Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Landau quantization and highly mobile fermions in an insulator

An Author Correction to this article was published on 02 March 2021

This article has been updated


In strongly correlated materials, quasiparticle excitations can carry fractional quantum numbers. An intriguing possibility is the formation of fractionalized, charge-neutral fermions—for example, spinons1 and fermionic excitons2,3—that result in neutral Fermi surfaces and Landau quantization4,5 in an insulator. Although previous experiments in quantum spin liquids1, topological Kondo insulators6,7,8 and quantum Hall systems3,9 have hinted at charge-neutral Fermi surfaces, evidence for their existence remains inconclusive. Here we report experimental observation of Landau quantization in a two-dimensional insulator, monolayer tungsten ditelluride (WTe2), a large-gap topological insulator10,11,12,13. Using a detection scheme that avoids edge contributions, we find large quantum oscillations in the material’s magnetoresistance, with an onset field as small as about 0.5 tesla. Despite the huge resistance, the oscillation profile, which exhibits many periods, mimics the Shubnikov–de Haas oscillations in metals. At ultralow temperatures, the observed oscillations evolve into discrete peaks near 1.6 tesla, above which the Landau quantized regime is fully developed. Such a low onset field of quantization is comparable to the behaviour of high-mobility conventional two-dimensional electron gases. Our experiments call for further investigation of the unusual ground state of the WTe2 monolayer, including the influence of device components and the possible existence of mobile fermions and charge-neutral Fermi surfaces inside its insulating gap.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Device schematic and the insulating state of monolayer WTe2.
Fig. 2: Quantum oscillations in the monolayer insulator.
Fig. 3: Quantized regime and discrete Landau levels in the monolayer insulator.
Fig. 4: Gate dependence of the quantum oscillations.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available at Harvard Dataverse ( or from the corresponding author upon reasonable request.

Change history


  1. Zhou, Y., Kanoda, K. & Ng, T.-K. Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  2. Chowdhury, D., Sodemann, I. & Senthil, T. Mixed-valence insulators with neutral Fermi surfaces. Nat. Commun. 9, 1766 (2018).

    Article  ADS  Google Scholar 

  3. Barkeshli, M., Nayak, C., Papić, Z., Young, A. & Zaletel, M. Topological exciton Fermi surfaces in two-component fractional quantized Hall insulators. Phys. Rev. Lett. 121, 026603 (2018).

    Article  ADS  CAS  Google Scholar 

  4. Motrunich, O. I. Orbital magnetic field effects in spin liquid with spinon Fermi sea: possible application to κ-(ET)2Cu2(CN)3. Phys. Rev. B 73, 155115 (2006).

    Article  ADS  Google Scholar 

  5. Sodemann, I., Chowdhury, D. & Senthil, T. Quantum oscillations in insulators with neutral Fermi surfaces. Phys. Rev. B 97, 045152 (2018).

    Article  ADS  CAS  Google Scholar 

  6. Tan, B. S. et al. Unconventional Fermi surface in an insulating state. Science 349, 287–290 (2015).

    Article  ADS  CAS  Google Scholar 

  7. Xiang, Z. et al. Quantum oscillations of electrical resistivity in an insulator. Science 362, 65–69 (2018).

    Article  ADS  CAS  Google Scholar 

  8. Sato, Y. et al. Unconventional thermal metallic state of charge-neutral fermions in an insulator. Nat. Phys. 15, 954–959 (2019).

    Article  CAS  Google Scholar 

  9. Zibrov, A. A. et al. Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level. Nature 549, 360–364 (2017).

    Article  ADS  CAS  Google Scholar 

  10. Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in two-dimensional transition metal dichalcogenides. Science 346, 1344–1347 (2014).

    Article  ADS  CAS  Google Scholar 

  11. Fei, Z. et al. Edge conduction in monolayer WTe2. Nat. Phys. 13, 677–682 (2017).

    Article  CAS  Google Scholar 

  12. Tang, S. et al. Quantum spin Hall state in monolayer 1T′-WTe2. Nat. Phys. (2017).

  13. Wu, S. et al. Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76–79 (2018).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  14. Ali, M. N. et al. Large, non-saturating magnetoresistance in WTe2. Nature 514, 205–208 (2014).

    Article  ADS  CAS  Google Scholar 

  15. Fatemi, V. et al. Magnetoresistance and quantum oscillations of an electrostatically tuned semimetal-to-metal transition in ultrathin WTe2. Phys. Rev. B 95, 041410(R) (2017).

    Article  ADS  Google Scholar 

  16. Zheng, F. et al. On the quantum spin Hall gap of monolayer 1T′-WTe2. Adv. Mater. 28, 4845–4851 (2016).

    Article  CAS  Google Scholar 

  17. Song, Y.-H. et al. Observation of Coulomb gap in the quantum spin Hall candidate single-layer 1T′-WTe2. Nat. Commun. 9, 4071 (2018).

    Article  ADS  Google Scholar 

  18. Fatemi, V. et al. Electrically tunable low-density superconductivity in a monolayer topological insulator. Science 362, 926–929 (2018).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  19. Sajadi, E. et al. Gate-induced superconductivity in a monolayer topological insulator. Science 362, 922–925 (2018).

    Article  ADS  CAS  Google Scholar 

  20. Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 2009).

  21. Erten, O., Chang, P.-Y., Coleman, P. & Tsvelik, A. M. Skyrme insulators: insulators at the brink of superconductivity. Phys. Rev. Lett. 119, 057603 (2017).

    Article  ADS  Google Scholar 

  22. Motrunich, O. I. Variational study of triangular lattice spin-1/2 model with ring exchanges and spin liquid state in κ-(ET)2Cu2(CN)3. Phys. Rev. B 72, 045105 (2005).

    Article  ADS  Google Scholar 

  23. Lee, S.-S. & Lee, P. A. U(1) gauge theory of the Hubbard model: spin liquid states and possible application to κ-(BEDT-TTF)2(Cu)2(CN)3. Phys. Rev. Lett. 95, 036403 (2005).

    Article  ADS  Google Scholar 

  24. Ioffe, L. B. & Larkin, A. I. Gapless fermions and gauge fields in dielectrics. Phys. Rev. B 39, 8988–8999 (1989).

    Article  ADS  CAS  Google Scholar 

  25. Han, Z., Li, T., Zhang, L., Sullivan, G. & Du, R.-R. Anomalous conductance oscillations in the hybridization gap of InAs/GaSb quantum wells. Phys. Rev. Lett. 123, 126803 (2019).

    Article  ADS  CAS  Google Scholar 

  26. Xiao, D., Liu, C.-X., Samarth, N. & Hu, L.-H. Anomalous quantum oscillations of interacting electron-hole gases in inverted type-II InAs/GaSb quantum wells. Phys. Rev. Lett. 122, 186802 (2019).

    Article  ADS  CAS  Google Scholar 

  27. Shen, H. & Fu, L. Quantum oscillation from in-gap states and a non-Hermitian Landau level problem. Phys. Rev. Lett. 121, 026403 (2018).

    Article  ADS  CAS  Google Scholar 

  28. Zhang, L., Song, X.-Y. & Wang, F. Quantum oscillation in narrow-gap topological insulators. Phys. Rev. Lett. 116, 046404 (2016).

    Article  ADS  Google Scholar 

  29. Grubinskas, S. & Fritz, L. Modification of the Lifshitz-Kosevich formula for anomalous de Haas–van Alphen oscillations in inverted insulators. Phys. Rev. B 97, 115202 (2018).

    Article  ADS  CAS  Google Scholar 

  30. Ram, P. & Kumar, B. Theory of quantum oscillations of magnetization in Kondo insulators. Phys. Rev. B 96, 075115 (2017).

    Article  ADS  Google Scholar 

  31. Knolle, J. & Cooper, N. R. Quantum oscillations without a Fermi surface and the anomalous de Haas–van Alphen effect. Phys. Rev. Lett. 115, 146401 (2015).

    Article  ADS  Google Scholar 

  32. Knolle, J. & Cooper, N. R. Excitons in topological Kondo insulators: theory of thermodynamic and transport anomalies in SmB6. Phys. Rev. Lett. 118, 096604 (2017).

    Article  ADS  Google Scholar 

  33. Knolle, J. & Cooper, N. R. Anomalous de Haas–van Alphen effect in InAs/GaSb quantum wells. Phys. Rev. Lett. 118, 176801 (2017).

    Article  ADS  Google Scholar 

  34. Pal, H. K., Piéchon, F., Fuchs, J.-N., Goerbig, M. & Montambaux, G. Chemical potential asymmetry and quantum oscillations in insulators. Phys. Rev. B 94, 125140 (2016).

    Article  ADS  Google Scholar 

  35. Baskaran, G. Majorana Fermi sea in insulating SmB6: a proposal and a theory of quantum oscillations in Kondo insulators. Preprint at (2015).

  36. Jia, Y. et al. Evidence for a monolayer excitonic insulator. Preprint at (2020).

  37. Sutherland, B. Localization of electronic wave functions due to local topology. Phys. Rev. B 34, 5208–5211 (1986).

    Article  ADS  CAS  Google Scholar 

  38. Ali, M. N. et al. Correlation of crystal quality and extreme magnetoresistance of WTe2. Europhys. Lett. 110, 67002 (2015).

    Article  ADS  Google Scholar 

  39. Yin, J. et al. Dimensional reduction, quantum Hall effect and layer parity in graphite films. Nat. Phys. 15, 437–442 (2019).

    Article  CAS  Google Scholar 

  40. Zeng, Y. & MacDonald, A. H. Electrically controlled two-dimensional electron-hole fluids. Phys. Rev. B 102, 085154 (2020).

    Article  ADS  CAS  Google Scholar 

Download references


S.W. is indebted to N. P. Ong and A. Yazdani for their support for his new laboratory and their encouragement and discussions regarding this work. We acknowledge discussions with B. A. Bernevig, D. Cobden, F. D. M. Haldane, P. Jarillo-Herrero, P. A. Lee, T. Senthil, I. Sodemann, S. Sondhi and X. Xu. This research was supported primarily by NSF through the Princeton University Materials Research Science and Engineering Center (DMR-1420541 and DMR-2011750) and a CAREER award to S.W. (DMR-1942942). Early measurements were performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement no. DMR-1644779 and the State of Florida. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan, grant no. JPMXP0112101001, JSPS KAKENHI grant no. JP20H00354 and the CREST (JPMJCR15F3), JST. F.A.C. and R.J.C. acknowledge support from the ARO MURI on Topological Insulators (grant W911NF1210461). S.L, S.K. and L.M.S. acknowledge support from the Arnold and Mabel Beckman Foundation through a Beckman Young Investigator grant awarded to L.M.S and the Gordon and Betty Moore Foundation through grant GBMF9064 to L.M.S.

Author information

Authors and Affiliations



P.W., G.Y., Y.J. and M.O. fabricated devices and built measurement systems. P.W., G.Y., Y.J. and S.W. performed measurements and analysed data. S.W. designed and supervised the project. F.A.C., R.J.C., S.L., S.K. and L.M.S. grew and characterized bulk WTe2 crystals. K.W. and T.T. provided hBN crystals. S.W. and P.W. wrote the paper with input from all authors.

Corresponding author

Correspondence to Sanfeng Wu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Johannes Knolle, Feng Miao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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 Sample fabrication process.

a, An optical image of device 1. The monolayer WTe2 is highlighted by the red dashed line. b, Bottom hBN and graphite preparation. c, Deposition of metal contacts (3 nm Ti and 17 nm Pd). d, Transfer of thin hBN in a dry transfer system under ambient conditions. e, Etching thin hBN at the tips of the electrodes. f, Picking up the top graphite. g, Picking up the top hBN. h, Picking up monolayer WTe2. Images in g and h are taken by flipping over the stamp after picking up the flakes. No visible bubbles are observed. i, Transfer of monolayer WTe2 to the prefabricated bottom part of the sample prepared in be. j, k, Typical IV curves (d.c. measurement) of device 1 (j) and device 3 (k) between two contacts at 1.8 K and 7 K respectively, showing ohmic behaviour. All scale bars, 4 μm. Detailed information about the sample fabrication is presented in Supplementary Information section S1.

Extended Data Fig. 2 Summarized data from device 2.

a, An optical image of device 2. The red dashed line highlights monolayer WTe2. b, Gate-dependent two-probe resistance R. The inset shows the measurement geometry. The coloured arrows indicate the gate voltages used in cf. cf, Raw data of the magnetoresistance curves taken at selected Vbg: c, 2.50 V; d, −0.80 V; e, −2.50 V; f, −2.90 V. Strong QOs are observed in the insulating regime but disappear when the monolayer is doped. gj, The same data as in cf, but plotted in G.

Extended Data Fig. 3 Quantum oscillations observed in device 3 with high charge-carrier mobility.

a, An optical image of device 3. The red dashed line highlights monolayer WTe2. b, Comparison of the gate-tuned resistance of the three devices at 70 K (normalized R4p for devices 2 and 3, and normalized R for device 1). See Methods for definition of ng. c, Gate-tuned R4p of device 3 with a fixed D = −0.59 V nm−1 (D ≡ (Vbg/dbg − Vtg/dtg)εr/2)), showing the metal–insulator transition (MIT) at ng ≈ 2.4 × 1012 cm−2, indicated by the arrow. d, Schematic diagram of gate-tuning traces used in e (red) and f (black), along which lines D is fixed while ng is varied. The dashed line indicates the CNP. e, Normalized FFT amplitude of the QOs under varying ng at D = 0.15 V nm−1 (see Methods for discussion of D and of the normalization of FFT amplitude). In addition to the e-like and h-like QO branches, an additional branch α1 appears near CNP. f, ng-tuned FFT amplitude (normalized) of the QOs at D = −0.59 V nm−1. QOs in both the metallic (e-like and h-like branches) and the insulating regime (α2 and β branches) are clearly observed. g, Zoomed-in plot of the FFT features near the CNP in f, highlighting the correlations between the insulating and metallic branches. h, The magnetoresistance trace corresponding to the α1 branch in e (ng = 1.4 × 1012 cm−2 and D = 0.15 V nm−1). i, The magnetoresistance trace (R4p) of the metallic QO branch in f (ng = 5.0 × 1012 cm−2 and D = −0.59 V nm−1). j, Magnetoresistance (R) trace taken at CNP with D = −0.59 V nm−1, showing QOs in the insulating phase. k, FFT of the oscillation component in j. Discussions of these observations are presented in Methods.

Extended Data Fig. 4 Further investigation of the quantum oscillations reported in Extended Data Fig. 3.

a, D-tuned QO frequency map (normalized) at ng = 2.0 × 1012 cm−2 and T = 7 K. Visible branches (α3–α6, e-like and h-like) are noted. b, Vtg-tuned QO frequency map at Vbg = −2.5 V and T = 15 K. c, Vbg-tuned QO frequency map at Vtg = 3.3 V and T = 15 K. Three visible branches of QOs in b and c are labelled as α3, h-like and e-like. At T = 15 K, no QOs are observed in the insulator regime. d, Raw data of the QOs at Vtg = 3.3 V and Vbg = −2.5 V, corresponding to the dashed line in a. Inset, the FFT of the oscillating amplitude, showing three peaks as labelled. e, Illustration of monolayer WTe2, showing the geometry of top and bottom gates. The presence of the displacement field polarizes the electron distribution. This results in the separation of electrons and holes, which can then couple to the top and bottom gates, respectively. f, D-dependence of the insulator QOs (α2 and β branches) at the CNP (T = 2 K). Inset, the corresponding resistance at zero B (the resistance increases further with B), confirming the insulating nature. The behaviours of the α2 and β branches are also sketched (solid orange lines). g, A sketch summarizing the appearance of the gate-independent α branches (α1–α6) on the maps that we have presented. Grey dashed lines, traces that were examined. Blue solid lines, MIT. Black solid line, CNP. Grey area, bad contact region (where information about QOs is inaccessible in our measurement). Discussions of these observations are presented in Methods.

Extended Data Fig. 5 Quantum oscillations observed when using different contact configurations in device 2.

The QOs are robustly observed between various contact pairs connected to the monolayer WTe2 flake. In this device, 7 out of 8 contacts are working (contact 1 was broken during fabrication). Inset, the contact pairs for each data curve. For each coloured data curve, the source and drain contacts are indicated by connecting them with a solid line of the same colour. The data curves were taken individually, that is, all other contacts were disconnected when one pair of contacts was chosen to be measured. Data were recorded at 2 K.

Extended Data Fig. 6 Quantum oscillations observed in d.c. measurements.

The two-probe d.c. conductance (G) as a function of the magnetic field B was measured by applying a small d.c. excitation with an amplitude of 1.5 mV via a Keithley 2400 while monitoring the d.c. current. This strict d.c. measurement yields results consistent with the low-frequency a.c. measurement reported in the main text. Data were taken at 2 K from a pair of contacts in device 2.

Extended Data Fig. 7 The effect of an in-plane magnetic field on the resistance.

The two-probe resistance (R) as a function of the magnetic field B|| applied in the atomic plane of the monolayer WTe2. Data were taken in device 1 using the same two contacts used in Fig. 2, in the insulating plateau at T = 1.8 K.

Extended Data Fig. 8 Additional analysis of the quantum oscillations in devices 1 and 2.

a, Temperature dependence of conductance G ≡ 1/R for device 1, normalized by its zero-field conductance. The same data set was used in Fig. 2, in which R is plotted instead of G. b, Temperature-dependent oscillating amplitude at a selected field (7.13 T) for device 1. The solid line is the fit based on the modified LK formula described in the main text. Values of the fitting parameters extracted here are similar to those from the analysis in Fig. 2. c, d, Conductance data and LK fitting for device 2.

Extended Data Fig. 9 The effect of gating on the quantum oscillations in device 1.

ae, Magnetoresistance (MR ≡ (R(B) – R0)/R0) curves taken at selected Vtg (Vbg = 0 V). f, Colour map of the normalized oscillating component, Rosc/R0, under varying B and Vtg. g, FFT map of the same data. h, A zoomed-in plot of the FFT map at Vtg > 1 V, highlighting the weak peak that features a gate-tunable frequency. i, Gate-dependent FFT amplitudes of all visible peaks. j, A sketch summarizing the observed peaks. The solid black line (αH peak, corresponding to the α peak mentioned in the main text) is the most pronounced peak and the focus of our discussion. Dashed coloured lines are weak features, whose amplitudes are shown in i. In the insulating regime, a lower-frequency peak (αL) emerges together with the αH peak. Towards the hole side, the αH peak splits into two peaks (αH′ and s). Towards the electron side, a new peak (e) emerges from almost zero frequency with an amplitude that decreases monotonically. Coincident with the emergence of the e peak, the amplitudes of the αH and αL peaks drop abruptly. Careful studies of these weak modes under higher magnetic fields and lower temperatures may provide further information to help the understanding of this system. Data were recorded from a pair of contacts in device 1, at 1.8 K.

Extended Data Fig. 10 Magnetoresistance oscillations in non-graphite-gated devices.

a, Two-probe resistance (R) measured for device NG1 as a function of the bottom-gate voltage (Vbg) at a fixed top-gate voltage (Vtg). Inset, the device image and layer schematic. b, The magnetoresistance at a fixed Vbg, corresponding to a location indicated by the green arrow in a. The oscillations are visible at 200 mK but disappear at higher temperatures. c, The normalized oscillating component of the magnetoresistance (Rosc/R0) shown in b after background extraction. d, FFT of the 200 mK curve in c. e, Gate-dependent R and device information for another non-graphite-gated device, NG2. Inset, the device image and layer schematic. f, A magnetoresistance trace observed in the insulating regime (location indicated by the green curve in e). g, Its oscillation component together with the FFT (inset). hj, The same summary as eg but for device NG3. The top gate is replaced by a thin ZrTe2 flake, which is air-sensitive and its degradation could introduce inhomogeneities to the WTe2 channel. We note that in all three devices, the observed sample resistances are lower than our typical graphite devices. Higher top-gate voltages are also needed for the metal top-gated devices in order to perform measurements in the insulating regime (that is, we expect substantial contact resistance in the measured curves). Magnetoresistance oscillations only develop in the sub-kelvin regime, indicating that freezing out the excess charge carrier is perhaps important to their appearance. All scale bars in the device images are 3 μm.

Supplementary information

Supplementary Information

This file contains S1. Sample Fabrication Process, S2. Summary of Device Parameters and S3. Fitting Procedure Using a Modified Lifshitz-Kosevich (LK) Formula.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, P., Yu, G., Jia, Y. et al. Landau quantization and highly mobile fermions in an insulator. Nature 589, 225–229 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


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.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing