Introduction

Despite the complexity of strongly interacting quantum systems, the macroscopic dynamics of conserved quantities like spin and charge can often be described by emergent classical hydrodynamics. At low temperatures, transport properties are highly sensitive to the ground state and the nature of low-lying quasiparticles, long wavelength order parameter fluctuations, and the presence of superfluidity. At higher temperatures, transport can be understood in terms of the Nernst–Einstein relation arising from the diffusive spreading of conserved quantities. While both spin and charge are carried by electrons, the hydrodynamic coefficients governing spin and charge diffusion are generally distinct. Understanding how spin transport emerges from a microscopic model is an important question in studying quantum dynamics in strongly interacting quantum matter.

The moiré superlattice, formed by stacking two-dimensional (2D) materials with a relative twist angle or lattice mismatch, offers a unique platform for studying strongly correlated physics in a highly tunable setting1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17. Semiconductor transition metal dichalcogenide (TMD) moiré superlattices, such as WSe2/WS2, has emerged as a quantum simulator of the extended triangular lattice Hubbard model1,2. Crucially, the narrow moiré bandwidth allows access to a new parameter regime of doped Mott insulators, characterized by the separation of scales \(t\ll {k}_{{{\rm{B}}}}T\ll U\), typically inaccessible in traditional solid-state materials where \({k}_{{{\rm{B}}}}T\) is usually much lower than \(t\). Here \(t\) is the hopping amplitude, \({k}_{{{\rm{B}}}}T\) is the thermal energy, and \(U\) is the onsite Coulomb repulsion. Measurements of charge transport in the WSe2/WS2 moiré superlattice have revealed correlated insulators, including Mott1,2 and generalized Wigner crystal states1, at specific filling of the superlattice unit cell. Yanhao et al., provided direct transport measurements showing resistance peak at Mott state2. Emma et al., provided an alternative approach named optically detected resistance and capacitance (ODRC) and showed a Mott insulating state as well1. Theories predict that spin physics, such as quantum spin liquids and superfluid spin transport, can also emerge in correlated states of moiré heterostructures18,19,20. While the presence of spin moments with antiferromagnetic interaction has been observed, experimental characterization of spin transport phenomena in moiré heterostructures has been challenging so far.

Here we use spatial- and temporal-resolved circular dichroism spectroscopy to directly image non-equilibrium spin transport in WSe2/WS2 moiré heterostructures, giving insight into the behavior of spin in a doped 2D Hubbard model.

Results & Discussion

In the TMD moiré superlattice, the optical selection rules provide an opportunity to conveniently create and probe spin excitations using circularly polarized light21,22,23. A circularly polarized beam generates an exciton population in one valley, which is also spin-polarized due to the spin–valley locking in TMDs. Although this spin polarization is typically short-lived in TMD monolayers due to the electron-hole exchange interaction, long-lived spin lifetimes can be realized in type II heterostructures, like WSe2/WS2, where the electron and hole occupy separate layers24. Importantly, previous studies have shown that one can optically create a pure spin excitation, with no associated charge excitation, that has a lifetime of many microseconds in a hole-doped WSe2/WS2 heterostructure1,24.

Using this approach, we optically generate a local spin population in the hole-doped WSe2/WS2 moiré superlattice and image their spatial and temporal evolution (Fig. 1b). We observe diffusive spin transport that is reduced compared to the spin transport in heterostructures without a large wavelength moiré superlattice. The spin transport remains significant at the Mott state with one hole per moiré unit cell (\(p/{p}_{0}\)= 1) at 11 K, where charge transport is strongly suppressed1,2. This observation is consistent with spin-charge separation in the Mott state. Furthermore, the spin transport exhibit unusual temperature and doping dependence in the moiré heterostructure. This spin transport can be captured by a t–J model on a triangular lattice, where spin conduction occurs via nearest neighbor hopping \(t\) of spin-polarized free carriers and via exchange coupling \(J\). At the Mott state, holes are immobile, so antiferromagnetic exchange between charges on neighboring superlattice sites is responsible for the spin diffusion. When additional electrons or holes are doped into the Mott state, the resulting empty and doubly occupied sites allow for conduction of spin-polarized holes.

Fig. 1: Optical generation and detection of spin transport in a WSe2/WS2 moiré superlattice.
figure 1

a An optically-generated 1D spin–valley polarization (up arrows) drives a spin current in the doped moiré superlattice (red circles are holes) at the Mott insulator state (one hole per moiré unit cell, \(p/{p}_{0}\) = 1). b A circularly polarized pump beam (yellow) creates a local spin polarization in the hole-doped moiré superlattice. The time-resolved circular dichroism of a probe beam (orange) measures the evolution of the 1D spin polarization in space and time as the pump–probe spatial separation (Δx) is scanned. A voltage V on the capacitor formed by the graphite gate and the moiré superlattice is used to tune the equilibrium hole concentration, p. c Doping-dependent reflection contrast spectrum of device D1, a WSe2/WS2 moiré superlattice. The three moiré exciton states are labeled as I, II, and III. The probe beam is near-resonant with exciton I.

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Here we study the spin transport in a gated WSe2/WS2 moiré superlattice (device D1, see Supplementary Fig. 1). The WSe2 and WS2 monolayers are stacked with near-zero-degree twist angle and encapsulated in hBN. Few-layer graphite flakes form the electrostatic gate and contact, allowing for the injection of electrons and holes into the device. The ~4% lattice mismatch between the WSe2 and WS2 layers results in a moiré superlattice with a period of \({a}_{{{\rm{M}}}}\)~8 nm. Figure 1c shows the doping-dependent reflection contrast spectrum of device D1 in the energy range around the WSe2 intralayer excitons. The three absorption peaks are signatures of the moiré superlattice25, where the strong moiré potential results in three WSe2 intralayer excitons states with different spatial characters. The lowest-energy peak is strongly enhanced when the heterostructure is doped to one electron or hole per moiré unit cell (\({p}_{0}\) = 1.8\(\times\)1012 cm−2, see Methods) due to the formation of Mott insulator states.

We use a spatial- and temporal-resolved pump–probe technique24 to image spin transport in a hole-doped WSe2/WS2 moiré superlattice (Fig. 1b, Supplementary Fig. 2). We exploit the unusual ultrafast dynamic processes in the TMD heterostructures to achieve pure spin excitation with circularly polarized light in hole-doped WSe2/WS2 heterostructures24. Briefly, a circularly polarized pump at 1.807 eV selectively excites K valley excitons in the WSe2 layer. Due to the spin–valley locking in TMDs, the K-valley excitons are composed of spin-up electrons and holes. The electrons transfer to the WS2 layer within 100 fs, and their spins depolarize at the nanosecond time scale. The spin-depolarized electrons in the WS2 layer can then recombine with holes of either spin in the WSe2 layer over ~100 ns. After the recombination, net spin excitation of holes (i.e., more spin-up holes than spin-down holes) remains in the WSe2/WS2 moiré superlattice. The net spin polarization is equivalent to a net valley polarization in the TMD heterostructure due to the spin–valley locking. Therefore, the observed spin transport can also be viewed as valley transport. We focus the pump beam into a line to create a one-dimensional (1D) spin excitation with an FWHM of approximately 1 μm that propagates through the sample. We probe the evolution of the spin polarization via the pump-induced circular dichroism of a probe beam at 1.703 eV, near-resonant with the lowest-energy WSe2 exciton. By tuning the spatial separation (Δx) and the temporal delay between the pump and probe beams, we monitor the spin population in both space and time.

Using the spatial-temporal pump-probe technique, we generate and monitor a pure spin population in device D1 as we tune the charge concentration from near charge neutral to past the Mott state (\(p/{p}_{0}\) = 1) at 11 K, well below the Mott transition temperature of ~150 K2. Figure 2a shows the evolution of the spin-polarized hole population for a subset of the hole concentrations measured (see Supplementary Fig. 3 for full dataset). The horizontal and vertical axes represent the temporal and spatial separation between the pump and probe pulses, respectively. At zero-time delay, the optically-generated spin polarization is localized at the pump position (Δx < 1 μm), and the circular dichroism is negligible away from the pump. After a finite time delay, the spin-polarized holes move out of the excitation region, and the circular dichroism increases away from the pump. Notably, spin transport is observed at all doping levels, including at \(p/{p}_{0}\) = 1 where the system is electrically insulating. Therefore, the spin transport is decoupled from the charge transport in the Mott insulator at \(p/{p}_{0}\) = 1.

Fig. 2: Diffusive spin transport in a doped WSe2/WS2 moiré superlattice.
figure 2

a The spatial-temporal evolution of a spin excitation in the WSe2/WS2 heterostructure at various hole concentrations, labeled in units of \(p/{p}_{0}\). The horizontal and vertical axes represent the temporal and spatial separation between the pump and probe pulses, 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) for \(p/{p}_{0}\) = 0.5 (b) and \(p/{p}_{0}\) = 1 (c). The spin transport is well described by a diffusion-decay model for all hole concentrations. d Extracted spin diffusion constant at various initial hole concentrations in the moiré superlattice. Error bars represent 95% confidence intervals obtained when fitting the data to a diffusion-decay model.

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Like in large-twist angle heterostructures24, the spin transport in the moiré superlattice is diffusive for all measured hole concentrations. The spin-polarized hole density \(\Delta {p}_{{{\rm{s}}}}\left(x,t\right)\) can be described by a diffusion-decay model:

$$\Delta {p}_{{{\rm{s}}}}\left(x,t\right)=\frac{\Delta {p}_{0}}{\sqrt{\pi \left({\sigma }_{0}^{2}+4{D}_{{{\rm{s}}}}t\right)}}{e}^{-\frac{{x}^{2}}{{\sigma }_{0}^{2}+4{D}_{{{\rm{s}}}}t}}{e}^{-\frac{t}{\tau }}$$
(1)

where \(\varDelta {p}_{0}\) is the total number of pump-induced spin-polarized holes, and \({\sigma }_{0}\) is the FWHM of the pump beam, \({D}_{{{\rm{s}}}}\) is the spin diffusion constant, and \(\tau\) is the spin lifetime. x and t are the space and time coordinates respectively. We fit the experiment data according to the diffusion-decay model based on the nonlinear least squares method. From the fitting results, we get the fitted parameters along with their confidence intervals. We achieve > 0.9 R2 and <10−4 root-mean-square error (RMSE) levels across all fittings which convinces the quality of our 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-decay model (lines) for \(p/{p}_{0}\) = 0.5 and \(p/{p}_{0}\) = 1, respectively. The doping-dependent spin lifetime has been reported1, and our result is shown in Supplementary Fig. 6. At low hole concentration, the spin lifetime increases with doping because the spin lifetime becomes decoupled from the charge lifetime24. The spin lifetime is very long at the Mott insulator states1, followed by a rapid decrease at higher doping. The detailed spin lifetime behavior shows some variation between different samples1, possibly due to variations in defect density, residue strain, or inhomogeneities. Currently, there is no microscopic understanding of 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 various hole concentrations in the superlattice at 11 K, which shows a smooth dependence on the hole concentration in the WSe2/WS2 moiré heterostructure. Notably, the spin diffusion constant does not exhibit a noticeable decrease at the Mott insulator state. This is in striking contrast to the electrical transport behavior of the moiré heterostructure, where pronounced suppression of conductance is observed at the Mott insulator state in previous studies1,2. It provides signatures of the decoupling of the spin and charge transport in the Mott insulator state. The lack of suppression of spin diffusion constant at the 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 a strong increase of the spin diffusion constant from the higher itinerant hole concentration away from the Mott density, similar to the increase of the charge conductance from the higher itinerant hole concentration away from the Mott density.

Figure 3 compares the doping-dependent spin diffusion constant for device D1 to a second near-zero twist angle WSe2/WS2 heterostructure D2, and a large twist angle WSe2/WS2 heterostructure. The two near-zero twist angle moiré superlattice samples show similar spin diffusion constants for a given hole concentration, which are more than an order of magnitude smaller than that of the large twist angle heterostructure. The reduction of spin diffusion in the moiré superlattice is a manifestation of strongly correlated electron states in the moiré superlattice flat bands.

Fig. 3: Spin diffusion in aligned and misaligned WSe2/WS2 heterobilayers.
figure 3

Extracted spin diffusion constant from two near-zero-twist angle moiré superlattices (black and blue dots) and a large-twist-angle heterobilayer24 (green circles). The spin diffusion is strongly suppressed in a moiré superlattice.

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We measure the doping-dependent spin transport at two elevated temperatures of 22 K and 45 K. The extracted spin diffusion constants are shown in Fig. 4. At the Mott state, highlighted in grey, the spin diffusion depends weakly on temperature. When temperature changes from 11 K to 22 K (100% change) and 45 K (309% change), the diffusion constant increases only by 11% and 53% respectively for the Mott state (\(p/{p}_{0}\) = 1). When additional electrons or holes are doped to the Mott insulator, the spin diffusion becomes notably faster at elevated temperatures. For example, the spin diffusion constant increases by 130% and 317% respectively for \(p/{p}_{0}\) = 1.3 and 296% and 1078% respectively for \(p/{p}_{0}\) = 0.2 when the temperature increases from 11 K to 22 K and 45 K.

Fig. 4: Temperature-dependent spin diffusion.
figure 4

The doping-dependent diffusion constants are extracted from spatial-temporal pump-probe measurements at three temperatures: 11 K, 25 K, and 45 K. The circles are fits to the diffusion model, and the lines are guides to the eye. The Mott insulator at \(p/{p}_{0}\) = 1 is highlighted in grey. A weak temperature dependence is observed at the Mott insulator.

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We can understand this doping and temperature-dependent spin transport in the moiré heterostructures through an effective Hubbard model on the triangular lattice26. For WSe2/WS2, the hopping amplitude \(t\) is much smaller than the on-site Coulomb repulsion \(U\)27. In this regime, for \(\nu \le 1\), the Hubbard model reduces to the t–J model, \({H}_{{tJ}}=-t{\sum }_{\langle {ij}\rangle,\sigma }({c}_{i\sigma }^{{\dagger} }{c}_{j\sigma }+{{\mbox{h.c.}}})+J{\sum }_{\langle {ij}\rangle }({\vec{S}}_{i}\cdot {\vec{S}}_{j}-\frac{{n}_{i}{n}_{j}}{4})\) which captures the essential physics. Here \(\nu\) is the filling factor of the moiré superlattice, \({\vec{S}}_{i}\) is the spin-\(\frac{1}{2}\) operator given by \({\vec{S}}_{i}={\sum }_{\sigma {\sigma }^{{\prime} }}{c}_{i\sigma }\frac{{\sigma }_{\sigma {\sigma }^{{\prime} }}}{2}{c}_{i{\sigma }^{{\prime} }}\) and \({n}_{i}\) is the particle number operator given by \({n}_{i}={\sum }_{\sigma }{c}_{i\sigma }^{{\dagger} }{c}_{i{{\rm{\sigma }}}}\), where \(\sigma\) is the Pauli matrix, and \({c}_{i\sigma }\) are electron annihilation/creation operators on the site \(i\). At higher density \(\nu \, > \, 1\), the essential physics can again be captured by an effective t–J model at filling \(2-\nu\), but with modified parameters.

Theoretical calculation of the spin diffusion constant is highly challenging in such systems but is possible under certain assumptions and limits28,29,30,31,32,33,34,35,36,37. We consider the spin diffusion constant from the t–J model via the Nernst–Einstein relation \({D}_{{{\rm{s}}}}={\sigma }_{{{\rm{s}}}}/{\chi }_{{{\rm{s}}}}\), where \({\sigma }_{{{\rm{s}}}}\) and \({\chi }_{{{\rm{s}}}}\) are the d.c. spin conductivity and compressibility in equilibrium. We obtain analytic expressions for \({D}_{{{\rm{s}}}}\) in the high-temperature limit of the t–J model (corresponding to the realistic separation of scales \(t\ll {k}_{{{\rm{B}}}}T\ll U\) of the Hubbard model) in two limits, given by

$${D}_{{{\rm{s}}}}^{2}=\left\{\begin{array}{cc}\frac{9\pi {\delta }^{2}{t}^{2}{a}^{4}}{24-20\delta -4{\delta }^{2}} & J=0\\ \frac{9\pi {J}^{2}{a}^{4}\left(1-2\delta+{\delta }^{2}\right)}{16\left(7-5\delta \right)} & t=0\end{array}\right.$$
(2)

where \(\delta=\left|\nu -1\right|\) is the doping away from 1. Some physical insight can be gained from these expressions. For small \(\delta\), we find \({D}_{{{\rm{s}}}}/t\approx \sqrt{\frac{3\pi }{8}}\delta\) increases with \(\delta\) in the \(J=0\) limit, which can be understood as an increase in the number of itinerant spin-current carriers. In the \(t=0\) limit, \({D}_{{{\rm{s}}}}/J\approx \sqrt{\frac{9\pi }{112}}\left(1-\frac{9}{14}\delta \right)\) instead decreases with \(\delta\), which can be understood as a decrease in the number of local spin moments. At high temperatures, the observed increase of \({D}_{{{\rm{s}}}}\) away from the Mott insulator indicates that the dominant contribution to spin transport is via the increase of itinerant spin carriers, in agreement with the theoretical expectation that \(J/t=4t/U\ll 1\).

We further address the general temperature and doping dependence through exact diagonalization, and indeed find that the observed trends can be described by realistic model parameters. We extract an exchange coupling \(J\approx 0.1\) meV from the measured \({D}_{{{\rm{s}}}}\) at the Mott insulator. Here, the weak temperature dependence and Einstein relation implies a T-linear spin resistivity \({\rho }_{{{\rm{s}}}}={\left({D}_{{{\rm{s}}}}{\chi }_{{{\rm{s}}}}\right)}^{-1}\sim T\), since the spin compressibility \({\chi }_{{{\rm{s}}}}\approx 1/(4T)\) for \(T\gg J\). The asymmetric increase of \({D}_{{{\rm{s}}}}\) upon doping away from the Mott insulator implies that the effective hopping amplitude \(t\) is much larger for \(\nu > 1\) than for \(\nu < 1\). A possible explanation is that holes for \(\nu > 1\) fill a charge transfer band27, which can again be described by a t–J model with an enhanced hopping through the physics of Zhang and Rice38. An alternative possible explanation is electron-assisted hopping due to Coulomb repulsion within a moiré site that effectively increases the size of the localized orbital and therefore the hopping integral39.

The complex interplay of spin and charge physics in doped quantum magnets has received recent theoretical and experimental attention in the platform of ultracold atomic Hubbard model simulators40. Spin transport has been observed in 1D spin chains in the low41,42 and high43 temperature regimes, and in the 2D square lattice Hubbard model near half filling28,44. Here we have directly visualized spin in a highly non-equilibrium setting, approaching the level of control possible in ultracold atomic simulators but in a solid-state platform. Conclusive demonstration and quantitative understanding of the spin-charge separation behavior could be achieved with further experimental studies, where simultaneous electrical and spin transport measurements can be performed on TMD heterostructures with different combinations of TMDs and twist angles.

Methods

Device fabrication

To fabricate the WSe2/WS2 moiré superlattices, we first exfoliate WS2 and WSe2 monolayers from bulk crystal onto SiO2/Si substrates and do polarization-resolved second harmonic generation (SHG) measurements to determine the crystal orientation in each flake. We then assemble the WS2 and WSe2 monolayers into a heterostructure with their crystal axes aligned using a polycarbonate (PC) stamp. During the transfer process, the near-zero-degree-twist-angle WSe2/WS2 stack is contacted by few-layer graphite (FLG) and sandwiched between two hexagonal boron nitride (hBN) layers with thickness of 15-25 nm. Additional FLG flakes serve as electrostatic gates. The whole heterostructure is released onto a 90 nm SiO2/Si substrate. Electrodes (100 nm Au with 5 nm Cr adhesion layer) are fabricated using a photolithography system (Durham Magneto Optics, MicroWriter) and an electron-beam deposition system. After fabrication, we again perform polarization-resolved SHG measurements on the monolayer WS2 and WSe2 regions within the heterostructure to determine the exact twist angle and on the heterostructure region to distinguish between near-zero and near-sixty-degree samples.

Doping-dependent spatial-temporal pump–probe technique

A function generator (Siglent SDG6022X) is used to generate two synchronized electronic pulse trains that drive two RF-coupled laser diode modules (Thorlabs LDM56) with center wavelengths of ~690 nm (pump) and ~730 nm (probe). The probe laser wavelength is finetuned to be near-resonant with the lowest energy WSe2 A exciton using the temperature-controlled mount. The pump beam travels through a polarizer and shared quarter waveplate (QWP) to generate a left circularly polarized pump. The probe beam travels through a polarizer, rotating half-wave plate (HWP), and the shared QWP to switch between left- and right- circularly polarized probe. The pump and probe beams are combined with a dichroic mirror and focused onto the sample mounted in a Montana Instruments cryostation using an objective. An additional cylindrical lens in the pump path ensures that the pump beam is focused on a line on the sample. The pump–probe spatial separation is tuned via a piezo-controlled mirror in the pump path. For all pump-probe measurements, the pump power is set such that the photoexcited hole population is small compared to the electrostatically injected hole population.

The reflected beams are spectrally filtered with a 715 nm long-pass filter to isolate the probe, which is then monitored with an avalanche photodiode (APD). The APD output is sent to a Keithley 2400 SourceMeter (measures the static reflectivity, RC) and a SRS864A lock-in amplifier (measures the pump-induced change in the reflectivity, ΔRC). The lock-in amplifier is locked to the frequency of the optical chopper in the pump path.

Doping dependent spin transport measurements are conducted by applying voltages to the graphite gate using Keithley 2400 or 6482 SourceMeters. The charge density \(p\), is defined using a parallel plate capacitor model:

$$p=\pm \frac{1}{e}\frac{{\epsilon }_{{{\rm{hBN}}}}{\epsilon }_{0}}{d}\left(V-{V}_{0}\right)$$
(3)

where \({\epsilon }_{{{\rm{hBN}}}}\) is the dielectric constant of hBN (measured as \(4.2\pm 0.2\)), \({\epsilon }_{0}\) is the permittivity of free space, \(d\) is the thickness of the bottom hBN layer, and \(V\) is the voltage applied to the gate. We account for the quantum capacitance (voltage range where charges are not injected) using an offset voltage \({V}_{0}\) for electron and hole doping. The offsets are defined by the voltage where the reflection contrast spectrum begins to change with electron and hole doping.

The moiré density \({p}_{0}\) is defined to correspond to one hole per moiré unit cell and is determined through the relation \({p}_{0}=1/[{L}_{{{\rm{M}}}}^{2}\sin (\pi /3)]\) where \({L}_{{{\rm{M}}}}\) is the moiré superlattice constant. The twist angle (\(\theta\)) and lattice mismatch (\(\delta=(a-{a}^{{\prime} })/a\)) between the two layers determines \({L}_{{{\rm{M}}}}\) via \({L}_{{{\rm{M}}}}=a/\sqrt{{\delta }^{2}+{\theta }^{2}}.\) STM measurements of a near-zero degree WSe2/WS2 give \({L}_{{{\rm{M}}}} \sim\) \(8\) \({\mbox{nm}}\), which is consistent with that expected due to the lattice constant mismatch between the layers. Therefore, for zero-degree-twist angle samples, \({p}_{0}=1.80\times {10}^{12}\) cm−2.