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Light-Induced Ferromagnetism in Moir\'e Superlattices

Many-body interactions between carriers lie at the heart of correlated physics. The ability to tune such interactions would open the possibility to access and control complex electronic phase diagrams on demand. Recently, moir\'e superlattices formed by two-dimensional materials have emerged as a promising platform for quantum engineering such phenomena. The power of the moir\'e system lies in the high tunability of its physical parameters by tweaking layer twist angle, electrical field, moir\'e carrier filling, and interlayer coupling. Here, we report that optical excitation can drastically tune the spin-spin interactions between moir\'e trapped carriers, resulting in ferromagnetic order in WS2/WSe2 moir\'e superlattices over a small range of doping at elevated temperatures. Near the filling factor v = -1/3 (i.e., one hole per three moir\'e unit cells), as the excitation power at the exciton resonance increases, a well-developed hysteresis loop emerges in the reflective magnetic circular dichroism (RMCD) signal as a function of magnetic field, a hallmark of ferromagnetism. The hysteresis loop persists down to charge neutrality, and its shape evolves as the moir\'e superlattice is gradually filled, indicating changes of magnetic ground state properties. The observed phenomenon points to a mechanism in which itinerant photo-excited excitons mediate exchange coupling between moir\'e trapped holes. This exciton-mediated interaction can be of longer range than direct coupling between moir\'e trapped holes, and thus magnetic order can arise even in the dilute hole regime under optical excitation. This discovery adds a new and dynamic tuning knob to the rich many-body Hamiltonian of moir\'e quantum matter.

To experimentally realize these states, the ability to tailor the interactions between moiré trapped spins in situ, and thus the many-body Hamiltonian is highly desirable. One possible approach is to utilize optical excitation. In fact, optical excitation of strongly correlated quantum materials can have profound effects on the many-body electronic states, such as light induced superconductivity 24 , hidden charge density wave order 25 , etc. In the context of TMD heterobilayers, the exciton stands out as a promising candidate, which is itself of great interest for the strong lightmatter interactions, unique spin-valley coupled optical effects 26 , and moiré superlattice physics [27][28][29][30][31] . These optical excitations are already exploited as effective probes of correlated insulating and generalized Wigner crystal states in moiré superlattices [7][8][9][20][21][22][23] . However, the impact of optical excitation on many-body electronic states, i.e., non-equilibrium effects, in the strongly correlated moiré superlattice, remains unexplored.
Here, we report the discovery that optical excitation can drastically enhance spin-spin exchange interaction J between moiré trapped carriers and thus triggers the emergence of ferromagnetic order in the WS2/WSe2 moiré superlattice. This finding points to a profound and dynamic control of the moiré many-body Hamiltonian. Dual gated WS2/WSe2 samples were fabricated to allow independent control of carrier doping and vertical electrical field (Figure 1a, see methods). Figure  1b shows a piezoresponse force microcopy image of a representative device. A homogeneous triangular moiré superlattice is observed with a moiré wavelength of about 7.5 nm, consistent with near zero-degree twist angle alignment. Optical reflection measurements as a function of doping show typical moiré excitons with moiré miniband filling effects (Fig. 1c), as reported previously 7,9,29 . The filling factor v is labeled on the right axis (see Extended Data Fig. 1 for the filling factor assignment).
To probe the magnetic properties, we performed reflective magnetic circular dichroism (RMCD) measurements. Figure 1d plots the RMCD intensity as a function of v and magnetic field µoH applied in the out of plane direction. We mainly focus on − 3 2 ≤ ≤ 2 3 , where the magnetic response is most appreciable. The data are taken with the laser excitation in resonance with the WSe2 A exciton (see wavelength dependent data in Extended Data Fig. 2). The excitation power is 200 nW and the experimental temperature is 1.6 K (see Methods). Fig. 1d displays a rich v dependent magneto-optical response. These effects are illustrated by the RMCD line traces at selected filling factors in Fig. 1e. At v = -1, i.e., one hole per moiré unit cell, RMCD shows a superparamagnetic-like response and saturates for |µoH| above 1 T. This demonstrates the existence of local magnetic moments. The superparamagnetic-like response is robust for v near -1 (see Extended Data Fig. 3 for selected RMCD line traces). The observation is consistent with previous report of intralayer exciton Zeeman splittings 9 .
Remarkably, when hole doping becomes significantly smaller, a magnetic hysteresis loop starts to appear in the RMCD signal vs µoH, a signature of ferromagnetic order (See Extended Data Fig.  4 for filling factor dependent loop width). The RMCD lineshape is very sensitive to doping. In a narrow doping regime near v = -1/3, the RMCD signal vs µoH shows typical ferromagnetic behavior, distinct from linear valley Zeeman effect. It possesses a hysteresis loop with spin flip transitions around µoHc = ±11 mT, above which the signal saturates ( Fig. 1e and Extended Data Fig. 5). When the doping density further reduces, the RMCD line traces evolves into a heartbeat lineshape, and eventually vanishes as electron doping dominates. The emergent sharp RMCD features near zero magnetic field, which are associated with ferromagnetic states at low hole doping regime, are the focus of this paper.
We first present the results near v = -1/3. Figure 2a shows the RMCD signal as a function of excitation power P. When P is smaller than 16 nW, the RMCD signal vs µoH vanishes (after subtracting a constant background) and behaves as a featureless straight line. A hysteresis loop emerges as P is increased above a critical threshold. The amplitude of the remnant RMCD signal at zero field grows as P increases, and eventually saturates (Fig. 2b). In comparison, the RMCD response and its saturation value at v = -1 have little excitation power dependence (Extended Data Fig. 3). This is consistent with the attribution of the RMCD response at v = -1 to the intrinsic magnetic interactions 9 . At low filling factors, however, the intrinsic magnetic interaction is significantly weaker because of larger hole-hole distances compared to the v = -1 case. Thus, the peculiar power dependent RMCD responses at the fractional filling factor v = -1/3 signify the emergence of ferromagnetic order by optically induced long-range spin-spin interaction.
The hysteresis loop width exhibits negligible dependence on the optical excitation power. This can be understood that at temperatures much lower than the Curie temperature, the loop width is mainly determined by the magnetic anisotropy. Our measurements thus suggest that magnetic anisotropy is intrinsic to this moiré system. We note that the width of the hysteresis loop depends on the magnetic field sweeping rate (Extended data Fig. 5). Slower sweeping leads to a narrower hysteresis loop width. This result reflects slow magnetic domain dynamics in the magnetization flipping process. Here, all magnetic field sweep results are taken with a sweep rate of 14 Gauss/second, below which the hysteresis loop width is nearly independent of the sweep rate.
The ferromagnetic order is further confirmed by temperature dependent RMCD measurements, shown in Fig. 2c, measured at P = 103 nW. The hysteresis loop width reduces as the temperature increases, as shown in Fig. 2d. There is a 5 mT instrument offset above 10 K until the noise level exceeds the signal above ~45K. Similar temperature dependence is also observed at other filling factors (Extended Data Fig. 4). The effective Curie temperature is determined to be about 8 K. In the mean field limit, this gives rise to an estimated exchange interaction J on the order of -1 meV (with the Hamiltonian H in the form of JS1·S2, see methods). Note that the main experimental results are reproduced in an additional device (Extended Data Fig. 6).
At v = -1/3, photoluminescence measurements show a correlated insulating state (Extended Data Fig. 1). This is consistent with previous reports that a generalized Wigner crystal forms with holes arranged in a triangular lattice 7-10,20-23 . Monte Carlo simulations also suggest that this v = -1/3 state is relatively robust compared to other fractional filling charge orders (Extended Data Fig.  1). The period of the charge order at v = -1/3 (~13 nm) is √3 times larger than that at v = -1. Thus, the intrinsic spin-spin interaction of the former should be much smaller compared to the latter. This is manifested by the vanishing RMCD signal taken at very low excitation power (Fig. 2a), which implies negligible intrinsic magnetic interaction at such a low filling. The power dependent RMCD signal highlights the drastic optical tuning of the spin-spin interaction strength J, and hence of the magnetic phase in the moiré superlattice (Fig. 2d, inset).
When the hole doping density reduces and v deviates from -1/3, the line trace of RMCD signal versus µoH quickly evolves into heartbeat-like shapes with a narrow width (<100 mT), as shown in Fig. 1e. Figure 3a plots the zoom-in RMCD intensity in the low doping regime. The signal reaches maximum near v = -1/7, and is still appreciable at nominal charge neutrality. Although the total net charge of the system vanishes at the charge neutrality point, inevitable defects as well as optically created photo-carriers can give rise to puddles of dilute holes trapped in the moiré potential of the WSe2 layer. As electron doping increases, the RMCD signal reduces and eventually vanishes. The signal peaking at v = -1/7 is likely due to the formation of a correlated insulating state, formed by holes arranged in a triangular lattice (Extended Data Fig. 1). Figure 3b shows the RMCD signal as a function of excitation power P at v = -1/7. It starts with a slightly tilted straight line without any hysteresis, which is the typical magnetic-optical response of TMDs without magnetic order. As P increases, the heartbeat lineshape emerges with a hysteresis loop as the magnetic field is swept back and forth. The RMCD amplitude, defined as the peak-tovalley value of the heartbeat lineshape, increases with P and eventually saturates (Fig. 3c). These results demonstrate optically induced ferromagnetism in the dilute hole doping regime.
The above experimental observations establish optical creation and tuning of the magnetic order, otherwise absent in dark conditions, in the WS2/WSe2 moiré superlattices. We note that the RMCD probe of the magnetic response is through resonant excitation of the WSe2 exciton, upon which ultrafast interlayer charge transfer leads to formation of long-lived interlayer excitons 32,33 . Experiments on MoSe2/WSe2 moiré superlattices have shown that at the low excitation limit (< 10 nW), these excitons are mostly trapped by the moiré potentials, giving rise to narrow photoluminescence lines. With increasing excitation power, the steady state population of itinerant (or untrapped) interlayer excitons grows significantly, manifested by a broad luminescence background 27 . Our observations therefore point to the role of these optically injected itinerant excitons in aiding the magnetic interactions between holes trapped in moiré potentials. Indeed, spin-spin interactions between two spatially separated holes can arise as a second-order process mediated by their exchange interaction with a common itinerant exciton, as schematically illustrated in Fig. 3d. Similar role of exciton has been theoretically explored in a virtual excitation regime, where a fourth-order perturbation process leads to short-ranged exchange interaction of localized spins in III-V semiconductors 34 , which may also result in magnetic order 35 .
We found that this exciton mediated hole-hole exchange interaction is ferromagnetic (see Methods). The strength of the exciton mediated exchange interaction depends on the separation of moiré trapped holes, exciton density, and effective temperature of the exciton gas which can in general be different from the lattice temperature. Figure 3e plots the calculated J as a function of hole separation and exciton density, at an exciton temperature of 20 K (see Extended Data Fig. 7 for other exciton temperatures). We found that J remains significant at a hole-hole distance of 20 nm. At a reasonable exciton density of 10 11 cm -2 (see Methods), its strength can reach the order of 0.5 meV (Fig. 3e). We note that a quantitative comparison between the calculation and experiment is challenging for non-equilibrium many-body problems. For instance, the RMCD response does not resemble the typical magnetization curve even qualitatively, subject to the influence of the excitonic lineshape. The exact origin of the lineshapes, for example the heartbeat at dilute hole doping regime, is not clear. It might reflect the change of the magnetic ground state properties upon doping. Nevertheless, the observed RMCD hysteresis upon the forward and backward sweeping of magnetic field is unambiguous evidence of magnetic order, which is consistent with our calculation that excitons can effectively establish ferromagnetic exchange interactions in dilute moiré hole gas.
We further performed temperature dependent RMCD measurements at v = -1/7 as a function of optical excitation power, ranging from 26 to 253 nW. Figure 4a illustrates the results at select excitation powers. We define a critical temperature TC, above which the heartbeat amplitude vanishes (below the noise level 0.2%), as for the magnetic response. Using 253 nW optical excitation as an example, the heartbeat lineshape remains strong up to about 40 K. Fig. 4a displays a strong dependence of TC on the optical excitation power. To further highlight this effect, we plot the extracted amplitude of the RMCD signal as a function of excitation power and temperature in Fig. 4b (see Extended Data Fig. 8 for line cuts). These data show that once the optical excitation power is large enough to introduce magnetic order, TC can be optically tuned from about 20 K to 45 K.
TC also exhibits strong dependence on the filling factors. Fig. 4c shows RMCD signal versus magnetic field as a function of temperature at select v (Extended Data Fig. 9). We then plot the extracted RMCD amplitude in Fig. 4d. This figure highlights that for the magnetic states within -2/3 < v < 1/3, TC peaks near correlated insulating states at fractional fillings and drops sharply once v deviates from these states (e.g., -1/2 and -1/7, and see -1/3 in Extended Data Fig. 10). While the correlated insulating states are not required for the formation of light induced magnetic states, these observations suggest that the latter is further stabilized with the formation of charge order, where the spin fluctuations among moiré spins are minimized. The optical excitation power and filling factor dependent results underscore the broad tunability of the moiré many-body Hamiltonian, which may lead to the creation of exotic states unique to moiré superlattices. We envision several immediate exciting possibilities, such as exploring the effect of optical orientation of exciton spins on the formation of magnetic states 36,37 , electric field tuning effects, transient magnetism, optically controlled topological phase transitions, magneto-exciton polaritons in the absence of magnetic field, and many-spin entangled states for optically driven quantum information 34 .

Sample fabrication.
Mechanically exfoliated monolayers of WS2 and WSe2 are stacked using a dry-transfer technique on hexagonal boron nitride (BN) and graphene back gate, with prefabricated Platinum contacts. The crystal orientation of the individual monolayers is first determined by linear-polarization resolved second-harmonic generation before transfer. The alignment angle of WS2 and WSe2 is double checked by piezoresponse force microscopy (PFM) during transfer before encapsulating with top BN and graphite. The BN encapsulation (20 -40 nm) provides an atomically smooth substrate. PFM is performed on a Bruker Dimension Icon AFM. Platinum-Iridium coated, electrically conductive AFM probes (Bruker's SCM-PIT-V2) with a force constant of ~3 N m −1 are used. An AC bias is applied between the tip and sample that induces a periodic deformation of the sample whose amplitude and phase give local information on the electromechanical response. Small AC bias magnitudes (<300 mV) with resonance frequencies in the range of ~ 700 kHz are applied for lateral PFM.

Optical measurements.
All PL and differential optical reflection measurements are performed in a home-built confocal optical microscope in the reflection geometry. The sample is mounted in an exchange-gas cooled cryostat (attoDRY 2100) equipped with a 9 T superconducting magnet in Faraday configuration. The sample temperature is kept at 1.6 K unless otherwise specified. SuperK Extreme Supercontinuum White Light laser (pulse duration around 2 ns; repetition rate around 78 MHz) and LLTF tunable high contrast filter with selected wavelength 739.2 nm are used to excite the sample at normal incidence, unless otherwise specified. The power is specified in the manuscript and corresponding figure captions. The Gaussian profile of the excitation beam has FWHM around 700 nm. We also used the same laser with white light emission for ΔR/R with 600-nm long pass and 800-nm short pass (average power 15 nW). Reflectance and PL signals are dispersed by a diffraction grating (600 grooves per mm) and detected on a silicon CCD camera. The photoluminescence is spectrally filtered from the laser using a long-pass filter before being directed into a spectrometer.
The reflective magnetic circular dichroism (RMCD) measurement was performed with an AC lock-in measurement technique. In brief, an intensity modulated laser beam (at 800 Hz) was linearly polarized 45° to the photoelastic modulator (PEM) slow axis. Transmitting through the PEM, the light was sinusoidally phase modulated at 50.1 kHz, with a maximum retardance of λ/4. This produces an alternating left and right circularly polarized light at the phase modulation frequency. The modulated light was focused down onto the sample at normal incidence. The optical reflection was then separated from the incidence path via a laser line nonpolarizing beam splitter and projected onto a photodiode, whose signal is sent into two lock-in amplifiers: one is tuned at 50.1 kHz to detect the RMCD, and the other is tuned at the chopper frequency, 800 Hz, to normalize the RMCD signal to laser intensity, reflected off the sample.

Estimation of interlayer exciton (IX) density.
We estimate the upper bound of the exciton density under 100 nW pulse excitation (78 MHz, pulse duration 2 ns) with photon energy in resonance with WSe2 A exciton (1.677 eV). The optical absorption is mainly from the WSe2 layer which is about 10%. 38 Approximately 500 photons, and thus 500 excitons, are excited per pulse. We normalize the exciton population over the Gaussian profile of the excitation beam (FWHM 700 nm). Assuming near unity intralayer to interlayer exciton conversion efficiency, and due to the short duration of the pulse compared to the exciton lifetime, the IX density is about 10 11 cm -2 immediately after a pulse.

Estimation of filling factor based on doping density.
The doping densities in the heterobilayer are determined from the applied gate voltages based on the parallel-plate capacitor model. The thickness of BN is determined by atomic force microscopy. Both top and bottom BN flakes of the device presented in the main text are 37 nm (top BN) and 38 nm (bottom BN). The doping density is calculated as CtΔVt + CbΔVb, where Ct and Cb are the capacitance of the top and bottom gates. ΔVt and ΔVb are the applied gate voltages relative to valence/conduction band edges. The geometric capacitance is calculated with dielectric constant εhBN ~ 3. The moiré lattice constant is obtained directly from PFM. The filling factor is estimated based on the doping density and size of the moiré unit cell, which is then compared with the assignment of integer filling factors based on differential reflectance and PL.

Calculation of exciton medicated magnetic exchange interaction.
Coulomb exchange between exciton and trapped hole. The Coulomb interactions between electrons, between holes, and between electron and hole are expressed as 39 Here, † and ℎ † = denote electron and hole creation operators, respectively. Their matrix elements between three-particle states consisting of one electron and two holes are , , , Here the bold character denotes all the degrees of freedom including momentum and spin-valley index , e.g. ≡ ( , )). ⟨ 1 , 1 2 , 2 | | 3 , 3 4 , 4 ⟩ and ⟨ 1 , 1 2 , 2 | | 3 , 3 4 , 4 ⟩ are the two-body matrix elements in momentum space with the following expression 40 4 , 4 , * where | ⟩ is the periodical part of the Bloch function, ( ) = ∫ ⋅ ( ) is the Fourier transformation of 2D Coulomb potential ( ) and is the area of the 2D plane (for box normalization).
The Coulomb exchange interaction between the exciton and the trapped hole is evaluated, in the basis � , |0⟩ is the momentum eigenstate of the interlayer exciton with electron (hole) spin/valley index ( ℎ ), and � ℎ, � = † |0⟩ denotes a trapped hole wave packet centered at . We have � , with the normalization condition: For the interlayer exciton and trapped hole in the type-II heterobilayer, holes are layer separated from the electron component of the exciton. Therefore, we can neglect the terms that relies on wavefunction overlap between the electron and hole (second, fourth, sixth and eighth term in Eq. (3)). The Coulomb exchange interaction between the exciton and the trapped hole are then contributed by the second term in Eq. (2) from the hole-hole interaction, and the fifth and seventh terms in Eq. (3) from the electron-hole interaction, Exciton mediated spin-spin interaction between trapped holes. The Hamiltonian for the exciton and hole in the basis of exciton momentum eigenstate and moiré trapped hole is, where only the exchange part of the exciton hole interaction is included as . The exciton mediated spin-spin interaction between trapped holes arises as the second order perturbation of the off-diagonal exciton-hole exchange. And the exciton-hole exchange coefficient in contains three contributions given in Eq. (7), (8) and (9) respectively. The term from the intralayer holehole interaction is given as, In the second line, the dependence on and ′ is neglected, because the exciton states involved are the small momentum ones (much smaller compared to the momentum space width of ℎ and ). In such limit, both terms from the interlayer electron-hole interaction have the same form, 4πε r ε 0 (k TF +| X − h |) e −d| X − h | (12) With the selection of the following parameters, = 2 nm, ℎ = 2 nm, = 10 0 ≈ 100 7 m −1 , = 0.7 nm and = 4, we obtain ℎℎ A ≈ −0.71 eV ⋅ nm 2 and ℎ A ≈ 0.5 eV ⋅ nm 2 . Thus, = ℎℎ + 2 ℎ ≈ 0.29 eV⋅nm 2 (13) To find the form of the exciton mediated spin-spin interaction between trapped holes, Schrieffer-Wolff transformation is used to eliminate the off-diagonal . Introducing, Tracing out the exciton degrees of freedom with a diagonal density matrix, i.e. 〈 † 〉 = ,  16) where the last term is the mediated spin-spin interaction between the holes (as illustrated in Fig. 3d). We note that the exchange with the trapped hole can flip an exciton between the spin triplet and singlet configurations (of the intravalley and intervalley configurations respectively in the R type WS2/WSe2). So, the denominator involves the energy of the species − ′ ′ . As electronhole exchange is quenched by layer separation, there is no splitting between the intravalley spin triplet and intervalley singlet exciton dispersion, i.e. ≡ = ‾ . Thus, the spin-spin interaction terms in (2) between the trapped holes is of the form, where � − ′ � = ∑