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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Data measured or analysed during this study are available from the corresponding author on reasonable request. Source data are provided with this paper.
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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.
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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.
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Extended data figures and tables
Extended Data Fig. 1 Comparison of lattice distortion among different regions of LiFeAs.
a–c, 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. d–f, Fourier transform images of a–c, 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.
a–p, 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 (e–j). 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.
Extended Data Fig. 5 Statistics of MZMs under different magnetic fields.
a–f, 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.
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.
Extended Data Fig. 10 Correlation between the vortex spacing and the dI/dV spectra of the MZM vortices.
a–f, 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.
Supplementary information
Supplementary Information
This file contains Supplementary Table 1 and Supplementary Figs. 1–9
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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DOI: https://doi.org/10.1038/s41586-022-04744-8
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