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Ordered and tunable Majorana-zero-mode lattice in naturally strained LiFeAs

Abstract

Majorana zero modes (MZMs) obey non-Abelian statistics and are considered building blocks for constructing topological qubits1,2. Iron-based superconductors with topological bandstructures have emerged as promising hosting materials, because isolated candidate MZMs in the quantum limit have been observed inside the topological vortex cores3,4,5,6,7,8,9. However, these materials suffer from issues related to alloying induced disorder, uncontrolled vortex lattices10,11,12,13 and a low yield of topological vortices5,6,7,8. Here we report the formation of an ordered and tunable MZM lattice in naturally strained stoichiometric LiFeAs by scanning tunnelling microscopy/spectroscopy. We observe biaxial charge density wave (CDW) stripes along the Fe–Fe and As–As directions in the strained regions. The vortices are pinned on the CDW stripes in the As–As direction and form an ordered lattice. We detect that more than 90 per cent of the vortices are topological and possess the characteristics of isolated MZMs at the vortex centre, forming an ordered MZM lattice with the density and the geometry tunable by an external magnetic field. Notably, with decreasing the spacing of neighbouring vortices, the MZMs start to couple with each other. Our findings provide a pathway towards tunable and ordered MZM lattices as a platform for future topological quantum computation.

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Fig. 1: Crystalline structure, topographic image and superconducting behaviour of the unstrained and strained regions.
Fig. 2: dI/dV map of the vortices and the analysis of the vortex bound states under 0.5 T.
Fig. 3: Analysis of the dI/dV spectra along a linecut across two neighbouring vortices and the origin of the MZMs.
Fig. 4: Tuning the MZM lattice with external magnetic fields.

Data availability

Data measured or analysed during this study are available from the corresponding author on reasonable request. Source data are provided with this paper.

References

  1. Kitaev, A. Y. U. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    MathSciNet  CAS  MATH  Article  ADS  Google Scholar 

  2. Nayak, C. et al. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    MathSciNet  CAS  MATH  Article  ADS  Google Scholar 

  3. Zhang, P. et al. Observation of topological superconductivity on the surface of an iron-based superconductor. Science 360, 182–186 (2018).

    PubMed  Article  ADS  CAS  Google Scholar 

  4. Wang, D. F. et al. Evidence for Majorana bound states in an iron-based superconductor. Science 362, 333–335 (2018).

    CAS  PubMed  Article  ADS  Google Scholar 

  5. Machida, T. et al. Zero-energy vortex bound state in the superconducting topological surface state of Fe(Se,Te). Nat. Mater. 18, 811–815 (2019).

    CAS  PubMed  Article  ADS  Google Scholar 

  6. Kong, L. Y. et al. Majorana zero modes in impurity-assisted vortex of LiFeAs superconductor. Nat. Commun. 12, 4146 (2021).

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  7. Liu, Q. et al. Robust and clean Majorana zero mode in the vortex core of high-temperature superconductor (Li0.84Fe0.16)OHFeSe. Phys. Rev. X 8, 041056 (2018).

    CAS  Google Scholar 

  8. Liu, W. Y. et al. A new Majorana platform in an Fe–As bilayer superconductor. Nat. Commun. 11, 5688 (2020).

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  9. Zhu, S. Y. et al. Nearly quantized conductance plateau of vortex zero mode in an iron-based superconductor. Science 367, 189–192 (2020).

    CAS  PubMed  Article  ADS  Google Scholar 

  10. Hanaguri, T. et al. Scanning tunneling microscopy/spectroscopy of vortices in LiFeAs. Phys. Rev. B 85, 214505 (2012).

    Article  ADS  CAS  Google Scholar 

  11. Fente, A. et al. Influence of multiband sign-changing superconductivity on vortex cores and vortex pinning in stoichiometric high-Tc CaKFe4As4. Phys. Rev. B 97, 134501 (2018).

    Article  ADS  Google Scholar 

  12. Chiu, C. K. et al. Scalable Majorana vortex modes in iron-based superconductors. Sci. Adv. 6, eaay0443 (2020).

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  13. Zhang, S. T. S. et al. Vector field controlled vortex lattice symmetry in LiFeAs using scanning tunneling microscopy. Phys. Rev. B 99, 161103 (2019).

    CAS  Article  ADS  Google Scholar 

  14. Kitaev, A. Y. U. Unpaired Majorana fermions in quantum wires. Phys. Uspekhi 44, 131–136 (2001).

    Article  ADS  Google Scholar 

  15. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    CAS  Article  ADS  Google Scholar 

  16. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).

    PubMed  Article  ADS  CAS  Google Scholar 

  17. Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductor–superconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).

    PubMed  Article  ADS  CAS  Google Scholar 

  18. Oreg, Y., Refael, G. & von Oppen, F. Helical liquids and Majorana bound states in quantum wires. Phys. Rev. Lett. 105, 177002 (2010).

    PubMed  Article  ADS  CAS  Google Scholar 

  19. Potter, A. C. & Lee, P. A. Multichannel generalization of Kitaev’s Majorana end states and a practical route to realize them in thin films. Phys. Rev. Lett. 105, 227003 (2010).

    PubMed  Article  ADS  CAS  Google Scholar 

  20. Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor–semiconductor nanowire devices. Science 336, 1003–1007 (2012).

    CAS  PubMed  Article  ADS  Google Scholar 

  21. Nadj-Perge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).

    CAS  PubMed  Article  ADS  Google Scholar 

  22. Xu, J. P. et al. Experimental detection of a Majorana mode in the core of a magnetic vortex inside a topological insulator–superconductor Bi2Te3/NbSe2 heterostructure. Phys. Rev. Lett. 114, 017001 (2015).

    CAS  PubMed  Article  ADS  Google Scholar 

  23. Deng, M. T. et al. Majorana bound state in a coupled quantum-dot hybrid-nanowire system. Science 354, 1557–1562 (2016).

    CAS  PubMed  Article  ADS  Google Scholar 

  24. Wang, Z. J. et al. Topological nature of the FeSe0.5Te0.5 superconductor. Phys. Rev. B 92, 115119 (2015).

    Article  ADS  CAS  Google Scholar 

  25. Wu, X. X. et al. Topological characters in Fe(Te1−xSex) thin films. Phys. Rev. B 93, 115129 (2016).

    Article  ADS  CAS  Google Scholar 

  26. Yuan, Y. H. et al. Evidence of anisotropic Majorana bound states in 2M-WS2. Nat. Phys. 15, 1046–1051 (2019).

    CAS  Article  Google Scholar 

  27. Kezilebieke, S. et al. Topological superconductivity in a van der Waals heterostructure. Nature 588, 424–428 (2020).

    CAS  PubMed  Article  ADS  Google Scholar 

  28. Nayak, A. K. et al. Evidence of topological boundary modes with topological nodal-point superconductivity. Nat. Phys. 17, 1413–1419 (2021).

    CAS  Article  Google Scholar 

  29. Zhang, P. et al. Multiple topological states in iron-based superconductors. Nat. Phys. 15, 41–47 (2019).

    CAS  Article  Google Scholar 

  30. Cao, L. et al. Two distinct superconducting states controlled by orientation of local wrinkles in LiFeAs. Nat. Commun. 12, 6312 (2021).

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  31. Yim, C. M. et al. Discovery of a strain-stabilised smectic electronic order in LiFeAs. Nat. Commun. 9, 2602 (2018).

    PubMed  PubMed Central  Article  ADS  CAS  Google Scholar 

  32. Allan, M. P. et al. Anisotropic energy gaps of iron-based superconductivity from intraband quasiparticle interference in LiFeAs. Science 336, 563–567 (2012).

    CAS  PubMed  Article  ADS  Google Scholar 

  33. Allan, M. P. et al. Identifying the ‘fingerprint’ of antiferromagnetic spin fluctuations in iron pnictide superconductors. Nat. Phys. 11, 177–182 (2015).

    CAS  Article  Google Scholar 

  34. Zhang, J. L. et al. Upper critical field and its anisotropy in LiFeAs. Phys. Rev. B 83, 174506 (2011).

    Article  ADS  CAS  Google Scholar 

  35. Hayashi, N., Isoshima, T., Ichioka, M. & Machida, K. Low-lying quasiparticle excitations around a vortex core in quantum limit. Phys. Rev. Lett. 80, 2921–2924 (1998).

    CAS  Article  ADS  Google Scholar 

  36. Caroli, C., Degennes, P. G. & Matricon, J. Bound fermion states on a vortex line in a type-II superconductor. Phys. Lett. 9, 307–309 (1964).

    MATH  Article  ADS  Google Scholar 

  37. Fan, P. et al. Observation of magnetic adatom-induced Majorana vortex and its hybridization with field-induced Majorana vortex in an iron-based superconductor. Nat. Commun. 12, 1348 (2021).

    CAS  PubMed  PubMed Central  Article  ADS  Google Scholar 

  38. Kong, L. Y. et al. Half-integer level shift of vortex bound states in an iron-based superconductor. Nat. Phys. 15, 1181–1187 (2019).

    CAS  Article  Google Scholar 

  39. Hu, L. H., Wu, X. X., Liu, C. X. & Zhang, R. X. Competing vortex topologies in iron-based superconductors. Preprint at https://arxiv.org/abs/2110.11357 (2021).

  40. Cheng, M., Lutchyn, R. M., Galitski, V. & Das Sarma, S. Tunneling of anyonic Majorana excitations in topological superconductors. Phys. Rev. B 82, 094504 (2010).

    Article  ADS  CAS  Google Scholar 

  41. Kawakami, T. & Hu, X. Evolution of density of states and a spin-resolved checkerboard-type pattern associated with the Majorana bound state. Phys. Rev. Lett. 115, 177001 (2015).

    PubMed  Article  ADS  CAS  Google Scholar 

  42. Bonderson, P., Freedman, M. & Nayak, C. Measurement-only topological quantum computation via anyonic interferometry. Ann. Phys. 324, 787–826 (2009).

    MathSciNet  CAS  MATH  Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank G. Su and H. Ding for discussions. The work is supported by the Ministry of Science and Technology of China (2019YFA0308500, 2018YFA0305700, 2017YFA0206303), the National Natural Science Foundation of China (61888102, 51991340, 52072401), the Chinese Academy of Sciences (XDB28000000, XDB30000000, 112111KYSB20160061), and the CAS Project for Young Scientists in Basic Research (YSBR-003). Z.W. is supported by the US DOE, Basic Energy Sciences grant no. DE-FG02-99ER45747.

Author information

Authors and Affiliations

Authors

Contributions

H.-J.G. designed the experiments and supervised the project. X.W. and C.J. prepared samples. M.L., G.L., L.C. and X.Z. performed STM experiments with the guidance of H.-J.G. G.L., C.-K.C., S.J.P., Z.W. and H.-J.G. did data analysis and wrote the manuscript. All of the authors participated in analysing experimental data, plotting figures and writing the manuscript.

Corresponding authors

Correspondence to Ziqiang Wang or Hong-Jun Gao.

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The authors declare no competing interests.

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Nature thanks Andreas Kreisel, Peter Wahl and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Comparison of lattice distortion among different regions of LiFeAs.

ac, Atomic-resolution topographic images of the unstrained region, the uniaxial CDWFe–Fe region and the biaxial CDW region. The crystallographic directions are shown in the lower right. df, Fourier transform images of ac, respectively. The angles of the crystallographic directions are marked by coloured arrows.

Extended Data Fig. 2 Evolution of the CDWAs–As and CDWFe–Fe stripes with bias voltages.

ap, dI/dV maps of a 70 nm × 70 nm biaxial CDW region under bias voltages of 0, 1.0, 1.6, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 and 12.0 mV, respectively. The red dot markers in c and k outline the π phase shift of the CDWAs–As pattern below and above the superconducting gap. The green cross markers in d and p outline the π phase shift of the CDWFe–Fe pattern below and above the superconducting gap. The CDWAs–As pattern shows a splitting behaviour of the stripes, which starts at an energy of approximately 2.2 mV (e). The stripes split into two sets, as highlighted by red and blue dashed lines (ej). Each set of the stripes keeps the same periodicity with the original CDWAs–As stripes. However, both sets show dynamical behaviour with energy by moving in opposite directions, as highlighted by the red and blue arrows (g, j). At an energy of ~4 mV, the split stripes recombine, returning back into a single set of stripes (j, k), with a π phase shift (c, k). Similar splitting–recombining behaviour exists in the CDWFe–Fe stripes. The splitting starts at an energy of approximately 2.8 mV, as highlighted by green and brown dotted lines (h). The recombination happens at approximately 9 mV, which is accompanied by a π phase shift (d, p). The same behaviour happens on the negative-bias side for the CDWAs–As and CDWFe–Fe stripes. The full set of dI/dV maps can be found in Supplementary Video 1. The evolution of the uniaxial CDWFe–Fe stripes can be found in Supplementary Video 2.

Extended Data Fig. 3 Detailed analysis of the negative second derivative of the intensity plot of the dI/dV spectra across vortex #2.

a, Left, Intensity map of −d3I/dV3 spectra across a topological vortex (vortex #2) along the As–As direction. The vertical black dashed lines outline the positions of the discrete vortex core states. The bound states show spatial dispersion of peak positions, similar to that observed in vortex #1. The horizontal white dashed lines outline the positions of dark stripes of CDWFe–Fe, in accordance with the height profile at right. The spatial variations of the vortex bound states happen in the vicinity of the positions of the dark stripes of CDWFe–Fe, suggesting a signature of the hybridization of the core states, which is a consequence of C4 rotation and reflection-symmetry breaking. b, Statistical analysis of the peak positions in a. Five energy bound states at −1.81 ± 0.08 (L−2), −0.87 ± 0.15 (L−1), 0.03 ± 0.04 (L0), 0.92 ± 0.20 (L1) and 1.95 ± 0.19 (L2) meV are extracted.

Extended Data Fig. 4 Statistics of the topological and ordinary vortices in the MZM lattice.

a, dI/dV map of the large-scale vortices at 0 mV. The red dots mark the vortices with sharp ZBCP (topological) of the MZMs, and the yellow dots mark the vortices without sharp ZBCP (ordinary). The spectra are calibrated by the multi-Gaussian peak fitting to extract the accurate energy positions of the vortex bound states. 48 out of 51 vortices show the sharp ZBCP at the centres. The scanning area is 200 nm × 200 nm. b, c, Individual dI/dV spectrum at each of the topological (b) and ordinary (c) vortices. More than  90% of the vortices have the characteristics of the MZM.

Source data

Extended Data Fig. 5 Statistics of MZMs under different magnetic fields.

af, dI/dV spectra taken at the centres of different vortex cores under different magnetic fields. g, Histogram and percentage of topological vortices under different magnetic fields. The percentage of topological vortices is above 90% at all the magnetic fields up to 6 T.

Source data

Extended Data Fig. 6 Possible defect-induced topological–trivial vortex transition.

a, Vortex lattice in the first round of measurement under a magnetic field of 3 T. The red and yellow dots represent topological and trivial vortices, respectively. b, Topographic image of the same region in a. The positions of the topological and trivial vortices are overlaid. c, Vortex lattice of the second round of measurement after the field is ramped down to 0 and then back to 3 T. The red and yellow dots represent topological and trivial vortices, respectively. d, Topographic image of the same region in c. The positions of the topological and trivial vortices are overlaid. The white dashed circles in b and d mark the positions of the impurities. Although the trivial vortices appear in different regions for the two rounds of measurements, they are all located in the vicinity of the impurities (‘brighter dots’), as outlined by the white dashed circles in b and d.

Extended Data Fig. 7 Large-scale STM image of the strained and unstrained regions of the LiFeAs.

The large-scale image is stitched together from 12 independent STM topographic images. The strained region locates between two big steps with heights of approximately 7 nm, consisting of two kinds of regions, the uniaxial CDWFe–Fe (upper left) and the biaxial CDW (upper right) regions, Vs = −20 mV, It = 30 pA.

Extended Data Fig. 8 Spatially dependent modulation of the CDW gap in the strained region.

a, b, dI/dV spectra across the strained region (a) and the corresponding Fourier transform image (b). The hump features at energies of approximately 13 mV and roughly 22 mV are modulated by the As–As stripes. c, d, STM image (c) and CDWAs–As gap map (d) of the biaxial CDW region. The CDW gap is extracted by calculating the peak-to-peak values of the CDW coherence peaks, as labelled in Supplementary Fig. 7. The gap value is strongly modulated by the As–As stripes in a way that the gap sizes on the As–As stripes are lower than off the stripes (in c: Vs = −15 mV, It = 200 pA).

Extended Data Fig. 9 Comparison of the spectral feature of the vortices in the unstrained, uniaxial CDWFe–Fe and biaxial CDW regions.

a, dI/dV spectra taken at the centres of ordinary vortices in the unstrained (black), the uniaxial CDWFe–Fe region (brown) and a topological vortex in the biaxial CDW region (red). b, Intensity map of the dI/dV linecut across the ordinary vortex in the unstrained region. c, Intensity map of the dI/dV linecut across the ordinary vortex in the uniaxial CDWFe–Fe region. d, Intensity map of the dI/dV linecut across the topological vortex in the biaxial CDW region. The topological vortices exist only in the biaxial CDW regions.

Source data

Extended Data Fig. 10 Correlation between the vortex spacing and the dI/dV spectra of the MZM vortices.

af, dI/dV maps of MZM lattices at 0 mV under magnetic fields from 0.5 T to 6 T, respectively. g, Averaged dI/dV spectra under different magnetic fields. The spectra are taken under the same scanning settings. With increasing magnetic fields, the ZBCPs of the dI/dV spectra get lower and broader. This phenomenon indicates that a coupling of the MZMs appears when the vortices get closer to each other under higher magnetic fields. The averaged dI/dV spectra under different fields can be found in Supplementary Video 3.

Source data

Supplementary information

Supplementary Information

This file contains Supplementary Table 1 and Supplementary Figs. 1–9

Peer Review File

41586_2022_4744_MOESM3_ESM.mp4

Supplementary Video 1 Energy-dependent dI/dV maps of the biaxial CDW region. The CDWAs–As pattern shows a splitting behaviour of the stripes which starts at ~2.2 mV. Each set of the stripes keeps the same periodicity with the original CDWAs–As stripes but both sets show dynamical behaviour with energy by moving in opposite directions. At ~4 mV, the split stripes recombine into a single set of stripes, with a π phase shift. The same behaviour happens on the negative bias side.

41586_2022_4744_MOESM4_ESM.mp4

Supplementary Video 2 Energy-dependent dI/dV maps of the CDWFe–Fe region. dI/dV maps of a 26 nm × 26 nm CDWFe–Fe region under bias voltages from 0 to 16.2 mV. The π phase shift of the CDWFe–Fe pattern below and above the superconducting gap is displayed.

41586_2022_4744_MOESM5_ESM.mp4

Supplementary Video 3 dI/dV maps of the MZM lattice and the averaged dI/dV spectra under different fields. Left, dI/dV maps of MZM lattices at 0 mV under magnetic fields from 0.5 T to 6 T, respectively. Right, averaged dI/dV spectra at vortex centres under corresponding magnetic fields.

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Li, M., Li, G., Cao, L. et al. Ordered and tunable Majorana-zero-mode lattice in naturally strained LiFeAs. Nature 606, 890–895 (2022). https://doi.org/10.1038/s41586-022-04744-8

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