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Realization of the Haldane Chern insulator in a moiré lattice

Nature Physics volume 20, pages 275–280 (2024)Cite this article

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Abstract

The Chern insulator displays a quantized Hall effect without Landau levels. Theoretically, this state can be realized by engineering complex next-nearest-neighbour hopping in a honeycomb lattice—the so-called Haldane model. Despite its profound effect on the field of topological physics and recent implementation in cold-atom experiments, the Haldane model has not yet been realized in solid-state materials. Here we report the experimental realization of a Haldane Chern insulator in AB-stacked MoTe2/WSe2 moiré bilayers, which form a honeycomb moiré lattice with two sublattices residing in different layers. We show that the moiré bilayer filled with two holes per unit cell is a quantum spin Hall insulator with a tunable charge gap. Under a small out-of-plane magnetic field, it becomes a Chern insulator with a finite Chern number because the Zeeman field splits the quantum spin Hall insulator into two halves with opposite valleys: one with a positive and the other with a negative moiré band gap. We also demonstrate experimental evidence of the Haldane model at zero external magnetic field by proximity coupling the moiré bilayer to a ferromagnetic insulator.

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Fig. 1: AB-stacked MoTe2/WSe2 moiré bilayer.
Fig. 2: QSH insulator at ν = 2.
Fig. 3: Magnetic-field-induced Haldane Chern insulator.
Fig. 4: Temperature dependence of the Chern insulator.
Fig. 5: Proximity-coupled CrBr3–MoTe2/WSe2 device.

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All other data are available from the corresponding authors on reasonable request. Source data are provided with this paper.

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Acknowledgements

We thank L. Fu, Y. Zhang and A. H. MacDonald for fruitful discussions. The work at Cornell was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under award no. DE-SC0019481 (transport studies); the Air Force Office of Scientific Research under award nos. FA9550-19-1-0390 (development of moiré superlattices proximity coupled to a 2D magnet) and FA9550-20-1-0219 (magneto-optical studies); and the National Science Foundation (NSF) under award no. DMR-1807810 (analysis). The work at Cornell was also funded in part by the Gordon and Betty Moore Foundation (grant https://doi.org/10.37807/GBMF11563). The work at the University of California at Santa Barbara was primarily supported by the Army Research Office under award no. W911NF-20-2-0166 and by the Gordon and Betty Moore Foundation EPIQS programme under award no. GBMF9471. The growth of the hBN 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 no. NNCI-2025233. We also acknowledge support from the David and Lucille Packard Fellowship (K.F.M.), the Kavli Postdoctoral Fellowship (W.Z.) and the Swiss National Science Foundation (P.K.).

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Author notes

  1. These authors contributed equally: Wenjin Zhao, Kaifei Kang.

Authors and Affiliations

  1. Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA

    Wenjin Zhao, Jie Shan & Kin Fai Mak

  2. School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA

    Kaifei Kang, Zui Tao, Lizhong Li, Jie Shan & Kin Fai Mak

  3. Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA

    Yichi Zhang, Patrick Knüppel, Jie Shan & Kin Fai Mak

  4. Department of Physics, University of California at Santa Barbara, Santa Barbara, CA, USA

    Charles L. Tschirhart, Evgeny Redekop & Andrea F. Young

  5. National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe & Takashi Taniguchi

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Contributions

W.Z., Y.Z. and L.L. fabricated the devices. W.Z., K.K., Y.Z., L.L., C.L.T. and E.R. performed the electrical measurements and analysed the data. P.K., W.Z. and Z.T. carried out the optical measurements. K.K. performed the tight-binding model calculations. K.W. and T.T. grew the bulk hBN crystals. W.Z., K.K., J.S. and K.F.M. designed the scientific objectives and oversaw the project. All authors discussed the results and commented on the manuscript.

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Correspondence to Wenjin Zhao, Jie Shan or Kin Fai Mak.

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Extended data

Extended Data Fig. 1 Nonlocal transport supporting the QSH insulator.

a, Nonlocal resistance \({R}_{{\rm{nl}}}\) as a function of top and bottom gate voltages at \(B=0\) and \(T=\)330 mK. The green and orange dashed lines denote, respectively, the electric field direction at fixed \(\nu =2\) and the filling factor direction at fixed \(E={E}_{c}\). b, Optical micrograph of Device 1. The top gate (TG) and bottom gate (BG) are outlined by a black and red dashed line, respectively. The scale bar is 5 \({\rm{\mu }}{\rm{m}}\). c, Schematics of the measurement geometry. The device is current (I) biased along the arrow direction. The voltage drop \({V}_{{\rm{nl}}}\) is measured at the other end of the device. \({R}_{{\rm{nl}}}\) is negligible except at \(\nu =2\). Before band inversion, \({R}_{{\rm{nl}}}\) cannot be probed appropriately because the current is practically zero; after band inversion, it grows with increasing electric field. This is consistent with the helical edge transport in a QSH insulator.

Extended Data Fig. 2 QSH effect in devices of different twist angles.

a, b, Longitudinal resistance as a function of top and bottom gate voltages of Device 3 (2-degree twisted) at T = 300 mK (a) and Device 2 (near 0-degree twisted) at T = 10 mK (b). c, Two-terminal resistance measured by adjacent contacts as a function of top and bottom gate voltages of a bilayer MoTe2/monolayer WSe2 Device 4 (near 0-degree twisted) at T = 1.6 K. d-f, Line-cuts of a-c at varying top gate voltages. The grey dashed lines mark the quantized resistance \(\frac{h}{{2e}^{2}}(\)≈12.9 kΩ). For Device 3, \({R}_{{\rm{xx}}}\) at \(\nu =2\) is the smallest right after band inversion and plateaus with further increase of the electric field. \({R}_{{\rm{xx}}}\) and \({R}_{2{\rm{p}}}\) increase continuously with electric field after band inversion for Device 2 and 4, respectively.

Extended Data Fig. 3 Magneto-transport in Device 1 at 1.6 K.

a, b, Hall resistance as a function of the top and bottom gate voltages at B = 3 T (a) and of the magnetic field and filling factor at \(E={E}_{c}\) (b). The orange dashed line in a corresponds to \(E={E}_{c}\). The green dashed line in b marks the \({R}_{{\rm{xy}}}\) maximum. We determine the slope of the dashed lines to be \({n}_{{\rm{M}}}\frac{dv}{dB}=c\frac{e}{h}\) with c = 1.1 ± 0.1. Right inset shows the magnetic-field dependence of \({R}_{{\rm{xy}}}\) along the green dashed line.

Extended Data Fig. 4 Transport under different magnetic fields for Device 2.

a-e, Hall resistance (left) and longitudinal resistance (right) at 10 mK (lattice temperature) as a function of the top and bottom gate voltages. The magnetic field is 0.5 T (a), 1 T (b), 2 T (c, same as Fig. 3a of the main text), 3 T (d), 4 T (e).

Extended Data Fig. 5 High-magnetic-field transport.

a, \({R}_{{\rm{xy}}}\) (top) and \({R}_{{\rm{xx}}}\) (bottom) of Device 2 as a function of top and bottom gate voltages at B = 11.8 T and T = 10 \({\rm{m}}\)K (lattice temperature). The Landau levels (identified by the resistance minimum) are indexed. b, c, \({R}_{{\rm{xy}}}\) (top) and \({R}_{{\rm{xx}}}\) (bottom) as a function of filling factor and magnetic field at E = 0.435 V/nm (before band inversion, b) and at E = 0.463 V/nm (near band inversion, c). The dashed lines show the Landau fan. The white oval shows the Landau level that disperses with a negative slope.

Extended Data Fig. 6 The electric-field span of the Chern state as a function of magnetic field.

a, \({R}_{{\rm{xy}}}\) and \({R}_{{\rm{xx}}}\) of Device 2 at \(\nu =2\) and T = 10 mK (lattice temperature) as a function of electric field. The data are extracted from Extended Data Fig. 4. b, The electric-field span, \(\Delta E\), of the Chern state as a function of magnetic field. For each magnetic field, the electric-field dependence of \({R}_{{\rm{xy}}}\) is fit with a Gaussian function and \(\Delta E\) is determined as half of the variance. The error bars are one-sigma uncertainty of \(\Delta E\) in the Gaussian function fitting. The red dashed line is the best fit to \(\Delta E=\sqrt{{a}^{2}+{({bB})}^{2}}\) with fitting parameters a = 2.2 ± 0.1 mV/nm and b = 2.0 ± 0.1 mV/nmT.

Extended Data Fig. 7 Tight-binding model calculations.

a-c, Band structure simulated using the tight-binding Hamiltonian described in Methods. Red and blue curves denote the spin-up and spin-down bands, respectively. Under zero magnetic field (a), the bands are inverted at both the \({\rm{K}}\) and \({{\rm{K}}}^{{\prime} }\) valleys. This is a QSH insulator (QSHI). Under a small magnetic field (b), the bands in the \({{\rm{K}}}^{{\prime} }\) valley cross at one momentum. Under a sufficiently high magnetic field (c), the gap changes sign for the \({{\rm{K}}}^{{\prime} }\) valley; this is a Chern insulator. d, The direct band gap near the \({\rm{K}}/{{\rm{K}}}^{{\prime} }\) valleys as a function of the Zeeman energy and the sublattice/interlayer potential difference. The three symbols mark the phase space for which the electronic band structure is represented in a-c, respectively.

Extended Data Fig. 8 RC and MCD spectrum in AB-stacked MoTe2/WSe2 proximity-coupled to bilayer CrBr3.

a, Schematic side view of a dual-gated device of AB-stacked MoTe2/WSe2 moiré bilayer proximity-coupled to bilayer CrBr3. b, Bottom gate voltage dependence of the RC spectrum near the WSe2 intralayer exciton resonance for Vtg = −3.6 V. The strong feature near 1.67 eV is the neutral exciton resonance. It evolves into much weaker charged exciton resonances for \({V}_{{\rm{bg}}} < -1{\rm{V}}\). c, Bottom gate voltage dependence of the spontaneous MCD spectrum near the WSe2 exciton resonance at Vtg = −3.9 V. At this value of \({V}_{{\rm{tg}}}\), the WSe2 layer is doped for the entire range of bottom gate voltage. The MCD reported in the main text is spectrally integrated over the spectral window given by the black dashed lines (1.661–1.666 eV), which covers the attractive polaron resonance of WSe2.

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Zhao, W., Kang, K., Zhang, Y. et al. Realization of the Haldane Chern insulator in a moiré lattice. Nat. Phys. 20, 275–280 (2024). https://doi.org/10.1038/s41567-023-02284-0

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