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Cooper-pair density modulation state in an iron-based superconductor

Nature volume 640, pages 55–61 (2025)Cite this article

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Abstract

Superconducting (SC) states that break space-group symmetries of the underlying crystal can exhibit nontrivial spatial modulation of the order parameter. Previously, such states were intimately associated with the breaking of translational symmetry1,2, resulting in the density-wave orders3,4,5,6,7,8, with wavelengths spanning several unit cells9,10,11,12,13,14,15,16,17,18,19. However, a related basic concept has long been overlooked20: when only intra-unit-cell symmetries of the space group are broken, the SC states can show a distinct type of nontrivial modulation preserving long-range lattice translation. Here we refer to this new concept as the pair density modulation (PDM) and report the first observation of a PDM state in exfoliated thin flakes of the iron-based superconductor FeTe0.55Se0.45. Using scanning tunnelling microscopy (STM), we discover robust SC gap modulation with the wavelength corresponding to the lattice periodicity and the amplitude exceeding 30% of the gap average. Notably, we find that the observed modulation originates from the large difference in SC gaps on the two nominally equivalent iron sublattices. The experimental findings, backed up by model calculations, suggest that, in contrast to the density-wave orders, the PDM state is driven by the interplay of sublattice symmetry breaking and a peculiar nematic distortion specific to the thin flakes. Our results establish new frontiers for exploring the intertwined orders in strong-correlated electronic systems and open a new chapter for iron-based superconductors.

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Fig. 1: Electronic structure and superconductivity of Fe(Te,Se) thin flakes.
Fig. 2: Periodic modulation of SC gap.
Fig. 3: Imaging the PDM state.
Fig. 4: Origin of the PDM state.

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Data availability

The data presented in the main text are available from Zenodo62. All other data are available from the corresponding authors on reasonable request.

Code availability

The code that supports the findings of this study is available from the corresponding authors on reasonable request.

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Acknowledgements

We thank S. Das Sarma, R. Zhang, Y. Zhang, C. Schrade, F. Tang, H. Zhao, S. Wu, X. Huang, H. Miao, J. Cai, J. Ge, J.-X. Yin and Z. Wu for the helpful discussions. We thank H. Zhou and A. Mukherjee for the technical support on sample fabrication and AFM measurement. This work has been primarily supported by the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (PHY-2317110). Part of the work was supported by the National Science Foundation (grant no. DMR-2005129); P.A.L. acknowledges support by the U.S. Department of Energy, Office of Basic Energy Sciences grant no. DE-FG02-03ER46076; L.K. acknowledges support from an IQIM-AWS Quantum postdoctoral fellowship. Parts of this work were supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center (M.P.). M.P. received further fellowship support from the Emergent Phenomena in Quantum Systems programme of the Gordon and Betty Moore Foundation. H.K. acknowledges support from the Kwanjeong Fellowship. The work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, contract no. DOE-sc0012704.

Author information

Authors and Affiliations

  1. T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA

    Lingyuan Kong, Hyunjin Kim, Yiran Zhang, Eli Baum & Stevan Nadj-Perge

  2. Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA

    Lingyuan Kong, Hyunjin Kim, Yiran Zhang, Eli Baum & Stevan Nadj-Perge

  3. Department of Physics, University of Houston, Houston, TX, USA

    Michał Papaj

  4. Department of Physics, University of California, Berkeley, Berkeley, CA, USA

    Michał Papaj

  5. Department of Physics, California Institute of Technology, Pasadena, CA, USA

    Hyunjin Kim & Yiran Zhang

  6. Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA

    Hui Li

  7. National Institute for Materials Science, Tsukuba, Japan

    Kenji Watanabe & Takashi Taniguchi

  8. Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, NY, USA

    Genda Gu

  9. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Patrick A. Lee

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  1. Lingyuan Kong
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Contributions

L.K. designed the experiments; L.K. fabricated samples with the assistance of H.K., Y.Z. and S.N.-P.; L.K. performed STM measurements with the assistance of H.K., E.B. and S.N.-P.; L.K. analysed data with the assistance of M.P., H.L. and S.N.-P.; L.K. developed data analysis codes on Igor Pro; M.P. and P.A.L. provided theoretical explanations and model simulations; G.G. provided FeTe0.55Se0.45 single crystals; K.W. and T.T. provided the h-BN crystals; Y.Z. performed AFM measurements; L.K., M.P., P.A.L. and S.N.-P. wrote the manuscript with input from the other authors. S.N.-P. supervised the project.

Corresponding authors

Correspondence to Lingyuan Kong or Stevan Nadj-Perge.

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Extended data figures and tables

Extended Data Fig. 1 Sample details.

a, Optical image of device #1. b, Schematic of the van der Waals stacks and the circuits. The FeTe0.55Se0.45 thin flakes were stacked on top of the h-BN dielectric layer, with a graphite gate at the bottom. Bias voltage (Vbias) was applied to the Fe(Te,Se) flakes and the graphite gate was grounded. Top-right inset, atom-resolved STM topography of the graphite top contact (set points: Vbias = 100 mV, It = 20 pA). The yellow spot on the bottom panel indicates the position at which the graphite topography was measured. Top-left inset, side view of the STM device circuit. c,d,f, AFM images of the regions of interest. Panels c and d show the zoomed-out view of the region in which STM data has been taken (shown in e). f, Region of top graphite contact. The green dot in e indicates the position at which the SI-STM data for the main figures was measured. g–i, AFM line profiles along the traces in c, showing the thicknesses of the different regions of the Fe(Te,Se) flake. From left to right, the thickness increases from 25 nm to 50 nm (see also Supplementary Fig. 5, the STM topographies mapping through the Fe(Te,Se) flakes). A 210-nm-thick flake is attached in the left-bottom corner. j, AFM line profile along the trace in f, used to determine the thickness of the h-BN dielectric layer (50 nm). Note that the AFM measurements were performed after the STM measurements were completed.

Extended Data Fig. 2 Examples of dI/dV spectra measured on Fe(Te,Se) thin flakes.

dI/dV spectra taken at different positions showing one (a,c,f), two (b,e,g,i,k,l,o) or three (d,h,j,m,n) peak-like spectral features (flake thickness dt = 40 nm; data were measured on device #6). Panel a is also shown in Fig. 1e.

Extended Data Fig. 3 Two more examples of periodic gap modulation.

a1, False-colour plot of a dI/dV linecut measured along the y axis (dashed blue line in a2, inset). The SC gap is minimized at Se+ atom sites (dashed white lines). a2, Spatially averaged dI/dV spectra at gap maximum (blue) and minimum (black) sites of the linecut. Inset, atom-resolved topography at the same area in which dI/dV spectra were measured. a3, The waterfall plot of a1. The curves are offset for clarity. a4–a6, The spatial variation of SC gap, Δ(y), the zero-bias conductance, g0(y), and the coherence peak height, H(y), respectively. The semi-transparent solid lines are obtained by FT filtering of the raw data. The horizontal dashed line in a4 indicates the average value of SC gap in the linecut (\(\bar{\Delta }\)). a7–a9, Magnitude of FT of raw data in a4–a6. Flake thickness dt is 50 nm. The modulation ratio \(| {\Delta }_{{{\bf{P}}}_{{\rm{Y}}}}| /\bar{\Delta }\) is 13.4%. b1–b9, Same as a1–a9 but for another linecut. Flake thickness dt is 35 nm. The modulation ratio \(| {\Delta }_{{{\bf{P}}}_{{\rm{X}}}}| /\bar{\Delta }\) is 6.7%. The scale bars in the insets of a2 and b2 are 1 nm.

Extended Data Fig. 4 Gap modulation at a lower electron temperature.

a1–a4, The raw data measured at 0.4 K. b1,b2, The simulation results obtained by convoluting the raw data with the Fermi–Dirac function at 2 K. c1,c2, The simulation results obtained by convoluting the raw data with the Fermi–Dirac function at 4 K. a1, False-colour plot of a dI/dV linecut measured along the x axis, across the four nearest Se+ atoms. The SC gaps are minimized at Se+ sites (dashed white lines). a2, The waterfall plot of a1. The curves are offset for clarity. a3, Typical dI/dV spectra at gap maximum (blue) and minimum (black) sites. Inset, the corresponding atom-resolved topography (scale bar, 6 Å). On each dI/dV spectrum, three features can be identified by fitting the second derivative of the dI/dV curves. The fitting results are shown in a1 and a2 as black dots. All three features (located at energies E1, E2 and E3) modulate in phase and are plotted in a4. The semi-transparent solid lines are extracted by FT filtering of the raw data. The horizontal dashed lines indicate the average energy of the spectral features (Ēi = 1.26, 1.68 and 2.23 meV), The amplitudes of the three features are |δEi| = 0.24, 0.27 and 0.25 meV, leading to the modulation ratio (|δEi|/Ēi with i = 1,2,3) of these features being 18.9%, 16.0% and 11.1%, respectively. d, Comparison of typical spectrum with and without convolution. The convolution smears fine features into a single peak and changes the peak positions. e, The peak positions extracted from b (2 K) and c (4 K) are plotted as green and purple symbols, respectively. The average energy of the peak at 2 K is Ē2K = 1.65 meV and that at 4 K is Ē4K = 2.06 meV; the modulation amplitude |δE2K| = 0.23 meV, |δE4K| = 0.16 meV and the modulation ratio is 14.1% or 7.8%. The faint dashed curves are the modulation of the three spectral features measured at 0.4 K. The data were measured on device #6, flake thickness: dt = 40 nm.

Extended Data Fig. 5 Presence of the PDM state in high magnetic fields.

a, Zero-bias conductance map of a single vortex (15 × 15 nm2; see also a vortex lattice in Supplementary Fig. 31). The left and right panels are measured at the same field of view but under different magnetic fields: 6 T (out of plane) and 0 T, respectively. The left panel is also shown in Fig. 3j. We measured two dI/dV linecuts in this field of view and both of them follow the Se+ lattice. The linecut #1 (along the x axis) starts far away from the vortex and goes across the edge of the vortex core, with the corresponding dI/dV data shown in b. The periodic modulation of SC gap was observed outside the vortex core (dashed cyan box). The waterfall plot of the cyan region is shown in c. Spatially averaged dI/dV spectra at the gap maximum (blue) and minimum (black) sites are shown in d. The extraction of the gap modulation is shown in e and its FT magnitude is shown in f. At the edge of the vortex core, in-gap bound states were observed around 1 meV and they exhibit modulation with the same periodicity (dashed purple box in b). g, Negative curvature plot (see Supplementary information section 6) of the dashed purple box in b). The extraction of spatial modulation of the in-gap bound state is shown in h and its FT magnitude is shown in i. The linecut #2 (along the y axis) was measured at the vortex centre (blue arrow in a). j, False-colour plot of a dI/dV linecut (left) and waterfall spectrum plot (right) of linecut #2. k, Typical spectra inside the vortex core. The positions of each curve are indicated by the colour bar in j. l, Energy-dependent magnitude of one-dimensional FT of j. Colour scale, black for high intensity and white for low intensity. Inside the vortex core, the bound states oscillate from 0 meV to around ±1 meV with a similar wavelength as the PDM state. This observation is consistent with the signal that appears around (π, π) in l.

Extended Data Fig. 6 Periodic modulation in the areas with high impurity concentration.

A large area (50 × 50 nm2) contains regions with low and high impurity concentrations. a, Atom-resolved STM topography. Some impurities can be identified in the upper region, resulting in a high zero-bias conductance (b) and lower coherence peaks (c). In contrast to the interstitial iron impurities on the bulk single crystals without annealing, the impurities observed here are mobile and can be removed by constant-current-mode scanning (Supplementary Fig. 30). b,c, Differential conductance map at zero bias (0 meV) and the energy of the SC gap coherence peak (+2 meV). d–i, Three dI/dV linecuts measured in this area. Their locations are marked in a–c. They go gradually deeper inside the region with higher impurity concentration from #1 to #3. d,f,h, False-colour plots of the dI/dV linecuts. e,g,i, The corresponding negative curvature plots (see Supplementary information section 6). The periodic gap modulation exists throughout. Note that sharp bound states appear in the dI/dV spectrum when measured near the impurities. These states exhibit the same modulation periodicity as the PDM state. j, Selected spectra from f. The positions of the curves are indicated in f by the colour bar. k, Selected spectra from h feature the spatial modulation of the bound states. The positions of these curves are indicated in h by the colour bar.

Extended Data Fig. 7 Gap modulation on device #2.

a, Optical image of device #2. Inset, AFM line profile measured along the dashed white line in b, showing the thickness of the flake (30 nm). b, AFM image of device #2. c, SC gap map on a 1.7 × 1.7-nm2 area. Inset, the FT magnitude of the SC gap map. d, The STM topography acquired simultaneously with the gap map in c. The white circles mark the positions of gap maxima appearing at the Fey sites. e–j, dI/dV linecuts measured on device #2 along either the x axis or the y axis. The average gaps are \(\bar{\Delta }=1.30,\,2.00,\,2.09,\,2.03,\,1.60\,{\rm{and}}\,1.76\,{\rm{meV}}\) and the PDM modulation amplitudes are |ΔP| = 0.40, 0.50, 0.61, 0.69, 0.56 and 0.72 meV for each corresponding linecut. The average modulation ratio is 35% in this device, with individual ratios being 31%, 40%, 29%, 34%, 35% and 41%, respectively. Set points: Vbias = −5 mV, It = 700 pA for e, 1.3 nA for (c,d,f), 1.5 nA for g–i.

Extended Data Fig. 8 Fractional and integer topological defects on the PDM state.

a,b, PDM modulation phase at the ±PX and ±PY, calculated by the two-dimensional lock-in method (Supplementary information section 4). c,d, Separate visualization of the modulation at ±PX and ±PY, calculated by FT filtering with only unidirectional vectors selected (see Methods). Several topological defects are identified (#1–#3). e,f, Phase winding around fractional and integer topological defects. For clarity, the curves are shifted to match their phase minima at zero. In f, a dislocation–antidislocation pair is demonstrated by their opposite phase winding (see also a simulation in Supplementary Fig. 28). g–i, Zoom-in of a half-integer quantized topological defect. g, A zoom-in image of the dashed blue square area in c, which contains the topological defect #1. h,i, Simulation of the fractional topological defect #1 (see Supplementary information section 7). A misaligned half dislocation on the ±PX modulation of the PDM state (h) appears as an equivalent half vortex in its phase field (i). j–l, Same as g–i, showing an integer topological defect #2. A single dislocation on the ±PX modulation (k) appears as an equivalent single vortex in its phase field (l). The Gaussian cut-off length (σ) of FT was set to 1.3 nm in a–d.

Extended Data Fig. 9 Lattice segregation and LDOS imbalance.

a–d, Energy-integrated differential conductance map of segregated iron sublattice; both Fex and Fey sites are selected (see Methods). The integration energy window was selected to avoid the influence of the SC coherence peak (see Supplementary information section 8 and Supplementary Fig. 27). The grey lines show the PDM domain wall, which traces pLL = 0. The segregated LDOS at iron sites, Σg(rFe), shows the periodic modulation with wavelength equal to the lattice constant, regardless of whether a positive (a) or a negative (b) energy window is selected. The same modulation can be seen from the FT magnitude (b and d), in which the Bragg peaks can be resolved. e–h, Same as a–d but on segregated Se+ sublattice. No modulation is seen, as expected. The blue circles indicate the positions of Se+ sites. See also Supplementary Fig. 20 for an example of the lattice segregation procedure performed on simulated images with and without LDOS imbalance between Fex/Fey sites.

Extended Data Fig. 10 Gap modulation ratio.

a, Literature survey of the materials showing gap modulation9,11,12,13,14,15,16,17,18,19. Unidirectional gap modulation ratio, |ΔQ|/Δ0, versus SC critical temperature, Tc, is plotted for several materials. The solid symbols are extracted from SC gap measurements, Δ(r), whereas the open symbols are converted from Cooper pair density map, ns(r), which was measured by Josephson SI-STM experiments in the literature (see details in Supplementary information section 11). The current work (dark-red filled symbols) exhibits an unprecedentedly large gap modulation ratio (see also Extended Data Fig. 7). b,c, Spatial distribution of the unidirectional gap modulation ratio of the SI-STM data in device #1 (Fig. 3 and Supplementary Fig. 18a,b). The ratio is calculated individually along the ±PX and ±PY directions. The grey lines are the lattice-lock-in domain walls, which trace pLL = 0. The ratio is greater than 20% inside the domain. d, The gap difference between the neighbouring Fex and Fey atoms (pΔ(r)), which is derived by the sum of b and c; see details in Supplementary information section 11 and Supplementary Table 1. e, Map of the inhomogeneous non-modulating component, Δ0(r). In practice, it was defined as the polynomial background of the total gap, Δ(r) (Fig. 3a) and extracted by function fitting. The Δ0(r) was also depicted as the semi-transparent plane in the three-dimensional plot of Fig. 3a.

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Kong, L., Papaj, M., Kim, H. et al. Cooper-pair density modulation state in an iron-based superconductor. Nature 640, 55–61 (2025). https://doi.org/10.1038/s41586-025-08703-x

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