Abstract
The electron’s kinetic energy plays a pivotal role in magnetism. While virtual electron hopping promotes antiferromagnetism in an insulator, real hopping processes usually favour ferromagnetism. However, in kinetically frustrated systems such as hole-doped triangular lattice Mott insulators, real hopping has instead been shown to favour antiferromagnetism. Kinetic frustration has also been predicted to induce a new quasiparticle, a bound state of the doped hole and a spin flip called a spin polaron, at intermediate magnetic fields, which could form an unusual metallic state. Here we report the direct observation of spin polarons in triangular lattice MoTe2/WSe2 moiré bilayers. A spin polaron phase emerges at a lattice filling factor just below 1 and is separated from the fully spin-polarized phase by a metamagnetic transition. We determine that the spin polaron is a spin-3/2 particle and that its binding energy is commensurate with the kinetic hopping energy. Our results will enable the exploration of spin polaron pseudogap metals, spin polaron pairing and other new phenomena in triangular lattice moiré materials.
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Acknowledgements
We thank L. Fu, Y. Zhang, M. Davydova and J. Dong for fruitful discussions. The work was supported by the Air Force Office of Scientific Research under award number FA9550-19-1-0390 (magneto-optical studies), the US Department of Energy, Office of Science, Basic Energy Sciences, under award number DE-SC0019481 (device fabrication) and the National Science Foundation under DMR-1807810 (analysis). The work was also funded in part by the Gordon and Betty Moore Foundation (https://doi.org/10.37807/GBMF11563). The growth of the h-BN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan and CREST (JPMJCR15F3), JST. We used the Cornell NanoScale Facility, an NNCI member supported by NSF grant NNCI-2025233 for device fabrication. We also acknowledge support from the Kavli Postdoctoral Fellowship (W.Z.) and the Swiss National Science Foundation (P.K.).
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Z.T. and W.Z. performed the optical and electrical measurements and the analysis with the help of P.K. and B.S.; B.S. and T.L. fabricated the devices. K.W. and T.T. grew the bulk h-BN crystals. Z.T., K.F.M. and J.S. designed the scientific objectives and oversaw the project. All authors discussed the results and commented on the paper.
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Extended data
Extended Data Fig. 1 Schematic superlattice structure and moiré bands in AB-stacked MoTe2/WSe2.
a, Moiré pattern (left) formed in AB-stack MoTe2/WSe2. MM, MX and XX (M = Mo, W; X = Se, Te) are the high-symmetry stacking sites (right). The Wannier orbitals of the topmost moiré valence band occupy the MM sites, forming a triangular lattice. The arrow labels the electric field direction in our experiment. b, Schematic illustration of the electric field effect on the band alignment at \(\nu =1\). Grey dashed line indicates the Fermi level at \(\nu =1.\) Left: The Mo-moiré band is split into the lower and upper Hubbard bands by the on-site Coulomb \(U\). The W-moiré band lies well below the Mott gap, resulting in a single-band Hubbard model. Right: With increasing electric field, the W-moiré band is pushed up but remains well separated from the Mo-Hubbard band in our experiment.
Extended Data Fig. 2 Additional results at E = 0.3 V/nm.
a, Magnetic-field dependent MCD at representative filling factors at 1.6 K for both field polarities. The intermediate MCD plateau is observed for \(0.83\lesssim \nu \lesssim 0.93.\) b, Filling factor dependent magnetization ratio between the intermediate plateau and full saturation. The dashed line describes \(\frac{1-3x}{1-x}\). c, Filling factor dependent saturation field \({B}_{s}\) and metamagnetic transition field \({B}_{m}\) extracted from experiment (symbols). The lines are guide to the eye. The shaded area represents the region for the spin polaron phase.
Extended Data Fig. 3 Filling factor calibration.
a,b, Filling factor dependence of the two-terminal longitudinal resistance (a) and the MCD spectrum (at 0.6 T, b) simultaneously measured at 1.6 K. The resistance peak identifies the \(\nu =1\) Mott insulating state (dashed line). The filling factor and moiré density calibration is further confirmed by quantum oscillation measurements and the known Landau level degeneracy in the WSe2 layer. The small shift between the resistance and the MCD peak is reproducible at different electric fields and in different devices.
Extended Data Fig. 4 Magnetic field dependent MCD spectrum at selected filling factors.
The electric field is fixed at 0.5 V/nm; the temperature is 1.6 K. Three intralayer moiré exciton resonances from MoTe2 can be identified. The dashed lines show the spectral range for MCD integration. The arrows indicate the metamagnetic transition field \({B}_{m}\), where a kink in the MCD spectrum is observed.
Extended Data Fig. 5 Magnetic field dependent MCD at varying fillings.
a,b, Magnetic field dependence of the MCD (a) and the MCD derivative (b) at representative filling factors and at 1.6 K in the positive field direction. The electric field is fixed at 0.5 V/nm. The curves are vertically displaced for clarity. An intermediate MCD plateau at magnetic field between 2 - 4 T is observed for filling factor between 0.8 – 1. The MCD plateau manifests as a local minimum in MCD derivative.
Extended Data Fig. 6 Filling factor dependent susceptibility and Curie-Weiss temperature θ.
a,b, Filling factor dependence of the magnetic susceptibility \(\chi\) at 1.6 K (a) and the Curie-Weiss temperature \(\theta\) (b). The magnetic susceptibility is obtained by fitting the MCD slope at zero magnetic field. The analysis is described in the main text, and the result near \(\nu =1\) is included in Fig. 3d. A suppression of the AF interaction is observed, as evidenced by a peak in \(\chi\) and a dip in \(\theta\) for the Wigner-Mott insulator at \(\nu\) = 2/3 (arrows). Inset in a: Filling factor dependence of \(\chi\) at 1.6 K over a broader filling factor range. No sign of FM interaction is observed above \(\nu =1.2\).
Extended Data Fig. 7 Analysis of the magnetization ratio.
a,b, Examples of determining the magnetization ratio at \(\nu =0.91\) (a) and \(\nu =0.89\) (b). The magnetization ratio is obtained by the MCD ratio between the spin polaron magnetization \({m}_{1}\) and the saturation magnetization \({m}_{2}\), that is \(\frac{{m}_{1}}{{m}_{2}}\) (horizontal dashed lines). The spin polaron magnetization \({m}_{1}\), which corresponds to the intermediate plateau in MCD, is determined by the local minimum of the MCD derivative (vertical dashed lines).
Extended Data Fig. 8 Reproducibility in another device.
a, MCD spectrum at B = 0.6 T as a function of filling factor ν. The out-of-plane electric field is fixed at 0.5 V/nm. Three intralayer moiré exciton resonances from MoTe2 can be identified. The dashed lines show the spectral range for MCD integration. b,c, Magnetic-field dependence of the MCD (b) and MCD derivative (c) at representative filling factors. An intermediate MCD plateau at magnetic field between 2 - 4 T is observed for filling factor between 0.8 – 1. Two examples are shown for \(\nu\) = 0.92 and 0.89. The intermediate MCD plateau manifests a local minimum in the MCD derivative. The dashed lines denote the two ends of the plateau, \({B}_{s}\) and \({B}_{m}\). For filling above 1 and below 0.8, only one of the fields can be identified. d, Filling dependent ratio of the intermediate magnetization plateau to the saturation magnetization (symbols). It follows \(\frac{1-3x}{1-x}\) (dashed line).
Extended Data Fig. 9 Magnetic field dependent MCD at varying probe light intensities.
The MCD profile does not change with the probe light intensity over one order of magnitude, showing that the probe light does not perturb the spin polaron phase.
Extended Data Fig. 10 Additional MCD analysis.
a, MCD spectrum at B = 0.6 T as a function of filling factor ν. The out-of-plane electric field is fixed at 0.5 V/nm. The dashed lines show the spectral range for MCD integration, which covers the strongest moiré exciton resonance. b,c, Magnetic-field dependence of the MCD (b) and MCD derivative (c) at representative filling factors. The results are qualitatively the same as those from full spectral integration in the main text. In particular, an intermediate MCD plateau at magnetic field between 2 - 4 T is observed for filling factor between 0.8 – 1. d, Filling dependent ratio of the intermediate magnetization plateau to the saturation magnetization (symbols). It follows \(\frac{1-3x}{1-x}\) (dashed line).
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Tao, Z., Zhao, W., Shen, B. et al. Observation of spin polarons in a frustrated moiré Hubbard system. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02434-y
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DOI: https://doi.org/10.1038/s41567-024-02434-y
