Abstract
Topology1,2,3 and interactions are foundational concepts in the modern understanding of quantum matter. Their nexus yields three important research directions: (1) the competition between distinct interactions, as in several intertwined phases, (2) the interplay between interactions and topology that drives the phenomena in twisted layered materials and topological magnets, and (3) the coalescence of several topological orders to generate distinct novel phases. The first two examples have grown into major areas of research, although the last example remains mostly unexplored, mainly because of the lack of a material platform for experimental studies. Here, using tunnelling microscopy, photoemission spectroscopy and a theoretical analysis, we unveil a ‘hybrid’ topological phase of matter in the simple elemental-solid arsenic. Through a unique bulk-surface-edge correspondence, we uncover that arsenic features a conjoined strong and higher-order topology that stabilizes a hybrid topological phase. Although momentum-space spectroscopy measurements show signs of topological surface states, real-space microscopy measurements unravel a unique geometry of topologically induced step-edge conduction channels revealed on various natural nanostructures on the surface. Using theoretical models, we show that the existence of gapless step-edge states in arsenic relies on the simultaneous presence of both a non-trivial strong Z2 invariant and a non-trivial higher-order topological invariant, which provide experimental evidence for hybrid topology. Our study highlights pathways for exploring the interplay of different band topologies and harnessing the associated topological conduction channels in engineered quantum or nano-devices.
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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.
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Acknowledgements
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 Oak Ridge National Laboratory) and Princeton University, support from the Gordon and Betty Moore Foundation for scanning tunneling microscopy instrumentation and theoretical work (Grant No. GBMF9461) and support from the DOE under the Basic Energy Sciences (DOE-BES) programme for theoretical work and photoemission spectroscopy (Grant No. DOE/BES DE-FG-02-05ER46200). R.I. acknowledges support from the Foundation for Polish Science through the international research agendas programme co-financed by the European Union within the smart growth operational programme and the National Science Foundation (NSF; Grant No. OIA-2229498), University of Alabama at Birmingham (UAB) internal startup funds, and a UAB Faculty Development Grant Program, Office of the Provost. L.B. is supported by the DOE-BES (Award No. DE-SC0002613). The National High Magnetic Field Laboratory acknowledges support from the NSF (Cooperative Agreement Grant No. DMR-1644779) and the state of Florida. T.N. acknowledges support from the Swiss National Science Foundation through a consolidator grant (Grant No. TMCG-2_213805). Z.M. and W.Z. acknowledge support from the National Natural Science Foundation of China (Grant No. 62150410438), the Beihang Hefei Innovation Research Institute (Project No. BHKX-19-02) for growth and beamline 13U of the National Synchrotron Radiation Laboratory for the ARPES experiments. Work at Nanyang Technological University was supported by the National Research Foundation, Singapore, under its Fellowship Award (NRF-NRFF13-2021-0010), the Agency for Science, Technology and Research (A*STAR) under its Manufacturing, Trade and Connectivity (MTC) Individual Research Grant (IRG) (Grant No. M23M6c0100), and the Nanyang Assistant Professorship grant (NTU-SUG).
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M.S.H. conceived the project. The STM experiments were performed by M.S.H. and Y.X.J. in consultation with M.Z.H. F.S. performed the tight-binding calculations. R.I. performed the first-principles calculations. The crystals were grown by Z.M. and W.Z. The ARPES measurements were performed by Z.J.C. and Z.M. The Landau-level calculations were performed by T.H. and M.Y. under the supervision of G.C. The quasiparticle interference calculations were carried out by H.C. under the supervision of G.C. M.S.H., F.S., R.I., T.N., L.B. and M.Z.H. developed the figures and wrote the paper. M.Z.H. supervised the project. All authors discussed the results and participated in the making the interpretation and drawing the conclusion.
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Extended data figures and tables
Extended Data Fig. 1 First-principles calculations and angle-resolved photoemission spectroscopy determination of the electronic band structures.
a, First Brillouin zone and the projected surface along (111). Note that L is part of a group of three inequivalent L-points that are related by C3 rotation symmetry around the z-axis. In the caption of Fig. 4 and the Supplementary Information, these are referred to as L, L′, and L″. b, Electronic band structure of bulk α-As calculated using density functional theory over the whole bulk Brillouin zone, taking into consideration the presence of spin-orbit coupling. The bands crossing the Fermi energy are highlighted in magenta (valence band) and cyan (conduction band). c, Left: Calculated surface bands connecting the conduction and valence bands with Rashba-like features. These bands, depicted using bright orange curves, are projected on the (111) surface (also shown in Fig. 1g). Right: Calculated spin texture of the Rashba-like surface bands. d-f, Angle-resolved photoemission spectroscopy energy-momentum cuts encompassing high symmetry points- \(T\) (panel d), \(\varGamma \) (panel e), and \({L}_{3}\) (panel f)- of the bulk Brillouin zone. The directions are marked in panel a with dashed blue lines. The cuts are measured with a linear -horizontal polarized light with the photon energies of 67 eV (panel d), 40 eV (panel e), and 97 eV (panel f), respectively. The right panels in d-f show the corresponding band structures obtained via first-principles calculations. The bulk bands obtained from photoemission spectroscopy qualitatively match the first-principles results. g, Constant energy contour of the photon energy dependence measurement at the Fermi energy. The size of the electron pocket near \(\varGamma \) does not depend on the photon energy, which confirms its surface-state nature. h, Energy-momentum cut (left) and its second derivative (right) taken along \(\bar{\varGamma }-\bar{M}\) direction of the surface Brillouin zone, obtained from photoemission spectroscopy on the cleaved (111) surface (ab plane). The spectra were acquired using 22 eV, linear-horizontally polarized light. The topological surface state is marked by the white arrow. The small Rashba splitting is visualized in the second derivative plot (also shown in Fig. 1f). i, Extracted dispersion of the Rashba-split surface states, shown using red and blue points. Red and blue curves denote parabolic fits to the photoemission data. These photoemission spectroscopy results are consistent with the first-principles calculations presented in panel c.
Extended Data Fig. 2 Fermi surface of α-As obtained via first-principles calculations and photoemission spectroscopy.
a and b, Three-dimensional Fermi surface of α-As featuring a hole-like (α) pocket shown in magenta and an electron-like (β) pocket shown in cyan, originating from the magenta and cyan bands in Extended Data Fig. 1b, respectively. c, Fermi surface projected on the (111) surface obtained using first-principles tight-binding calculations. It exhibits two additional Fermi pockets centered at k = 0, due to the Rashba-like surface states. d, Fermi surface map taken on the As (111) surface. The map is obtained using 20 eV photon energy and with linear-horizontally polarized light. Photoemission data also show ring-shaped Fermi pockets centering at k = 0 in addition to the three-dimensional bulk state.
Extended Data Fig. 3 Quasiparticle interference signature of the surface state.
a, Large-scale spectroscopic maps (left) and the corresponding Fourier transform at different tip-sample biases; the respective topography is also shown. Clear quasiparticle interference patterns are seen at all the bias voltages. b, Calculated quasiparticle interference patterns. The joint density of states method, which takes into account all possible scattering vectors, has been employed for the calculation. c, Experimental quasiparticle interference spectrum (energy resolution ≃25 meV which is determined by the number of bias points in the tunneling spectrum at each spatial point; q resolution = 0.0015 Å−1 which comes from the dimensions of the scanned area) taken along a high-symmetry direction, marked with an arrow in panel a, right column. The spectrum demonstrates several interference branches. The parabolic-shaped branch, which is the most prominent scattering branch in the quasiparticle interference spectrum, originates from \(\sim -230\pm 15\) meV. This energy position matches the location of the surface band bottom in the photoemission spectroscopy (Fig. 1f) and the first-principles calculation results (Fig. 1g). (A quantitative comparison between the three sets of data is provided in Supplementary Table 1). Therefore, it is most likely that this quasiparticle interference branch stems from the surface state. Tunneling junction set-up for dI/dV maps: Vset = 200 mV, Iset = 1 nA, Vmod = 5 mV. d, Calculated quasiparticle interference spectrum, showing a clear parabolic-shaped scattering branch, which is in excellent agreement with the experimental result presented in panel c.
Extended Data Fig. 4 Landau level spectroscopy of α-As.
a, dI/dV line map (top) and its derivative (bottom) under B = 8 T applied perpendicularly to the cleaved surface. The intense modulation in differential conductance is due to Landau quantization. b, Averaged dI/dV spectra at B = 0 T, and 8 T taken from the same region clarifying that the dI/dV modulation is due to Landau quantization. c, Derivative (to remove the slowly varying background) of the 8 T data and its Fourier transform (inset) showing two (close-by) peaks, suggesting two sets of Landau fans. d, Field-dependent differential conductance data obtained from a line cut of the Landau fan (shown in Fig. 1h) at the Fermi energy. e, Fourier transform of panel d data (after replotting the data as a function of \(\frac{1}{B}\)). Fourier transform magnitude as a function of cyclotron frequency reveals two well-developed peaks at 175 T and 187 T. The two peaks correspond to the Fermi surface areas (in units of cyclotron frequency) of the two Fermi pockets stemming from the Rashba-split surface state. f, Schematic illustration of the Fermi surface stemming from the Rashba bands (see Extended Data Fig. 1c for numerically computed bands). Left: Schematic energy-momentum plot of the parabolic Rashba bands. A cut at the Fermi energy yields two concentric, ring-shaped Fermi pockets. Right: Schematic Fermi surface plot containing the two concentric ring-shaped pockets (see Extended Data Fig. 2c for numerically computed Fermi surfaces). The experimentally obtained 175 T and 187 T peaks (panel e) represent the area of the inner and outer ring-shaped Fermi pockets, respectively. Tunneling junction set-up: Vset = 50 mV, Iset = 0.5 nA, Vmod = 0.5 mV.
Extended Data Fig. 5 Line spectroscopy with high spatial and energy resolutions and the corresponding Fourier transform taken along a bilayer step edge.
This bilayer step edge, whose orientation is identical to that of the step edges displayed in Fig. 3a and d, manifests a step edge state. a, Intensity plot revealing pronounced quantum interference patterns. The green dotted line on the topographic image (shown at the top) indicates the location where the dI/dV line spectroscopy is performed. The direction of the scan is marked by an arrow. b, Corresponding one-dimensional Fourier transform, highlighting a clear edge state dispersion. Tunneling junction set-up: Vset = 100 mV, Iset = 0.5 nA, Vmod = 0.5 mV.
Extended Data Fig. 6 Orientation-dependence of the monolayer step edge states.
a, b, Topographic images, height profiles, and the corresponding differential conductance maps around two monolayer step edges (1 and 2 as marked). The color-coded arrows in the topographies and the height profiles indicate the directions from the bottom to the top terraces. Depending on this direction, one monolayer step edge (monolayer step edge 2) exhibits a step edge state while the other monolayer step edge (monolayer step edge 1) does not. c, Differential spectra, taken at the monolayer step edge 1 (orange), monolayer step edge 2 (green), and away from the step edges (violet), revealing striking differences between the two step edges. Orange and green dots in the topographic images in panels a and b denote the respective positions on the monolayer step edges 1 and 2 where the differential spectra are taken. While the monolayer step edge 2 exhibits a pronounced step edge state, the monolayer step edge 1 features a largely suppressed differential conductance within the soft gap. Note that the monolayer step edge 2 has the same orientation as the monolayer step edge in Fig. 2a, and thereby both exhibit the same electronic structure. On the other hand, resembling the two bilayer step edges shown in Fig. 3a,b, the two monolayer step edges also show a dramatic contrast—one step edge exhibits a pronounced step edge state while the other step edge along the same crystallographic axis, but with a different orientation, does not. Here, however, the orientation favoring a step edge state is opposite to that of the bilayer case (Fig. 3a,b). Tunneling junction set-up for the differential spectra: Vset = 100 mV, Iset = 0.5 nA, Vmod = 0.5 mV. Tunneling junction set-up for dI/dV maps: Vset = 100 mV, Iset = 0.5 nA, Vmod = 1 mV.
Extended Data Fig. 7 Low-energy spectra of the tight binding model in absence of first-order topology.
Here we delineate the step edge dispersion in a tight-binding system that differs from α-As in that it only has a double band inversion at the Γ point of the bulk Brillouin zone, and not an extra single band inversion at each of the three L, L′, L″ points as is the case for α-As. Correspondingly, the model only has nontrivial higher-order topology, but no first-order topology. As derived in our Supplementary Information, such a situation implies a gapped step edge dispersion, however, the gapped modes are precursors of hinge modes localized near the step edges. This situation is schematically depicted in Fig. 4c. a, Monolayer step edge geometry. We preserve periodic boundary conditions in the a1-direction (the out-of-plane direction, with lattice spacing a1), so that k1 is a conserved crystal momentum. There are two step edges, one of type A and one of type B (see Supplementary Information), on each of the top and bottom surfaces. This is the minimal configuration of step edges that preserves inversion symmetry as well as periodic boundary conditions along the a2-direction. b, Monolayer step edge dispersion with k1 using the lattice shown in panel a. We only show the momentum range k1∈ [0, π] because the spectrum in the range k1∈ [π, 2π] is related by time-reversal symmetry. There is no nontrivial spectral flow (the spectrum is gapped). c, Local density of states (LDOS) for the A step edge. The LDOS is large for the low-energy bands closest to the gap, implying that they are well-localized at the step edge. d, Local density of states for the B step edge. This LDOS is small, implying that there are no low-energy states at the step edge. e, Bilayer step edge geometry. f, Bilayer step edge dispersion. There is still a gap, but the low-energy bands are slightly different from the monolayer case. g, Local density of states for the A step edge. The LDOS is again large for the low-energy bands, implying that they are well-localized at the same step edge as for the monolayer. h, Local density of states for the B step edge. Like the monolayer case, there are no step edge states close to the Fermi level.
Extended Data Fig. 8 Step edge states at the monolayer step edges along the b-axis and rotated by 120o with respect to it.
a, Topographic image of two monolayer step edges along the b-axis and rotated by 120o with respect to it. The orientation of the two step edges is opposite to those of the two monolayer step edges in Fig. 3d. Here the crystal side lies between the two step edges contrary to Fig. 3d, where the crystal side extends away from the pit. b, Height profile taken perpendicular to the b-axis direction. The corresponding location is marked on the topographic image in panel a with a green line; the direction of the scan is marked with an arrow. c, Differential conductance maps at different bias voltages, taken in the region shown in panel a. The two monolayer step edges, along the b-axis and rotated by 120o with respect to it, harbor step edge states that are manifested via an enhanced differential conductance. This behavior is in stark contrast to the monolayer step edges along the same directions, but with opposite orientations (Fig. 3d). Such an orientation dependence is consistent with our other data on different step edges along different crystallographic directions and with different orientations and layer thicknesses. Tunneling junction set-up for the differential spectra: Vset = 100 mV, Iset = 0.5 nA, Vmod = 0.5 mV. Tunneling junction set-up for dI/dV maps: Vset = 100 mV, Iset = 0.5 nA, Vmod = 1 mV.
Extended Data Fig. 9 Orientation-dependence of the three-, four-, and six-layer-thick step edge states.
Topographic images, height profiles, and the corresponding differential conductance maps at V = −5 mV around two tri-layer (a and b), four-layer (c and d), and six-layer (e and f) step edges. The color-coded arrows in the topographies and the height profiles indicate the directions from the bottom to the top terraces. Depending on this direction and the layer thickness, certain step edge orientations (such as tri-layer step edge 2, four-layer step edge 1, and six-layer step edge 1) exhibit step edge states, while other orientations (like tri-layer step edge 1, four-layer step edge 2, and six-layer step edge 2) do not. g-i, Differential spectra, taken at the two tri-layer (panel g), four-layer (panel h), and six-layer (panel i) step edges, revealing striking differences between the two step edge orientations. Consistent with the orientation dependence of the monolayer step edges in Extended Data Fig. 6, and opposite to that of the bilayer step edges in Fig. 3a,b, the tri-layer step edge 2 (green curve in panel g; dI/dV map in panel b) shows a pronounced step edge state, while the tri-layer step edge 1 (orange curve in panel g; dI/dV map in panel a) exhibits largely suppressed differential conductance within the soft gap of the surface. Interestingly, this orientation dependence switches for the four-layer step edge, where the four-layer step edge 2 (green curve in panel h; dI/dV map in panel d) lacks a step edge state, while the four-layer step edge 1 (orange curve in panel h; dI/dV map in panel c) carries a strong step edge state. This observation is consistent with the even-odd effect observed between the step-edge preferred orientations in mono and bilayer step edges. The six-layer step edge displays a similar geometry dependence as the four-layer step edge, with the six-layer step edge 2 (green curve in panel i; dI/dV map in panel f) lacking a step edge state, while the six-layer step edge 1 (orange curve in panel i; dI/dV map in panel e) exhibits a pronounced step edge state. Tunneling junction set-up for the differential spectra: Vset = 100 mV, Iset = 0.5 nA, Vmod = 0.5 mV. Tunneling junction set-up for dI/dV maps: Vset = 100 mV, Iset = 0.5 nA, Vmod = 1 mV.
Extended Data Fig. 10 Atomic structure for two different geometric orientations of the mono and bilayer step edges, highlighting the asymmetry between them.
a, Side view (along the bc plane) of the two monolayer step edges. b and c, Atomically resolved topographic images of the corresponding monolayer step edges along the a-direction. d, Side view (along the bc plane) of the two bilayer step edges. e and f, Atomically resolved topographic images (Vgap = 100 mV, It = 3 nA) of the corresponding bilayer step edges along the a direction. Notably, there is an asymmetry between the two orientations in both monolayer and bilayer step edges, with one orientation featuring a sharper step edge. Interestingly, this preferred orientation alternates between the mono and bilayer cases, in accordance with the even-odd effect observed in the step edge states, where the step edge state appears to favor the smoother edge. We note that visualizing atoms directly at the step edge poses challenges due to the abrupt change in height at that location and the small inter-atomic distance (\(\simeq \)3.7 Å) in α-As. As a result, the atom arrays at the step edge may not be as clearly visualized in our scanning tunneling microscopy compared to those located away from the step edge. g, Extended scanning tunneling microscopy data: Topography (bottom) and the corresponding dI/dV maps (V = −5 mV) acquired at B = 0 T and 2 T. At B = 2 T, the spectral weight at the step edge is suppressed. Such a suppression highlights the impact of time-reversal symmetry breaking on the step edge state. Tunneling junction set-up: Vset = 100 mV, Iset = 0.5 nA, Vmod = 1 mV.
Supplementary information
Supplementary Information
This Supplementary Information file contains the following sections: (I) Theory of step edge states in α-As; (II) Tight-binding model with hybrid topology; (III) Lattice-based model of step edge states in the context of quasiparticle interference at step edges; (IV) First-principles calculations showing band hybridization leading to gapless edge states in α-As; and Supplementary References.
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Hossain, M.S., Schindler, F., Islam, R. et al. A hybrid topological quantum state in an elemental solid. Nature 628, 527–533 (2024). https://doi.org/10.1038/s41586-024-07203-8
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DOI: https://doi.org/10.1038/s41586-024-07203-8
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