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.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
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
In the version of the article originally published, an error in the sizing of the main figures and extended data figures caused these to render as tiny images in the HTML; this has now been corrected.
References
Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
Georges, A., Kotliar, G., Krauth, W. & Rozenberg, A. J. Dynamical mean-field theory of strongly correlated fermion system and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996).
Dobrosavljević, V., Trivedi, N. & Valles, J. M. Jr Conductor-Insulator Quantum Phase Transitions (Oxford Univ. Press, 2012).
Senthil, T. Theory of a continuous Mott transition in two dimensions. Phys. Rev. B 78, 045109 (2008).
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
Mishmash, R. V., González, I., Melko, R. G., Motrunich, O. I. & Fisher, M. P. A. Continuous Mott transition between a metal and a quantum spin liquid. Phys. Rev. B 91, 235140 (2015).
Szasz, A., Motruk, J., Zaletel, M. P. & Moore, J. E. Chiral spin liquid phase of the triangular lattice hubbard model: a density matrix renormalization group study. Phys. Rev. X 10, 021042 (2020).
Qi, Y. & Sachdev, S. Insulator-metal transition on the triangular lattice. Phys. Rev. B 77, 165112 (2008).
Wu, F., Lovorn, T. Tutuc, E. & MacDonald, A. H. Hubbard Model physics in transition metal dichalcogenide moiré bands. Phys. Rev. Lett. 121, 026402 (2018).
Regan, E. C. et al. Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices. Nature 579, 359–363 (2020).
Tang, Y. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020).
Shimazaki, Y. et al. Strong correlated electrons and hybrid excitons in a moiré heterostructure. Nature 580, 472–477 (2020).
Wang, L. et al. Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 19, 861–866 (2020).
Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature 587, 214–218 (2020).
Jin, C. et al. Stripe phases in WSe2/WS2 moiré superlattices. Nat. Mater. 20, 940–944 (2021).
Huang, X. et al. Correlated insulating states at fractional fillings of the WS2/WSe2 moiré lattice. Nat. Phys. 17, 715–719 (2021).
Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).
Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).
Zhang, Y., Yuan, N. F. Q. & Fu, L. Moiré quantum chemistry: charge transfer in transition metal dichalcogenide superlattices. Phys. Rev. B 102, 201115(R) (2020).
Pan, H., Wu, F. & Das Sarma, S. Quantum phase diagram of a moiré-Hubbard model. Phys. Rev. B 102, 201104(R) (2020).
Morales-Durán, N., Potasz, P. & MacDonald, A. H. Metal-insulator transition in transition metal dichalcogenide heterobilayer moiré superlattices. Phys. Rev. B 103, L241110 (2021).
Pan, H. & Das Sarma, S. Interaction-driven filling-induced metal-insulator transition in 2D moiré lattices. Phys. Rev. Lett. 127, 096802 (2021).
Wietek, A. et al. Mott insulating states with competing orders in the triangular lattice Hubbard model. Preprint at https://arxiv.org/abs/2102.12904 (2021).
Brinkman, W. F. & Rice, T. M. Application of Gutzwiller’s variational method to the metal-insulator transition. Phys. Rev. B 2, 4302–4304 (1970).
Florens, S. & Georges, A. Slave-rotor mean-field theories of strongly correlated systems and the Mott transition in finite dimensions. Phys. Rev. B 70, 035114 (2004).
Yang, H. Y., Läuchli, A. M., Mila, F. & Schmidt, K. P. Effective spin model for the spin-liquid phase of the Hubbard model on the triangular lattice. Phys. Rev. Lett 105, 267204 (2010).
Terletska, H., Vučičević, J., Tanasković, D. & Dobrosavljević, V. Quantum critical transport near the Mott transition. Phys. Rev. Lett 107, 026401 (2011).
Spivak, B., Kravchenko, S. V., Kivelson, S. A. & Gao, X. P. A. Colloquium: Transport in strongly correlated two dimensional electron fluids. Rev. Mod. Phys. 82, 1743–1766 (2010).
Vučičević, J., Terletska, H., Tanasković, D. & Dobrosavljević, V. Finite-temperature crossover and the quantum Widom line near the Mott transition. Phys. Rev. B 88, 075143 (2013).
Furukawa, T., Miyagawa, K., Taniguchi, H., Kato, R. & Kanoda, K. Quantum criticality of Mott transition in organic materials. Nat. Phys. 11, 221–242 (2015).
Pustogow, A. et al. Quantum spin liquids unveil the genuine Mott state. Nat. Mater. 17, 773–777 (2018).
Kadowaki, K. & Woods, S. B. Universal relationship of the resistivity and specific heat in heavy-Fermion compounds. Solid State Comm. 58, 507–509 (1986).
Emery, V. J. & Kivelson, S. A. Superconductivity in bad metals. Phys. Rev. Lett. 74, 3253–3256 (1995).
Xiao, D., Liu, G. B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).
Werner, F., Parcollet, O., Georges, A. & Hassan S. R. Interaction-induced adiabatic cooling and antiferromagnetism of cold fermions in optical lattices. Phys. Rev. Lett. 95, 056401 (2005).
MacNeill, D. et al. Breaking of valley degeneracy by magnetic field in monolayer MoSe2. Phys. Rev. Lett. 114, 037401 (2015).
Abrahams, E., Kravchenko, S. V. & Sarachik, M. P. Metallic behavior and related phenomena in two dimensions. Rev. Mod. Phys. 73, 251–266 (2001).
Wang, Z., Shan, J. & Mak, K. F. Valley- and spin-polarized Landau levels in monolayer WSe2. Nat. Nanotechnol. 12, 144–149 (2017).
Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614–617 (2013).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. 132, 154104 (2010).
Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comp. Mat. Sci. 6, 15–50 (1996).
Shabani, S. et al. Deep moiré potentials in twisted transition metal dichalcogenide bilayers. Nat. Phys. 17, 720–725 (2021).
Li, H. et al. Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices. Nat. Mater. 20, 945–950 (2021).
Li, T. et al. Charge-order-enhanced capacitance in semiconductor moiré superlattices. Nature Nanotechnol. https://doi.org/10.1038/s41565-021-00955-8 (2021).
Zhang, F. C. & Rice, T. M. Effective Hamiltonian for the superconducting Cu oxides. Phys. Rev. B 37, 3759(R) (1988).
Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Yin, J. et al. Dimensional reduction, quantum Hall effect and layer parity in graphite films. Nat. Phys. 15, 437–442 (2019).
Skinner, B. & Shklovskii, B. I. Anomalously large capacitance of a plane capacitor with a two-dimensional electron gas. Phys. Rev. B 82, 155111 (2010).
Xu, Y. et al. Creation of moiré bands in a monolayer semiconductor by spatially periodic dielectric screening. Nat. Mater. 20, 645–649 (2021).
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 and Affiliations
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
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
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.
Source data
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41586-021-03853-0
This article is cited by
-
Valley-polarized excitonic Mott insulator in WS2/WSe2 moiré superlattice
Nature Physics (2024)
-
Giant spin Hall effect in AB-stacked MoTe2/WSe2 bilayers
Nature Nanotechnology (2024)
-
Realization of the Haldane Chern insulator in a moiré lattice
Nature Physics (2024)
-
Excitonic Mott insulator in a Bose-Fermi-Hubbard system of moiré WS2/WSe2 heterobilayer
Nature Communications (2024)
-
The interplay of field-tunable strongly correlated states in a multi-orbital moiré system
Nature Physics (2024)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.