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Competing correlated states around the zero-field Wigner crystallization transition of electrons in two dimensions


The competition between kinetic energy and Coulomb interactions in electronic systems leads to complex many-body ground states with competing orders. Here we present zinc oxide-based two-dimensional electron systems as a high-mobility system to study the low-temperature phases of strongly interacting electrons. An analysis of the electronic transport provides evidence for competing correlated metallic and insulating states with varying degrees of spin polarization. Some features bear quantitative resemblance to quantum Monte Carlo simulation results, including the transition point from the paramagnetic Fermi liquid to Wigner crystal and the absence of a Stoner transition. At very low temperatures, we resolve a non-monotonic spin polarizability of electrons across the phase transition, pointing towards a low spin phase of electrons, and a two-order-of-magnitude positive magnetoresistance that is challenging to understand within traditional metallic transport paradigms. This work establishes zinc oxide as a platform for studying strongly correlated electrons in two dimensions.

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Fig. 1: The device and quantum transport.
Fig. 2: The metal–insulator transition.
Fig. 3: Spin polarization in an in-plane magnetic field.
Fig. 4: Nonlinear transport characteristics and phase diagram.

Data availability

The data that support the findings of this study are available from the corresponding author on request.


  1. Tanatar, B. & Ceperley, D. M. Ground state of the two-dimensional electron gas. Phys. Rev. B 39, 5005–5016 (1989).

    Article  CAS  Google Scholar 

  2. Rapisarda, F. & Senatore, G. Diffusion Monte Carlo study of electrons in two-dimensional layers. Aust. J. Phys. 49, 161–182 (1996).

    Article  CAS  Google Scholar 

  3. Phillips, P., Wan, Y., Martin, I., Knysh, S. & Dalidovich, D. Superconductivity in a two-dimensional electron gas. Nature 395, 253–257 (1998).

    Article  CAS  Google Scholar 

  4. Chamon, C., Mucciolo, E. R. & Castro Neto, A. H. P-wave pairing and ferromagnetism in the metal-insulator transition in two dimensions. Phys. Rev. B 64, 245115 (2001).

    Article  Google Scholar 

  5. Attaccalite, C., Moroni, S., Gori-Giorgi, P. & Bachelet, G. B. Correlation energy and spin polarization in the 2D electron gas. Phys. Rev. Lett. 88, 256601 (2002).

    Article  Google Scholar 

  6. Spivak, B. & Kivelson, S. A. Phases intermediate between a two-dimensional electron liquid and Wigner crystal. Phys. Rev. B 70, 155114 (2004).

    Article  Google Scholar 

  7. Drummond, N. D. & Needs, R. J. Phase diagram of the low-density two-dimensional homogeneous electron gas. Phys. Rev. Lett. 102, 126402 (2009).

    Article  CAS  Google Scholar 

  8. 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).

    Article  CAS  Google Scholar 

  9. Abrahams, E., Kravchenko, S. V. & Sarachik, M. P. Metallic behavior and related phenomena in two dimensions. Rev. Mod. Phys. 73, 251–266 (2001).

    Article  CAS  Google Scholar 

  10. Kravchenko, S. V. & Sarachik, M. P. Metal-insulator transition in two-dimensional electron systems. Rep. Prog. Phys. 67, 1–44 (2003).

    Article  Google Scholar 

  11. Shashkin, A. A. & Kravchenko, S. V. Recent developments in the field of the metal–insulator transition in two dimensions. Appl. Sci. (2019).

  12. Dolgopolov, V. T. Two-dimensional system of strongly interacting electrons in silicon (100) structures. Phys. Usp. 62, 633–648 (2019).

    Article  CAS  Google Scholar 

  13. Yoon, J., Li, C. C., Shahar, D., Tsui, D. C. & Shayegan, M. Wigner crystallization and metal-insulator transition of two-dimensional holes in GaAs at B = 0. Phys. Rev. Lett. 82, 1744–1747 (1999).

    Article  CAS  Google Scholar 

  14. Hossain, M. S. et al. Observation of spontaneous ferromagnetism in a two-dimensional electron system. Proc. Natl Acad. Sci. USA 117, 32244–32250 (2020).

    Article  CAS  Google Scholar 

  15. Falson, J. et al. MgZnO/ZnO heterostructures with electron mobility exceeding 1 × 106 cm2/Vs. Sci. Rep. 6, 26598 (2016).

    Article  CAS  Google Scholar 

  16. Falson, J. & Kawasaki, M. A review of the quantum Hall effects in MgZnO/ZnO heterostructures. Rep. Prog. Phys. 81, 056501 (2018).

    Article  Google Scholar 

  17. Maryenko, D. et al. Interplay of spin–orbit coupling and Coulomb interaction in ZnO-based electron system. Nat. Commun. 12, 3180 (2021).

    Article  CAS  Google Scholar 

  18. Kozuka, Y. et al. Rashba spin-orbit interaction in a MgxZn1−xO/ZnO two-dimensional electron gas studied by electrically detected electron spin resonance. Phys. Rev. B 87, 205411 (2013).

    Article  Google Scholar 

  19. Falson, J. et al. Even denominator fractional quantum Hall physics in ZnO. Nat. Phys 11, 347–351 (2015).

    Article  CAS  Google Scholar 

  20. Falson, J. et al. A cascade of phase transitions in an orbitally mixed half-filled Landau level. Sci. Adv. 4, eaat8742 (2018).

    Article  Google Scholar 

  21. Shklovskii, B. I. & Efros, A. L. Electronic Properties of Doped Semiconductors (Springer, 1984).

  22. Shklovskii, B. I. Coulomb gap and variable range hopping in a pinned Wigner crystal. Phys. Status Solidi C 1, 46–50 (2004).

    Article  Google Scholar 

  23. Zala, G., Narozhny, B. N. & Aleiner, I. L. Interaction corrections at intermediate temperatures: magnetoresistance in a parallel field. Phys. Rev. B 65, 020201 (2001).

    Article  Google Scholar 

  24. Zala, G., Narozhny, B. N. & Aleiner, I. L. Interaction corrections at intermediate temperatures: longitudinal conductivity and kinetic equation. Phys. Rev. B 64, 214204 (2001).

    Article  Google Scholar 

  25. Knighton, T. et al. Evidence of two-stage melting of Wigner solids. Phys. Rev. B 97, 085135 (2018).

    Article  CAS  Google Scholar 

  26. Kravchenko, S. V., Simonian, D., Sarachik, M. P., Mason, W. & Furneaux, J. E. Electric field scaling at a B = 0 metal-insulator transition in two dimensions. Phys. Rev. Lett. 77, 4938–4941 (1996).

  27. Das Sarma, S. & Hwang, E. H. Low-density finite-temperature apparent insulating phase in two-dimensional semiconductor systems. Phys. Rev. B 68, 195315 (2003).

    Article  Google Scholar 

  28. Matveev, K. A., Glazman, L. I., Clarke, P., Ephron, D. & Beasley, M. R. Theory of hopping magnetoresistance induced by Zeeman splitting. Phys. Rev. B 52, 5289–5297 (1995).

    Article  CAS  Google Scholar 

  29. Chakravarty, S., Kivelson, S., Nayak, C. & Voelker, K. Wigner glass, spin liquids and the metal-insulator transition. Phil. Mag. B 79, 859–868 (1999).

    Article  CAS  Google Scholar 

  30. Bernu, B., Cândido, L. & Ceperley, D. M. Exchange frequencies in the 2D Wigner crystal. Phys. Rev. Lett. 86, 870–873 (2001).

    Article  CAS  Google Scholar 

  31. Bernu, B., Delyon, F., Holzmann, M. & Baguet, L. Hartree-Fock phase diagram of the two-dimensional electron gas. Phys. Rev. B 84, 115115 (2011).

    Article  Google Scholar 

  32. Bernu, B., Delyon, F., Baguet, L. & Holzmann, M. Periodic ground states of the electron gas in two and three dimensions. Contrib. Plasma Phys. 57, 524–531 (2017).

    Article  CAS  Google Scholar 

  33. Jamei, R., Kivelson, S. & Spivak, B. Universal aspects of Coulomb-frustrated phase separation. Phys. Rev. Lett. 94, 056805 (2005).

    Article  Google Scholar 

  34. Spivak, B. & Kivelson, S. A. Transport in two dimensional electronic micro-emulsions. Ann. Phys. 321, 2071–2115 (2006).

    Article  CAS  Google Scholar 

  35. Li, S., Zhang, Q., Ghaemi, P. & Sarachik, M. P. Evidence for mixed phases and percolation at the metal-insulator transition in two dimensions. Phys. Rev. B 99, 155302 (2019).

    Article  CAS  Google Scholar 

  36. Andrei, E. Y. et al. Observation of a magnetically induced Wigner solid. Phys. Rev. Lett. 60, 2765–2768 (1988).

    Article  CAS  Google Scholar 

  37. Ye, P. D. et al. Correlation lengths of the Wigner-crystal order in a two-dimensional electron system at high magnetic fields. Phys. Rev. Lett. 89, 176802 (2002).

    Article  CAS  Google Scholar 

  38. Smolenski, T. et al. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature 595, 53–57 (2021).

    Article  CAS  Google Scholar 

  39. Zhou, Y. et al. Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Nature 595, 48–52 (2021).

    Article  CAS  Google Scholar 

  40. Jang, J., Hunt, B. M., Pfeiffer, L. N., West, K. W. & Ashoori, R. C. Sharp tunnelling resonance from the vibrations of an electronic Wigner crystal. Nat. Phys. 13, 340–344 (2017).

    Article  CAS  Google Scholar 

  41. Pan, W. et al. Exact quantization of the even-denominator fractional quantum Hall state at ν = 5/2 Landau level filling factor. Phys. Rev. Lett. 83, 3530–3533 (1999).

    Article  CAS  Google Scholar 

  42. Samkharadze, N. et al. Integrated electronic transport and thermometry at millikelvin temperatures and in strong magnetic fields. Rev. Sci. Instrum. 82, 053902 (2011).

    Article  CAS  Google Scholar 

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We appreciate discussions with I. Aleiner, J. Checkelsky, S. Das Sarma, N. Drummond, J. Eisenstein, S. Kivelson, C. Murthy, B. Narozhny, B. Spivak and A. Young, along with technical support from J.-S. Xia, N. Sullivan, G. Euchner and S. Wahl. J.F. acknowledges support from the Max Planck Institute, University of British Columbia and University of Tokyo Center for Quantum Materials; the Deutsche Forschungsgemeinschaft (FA 1392/2-1); and the Institute for Quantum Information and Matter, a National Science Foundation Physics Frontiers Center (grant PHY-1733907). B.S. acknowledges support from the National Science Foundation under grant DMR-2045742. Y.K. acknowledges the Japan Science and Technology Agency, PRESTO grant number JPMJPR1763, Japan. M.K. acknowledges the financial support of the Japan Science and Technology Agency, CREST grant number JPMJCR16F1, Japan.

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Authors and Affiliations



J.F. and D.T. gathered experimental data. J.F. performed the molecular beam epitaxy with assistance from Y.K., A.T. and M.K. J.F., I.S. and B.S. wrote the manuscript. All authors discussed the results and commented on the manuscript.

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Correspondence to J. Falson.

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Peer review information Nature Materials thanks Rui-Rui Du, Raymond Ashoori and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Figs. 1–15, Table 1 and Discussion.

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Falson, J., Sodemann, I., Skinner, B. et al. Competing correlated states around the zero-field Wigner crystallization transition of electrons in two dimensions. Nat. Mater. 21, 311–316 (2022).

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