Abstract
Localized or propagating Majorana boundary modes are the key feature of topological superconductors. They are rare in naturally occurring compounds, but the tailored manipulation of quantum matter offers opportunities for their realization. Specifically, lattices of Yu–Shiba–Rusinov bound states—Shiba lattices—that arise when magnetic adatoms are placed on the surface of a conventional superconductor can be used to create topological bands within the superconducting gap of the substrate. Here we reveal two signatures consistent with the realization of two types of mirror-symmetry-protected topological superconductor using scanning tunnelling microscopy to create and probe adatom lattices with single-atom precision. The first has edge modes as well as higher-order corner states, and the second has symmetry-protected bulk nodal points. In principle, their topological character and boundary modes should be protected by the spatial symmetries of the adatom lattice. Our results highlight the potential of Shiba lattices as a platform to design the topology and sample geometry of two-dimensional superconductors.
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Data availability
The data underlying Figs. 1, 2 and 4 are available at https://doi.org/10.6084/m9.figshare.22794413. Source data are provided with this paper.
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Acknowledgements
A.A. and S.L. thank M. dos Santos Dias and S. Brinker for fruitful discussions. G.W. acknowledges NCCR MARVEL funding from the Swiss National Science Foundation. T.N. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC-StG-Neupert-757867-PARATOP). M.O.S. acknowledges funding from the Swiss National Science Foundation (Project 200021E_198011) as part of the FOR 5249 (QUAST) lead by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation). A.A. is funded by the the Palestinian-German Science Bridge (BMBF grant no. 01DH16027) and her DFT simulations were made on the supercomputer JURECA at Forschungszentrum Jülich with computing time granted through JARA. R.T. acknowledges the Wuerzburg-Dresden Cluster of Excellence ct.qmat.
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F.K., S.D. and P.S. conceived and performed the experiment. M.O.S., G.W., R.T. and T.N. carried out the theoretical analysis. A.A. and S.L. performed the first-principles calculations. M.O.S., G.W., F.K., P.S., T.N. and S.S.P.P. wrote the paper with input from all the authors.
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Extended data
Extended Data Fig. 1 Scanning tunneling spectroscopy measured with a superconducting Nb tip on clean Nb(110).
A bulk superconducting Nb cluster at the tip was obtained by deep indentation into the sample. a Tunneling spectroscopy shows the convolution of tip and sample density of states resulting in a large gap with size 2(Δtip + Δsample)/e. From the measured convoluted gap size of 3.05 mV and the database value for bulk Nb (ΔNb = 1.52 meV), we obtain a tip gap approximately equal to the bulk value, that is Δtip ≊ ΔNb, which is indicated by the gray area. Measurement parameters: stabilized at sample bias -5 mV b, tunneling current 500 pA, bias AC modulation amplitude 40 μV, temperature 500 mK. b Zero bias differential conductance peak. The measured FWHM of 135 μV provides a reference for our experimental energy resolution. Measurement parameters: stabilized at sample bias -5 mV, tunneling current 30 nA, bias AC modulation amplitude 10 μV, temperature 500 mK.
Extended Data Fig. 2 Spectroscopy with spin-sensitive tip used for Fig. 1e,g of the main text.
Both measurements are acquired on the clean Nb substrate with a superconducting tip featuring single Cr atoms at the apex that induce Shiba states indicated by green arrows for the zero-field measurement (black curve). The applied external magnetic field of 0.7 T in-plane overcomes the second critical field for the Nb sample while superconductivity is maintained in the Nb cluster at the tip (blue curve) where Shiba states are still visible as shoulders on the flank of the gap. The observed spin contrast in constant-height dI/dU maps was strongest for the indicated bias at 0.33 mV.
Extended Data Fig. 3 Deconvoluted tunneling spectroscopy data.
Tunneling spectroscopy data corresponding to Fig. 2 of the main article after performing numerical deconvolution, and the panels correspond to the terminations A1 (a), A2 (b), A3 (c), Β1 (d) and Β2 (e). Respective lattice positions are indicated by colored circles corresponding to STS lines. Numerical deconvolution was carried out following the process described in Ref. 15.
Extended Data Fig. 4 Formation of two dimensional Shiba bands inside the bulk of artificial spin structures.
a, Topographic image from the full spectroscopic measurement grid on structure A1. b–d, Two representative dI/dU maps for bulk Shiba bands at sample biases -2.37 mV (b), -2.21 mV (c) and -2.09 mV (d), demonstrating their 2D character by intensity modulations in both orthogonal directions ([001] and [\(1\overline{1}0\)]).
Extended Data Fig. 5 Analysis of Shiba orbital characters.
a, Topographic image of the A1 lattice analyzed in the main text. b–c, dI/dU maps acquired at biases corresponding to the two most prominent peaks in the dI/dU spectrum reported in d. The spectrum has been obtained by averaging over the region identified by a dashed black box in a. Two peaks are dominating the scene inside the superconducting gap, centered at U = −2.24mV and U = −2.24mV (see black lines). Their strong intensity allow assigning them to d\({}_{{{{{\rm{z}}}}}^{2}}\) -derived states which can be effectively measured by STS because of their large extension into the vacuum. This orbital assignment is corroborated by the respective dI/dU maps, which show two distinct patterns, both characterized by intensity maxima centered around the position of the adatoms, as expected for d\({}_{{{{{\rm{z}}}}}^{2}}\)-derived states.
Extended Data Fig. 6 Averaged bulk signals of structures A and B.
a,b, Topography images of structures A and B with the respective bulk area indicated by a dashed rectangle used for creating a bulk STS signal which is shown in b. c, Plots each signal after deconvolution by a model tip DOS corresponding to a Nb superconducting gap in order to get a good approximation of the sample DOS. Structure B has clearly no gap in the bulk while A features a dip at zero energy which is very close to zero intensity.
Extended Data Fig. 7 Tunneling spectroscopy data acquired in several points along straight lines over the A3 termination.
a, Topography image of structure A3 with two dashed lines indicating STS line paths A (through bulk) and B (along edge) as well as the corner C. b,c Corresponding dI/dU spectra along those lines. At the corner C, indicated by a black solid arrow, we observe the strongest residual intensity at -Δtip (dashed black line) corresponding to zero energy.
Extended Data Fig. 8 Shiba bands along [001] in momentum space.
Evolution of Shiba bands from 1D to 2D comparing a long 1D wire of Cr adatoms along the [001] direction and 0.66 nm inter-atomic distance to a two dimensional arrangement according to structure A. a,b, Topography images. c,d, FFT calculated from the STS data taken along the [001] direction. For better visibility, the plots are separated to show the energy range −4mV… − Δtip and Δtip…4mV with adjusted dI/dU colorscale contrast. The momentum space plots suggest dispersing bands. The signal with highest intensity around ± 2mV is assigned to the dz2 atomic orbital (white dashed lines). Other bands arise from different single-atomic orbitals (red dashed line). From 1D to 2D, a change of sign in the dispersion relation is observed. This demonstrates the potential to engineer the band structure by advancement into two dimensions.
Supplementary information
Supplementary Information
Supplementary Sections I and II.
Source data
Source Data Fig. 2
Spectroscopic data for lattice terminations A1, A2, A3, B1 and B2. Zeroth column: sample bias (meV). Columns 1–4: dI/dU (nS) data for termination A1 (Fig. 2a). The four columns correspond to corner lower left, edge bottom, edge left and bulk. Columns 5–8: dI/dU (nS) data for termination A2 (Fig. 2b). The four columns correspond to corner lower, bulk, corner left and edge lower left. Columns 9–12: dI/dU (nS) data for termination A3 (Fig. 2c). The four columns correspond to edge lower left, bulk, corner left and corner lower. Columns 13–16: dI/dU (nS) data for termination B1 (Fig. 2d). The four columns correspond to bulk, lower corner, lower left edge and left corner. Columns 17–20: dI/dU (nS) data for termination B2 (Fig. 2e). The four columns correspond to lower edge, lower left corner, left edge and bulk.
Source Data Fig. 3
Magnetic coupling extracted from ab initio calculations. Column 0: distance from the reference site. Column 1: distance along the x direction from the reference site. Column 2: distance along the y direction from the reference site. Column 3: ab initio Heisenberg exchange interaction J (meV). Rows 0–21: positions compatible with the rectangular structure (A). Rows 22–39: positions compatible with the rhombic structure (B).
Source Data Fig. 4
Source data for Fig. 4b–d,j. It comprises the following files: data_ribbons_Fig4.txt (for Fig. 4b,c,j) and Spectrum_A3_Fig4d_lower.txt and Spectrum_A3_Fig4d_upper.txt (for Fig. 4d). File data_ribbons_Fig4.txt: ribbon spectra for structures A1 (Fig. 4b), A2 (Fig. 4c) and B1 (Fig. 4j). Ribbon spectra shown in Fig. 4b,c,j. The rows correspond to different values of the momentum. The nth row: k = –π + 2π × n/200. Columns 0–959: eigenvalues of the ribbon structure shown in Fig. 4b. Columns 960–1,439: eigenvalues of the ribbon structure shown in Fig. 4c. Columns 1,440–1,915: eigenvalues of the ribbon structure shown in Fig. 4j. File Spectrum_A3_Fig4d_lower.txt: energy spectrum for A3 termination with n = 101 rows. File Spectrum_A3_Fig4d_upper.txt: energy spectrum for A3 termination with n = 17 rows.
Source Data Extended Data Fig. 1
Source data for Extended Data Fig. 1a,b. The zip files contains ED1_a.txt (Extended Data Fig. 1a) and ED1_b.txt (Extended Data Fig. 1b). File ED1_a.txt: scanning tunnelling spectroscopy measured with a superconducting Nb tip on clean Nb(110). Zeroth column: sample bias (mV). First column: dI/dU (µS). File ED1_b.txt: scanning tunnelling spectroscopy measured with a superconducting Nb tip on clean Nb(110). Zeroth column: sample bias (mV). First column: dI/dU (µS).
Source Data Extended Data Fig. 2
Spectroscopy with spin-sensitive tip used for Extended Data Fig. 1e,g. Zeroth column: sample bias (mV). First column: dI/dU (µS) with 0 T external magnetic field. Second column: dI/dU (µS) with 0.7 T external magnetic field applied in plane.
Source Data Extended Data Fig. 3
Deconvoluted spectroscopic data for lattice terminations A1, A2, A3, B1 and B2. Zeroth column: sample bias (meV). Columns 1–4: density of states (DOS) (arb. units) data for termination A1 (Extended Data Fig. 3a). The four columns correspond to corner lower left, edge bottom, edge left and bulk. Columns 5–8: DOS (arb. units) data for termination A2 (Extended Data Fig. 3b). The four columns correspond to corner lower, bulk, corner left and edge lower left. Columns 9–12: DOS (arb. units) data for termination A3 (Extended Data Fig. 3c). The four columns correspond to edge lower left, bulk, corner left and corner lower. Columns 13–16: DOS (arb. units) data for termination B1 (Extended Data Fig. 3d). The four columns correspond to bulk, lower corner, lower left edge and left corner. Columns 17–20: DOS (arb. units) data for termination B2 (Extended Data Fig. 3e). The four columns correspond to lower edge, lower left corner, left edge and bulk.
Source Data Extended Data Fig. 5
Analysis of Shiba orbital characters. Zeroth column: sample bias (mV). First column: dI/dU (nS).
Source Data Extended Data Fig. 6
Two files with source data for Extended Data Fig. 6b,c, namely, files ED_6b.txt and ED_6c.txt, respectively. File ED_6b.txt: averaged bulk signals of structures A and B. Zeroth column: sample bias (mV). First column: dI/dU (nS), structure A. Second column: dI/dU (nS), structure B. File ED_6c.txt: averaged bulk signals of structures A and B deconvoluted. Zeroth column: energy with respect to the Fermi energy (meV). First column: sample LDOS (arb. units), structure A. Second column: sample LDOS (arb. units), structure B.
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Soldini, M.O., Küster, F., Wagner, G. et al. Two-dimensional Shiba lattices as a possible platform for crystalline topological superconductivity. Nat. Phys. 19, 1848–1854 (2023). https://doi.org/10.1038/s41567-023-02104-5
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DOI: https://doi.org/10.1038/s41567-023-02104-5
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