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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The data shown in the plots and that support the findings of this study are available from the corresponding author on reasonable request.
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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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Extended data figures and tables
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 x–z 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 x–y 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 x–y 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 x–z cross-section, averaged over y ∈ [0.1Ly, 0.9Ly]; μ = 0.41. f, Local DOS in the x–y 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.
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 y–z 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.
a–c, 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 x–z 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 x–y cross sections, with the local DOS averaged across the thickness Lz. The chiral modes vanish completely in c.
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. c–d, 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.
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.
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.
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.
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.
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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