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Quantum transport response of topological hinge modes

Nature Physics (2024)Cite this article

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Abstract

Electronic topological phases are typified by the conducting surface states that exist on the boundary of an insulating three-dimensional bulk. While the transport response of the two-dimensional surface states has been studied, the quantum response of the one-dimensional hinge modes has not been demonstrated. Here we provide evidence for quantum transport in gapless topological hinge states existing within the insulating bulk and surface energy gaps in the intrinsic topological insulator α-Bi4Br4. Our magnetoresistance measurements reveal pronounced Aharonov–Bohm oscillations that are periodic in h/e (where h denotes Planck’s constant and e is the electron charge). The observed periodicity evinces quantum interference of electrons circumnavigating around the hinges. We also demonstrate that the h/e oscillations evolve as a function of magnetic field orientation, following the interference paths along the hinge modes that are allowed by topology and symmetry. Our findings reveal the quantum transport response of topological hinge modes with both topological nature and quantum coherence, which can eventually be applied to the development of efficient topological electronic devices.

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Fig. 1: Observation of Aharonov–Bohm oscillations of the topological hinge states.
Fig. 2: Angular dependence of the Aharonov–Bohm oscillations confirming hinge-state formation on the bc plane.
Fig. 3: Temperature dependence of Aharonov–Bohm oscillations.
Fig. 4: Observation of Aharonov–Bohm oscillations in a six-layer α-Bi4Br4 sample.

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All data needed to evaluate the conclusions in the paper are present in the paper. Additional data are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We acknowledge C. Yoon and F. Zhang for providing the calculated hinge states. M.Z.H. group acknowledges primary support from the US Department of Energy (DOE), Office of Science, National Quantum Information Science Research Centers, Quantum Science Center (at ORNL) and Princeton University; STM Instrumentation support from the Gordon and Betty Moore Foundation (GBMF9461) and the theory work; and support from the US DOE under the Basic Energy Sciences programme (grant number DOE/BES DE-FG-02-05ER46200) for the theory and sample characterization work including ARPES. L.B. is supported by the United States Department of Energy under the Basic Energy Sciences programme through grant no. DE-SC0002613. The National High Magnetic Field Laboratory acknowledges support from United States National Science Foundation cooperative agreement grant no. DMR-1644779 and the state of Florida. We thank T. Murphy, G. Jones, L. Jiao and R. Nowell at the National High Magnetic Field Laboratory for technical support. T.N. acknowledges supports from the European Union’s Horizon 2020 research and innovation programme (grant no. ERC-StG-Neupert-757867-PARATOP). The team at the University of Texas at Dallas acknowledges the support by the US Air Force Office of Scientific Research (AFOSR) (FA9550-19-1-0037), National Science Foundation (NSF) (DMREF-2324033) and Office of Naval Research (ONR) (N00014-23-1-2020). Crystal growth at Beijing Institute of Technology is supported by the National Science Foundation of China (NSFC) (Grant No: 92065109), the National Key R&D Program of China (Grant Nos: 2020YFA0308800, 2022YFA1403400), the Beijing Natural Science Foundation (Grant No. Z210006), Y.G.Y. is supported by the National Science Foundation of China (NSFC) (Grant Nos: 12321004, 12234003). Z.W. thanks the Analysis & Testing Center at BIT for assistance in facility support. The authors acknowledge the use of Princeton’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation (NSF)-MRSEC program (DMR-2011750).

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Author notes

  1. These authors contributed equally: Md Shafayat Hossain, Qi Zhang, Zhiwei Wang, Nikhil Dhale.

Authors and Affiliations

  1. Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, NJ, USA

    Md Shafayat Hossain, Qi Zhang, Maksim Litskevich, Nana Shumiya, Jia-Xin Yin, Tyler A. Cochran, Yu-Xiao Jiang, Zi-Jia Cheng, Xian P. Yang & M. Zahid Hasan

  2. Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing, China

    Zhiwei Wang, Yongkai Li, Yuqi Zhang & Yugui Yao

  3. Beijing Key Lab of Nanophotonics and Ultrafine Optoelectronic Systems, Beijing Institute of Technology, Beijing, China

    Zhiwei Wang, Yongkai Li, Yuqi Zhang & Yugui Yao

  4. Material Science Center, Yangtze Delta Region Academy of Beijing Institute of Technology, Jiaxing, China

    Zhiwei Wang & Yongkai Li

  5. Department of Physics, University of Texas at Dallas, Richardson, TX, USA

    Nikhil Dhale, Wenhao Liu & Bing Lv

  6. National High Magnetic Field Laboratory, Tallahassee, FL, USA

    Brian Casas & Luis Balicas

  7. Princeton Materials Institute, Princeton University, Princeton, NJ, USA

    Guangming Cheng & Nan Yao

  8. Department of Physics, University of Zurich, Zurich, Switzerland

    Titus Neupert

  9. Quantum Science Center, Oak Ridge National Laboratory, Oak Ridge, TN, USA

    M. Zahid Hasan

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  1. Md Shafayat Hossain
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  2. Qi Zhang
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Contributions

Transport experiments were performed by M.S.H. and Q.Z. in consultation with M.Z.H. Scanning tunneling microscopy experiments were performed by M.S.H., N.S. and M.L. in consultation with M.Z.H. Crystal growth was carried out by Z.W., N.D., W.L., Y.L., Y. Z., Y. Y. and B.L. Scanning transmission electron microscopy experiments and diffraction analysis were performed by G.C. and N.Y. in consultation with M.S.H. Figure development and writing of the paper were undertaken by M.S.H., L.B., T.N. and M.Z.H. M.Z.H. supervised the project. All the authors discussed the results, interpretation and conclusion.

Corresponding authors

Correspondence to Md Shafayat Hossain, Qi Zhang, Yugui Yao, Bing Lv or M. Zahid Hasan.

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Nature Physics thanks Xian-Lei Sheng, Piotr Wisniewski and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Crystal structure and experimental X-ray diffraction data of α-Bi4Br4.

a, Side view of the crystallographic structure of α-Bi4Br4, showing the interlayer AB stacking. b, Top view of the A-type and B-type monolayers. c, X-ray diffraction data obtained from a single crystal of α-Bi4Br4, featuring the (00 l) preferred peaks. The refined lattice parameters are: a=\(13.0667\)(12) Å, b=4.3359(4) Å, c=20.0676(19) Å, and β=107.336(2)°. The inset displays an optical microscopy image of a typical α-Bi4Br4 bulk single crystal, placed on millimeter grid paper, with the top surface corresponding to the (001) plane. This surface was used for the mechanical exfoliation process to fabricate the devices utilized in our transport measurements.

Source data

Extended Data Fig. 2 Electron diffraction analysis of a mechanically exfoliated α-Bi4Br4.

flake.a, Scanning electron microscopy image of rectangular-shaped α-Bi4Br4 thin flakes on a Si substrate. The crystallographic plane and the axes are indicated for the largest flake used for the diffraction analysis. b, Transmission electron microscopy image displaying the cross-section of the thin flake, which is perpendicular to the long axis of the flake. Inset: corresponding electron diffraction pattern from the [010] zone axis.

Extended Data Fig. 3 Raman spectroscopy of exfoliated and bulk α-Bi4Br4.

a, Raman spectrum obtained from a mechanically exfoliated α-Bi4Br4 flake consisting of four atomic layers. The inset shows an optical microscopy image of the corresponding α-Bi4Br4 flake on a SiO2/Si substrate, with the flake boundary outlined by a white dashed rectangle. b, Raman spectrum of bulk α-Bi4Br4, accompanied by the corresponding optical microscopy image in the inset. The Raman shift acquired from the four-layer flake matches the bulk Raman shift, indicating the preservation of the crystalline properties throughout the exfoliation process. The error bar associated with peak identification is approximately 1 cm−1.

Source data

Extended Data Fig. 4 Scanning tunneling microscopy evidence for a hinge state in a four-layer step edge.

a, Topographic images of two four-layer atomic step edges of opposite geometric orientations. b, Magnified images of the two sides of the four-layer atomic step edges, revealing A type surfaces on both sides. The corresponding regions are marked as 1 and 2 in the large-scale topographic image in panel a. c, Height profiles of the topographic images in panel a, taken perpendicular to the b axis. The corresponding locations are marked on the topographic images in panel a with color-coded lines; the scan directions are indicated with arrows. d, dI/dV spectra shown in color-coding, obtained at the left step edge, right step edge, and regions away from the edge. The positions where these spectra were acquired are marked with matching color-coded dots on the topographic images in panel a. The left step edge exhibits a pronounced in-gap state, while the right step edge shows notable suppression of the density of states at the bulk gap. e, Differential conductance maps of the two four-layer atomic step edges shown in panel a, taken at the Fermi energy (V = 0 mV). A pronounced edge state is observed on the left edge whereas on the right edge, no edge state is visible. Tunnelling junction set-up for spectroscopy: V = −600 mV, I = 0.5 nA, and a root mean square oscillation voltage of 1 mV. Tunnelling junction set-up for dI/dV maps: V = −600 mV, I = 0.3 nA, and a root mean square oscillation voltage of 10 mV. All data were obtained at T ≃ 4.2 K. f, Schematic of a four-layer edge showing quantum hybridization of the quantum spin Hall edge states for neighboring layers. The edges of A and B layers with a facing angle smaller than 180° exhibit a stronger hybridization of the monolayer edge states (depicted as purple spheres). The hybridization of quantum spin Hall edge state is destructive, as illustrated by the lighter color of the edge states (purple spheres). As a result, the left four-layer configuration hosts hinge states on the top and bottom hinges, while the right configuration lacks hinge states.

Source data

Extended Data Fig. 5 Reproducibility of the scanning tunneling microscopy data– evidence for a hinge state in a four-layer step edge, acquired using a different sample and tip.

a, Topographic images of two four-layer atomic step edges with opposite geometric orientations. The terraces on both sides of the step edges exhibit an A-type surface. b, Height profiles of the topographic images in panel a, taken perpendicularly to the b axis. The corresponding locations are marked on the topographic images in panel a with color-coded lines, and the directions of the scans are indicated by arrows. c, Differential conductance maps of the two four-layer atomic step edges shown in panel a, taken at the Fermi energy (V = 0 mV). A pronounced edge state is observed on the left edge, while no edge state is visible on the right edge. d, dI/dV spectra taken at the left step edge (orange), right step edge (green), and away from the edge (purple), revealing striking differences between the two step edges. Orange and green dots in panel a denote the respective positions, on the left and right step edges, where the differential spectra were collected. The left step edge exhibits a pronounced in-gap state, whereas on the right edge, the density of states at the bulk gap is notably suppressed. Tunnelling junction set-up for dI/dV maps: V = −600 mV, I = 0.3 nA, and a root mean square oscillation voltage of 10 mV. Tunnelling junction set-up for spectroscopy: V = −600 mV, I = 0.5 nA, and a root mean square oscillation voltage of 1 mV. All data were obtained at T≃4.2 K and are consistent with Extended Data Fig. 4 results.

Source data

Extended Data Fig. 6 Topological boundary states in a α-Bi4Br4 nanorod with an even number of layers where the inversion symmetry is broken.

a, Nanorod of α-Bi4Br4 featuring helical hinge states resulting from higher-order band topology and two surface Dirac cones on the (010) surface protected by the C2 rotation symmetry around the [010] axis. It is worth noting that the helical hinge states do not require the inversion symmetry and are robust against inversion symmetry breaking16. b, Propagation paths of the helical hinge states when the C2 symmetry is broken on the (010) surface.

Extended Data Fig. 7 Calculated hinge states for a four-layer α-Bi4Br4.

a, The edge-projected band structure for a (001) four-layer ribbon on the top surface of α-Bi4Br4. Purple bands represent gapless hinge states, cyan bands are from the bulk and (001) surfaces, and orange bands are from the (100) and (−100) side surfaces of the ribbon. Due to the inversion asymmetry inherent in even-layer systems, the bands are singly degenerate at each kb. The ribbon is infinitely long in the b-direction and 50-chain wide in the a-direction. For a four-layer system, the left bottom and the left top states (depicted in panel b) undergo hybridization, resulting in a small gap of 4.1396 meV. Note that the energy gap may be slightly overestimated in this calculation due to its derivation from three-dimensional bulk band structure calculations. In reality, our tunneling spectroscopy measurements (see Fig. 4) reveal the presence of a gapless state on the left edge. Additionally, there exists a quantitative disparity in the positioning of the Fermi energy when comparing the calculation with the tunneling spectroscopy measurements. Nevertheless, the calculation provides a qualitative depiction of well-defined hinge states within the energy gaps of both the bulk and surface states. b, Schematic representations in real space illustrating the bulk, surface, and hinge states of a four-layer α-Bi4Br4.

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Extended Data Fig. 8 Temperature dependence of sample resistance and Arrhenius plot of α-Bi4Br4 Sample 1.

Sample resistance, Rxx as a function of temperature, T (in Kelvin), exhibiting a sharp rise below T ≃ 20 K. Inset: The electrical resistivity ρ, normalized to the 300K-value \({\rho }_{T=300{\rm{K}}}\), (left axis, black) and its logarithmic derivative (right axis, red) as a function 1/T.

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Extended Data Fig. 9 Determination of the phase coherent diffusion length from temperature dependent Bohm oscillation data.

a, Temperature dependence of the Fourier transform amplitude of the h/e oscillations. Fitting the data to \(A{T}^{-0.5}\exp \left(P/{L}_{\phi }(T)\right)\) where \({L}_{\phi }\left(T\right)=B{T}^{-0.5}\), A and B are the fitting parameters, P is the perimeter of the loop along the bc plane, yields A=0.43 and \(\frac{P}{B}=0.04\pm 0.02\). The resulting fitting function is represented by the orange curve. b, Temperature dependence of the Fourier transform amplitude of the h/2e oscillations. Fitting the data to the same function yields A=0.2 and \(\frac{P}{B}=0.14\pm 0.06\). The resulting fitting function is represented by the orange curve. c, d, Temperature dependence of the Fourier transform amplitude of the h/e and h/2e oscillations, respectively, now fitted considering \({L}_{\phi }\left(T\right)=B{T}^{-1}\). The fitting yields A=0.44, \(\frac{P}{B}=0.004\pm 0.01\) for h/e oscillations and A=0.18, \(\frac{P}{B}=0.08\pm 0.04\) for h/2e oscillations. The resulting fitting functions are represented by the orange curves. These fits provide \({L}_{\phi }^{{\rm{AB}}}=(100\pm 50){T}^{-\frac{1}{2}}\mu {\rm{m}}\) (or \({L}_{\phi }^{{\rm{AB}}}=(1000\pm 2500){T}^{-1}\mu {\rm{m}}\)) for the Bohm oscillations, and \({L}_{\phi }^{{\rm{AAS}}}=(60\pm 26){T}^{-\frac{1}{2}}\) μm (or \({L}_{\phi }^{{\rm{AAS}}}={\left(100\pm 50\right)T}^{-1}\mu {\rm{m}})\) for the Altshuler–Aronov–Spivak oscillations.

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Hossain, M.S., Zhang, Q., Wang, Z. et al. Quantum transport response of topological hinge modes. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02388-1

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