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# Continuous Mott transition in semiconductor moiré superlattices

## Abstract

The evolution of a Landau Fermi liquid into a non-magnetic Mott insulator with increasing electronic interactions is one of the most puzzling quantum phase transitions in physics1,2,3,4,5,6. The vicinity of the transition is believed to host exotic states of matter such as quantum spin liquids4,5,6,7, exciton condensates8 and unconventional superconductivity1. Semiconductor moiré materials realize a highly controllable Hubbard model simulator on a triangular lattice9,10,11,12,13,14,15,16,17,18,19,20,21,22, providing a unique opportunity to drive a metal–insulator transition (MIT) via continuous tuning of the electronic interactions. Here, by electrically tuning the effective interaction strength in MoTe2/WSe2 moiré superlattices, we observe a continuous MIT at a fixed filling of one electron per unit cell. The existence of quantum criticality is supported by the scaling collapse of the resistance, a continuously vanishing charge gap as the critical point is approached from the insulating side, and a diverging quasiparticle effective mass from the metallic side. We also observe a smooth evolution of the magnetic susceptibility across the MIT and no evidence of long-range magnetic order down to ~5% of the Curie–Weiss temperature. This signals an abundance of low-energy spinful excitations on the insulating side that is further corroborated by the Pomeranchuk effect observed on the metallic side. Our results are consistent with the universal critical theory of a continuous Mott transition in two dimensions4,23.

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## Data availability

The source data that support the findings of this study are available with the paper. Source data are provided with this paper.

## Change history

• ### 10 August 2022

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## Acknowledgements

We thank V. Dobrosavljevic, E.-A. Kim, A. H. MacDonald, L. Rademaker and S. Todadri for fruitful discussions. Research was primarily supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award no. DE-SC0019481 (electrical measurements) and award no. DE-SC0018945 (band structure calculations). The study was partially supported by the National Science Foundation (NSF) under DMR-1807810 (magneto-optical measurements) and the US Army Research Office under grant number W911NF-17-1-0605 (device fabrication). Growth of the hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan and CREST (JPMJCR15F3), JST. This work made use of the Cornell Center for Materials Research Shared Facilities, which are supported through the NSF MRSEC program (DMR-1719875) and the Cornell NanoScale Facility, an NNCI member supported by NSF Grant NNCI-1542081. D.C. acknowledges support from faculty startup grants at Cornell University; K.F.M. acknowledges support from the David and Lucille Packard Fellowship.

## Author information

Authors

### Contributions

T.L., S.J. and L.L. fabricated the devices, performed the measurements and analysed the data. K.K. and J.Z. provided assistance in the device fabrication. Y.Z. and L.F. performed the DFT calculations and theoretical analysis. D.C. helped with the theoretical analysis. K.W. and T.T. grew the bulk hBN crystals. T.L., S.J., J.S. and K.F.M. designed the scientific objectives and oversaw the project. All authors discussed the results and commented on the manuscript.

### Corresponding authors

Correspondence to Jie Shan or Kin Fai Mak.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Lede Xian, You Zhou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Square resistance versus electric field and filling factor.

2D map of the square resistance (in log scale) as a function of electric field and filling factor at 300 mK, converted from the data in Fig. 1c. Electric-field-induced MITs are observed at both filling factor f = 1 and f = 2.

### Extended Data Fig. 2 Metal-insulator transition at f = 2.

a, Temperature dependence of square resistance at varying electric fields at f = 2. MIT is observed near 0.49 V nm–1. Compared to the MIT at f = 1, strong effective mass divergence and the Pomeranchuk effect on the metallic side are not observed. b, Magnetoresistance at varying electric fields at 300 mK. Compared to the MIT at f = 1, magnetic-field-induced metal–insulator transition is not observed.

### Extended Data Fig. 3 Extraction of activation gap at f = 1 and Landau Fermi liquid behaviour at low temperatures.

a, Temperature dependence of the square resistance (symbols) at varying electric fields in an Arrhenius plot. Thermal activation behaviour (dashed lines) is observed at high temperatures, from which the activation gaps are extracted. b, Square resistance (symbols) as a function of temperature squared at varying electric fields. The dashed lines are fits at low temperatures to $${R}_{{\rm{\square }}}={R}_{0}+A{T}^{2}$$ with fitting parameter $${R}_{0}$$ denoting the residual resistance and slope $$A\propto {(m* )}^{2}$$. The slope increases substantially near the critical electric field. The deviation from the Landau Fermi liquid behaviour at low temperatures very close to the critical point $$|E-{E}_{{\rm{c}}}| < 1$$ mV nm–1 is likely to be caused by sample disorders. Typical error bars for the applied electric field are ± 0.2 mV nm–1.

### Extended Data Fig. 4 Resistance scaling at f = 1 near the critical point.

a, Temperature dependence of square resistance at varying electric fields in a log–log plot. A power-law dependence $$\propto {T}^{-1.2}$$(dashed line) is observed at the critical electric field. b, Electric-field dependence of $${\rm{\log }}{R}_{{\rm{\square }}}$$ at different temperatures. The inflection points are marked by the colour symbols. The inset shows the temperature dependence of the electric field at the inflection point. The data shows that the Widom line is nearly a vertical line in Fig. 3c

### Extended Data Fig. 5 Absence of in-plane magnetic field dependence.

Square resistance as a function of bottom gate voltage at varying in-plane magnetic fields. The bottom gate voltage primarily changes the filling factor $$f$$. The electric field is fixed at 3.5 mV nm–1 (from $${E}_{{\rm{c}}}$$) near f = 1. No in-plane magnetic field dependence is observed due to the strong Ising spin–orbit coupling in monolayer TMDs

### Extended Data Fig. 6 Pomeranchuk effect at f = 1.

a, Temperature dependence of square resistance at f = 1 and near 3.5 mV nm–1 above the critical field. b, Temperature dependence of the inverse magnetic susceptibility under the same condition as a. The susceptibility saturates at low temperatures; it follows the Curie–Weiss dependence (dashed lines) above the crossover from a Fermi liquid to an incoherent metal (denoted by the arrow). c, Square resistance as a function of temperature and bottom gate voltage at a fixed top gate voltage. The bottom gate voltage mainly changes the filling factor. The electric field is fixed at 3.5 mV nm–1 near the f = 1 resistance peak (with deviations < 0.2 mV nm–1, the typical uncertain in applied electric fields). The f = 1 resistance peak is absent below ~7 K (horizontal dashed line), where the $${R}_{{\rm{\square }}}-T$$ dependence at f = 1 shows Fermi liquid behaviour (a). Above ~7 K but below $${T}^{\ast }\approx 16$$ K, the f = 1 resistance peak emerges and the $${R}_{{\rm{\square }}}-T$$ dependence deviates from the Fermi liquid behaviour (but still metallic $$\frac{{\rm{d}}{R}_{{\rm{\square }}}}{{\rm{d}}T} > 0$$). The emergence of the resistance peak and the deviation from the Fermi liquid behaviour are correlated with the emergence of local moments (b), demonstrating the Pomeranchuk effect. Above $${T}^{{\rm{* }}}\approx 16$$ K, the f = 1 resistance peak remains but the system displays insulating-like behaviour ($$\frac{{\rm{d}}{R}_{{\rm{\square }}}}{{\rm{d}}T} < 0$$). The result is fully consistent with the results presented in the main text, where the filling factor is kept constant at f = 1

### Extended Data Fig. 7 Spatial homogeneity of device 1.

Two-point current as a function of bottom gate voltage at fixed top gate voltage. The excitation bias voltage is 2 mV. The insulating states at f = 1 and f = 2 are seen at different source–drain pairs corresponding to the optical image in Fig. 1b. The slight shift of the insulating states in gate voltage manifests sample inhomogeneity. The two-point resistance also varies from pair to pair, reflecting the variation in contact/sample resistance

### Extended Data Fig. 8 Major results for device 2.

a, Temperature dependence of the longitudinal resistance at f = 1 under varying electric fields. The critical electric field is near $${E}_{{\rm{C}}}$$ = 0.63 V nm–1. A MIT similar to that in device 1 is observed. b, Longitudinal resistance at 1.6 K in logarithmic scale as a function of top and bottom gate voltages. The gate voltages relate to the hole filling factor f and the applied electric field E. Electric-field-induced MIT is observed at f = 1 and 2. Compared to device 1, there is a higher degree of spatial inhomogeneity in device 2, which prevents reliable scaling analysis near the critical point

### Extended Data Fig. 9 MCD spectrum under a perpendicular magnetic field of 3 T.

a, Electric-field dependence of the MCD spectrum near the WSe2 exciton resonance. Resonance enhancement is observed near 1.66 eV. The vertical dashed line marks the photon energy of the probe laser beam used for the MCD measurements in Fig. 4 and the horizontal dashed line marks the critical point for the MIT. b, MCD spectra at selected electric fields illustrating the resonance enhancement near the exciton peak

### Extended Data Fig. 10 Quantum oscillations in the insulating states.

a, Square resistance as a function of bottom gate voltage at 300 mK. The f = 2 insulating state is labeled. b, Magnetoresistance under a perpendicular magnetic field at selected bottom gate voltages marked by the arrows in a. Quantum oscillations due to the nearby graphite gate are observed near the insulating state. The oscillations disappear away from the f = 2 insulating state. c, Two-terminal magnetoresistance at the f = 2 insulating state with a graphite gate about 5 nm separated from the sample. d, The same as in c except the graphite gate is replaced by a few-layer metallic TaSe2 gate that is ~3 nm away from the sample. No quantum oscillations are developed in both the TaSe2 gate and in the sample under magnetic fields up to 9 T. The results verify that the quantum oscillations are originated from the high mobility graphite gate.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–12 and discussion.

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Li, T., Jiang, S., Li, L. et al. Continuous Mott transition in semiconductor moiré superlattices. Nature 597, 350–354 (2021). https://doi.org/10.1038/s41586-021-03853-0

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• DOI: https://doi.org/10.1038/s41586-021-03853-0

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