Landau quantization and highly mobile fermions in an insulator

Abstract

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.

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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.

Data availability

The data that support the findings of this study are available at Harvard Dataverse (https://doi.org/10.7910/DVN/KQNDDQ) or from the corresponding author upon reasonable request.

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Acknowledgements

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 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.

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Contributions

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.

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The authors declare no competing interests.

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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.

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Wang, P., Yu, G., Jia, Y. et al. Landau quantization and highly mobile fermions in an insulator. Nature 589, 225–229 (2021). https://doi.org/10.1038/s41586-020-03084-9

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