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Quantum Hall effect based on Weyl orbits in Cd3As2

Abstract

Discovered decades ago, the quantum Hall effect remains one of the most studied phenomena in condensed matter physics and is relevant for research areas such as topological phases, strong electron correlations and quantum computing1,2,3,4,5. The quantized electron transport that is characteristic of the quantum Hall effect typically originates from chiral edge states—ballistic conducting channels that emerge when two-dimensional electron systems are subjected to large magnetic fields2. However, whether the quantum Hall effect can be extended to higher dimensions without simply stacking two-dimensional systems is unknown. Here we report evidence of a new type of quantum Hall effect, based on Weyl orbits in nanostructures of the three-dimensional topological semimetal Cd3As2. The Weyl orbits consist of Fermi arcs (open arc-like surface states) on opposite surfaces of the sample connected by one-dimensional chiral Landau levels along the magnetic field through the bulk6,7. This transport through the bulk results in an additional contribution (compared to stacked two-dimensional systems and which depends on the sample thickness) to the quantum phase of the Weyl orbit. Consequently, chiral states can emerge even in the bulk. To measure these quantum phase shifts and search for the associated chiral modes in the bulk, we conduct transport experiments using wedge-shaped Cd3As2 nanostructures with variable thickness. We find that the quantum Hall transport is strongly modulated by the sample thickness. The dependence of the Landau levels on the magnitude and direction of the magnetic field and on the sample thickness agrees with theoretical predictions based on the modified Lifshitz–Onsager relation for the Weyl orbits. Nanostructures of topological semimetals thus provide a way of exploring quantum Hall physics in three-dimensional materials with enhanced tunability.

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Fig. 1: The quantum Hall effect in Weyl orbits.
Fig. 2: Quantum Hall effect in sample 1, which is wedge-shaped along the x axis.
Fig. 3: Quantum Hall effect in sample 2, which is wedge-shaped along the y axis.
Fig. 4: Asymmetry in the Hall resistance when tilting along two opposite directions within the yz plane.
Fig. 5: Analysis of the Landau-level shift in Weyl orbits.

Data availability

The data shown in the plots and that support the findings of this study are available from the corresponding author on reasonable request.

References

  1. 1.

    Klitzing, K. v., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    ADS  Article  Google Scholar 

  2. 2.

    Cage, M. E. et al. The Quantum Hall Effect (Springer, Berlin, 2012).

    Google Scholar 

  3. 3.

    Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    ADS  CAS  Article  Google Scholar 

  4. 4.

    Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    ADS  CAS  Article  Google Scholar 

  5. 5.

    Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    ADS  MathSciNet  CAS  Article  Google Scholar 

  6. 6.

    Potter, A. C., Kimchi, I. & Vishwanath, A. Quantum oscillations from surface Fermi arcs in Weyl and Dirac semimetals. Nat. Commun. 5, 5161 (2014).

    ADS  CAS  Article  Google Scholar 

  7. 7.

    Wang, C. M., Sun, H.-P., Lu, H.-Z. & Xie, X. C. 3D quantum Hall effect of Fermi arc in topological semimetals. Phys. Rev. Lett. 119, 136806 (2017).

    ADS  CAS  Article  Google Scholar 

  8. 8.

    Jeckelmann, B. & Jeanneret, B. The quantum Hall effect as an electrical resistance standard. Rep. Prog. Phys. 64, 1603–1655 (2001).

    ADS  CAS  Article  Google Scholar 

  9. 9.

    Halperin, B. I. Possible states for a three-dimensional electron gas in a strong magnetic field. Jpn. J. Appl. Phys. 26, 1913–1919 (1987).

    CAS  Article  Google Scholar 

  10. 10.

    Kohmoto, M., Halperin, B. I. & Wu, Y.-S. Diophantine equation for the three-dimensional quantum Hall effect. Phys. Rev. B 45, 13488 (1992).

    ADS  CAS  Article  Google Scholar 

  11. 11.

    Koshino, M., Aoki, H., Kuroki, K., Kagoshima, S. & Osada, T. Hofstadter butterfly and integer quantum Hall effect in three dimensions. Phys. Rev. Lett. 86, 1062–1065 (2001).

    ADS  CAS  Article  Google Scholar 

  12. 12.

    Bernevig, B. A., Hughes, T. L., Raghu, S. & Arovas, D. P. Theory of the three-dimensional quantum hall effect in graphite. Phys. Rev. Lett. 99, 146804 (2007).

    ADS  Article  Google Scholar 

  13. 13.

    Störmer, H., Eisenstein, J., Gossard, A., Wiegmann, W. & Baldwin, K. Quantization of the Hall effect in an anisotropic three-dimensional electronic system. Phys. Rev. Lett. 56, 85–88 (1986).

    ADS  Article  Google Scholar 

  14. 14.

    Cooper, J. et al. Quantized Hall effect and a new field-induced phase transition in the organic superconductor (TMTSF)2PF6. Phys. Rev. Lett. 63, 1984–1987 (1989).

    ADS  CAS  Article  Google Scholar 

  15. 15.

    Hannahs, S., Brooks, J., Kang, W., Chiang, L. & Chaikin, P. Quantum Hall effect in a bulk crystal. Phys. Rev. Lett. 63, 1988–1991 (1989).

    ADS  CAS  Article  Google Scholar 

  16. 16.

    Hill, S. et al. Bulk quantum Hall effect in η–Mo4O11. Phys. Rev. B 58, 10778–10783 (1998).

    ADS  CAS  Article  Google Scholar 

  17. 17.

    Cao, H. et al. Quantized Hall effect and Shubnikov–de Haas oscillations in highly doped Bi2Se3: evidence for layered transport of bulk carriers. Phys. Rev. Lett. 108, 216803 (2012).

    ADS  Article  Google Scholar 

  18. 18.

    Masuda, H. et al. Quantum Hall effect in a bulk antiferromagnet EuMnBi2 with magnetically confined two-dimensional Dirac fermions. Sci. Adv. 2, e1501117 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Liu, Y. et al. Zeeman splitting and dynamical mass generation in Dirac semimetal ZrTe5. Nat. Commun. 7, 12516 (2016).

    ADS  CAS  Article  Google Scholar 

  20. 20.

    Zhang, Y., Bulmash, D., Hosur, P., Potter, A. C. & Vishwanath, A. Quantum oscillations from generic surface Fermi arcs and bulk chiral modes in Weyl semimetals. Sci. Rep. 6, 23741 (2016).

    ADS  CAS  Article  Google Scholar 

  21. 21.

    Moll, P. J. et al. Transport evidence for Fermi-arc-mediated chirality transfer in the Dirac semimetal Cd3As2. Nature 535, 266–270 (2016).

    ADS  CAS  Article  Google Scholar 

  22. 22.

    Zhang, C. et al. Evolution of Weyl orbit and quantum Hall effect in Dirac semimetal Cd3As2. Nat. Commun. 8, 1272 (2017).

    ADS  Article  Google Scholar 

  23. 23.

    Borisenko, S. et al. Experimental realization of a three-dimensional Dirac semimetal. Phys. Rev. Lett. 113, 027603 (2014).

    ADS  Article  Google Scholar 

  24. 24.

    Wang, Z., Weng, H., Wu, Q., Dai, X. & Fang, Z. Three-dimensional Dirac semimetal and quantum transport in Cd3As2. Phys. Rev. B 88, 125427 (2013).

    ADS  Article  Google Scholar 

  25. 25.

    Neupane, M. et al. Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2. Nat. Commun. 5, 3786 (2014).

    CAS  Article  Google Scholar 

  26. 26.

    Liu, Z. K. et al. A stable three-dimensional topological Dirac semimetal Cd3As2. Nat. Mater. 13, 677–681 (2014).

    ADS  CAS  Article  Google Scholar 

  27. 27.

    Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    ADS  Article  Google Scholar 

  28. 28.

    Weng, H., Fang, C., Fang, Z., Bernevig, B. A. & Dai, X. Weyl semimetal phase in noncentrosymmetric transition-metal monophosphides. Phys. Rev. X 5, 011029 (2015).

    Google Scholar 

  29. 29.

    Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

    ADS  CAS  Article  Google Scholar 

  30. 30.

    Lv, B. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).

    Google Scholar 

  31. 31.

    Schumann, T. et al. Observation of the quantum Hall Effect in confined films of the three-dimensional Dirac semimetal Cd3As2. Phys. Rev. Lett. 120, 016801 (2018).

    ADS  Article  Google Scholar 

  32. 32.

    Datta, S. Electronic Transport in Mesoscopic Systems Ch. 2 (Cambridge Univ. Press, Cambridge, 1997).

    Google Scholar 

  33. 33.

    Feng, J. et al. Large linear magnetoresistance in Dirac semimetal Cd3As2 with Fermi surfaces close to the Dirac points. Phys. Rev. B 92, 081306 (2015).

    ADS  Article  Google Scholar 

  34. 34.

    Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).

    CAS  Article  Google Scholar 

  35. 35.

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    ADS  CAS  Article  Google Scholar 

  36. 36.

    Mostofi, A. A. et al. Wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 178, 685–699 (2008).

    ADS  CAS  Article  Google Scholar 

  37. 37.

    Zhang, E. et al. Magnetotransport properties of Cd3As2 nanostructures. ACS Nano 9, 8843–8850 (2015).

    CAS  Article  Google Scholar 

  38. 38.

    Zhang, C. et al. Room-temperature chiral charge pumping in Dirac semimetals. Nat. Commun. 8, 13741 (2017).

    ADS  CAS  Article  Google Scholar 

  39. 39.

    Chen, Z.-G. et al. Scalable growth of high mobility Dirac semimetal Cd3As2 microbelts. Nano Lett. 15, 5830–5834 (2015).

    ADS  CAS  Article  Google Scholar 

  40. 40.

    Ali, M. N. et al. The crystal and electronic structures of Cd3As2, the three-dimensional electronic analogue of graphene. Inorg. Chem. 53, 4062–4067 (2014).

    CAS  Article  Google Scholar 

  41. 41.

    Koch, C. C. Nanostructured Materials: Processing, Properties and Applications Ch. 2 (William Andrew, Norwich, 2006).

    Google Scholar 

  42. 42.

    Goyal, M. et al. Thickness dependence of the quantum Hall effect in films of the three-dimensional Dirac semimetal Cd3As2. APL Mater. 6, 026105 (2018).

    ADS  Article  Google Scholar 

  43. 43.

    Zhang, Y. et al. Crossover of the three-dimensional topological insulator Bi2Se3 to the two-dimensional limit. Nat. Phys. 6, 584–588 (2010).

    Article  Google Scholar 

  44. 44.

    Uchida, M. et al. Quantum Hall states observed in thin films of Dirac semimetal Cd3As2. Nat. Commun. 8, 2274 (2017).

    ADS  Article  Google Scholar 

  45. 45.

    Heying, B. et al. Optimization of the surface morphologies and electron mobilities in GaN grown by plasma-assisted molecular beam epitaxy. Appl. Phys. Lett. 77, 2885–2887 (2000).

    ADS  CAS  Article  Google Scholar 

  46. 46.

    Li, Y. Y. et al. Intrinsic topological insulator Bi2Te3 thin films on Si and their thickness limit. Adv. Mater. 22, 4002–4007 (2010).

    CAS  Article  Google Scholar 

Download references

Acknowledgements

F.X. was supported by the National Natural Science Foundation of China (grant numbers 61322407, 11474058, 61674040 and 11874116), the National Key Research and Development Program of China (grant numbers 2017YFA0303302 and 2018YFA0305601) and the National Young 1000 Talent Plan. Y.Z. was supported by NSF DMR-1308089 and a Bethe fellowship at Cornell University. J.Z. was supported by the Youth Innovation Promotion Association CAS (grant number 2018486) and the Users with Excellence Project of Hefei Science Center CAS (grant number 2018HSC-UE011). C.Z. and X.Y. were supported by the China Scholarships Council (CSC; grant numbers 201706100053 and 201706100054). We acknowledge discussions with Y.-M. Lu and Y. Yu. C.Z. thanks T. Song for helping with part of the atomic force microscope measurements. Part of the sample fabrication was performed at Fudan Nano-fabrication Laboratory. Part of the transport measurements was performed at the High Magnetic Field Laboratory (HMFL), CAS. A portion of this work was performed at the National High Magnetic Field Laboratory (USA), which is supported by NSF cooperative agreement numbers DMR-1644779 and DMR-1157490 and the state of Florida. H.L. was supported by the Guangdong Innovative and Entrepreneurial Research Team Program (grant number 2016ZT06D348), the National Key R&D Program (grant number 2016YFA0301700), the National Natural Science Foundation of China (grant number 11574127) and the Science, Technology, and Innovation Commission of Shenzhen Municipality (grant numbers ZDSYS20170303165926217 and JCYJ20170412152620376). A.N. acknowledges support from ETH Zurich. A.C.P. was supported by NSF DMR-1653007.

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Nature thanks T. Das and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Contributions

F.X. conceived the ideas and supervised the research. S.L., H.Z., Z.N. and R.L. synthetized the Cd3As2 nanobelts. C.Z. fabricated the devices. C.Z., X.Y., J.Z. and Y.L. carried out the high-magnetic-field transport measurements, with the help of L.P., E.S.C. and A.S. J.Z. designed the rotation probe and set up the measurement system for high-magnetic-field transport experiments in HMFL. C.Z. analysed the transport data. Y.Z., A.N., S.S., H.-Z.L. and A.C.P. provided theoretical support. C.Z., Y.Z. and F.X. wrote the paper, with input from all co-authors.

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Correspondence to Faxian Xiu.

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Extended data figures and tables

Extended Data Fig. 1 Numerical simulations of a wedge-shaped Weyl semimetal.

a, b, The local density of states (DOS; colour scale) in different cross-sections of a clean system with random impurities W = 0 (a) and a system with random impurities W = 0.1 (both in the bulk and on the surface; b), showing the location and extent of the modes. The system size is Lx = 120, Ly = 100 and Lz [21, 35], with sharp side walls at both ends. The upper panels show trapezoidal xz cross-sections, with the local DOS averaged across y [0.1Ly, 0.9Ly] to exclude contributions from the modes on the side walls at both ends in the y direction. The lower panels show the xy cross-sections, with the local DOS averaged across the thickness Lz. We set the Fermi energy to μ = 0.4. c, d, Real-space thickness profile in the xy plane (Lz, colour scale; c) and the local DOS (d). The settings are the same as in a, except that we consider a system with an uneven top surface (see c). e, Local DOS across the xz cross-section, averaged over y [0.1Ly, 0.9Ly]; μ = 0.41. f, Local DOS in the xy cross-section, averaged over the thickness Lz, for Fermi energies of μ = 0.41, μ = 0.40 and μ = 0.39. As the Fermi energy shifts, the x position of the bulk modes moves accordingly. g, Schematic of the Landau levels. A chiral mode appears where the Landau level crosses the Fermi energy μ (red dotted lines). The energy profile also illustrates the origin of the counter-propagating chiral modes on the right side wall. The system size is Lx = 90, Ly = 100 and Lz [21, 49]; W = 0.

Extended Data Fig. 2 Illustration of Weyl orbits beyond the Lifshitz energy.

a, b, Surface Fermi arcs and projection of the bulk Fermi surfaces (FSs). Whereas the Fermi surfaces are separate and carry topologically quantized Berry curvatures in a, they merge into a single Fermi surface in b. Still, parts of the surface Fermi arcs remain well defined because the bulk is gapped at these surface wavevectors. c, Local DOS (colour scale) in the yz cross-section of the Weyl semimetal lattice model with a slab geometry and a magnetic field at cyclotron resonance. The Fermi energy μ = 1.05 > 1.0 is beyond the Lifshitz transition in the bulk, yet we still observe clear signatures of the Weyl orbits consisting of both surface and bulk components.

Extended Data Fig. 3 Numerical simulations of a wedge-shaped Dirac semimetal.

ac, Local DOS (colour scale) of Dirac semimetal models showing the location and extent of the modes for different orbit hybridization amplitudes: Δ = 0.1 in the two surface layers at the top and bottom surfaces (a); Δ = 0.05 throughout the entire bulk (b); and Δ = 0.2 (c). The system size is Lx = 64, Ly = 100 and Lz [21, 35], with sharp side walls at both ends; μ = 0.4, W = 0. The upper panels are the trapezoidal xz cross sections, with the local DOS averaged across y [0.1Ly, 0.9Ly] to exclude contributions from the modes on the side walls at both ends in the y direction. The lower panels show the xy cross sections, with the local DOS averaged across the thickness Lz. The chiral modes vanish completely in c.

Extended Data Fig. 4 Finite-thickness effect in Cd3As2 slabs with (112) surface.

a, Bulk Landau-level spectrum for a Weyl cone without quantum confinement. The blue line denotes the gapless chiral Landau level. b, Bulk Landau-level spectrum for a Weyl cone with quantum confinement along the z axis. The yellow dots denote the discrete energy levels due to quantization in the finite-thickness quantum well. cd, Representative band structures of Cd3As2 slabs with thicknesses of about 40 nm (c) and about 60 nm (d). The red lines denote the Fermi arc surface states, which persist for the 40-nm-thick films. e, Energy gap as a function of thickness. The finite-size effect is only important for slabs with thicknesses of less than 10 nm.

Extended Data Fig. 5 SEM image of Cd3As2 nanoplates.

a, A typical SEM image of as-grown Cd3As2 nanostructures. Scale bar, 100 μm. b, X-ray diffraction pattern of a large Cd3As2 belt with the (112) plane.

Extended Data Fig. 6 Additional transport data for sample 1.

a, Low-field oscillations of Rxx. The inset shows the corresponding Landau fan diagrams, the fitting curves of which have the same slope. The oscillation frequency is unchanged when measured at different terminals, as indicated by the parallel dashed black lines, suggesting that the Landau-level shift is not induced by the change in the size of the Fermi surface. The low-field oscillations are used to extract the Landau-level index. b, Hall resistance as a function of 1/B. c, Comparison of the Hall resistance Rxy at positive and negative magnetic fields for terminals 1–2, which is field-symmetric. d, Temperature dependence of Rxx. Small oscillation features are observed at very low temperature (below 1 K). Considering their periodicity and carrier mobility, and given that there is no corresponding feature in Rxy, these oscillations are not from the fractional quantum Hall effect. They probably come from regimes with different thickness, the Landau levels of which are shifted by the thickness-dependent phase term. e, Stack view of bulk quantum oscillations with in-plane magnetic field measured at different terminals of sample 1.

Extended Data Fig. 7 Additional transport data for samples 2 and 3.

a, b, Bulk quantum oscillations in samples 2 (a) and 3 (b) with the magnetic field applied in-plane. The oscillation positions are symmetric in B and there is no shift between the two curves, which helps to exclude changes in the bulk band with thickness as the origin of the Landau-level shift. c, Rxx measured at terminals 2–4 at different temperatures, which is also asymmetric under magnetic fields, similarly to Fig. 3e. The arrow marks the large oscillation peak, which is detected in Rxy (Fig. 3e) at the same field. d, Hall resistance measured at two sets of Hall terminals along the x axis with the same thickness profile. The inset shows an optical image of the Hall bar device with terminals indexed; scale bar, 15 μm.

Extended Data Fig. 8 Transport data for samples 3 and 4.

a, b, Magnetotransport results for sample 3. c, d, Magnetotransport results for sample 4. Samples 3 and 4 both show strong asymmetric behaviour with magnetic field, similarly to sample 2.

Extended Data Fig. 9 Change in the Landau-level plateau width with sample thickness and field direction.

a, Change in plateau width in sample 1 at different terminals. The trends for even and odd plateaus are opposite. b, Change in the plateau width for positive and negative magnetic fields. Here ν is the filling factor. The error bars represent the deviations of the transition field.

Extended Data Table 1 Summary of physical parameters for the four samples

Supplementary information

Supplementary Information

This file contains detailed discussions of theories and models of the three-dimensional quantum Hall effect and chiral modes in Weyl/Dirac semimetals in the presence of thickness variations and orbital hybridization.

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Zhang, C., Zhang, Y., Yuan, X. et al. Quantum Hall effect based on Weyl orbits in Cd3As2. Nature 565, 331–336 (2019). https://doi.org/10.1038/s41586-018-0798-3

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