Abstract
Introducing the concept of topology has revolutionized materials classification, leading to the discovery of topological insulators and Dirac–Weyl semimetals1,2,3. One of the most fundamental theories underpinning topological materials is the Su–Schrieffer–Heeger (SSH) model4,5, which was developed in 1979—decades before the recognition of topological insulators—to describe conducting polymers. Distinct from the vast majority of known topological insulators with two and three dimensions1,2,3, the SSH model predicts a one-dimensional analogue of topological insulators, which hosts topological bound states at the endpoints of a chain4,5,6,7,8. To establish this unique and pivotal state, it is crucial to identify the low-energy excitations stemming from bound states, but this has remained unknown in solids because of the absence of suitable platforms. Here we report unusual electronic states that support the emergent bound states in elemental tellurium, the single helix of which was recently proposed to realize an extended version of the SSH chain9,10. Using spin- and angle-resolved photoemission spectroscopy with a micro-focused beam, we have shown spin-polarized in-gap states confined to the edges of the (0001) surface. Our density functional theory calculations indicate that these states are attributed to the interacting bound states originating from the one-dimensional array of SSH tellurium chains. Helices in solids offer a promising experimental platform for investigating exotic properties associated with the SSH chain and exploring topological phases through dimensionality control.
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Data availability
The data that support the findings of this study are available within the main text and the extended data. Any other relevant data are available from the corresponding author upon request.
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Acknowledgements
This work was supported by JST-PRESTO (no. JPMJPR18L7), JST-CREST (no. JPMJCR18T1), Grant-in-Aid for Scientific Research (JSPS KAKENHI grant no. JP21H04435) and KEK-PF (proposal no. 2021S2-001). T.K. acknowledges support from GP-Spin at Tohoku University, JSPS (no. 23KJ0099) and JST-SPRING (no. JPMJSP2114).
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The research was conceived by K.N. and proceeded by discussion among K.N., A.T. and T.S.; A.T. and K. Segawa carried out the growth of single crystals; K.N. and A.T. performed the sample surface preparation; K.N., A.T., A.M., T.K., K. Sugawara, S.S., M.K., K.H., H.K., T.T. and T.S. performed the ARPES measurements; K.Y. and T.O. carried out the band-structure calculations; K.N. wrote the paper with input from all the authors. All the authors discussed the results and commented on the paper.
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Extended data figures and tables
Extended Data Fig. 1 Normal-emission ARPES data and estimation of inner potential.
a, Bulk and surface Brillouin zone (BZ) (dark blue and orange, respectively) for Te(0001). b, ARPES spectra measured in the normal-emission set up with varying photon energy (hν) from 46 eV to 100 eV. Blue dotted curve is a guide for the eyes to trace the band dispersion as a function of hν. c, ARPES intensity as a function of kz and binding energy, generated from the data in b by using the inner potential of 10.5 eV. Red curves indicate the bulk bands along the ΓA line obtained from the first-principles band calculations. d-j, Plots of peak position of the energy band traced by the blue dashed line in b, as a function of kz calculated with V0 = 9.0, 9.5, 10.0, 10.5, 11.0, 11.5, and 12.0 eV, respectively. Red curves are the results of numerical fittings with a cosine curve. Blue arrows highlight a finite phase shift δ. k, Plot of δ as a function of V0.
Extended Data Fig. 2 Band structure at naturally occurring edge on the surface.
a, Optical microscope image of Te(0001) surface. b, Plot of ARPES intensity at EF measured at the edge of naturally occurring terraces (indicated by a red circle in a) with 86-eV photons. The ARPES intensity of the in-gap states is marked by an orange arrow. c, Band dispersion measured along the yellow dashed line in b. Light blue curves are the calculated bulk band structure at kz = π.
Extended Data Fig. 3 Photon-energy dependent study of the in-gap states.
a-e, ARPES intensity plots measured at T = 30 K with hν = 76, 86, 96, 106, and 116 eV, respectively, along the \(\overline{{\rm{K}}\Gamma {\rm{KM}}}\) cut (same as the yellow dashed line in Fig. 2e). The in-gap states are marked by orange arrows.
Extended Data Fig. 4 Observation of in-gap states at room temperature.
a, b, ARPES intensity plots at EF measured at room temperature on a large terrace and at the crystal edge of Te(0001) surface, respectively, using 86-eV photons. c, Band dispersion measured along the yellow dashed line in b. Light blue curves are the calculated bulk band structure at kz = π. The ARPES intensity originating from the in-gap states is marked by orange arrow in b and c.
Extended Data Fig. 5 Orbital character of the in-gap states.
a, Orbital-resolved band structure. Red circles in a-c represent the contributions from the px, py, and pz orbitals, respectively, of Te atoms at the end of open Te chains (orange circles in d). The size of circle indicates the magnitude of contribution. d, Model crystal structure constructed to reproduce the band structure of the edge states, reproduced from Fig. 4f.
Extended Data Fig. 6 Site-selective dimerization.
a, b, Side and top views, respectively, of an artificial lattice in which the central Te helix is one-atom longer than others. The endpoints of the central and surrounding helices are indicated by magenta and green spheres, respectively. c, Same as b, but after site-selective dimerization of the helices indexed as A and L. d, e, Magnified view of the central Te helix of b and c, respectively. The definition of Te bond angle θ1 and θ2 is displayed.
Extended Data Fig. 7 Surface tilting effects.
a, b, Side and top views, respectively, of an artificial lattice to illustrate an example of tilted Te(0001) surfaces. The endpoint of each helix is shown by either orange, green, or magenta sphere. c, Reconstructed structure of b, obtained by taking into account the site-selective dimer formation. 1D arrays of open Te chains are highlighted by dashed orange rectangle.
Extended Data Fig. 8 AFM image of Te(0001) surface.
3D view of edges on the Te(0001) surface separated by a surface scratch, visualized by AFM measurements.
Extended Data Fig. 9 Calculated band structure of fully dimerized chains.
a, [0001] projection of the crystal structure which consists of full dimerization of Te chains. The unit cell is indicated by black line. b, Calculated band dispersions for the model shown in a.
Extended Data Fig. 10 Calculated band structure for a slanted surface with full dimerization.
a, b, Top and side views, respectively, of a fully dimerized \((01\bar{1}2)\) surface. c, Band dispersions calculated by assuming the slab displayed in a and b.
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Nakayama, K., Tokuyama, A., Yamauchi, K. et al. Observation of edge states derived from topological helix chains. Nature 631, 54–59 (2024). https://doi.org/10.1038/s41586-024-07484-z
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DOI: https://doi.org/10.1038/s41586-024-07484-z
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