Abstract
Topological materials hosting metallic edges characterized by integer-quantized conductivity in an insulating bulk have revolutionized our understanding of transport in matter. The topological protection of these edge states is based on symmetries and dimensionality. While integer-dimensional effects on topological properties have been studied extensively, the interplay of topology and fractals, which may have a non-integer dimension, remains largely unexplored. Here we demonstrate that topological edge and corner modes arise in fractals formed upon depositing thin layers of bismuth on an indium antimonide substrate. Our scanning tunnelling microscopy results and theoretical calculations reveal the appearance and stability of nearly zero-energy modes at the corners of Sierpiński triangles, as well as the formation of outer and inner edge modes at higher energies. This work opens the perspective to extend electronic device applications in real materials at non-integer dimensions with robust and protected topological states.
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Data availability
All data supporting the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.
Code availability
The numerical codes used for solving the theoretical models (muffin-tin and tight-binding) are available upon request from the corresponding authors.
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Acknowledgements
We thank A. Moustaj for useful insights about the underlying symmetry of the fractal and for helping to set up the model to perform the multi-orbital tight-binding calculations. We are also grateful to M. Röntgen for insightful discussions about latent symmetry and to T. Cysne for useful discussions about spin–orbit coupling in honeycomb lattices. R.C., L.E. and C.M.S. acknowledge the research programme “Materials for the Quantum Age” (QuMat) for financial support. This programme (registration number 024.005.006) is part of the Gravitation programme financed by the Dutch Ministry of Education, Culture and Science (OCW). R.A. thanks the Knut and Alice Wallenberg Foundation for financial support. Authors from SJTU thank the Ministry of Science and Technology of China (Grants No. 2019YFA0308600, 2020YFA0309000), NSFC (Grant Nos. 11790313, 92065201, 11874256, 11874258, 12074247, 12104292, 12174252 and 11861161003), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), the Science and Technology Commission of Shanghai Municipality (Grants Nos. 2019SHZDZX01, 19JC1412701 and 20QA1405100), Innovation programme for Quantum Science and Technology (Grant No. 2021ZD0302500) and the China Postdoctoral Science Foundation (Grant BX2021184) for financial support.
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Contributions
C.M.S. and J.J. led the project. R.C. performed the theoretical calculations under the supervision of C.M.S. and R.A. R.C. and L.E. did the tight-binding calculations. Chen Liu, G.W. and Y.Y. carried out the experiments under the supervision of J.J., while D.G., Y.L., S.W., H.Z. and Canhua Liu did the data analysis. C.M.S. wrote the main paper; R.C., R.A. and C.L. wrote the Supplementary Information, with input from all authors. The manuscript reflects the contributions and ideas of all authors.
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Extended data
Extended Data Fig. 1 Effects of Disorder.
a-e Geometric disorder, f-j potential disorder, k-o displacement disorder. a,f, and k show the type of disorder studied, and the other plots are the LDOS in presence of the specific disorder. All different types of disorder have been simulated using the same parameters as before; a number of grid points nx = ny = 200, an effective electron mass meff = 0.42, a lattice parameter \({a}_{0}\) = 1 nm, an intrinsic spin-orbit parameter λISOC = 106, a potential height u = 0.9 eV, and FWHM of the Gaussian potential d = 0.62 nm. For geometric disorder, we used 1500 waves, but for the other two types of disorder only 750 waves. In the potential disorder, we introduced an error to the potential height taken from a uniform distribution between [-0.1 u, 0.1 u]. For the position disorder, the scatterers coordinates are modified using a uniform distribution between [-0.1\({a}_{0}\), 0.1\({a}_{0}\)] for both axes.
Supplementary information
Supplementary Information
Supplementary Figs. 1–22 and Discussion.
Source data
Source Data Fig. 1
Fig. 1e, dI/dV for substrate and wetting layer (0) for different bias voltages V. Fig 1f, dI/dV for different positions in the sample and bias voltage. The numbering in the data follows the positions in Fig. 1c.
Source Data Fig. 2
Fig. 2a, Calculated density of states for different positions in the muffin-tin simulation without spin–orbit coupling. The coordinates are indicated in the beginning of each row and are also shown in the inset of Fig. 2k. Fig. 2f, Calculated density of states for different positions in the muffin-tin simulation with intrinsic spin–orbit coupling. The coordinates are indicated in the beginning of each row and are also shown in the inset of Fig. 2k. Fig. 2k, Calculated density of states for different positions in the muffin-tin simulation with intrinsic and Rashba spin–orbit coupling. The coordinates are indicated in the beginning of each row and are also shown in the inset of Fig. 2k.
Source Data Fig. 4b
Calculated density of states for different positions in the muffin-tin simulation with straight edges. The coordinates are indicated in the beginning of each row and are also shown in Fig. 4c.
Source Data Extended Data Fig. 1
Extended Data Fig. 1g, Calculated density of states for different positions in the muffin-tin simulation with potential disorder. The coordinates are indicated in the beginning of each row and are also shown in Extended Data Fig. 1f. Extended Data Fig. 1l, Calculated density of states for different positions in the muffin-tin simulation with displacement disorder. The coordinates are indicated in the beginning of each row and are also shown in Extended Data Fig. 1k.
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Canyellas, R., Liu, C., Arouca, R. et al. Topological edge and corner states in bismuth fractal nanostructures. Nat. Phys. 20, 1421–1428 (2024). https://doi.org/10.1038/s41567-024-02551-8
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DOI: https://doi.org/10.1038/s41567-024-02551-8
