Abstract
Tin adatoms on a Si(111) substrate with a one-third monolayer coverage form a two-dimensional triangular lattice with one unpaired electron per site. These electrons order into an antiferromagnetic Mott-insulating state, but doping the Sn layer with holes creates a two-dimensional conductor that becomes superconducting at low temperatures. Although the pairing symmetry of the superconducting state is currently unknown, the combination of repulsive interactions and frustration inherent in the triangular adatom lattice opens up the possibility of a chiral order parameter. Here we study the superconducting state of Sn/Si(111) using scanning tunnelling microscopy, scanning tunnelling spectroscopy and quasiparticle interference imaging. We find evidence for a doping-dependent superconducting critical temperature with a fully gapped order parameter, the presence of time-reversal symmetry breaking and a strong enhancement in zero-bias conductance near the edges of the superconducting domains. Although each individual piece of evidence could have a more mundane interpretation, our combined results suggest the possibility that Sn/Si(111) is an unconventional chiral d-wave superconductor.
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Data availability
The data supporting this study are available via Zenodo at https://doi.org/10.5281/zenodo.7249821.
Code availability
The DCA++ code used for this project is available via GitHub at https://github.com/CompFUSE/DCA. The QUANTUM ESPRESSO code can be obtained from https://www.quantum-espresso.org/. The FIREBALL code is available via GitHub at https://github.com/fireball-QMD. Codes for performing the QPI and edge-state calculations are available via Zenodo at https://doi.org/10.5281/zenodo.7249821.
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Acknowledgements
We thank C. D. Batista, P. J. Hirschfeld, P. Kent, A. Tennant and R. Zhang for fruitful discussions. The experimental work and QPI calculations were supported by the Guangdong Basic and Applied Basic Research Foundation (ref no. 2021A1515012034) and by the Office of Naval Research under grant no. N00014-18-1-2675. F.M. acknowledges support from the NSFC (no. 12174456) and the Guangdong Basic and Applied Basic Research Foundation (grant no. 2020B1515020009). C.G. acknowledges financial support from the Community of Madrid through the project NANOMAGCOST CM-PS2018/NMT-4321 and the computer resources at Centro de Computación Científica at UAM (project Biofast) as well as Altamira, with the technical support provided by the Instituto de Física de Cantabria (IFCA) via project QHS-2021-3-0005. J.O. acknowledges financial support by the Spanish Ministry of Science and Innovation through grants MAT2017-88258-R and CEX2018-000805-M (María de Maeztu Programme for Units of Excellence in R&D). The DCA calculations were supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the US Department of Energy (DOE), Office of Science, Advanced Scientific Computing Research, and Basic Energy Sciences, Division of Materials Sciences and Engineering. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC05-00OR22725.
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F.M. and X.W. contributed equally to this work. F.M., X.W., C.C. and K.D.W. prepared the samples and performed the STM experiments. P.M. and T.A.M. performed the DCA calculations. J.S. and J.W.F.V. performed the edge-state calculations. C.G. and J.O. performed the DFT calculations and STM image simulations. S.J. performed the QPI calculations. S.J. and H.H.W. conceived and supervised the project and wrote the manuscript with input from all the authors.
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Extended data
Extended Data Fig. 1 Spectral features of the (\(\sqrt{3}\times \sqrt{3}\))-Sn surface grown on two different Si wafers and estimates of the hole concentration.
a dI/dV spectrum of the p = 0.08 surface at 0.5 K, featuring the lower Hubbard band, quasi-particle peak, and upper Hubbard band (LHB/QPP/UHB). b dI/dV/(I/V) spectrum obtained from the spectrum in panel a, fitted with six Gaussian peaks. From the fitting, we find that the area under the QPP represents 16.1 % of the total spectrum, excluding the peak on the far left which represents the contribution of the silicon valance band. This area fraction converts to a hole doping level of 8.05 %, i,e, p = 0.08; See Ref. 18 for more details. c dI/dV/(I/V) spectrum of the (\(\sqrt{3}\times \sqrt{3}\))-Sn surface (0.5 K), subject to the same fitting analysis as in b. The area fraction of the QPP is 12.1 %, which corresponds to hole doping level of 6.05 % (p = 0.06). Gaussians are used only for the purpose of spectral area determination.
Extended Data Fig. 2 Tunneling spectroscopy of the superconducting state.
a STM image (Vs = − 2 V, It = 0.01 nA) of the p = 0.08 surface with neighbouring (competing) (\(\sqrt{3}\times \sqrt{3}\))-Sn and (\(2\sqrt{3}\times 2\sqrt{3}\))-Sn domains. The former is superconducting while the latter is semiconducting. b-e STM tunneling spectra of the superconducting phase for p = 0.08. b, Field dependent dI/dV spectra measured at 0.5 K. c Temperature dependent dI/dV spectra measured in zero B-field. d Zero bias conductance (ZBC) extracted from panel b. The ZBC increases with the B field and saturates at ~ 13 T. e ZBC extracted from normalized dI/dV (most of the data are shown in Fig. 1e). The ZBC increases with the temperature and saturates around 7.8 K. f-k Tunneling spectra from the p = 0.06 surface. f Field dependent dI/dV spectra measured at 0.5 K. g Temperature dependent dI/dV spectra measured in zero B-field. h Temperature dependent dI/dV spectra measured in at 15 T. i, dI/dV spectra normalized by dividing the spectra in panel g with the corresponding spectra in panel h (same temperature), except for the 0.5 K and 2.0 K data in panel g, for which we used the 3.0 K data in panel h so as to avoid division by the very small signal at zero bias. Note the persistence of the gap feature up to 9 K. j ZBC extracted from panel f. The ZBC increases with the B-field and does not saturate at 15 T. k ZBC extracted from panel i. The ZBC increases with temperature and saturates around 9 K. The normal state spectra in b and c exhibit minor suppression of the conductance near zero bias, which is due to the slow dissipation of the tunneling charge from the surface into the bulk, see Ref. 56 for more details. Such effect becomes more significant for the p = 0.06 sample in panel f-g.
Extended Data Fig. 3 Fitting of the tunneling spectra.
To fit the full T-dependence, we performed a Dynes-like fit of the dI/dV spectra while adopting an angular-dependent gap function Δ(θ) as parameterized in Ref. 16. (The results obtained using this approach are consistent with those obtained by fitting the full momentum-dependence Green’s function in the superconducting state, see Fig. 2.) a-c Fitting results for the p = 0.08 system, assuming s-wave and \({d}_{{x}^{2}+{y}^{2}}+{{{\rm{i}}}}{d}_{xy}\) order parameters. The s-wave and \({d}_{{x}^{2}+{y}^{2}}+{{{\rm{i}}}}{d}_{xy}\)-wave fits only reveal minor differences. d Extracted values of Δ0 as a function of temperature. e The corresponding temperature dependence of the broadening parameter Λ. Error bars in d and e are estimated in a way similar to Ref. 16.
Extended Data Fig. 4 Experimental QPI results.
a-e QPI data and processing procedures. a STM image (Vs = 0.1 V, It = 0.1 nA) of a (\(\sqrt{3}\times \sqrt{3}\))-Sn surface (p = 0.1) with several surface defects appearing as dark spots. b Corresponding dI/dV image at T = 0.5 K. The bright star-like features are centered at the defect locations in panel a. c The power spectrum of panel b, symmetrized and rotated in panel d. The central region is subsequently suppressed to enhance the high frequency features, as shown in panel e [see Ref. 18 for more details]. f-h show 4, 3, and 2 sets of QPI results obtained from (\(\sqrt{3}\times \sqrt{3}\))-Sn surfaces for p=0.1, 0.08, and 0.06, respectively. Each column shows QPI images obtained in a fixed spatial region but with different biases, as indicated on the left. The measurement temperatures are labeled above each column, and data are shown for temperatures above and below Tc. The central flower leafs only appear when the sample is in superconducting state and when the measurement bias is within the superconducting gap (within ± 1.5 mV, ± 2.2 mV, and ± 3.6 mV, in f, g, and h, respectively). These QPI images are enclosed by the dashed red rectangles. Panel f shows QPI results obtained at T=5 K (slightly larger than Tc=4.7 K for this sample), or at 0.5 K in an 8 T B-field (H2c=3 T). These data have a significantly reduced flower leaf feature, which could come from superconducting fluctuations. In panel g, the “0.5 K \({( < {T}_{c})}^{* }\)” data are QPI results obtained from a sample with interstitial Sn adatoms, deposited at 120 K. The presence of interstitial Sn considerably enhances the flower-leaf features at the center of the Brillouin zone.
Extended Data Fig. 5 Simulated QPI spectra for different gap symmetries and scattering centers.
The top and bottom rows show results for nonmagnetic (\(\hat{V}={V}_{0}{\hat{\tau }}_{3}\)) and magnetic (\(\hat{V}={V}_{0}{\hat{\tau }}_{0}\)) scatterers, respectively, with V0 = 100 meV. Results are shown in the superconducting state assuming s-wave (left column), chiral p + ip (middle column), and d + id (right column) order parameters. The spectra are calculated at a bias voltage of 1 meV. In each case, the magnitude of the gap Δ0 and smearing parameter δ are obtained from fits of the dI/dV spectra shown in Fig. 2a (see Methods). Note the absence of the central flower-leaf feature for non-magnetic scattering combined with the s-wave order parameter.
Extended Data Fig. 6 Experimental and simulated STM images of the substitutional Si and interstitial Sn adatom defects.
a-d Experimental (It = 0.1 nA) and simulated STM images for the substitutional Si defect. The Si atom is invisible in filled state images, indicating that there are no occupied dangling bond states to tunneling from. The Si atom is visible in empty state images, but the atom appears to be dim due to its smaller covalent radius. e-h Experimental STM images (It = 0.1 nA) of the interstitial Sn adatom defect center. Panel e reveals three very bright adatoms that are part of the regular (\(\sqrt{3}\times \sqrt{3}\))-Sn lattice. The interstitial Sn atom is located at the center of this cluster and cannot be imaged within the accessible bias range, as the adatom moves at higher biases. i-l Simulated STM images. Note the slightly increased brightness of the Sn atoms indicated by the blue arrows in panel g. This subtle effect is captured by the theory simulation in panel k. m (9 × 9) supercell used in the DFT calculations for the interstitial Sn adatom defect. Sn adatom and Si substrate atoms are shown in green and gold, respectively. The interstitial Sn atom is placed near the center of the (9 × 9) unit cell, as indicated by the red arrow. n Experimental dI/dV spectra recorded on top of the interstitial adatom (red) and far away from the interstitial location (blue). The latter reveals the characteristic LHB/QPP/UHB features (see Extended Data Fig. 1). The strong peak at about -0.35 eV corresponds to the triangular adatom feature in panels a, e. It is captured by the DFT calculation (black line). The peak at +1.25 eV in the theoretical DOS mainly consists of the (empty) 5p orbitals of the interstitial adatom. Simulated images at this bias indeed visualize this atom (not shown), but it cannot be imaged at this tunneling bias.
Extended Data Fig. 7 Tunneling spectra (0.5 K) measured at defect locations on the superconducting (\(\sqrt{3}\times \sqrt{3}\))-Sn surface (p=0.08) and corresponding QPI data.
a STM image (Vs = 0.02 V, It = 0.1 nA) showing several intrinsic surface defects. S1, S2 and S3 are substitutional Si defects and V1 is a Sn vacancy, as judged by the appearance of a hole for a wide range of positive and negative tunneling biases (images at other biases are not shown). Defects with unknown structures are labelled On, with n = 1 − 6. b Tunneling spectra of the V1 and On defects in panel a. c Tunneling spectra of the interstitial Sn adatom defects A1, A2, and A3; see Supplementary Fig.. All defects produce a pair of in-gap states. The substitutional Si defects (S1, S2, and S3) exhibit a well defined double-peak structure at ± 0.6 meV. Interstitial Sn adatoms (A1 and A2) also possess a double peak structure, but with smaller energy splitting ( ± 0.2 meV). The A3 defect appears to have a single (unresolved) peak structure. d STM image (Vs = − 0.5 V, It = 0.1 nA) of a the (\(\sqrt{3}\times \sqrt{3}\))-Sn surface with interstitial Sn defects (indicated by arrows) and other intrinsic defects, mostly substitutional Si or adatom vacancies. e Corresponding dI/dV image obtained at a sample bias of Vs = − 0.4 mV. The interstitial Sn defects produce the strongest scattering features in the real space conductance maps, as compared to those of other intrinsic defects. f,g compares the QPI spectra from the same (\(\sqrt{3}\times \sqrt{3}\))-Sn surface with and without adsorbed Sn defects. IFC represents averaged scattering intensities for the segment of the Fermi contour enclosed by the magenta rectangle, while the averaged intensity for the flower leaf features near the center of the Brillouin zone (enclosed by the red rectangle) is represented in units of IFC. The relative scattering intensity of the flower leaf feature is strongest for the surface with the interstitial Sn adatoms defects.
Extended Data Fig. 8 The leading (degenerate) eigenvectors of the Bethe-Salpeter equation for the triangular lattice Hubbard model.
a, b The momentum space structure of the leading eigenvectors, which show a pairing symmetry consistent with \({d}_{{x}^{2}-{y}^{2}}+{{{\rm{i}}}}{d}_{xy}\) pairing. The size and color of the dots indicate the magnitude and sign of the eigenvector at the momentum points of the 3 × 3 cluster. The green hexagon shows the boundaries of the first Brillouin zone. c The doping dependence of the leading eigenvalues at an inverse temperature of β = 8/t1.
Extended Data Fig. 9 Magnetic vortices of the superconducting (\(\sqrt{3}\times \sqrt{3}\))-Sn surface observed at 0.5 K.
a-e Magnetic vortices for the p = 0.08 sample. a Topographic STM image. b, c dI/dV maps obtained with a tunneling bias inside the superconducting gap with a B-field of 1 T and 3 T, respectively. d dI/dV spectra measured along the line crossing a magnetic vortex in panel c. e ZBC obtained from panel d with an exponential fit for determining the superconducting coherence length ξ. f-i Magnetic vortices for the p = 0.06 sample. f Topographic image. g dI/dV map obtained with a tunneling bias inside the superconducting gap with a B-field of 4 T. This sample exhibits a very low zero bias dI/dV signal, presumably due to the very high series resistance of this lightly doped sample at 0.5 K. Therefore the vortex features are only resolved away from zero bias. h dI/dV spectra measured along the line crossing a magnetic vortex in panel g. i dI/dV (- 1.5 mV) along the line indicated in panel g with an exponential fit for determining the superconducting coherence length.
Extended Data Fig. 10 Electronic band dispersion for a chiral d-wave superconductor in the presence of edges.
Results were obtained by considering a mean-field \({d}_{{x}^{2}-{y}^{2}}+{{{\rm{i}}}}{d}_{xy}\) order parameter on a triangular lattice. The edge state spectrum is obtained by solving the tight-binding Hamiltonian on a cylinder, with open boundary conditions along the y-direction and periodic boundary conditions along the x-direction. Momentum kx remains a good quantum number and the resulting spectrum is shown here for a system of 400 chains (labeled by n) stacked in the y direction and a k-mesh of 400 points. To clearly show the edge states here we computed the spectrum for Δ0 = 1.5 meV. As expected for a chiral d-wave superconductor with Chern number C = ± 2, two chiral linearly dispersing in-gap states exist on each edge.
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Supplementary Information
Supplementary Figs. 1–3 and Notes 1–4.
Source data
Source Data Fig. 1
Source data for Fig. 1c,e–g.
Source Data Fig. 2
Source data for Fig. 2a.
Source Data Fig. 4
Source data for Fig. 4a,f–j.
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Ming, F., Wu, X., Chen, C. et al. Evidence for chiral superconductivity on a silicon surface. Nat. Phys. 19, 500–506 (2023). https://doi.org/10.1038/s41567-022-01889-1
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DOI: https://doi.org/10.1038/s41567-022-01889-1
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