and blue dots) and a large-twist-angle heterobilayer24 (green circles).The spin diffusion is strongly suppressed in a moiré superlattice.Fig. 4 | Temperature-dependent spin diffusion. The doping-dependent diffusionconstants are extracted from spatial-temporal pump-probemeasurements at threetemperatures: 11 K, 25K, and45K.The circles arefits to thediffusionmodel, and thelines are guides to the eye. The Mott insulator at p=p0 = 1 is highlighted in grey. Aweak temperature dependence is observed at the Mott insulator.Article https://doi.org/10.1038/s41467-024-54633-zNature Communications |        (2024) 15:10252 4www.nature.com/naturecommunicationsplatform. Conclusive demonstration and quantitative understandingof the spin-charge separation behavior could be achieved with furtherexperimental studies, where simultaneous electrical and spin trans-port measurements can be performed on TMD heterostructures withdifferent combinations of TMDs and twist angles.MethodsDevice fabricationTo fabricate the WSe2/WS2 moiré superlattices, we first exfoliate WS2andWSe2monolayers frombulk crystal onto SiO2/Si substrates and dopolarization-resolved second harmonic generation (SHG) measure-ments to determine the crystal orientation in each flake. We thenassemble the WS2 and WSe2 monolayers into a heterostructure withtheir crystal axes aligned using a polycarbonate (PC) stamp. During thetransfer process, the near-zero-degree-twist-angle WSe2/WS2 stack iscontacted by few-layer graphite (FLG) and sandwiched between twohexagonal boron nitride (hBN) layers with thickness of 15-25 nm.Additional FLG flakes serve as electrostatic gates. The whole hetero-structure is released onto a 90 nm SiO2/Si substrate. Electrodes(100nm Au with 5 nm Cr adhesion layer) are fabricated using a pho-tolithography system (Durham Magneto Optics, MicroWriter) and anelectron-beam deposition system. After fabrication, we again performpolarization-resolved SHG measurements on the monolayer WS2 andWSe2 regions within the heterostructure to determine the exact twistangle and on the heterostructure region to distinguish between near-zero and near-sixty-degree samples.Doping-dependent spatial-temporal pump–probe techniqueA function generator (Siglent SDG6022X) is used to generate twosynchronized electronic pulse trains that drive two RF-coupled laserdiodemodules (Thorlabs LDM56) with center wavelengths of ~690nm(pump) and ~730 nm (probe). The probe laser wavelength is finetunedto be near-resonant with the lowest energy WSe2 A exciton using thetemperature-controlled mount. The pump beam travels through apolarizer and shared quarter waveplate (QWP) to generate a left cir-cularly polarized pump. The probe beam travels through a polarizer,rotating half-wave plate (HWP), and the sharedQWP to switchbetweenleft- and right- circularly polarized probe. The pump and probe beamsare combined with a dichroic mirror and focused onto the samplemounted in a Montana Instruments cryostation using an objective. Anadditional cylindrical lens in the pump path ensures that the pumpbeam is focused on a line on the sample. The pump–probe spatialseparation is tuned via a piezo-controlledmirror in the pumppath. Forall pump-probe measurements, the pump power is set such that thephotoexcited hole population is small compared to the electro-statically injected hole population.The reflected beams are spectrally filteredwith a 715 nm long-passfilter to isolate the probe, which is then monitored with an avalanchephotodiode (APD). The APD output is sent to a Keithley 2400 Sour-ceMeter (measures the static reflectivity, RC) and a SRS864A lock-inamplifier (measures thepump-induced change in the reflectivity,ΔRC).The lock-in amplifier is locked to the frequency of the optical chopperin the pump path.Doping dependent spin transport measurements are conductedby applying voltages to the graphite gate using Keithley 2400 or 6482SourceMeters. The charge density p, is defined using a parallel platecapacitor model:p= ±1eϵhBNϵ0dV � V0� � ð3Þwhere ϵhBN is the dielectric constant of hBN (measured as 4:2 ±0:2), ϵ0is the permittivity of free space, d is the thickness of the bottom hBNlayer, and V is the voltage applied to the gate. We account for thequantum capacitance (voltage range where charges are not injected)using an offset voltageV0 for electron andhole doping. The offsets aredefined by the voltage where the reflection contrast spectrum beginsto change with electron and hole doping.The moiré density p0 is defined to correspond to one hole permoiré unit cell and is determined through the relationp0 = 1=½L2M sinðπ=3Þ� where LM is the moiré superlattice constant. Thetwist angle (θ) and lattice mismatch (δ = ða� a0Þ=a) between the twolayers determines LM via LM =a=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδ2 +θ2p: STM measurements of anear-zero degree WSe2/WS2 give LM � 8 nm, which is consistent withthat expected due to the lattice constantmismatch between the layers.Therefore, for zero-degree-twist angle samples, p0 = 1:80× 1012 cm−2.Data availabilityThe main data that support the findings of this study are availablewithin the article and its Supplementary Information files. More sup-porting data are available from the corresponding authors uponrequest. Source data are provided with this paper.References1. Regan, E. C. et al. 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Lett. 113, 147205 (2014).AcknowledgementsThis work was supported primarily by the Director, Office of Science,Office of Basic Energy Sciences, Materials Sciences and EngineeringDivision, of the US Department of Energy under contract no.DE-AC02−05-CH11231, within the van der Waals Heterostructures Pro-gram (KCWF16), which provided for the design of the project, devicefabrication, and spin transport experiments. Additional support wasprovided by the Director, Office of Science, Office of Basic Energy Sci-ences, Materials Sciences and Engineering Division, of the US Depart-ment of Energy under contract no. DE-AC02-05-CH11231, within theNanomachines Program (KC1203), which provided for supplementarymicroscopy. The work at Massachusetts Institute of Technology wassupported by the U.S. Department of Energy, Office of Science, BasicEnergy Sciences, under award no. DE-SC0020149. S.T. acknowledgessupport from DOE-SC0020653, NSF DMR 2111812, NSF DMR 1552220,NSF 2052527, DMR 1904716, and NSF CMMI 1933214 for WS2 bulkcrystal growth andanalysis. K.W. andT.T. acknowledge support from theJSPS KAKENHI (Grant Numbers 19H05790 and 20H00354).Author contributionsF.W. conceived the research. E.C.R., Z.L. and D.W. carried out opticalmeasurements. E.C.R., Z.L., D.W. and F.W. performed data analysis. Y.Z.,T.D. and L.F. performed theoretical calculations. E.C.R., Z.L., D.W., J.N.,Z.Z., W.Z. and A.Z. contributed to the fabrication of van der Waals het-erostructures. S.T. grewWSe2 andWS2 crystals. K.W. and T.T. grew hBNcrystals. All authors discussed the results and wrote the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-54633-z.Correspondence and requests for materials should be addressed toLiang Fu or Feng Wang.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 availableReprints 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.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-54633-zNature Communications |        (2024) 15:10252 6https://doi.org/10.1038/s41467-024-54633-zhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Spin transport of a doped Mott insulator in moiré heterostructures Results & Discussion Methods Device fabrication Doping-dependent spatial-temporal pump–probe technique Data availability References Acknowledgements Author contributions Competing interests Additional information