Pair density wave state in a monolayer high-tc iron-based superconductor

ABSTRACT The pair density wave (PDW) is an extraordinary superconducting state in which Cooper pairs carry non-zero momentum1,2. Evidence for the existence of intrinsic PDW order in

high-temperature (high-_T_c) cuprate superconductors3,4 and kagome superconductors5 has emerged recently. However, the PDW order in iron-based high-_T_c superconductors has not been observed

experimentally. Here, using scanning tunnelling microscopy and spectroscopy, we report the discovery of the PDW state in monolayer iron-based high-_T_c Fe(Te,Se) films grown on SrTiO3(001)

substrates. The PDW state with a period of _λ_ ≈ 3.6_a_Fe (_a_Fe is the distance between neighbouring Fe atoms) is observed at the domain walls by the spatial electronic modulations of the

local density of states, the superconducting gap and the π-phase shift boundaries of the PDW around the vortices of the intertwined charge density wave order. The discovery of the PDW state

in the monolayer Fe(Te,Se) film provides a low-dimensional platform to study the interplay between the correlated electronic states and unconventional Cooper pairing in high-_T_c

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BEING VIEWED BY OTHERS A NEW MAJORANA PLATFORM IN AN FE-AS BILAYER SUPERCONDUCTOR Article Open access 10 November 2020 UNUSUAL CROSSOVER FROM BARDEEN-COOPER-SCHRIEFFER TO

BOSE-EINSTEIN-CONDENSATE SUPERCONDUCTIVITY IN IRON CHALCOGENIDES Article Open access 19 July 2023 COOPER-PAIR DENSITY MODULATION STATE IN AN IRON-BASED SUPERCONDUCTOR Article 19 March 2025

DATA AVAILABILITY All data supporting the findings of this study are available from the corresponding authors on reasonable request. Source data are provided with this paper. CODE

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Wang, Z. Kramers Fulde–Ferrell state and superconducting spin diode effect. Preprint at https://arxiv.org/abs/2209.03520 (2022). Download references ACKNOWLEDGEMENTS We acknowledge technical

assistance from X. Xu, C. Chen and Y. Li and discussions with P. A. Lee and D. F. Agterberg. This work was supported by the National Key Research and Development Program of China (Grant No.

2018YFA0305604), the National Natural Science Foundation of China (Grant No. 11888101), the Beijing Natural Science Foundation (Z180010), the Innovation Program for Quantum Science and

Technology (2021ZD0302403), the Strategic Priority Research Program of Chinese Academy of Sciences (XDB28000000) and the China Postdoctoral Science Foundation (2021M700253). Z.W. is

supported by the US Department of Energy, Basic Energy Sciences Grant No. DE-FG02-99ER45747. AUTHOR INFORMATION Author notes * These authors contributed equally: Yanzhao Liu, Tianheng Wei

AUTHORS AND AFFILIATIONS * International Center for Quantum Materials, School of Physics, Peking University, Beijing, China Yanzhao Liu, Tianheng Wei, Guanyang He & Jian Wang *

Department of Physics, Shanghai University, Shanghai, China Yi Zhang * Department of Physics, Boston College, Chestnut Hill, MA, USA Ziqiang Wang * Collaborative Innovation Center of Quantum

Matter, Beijing, China Jian Wang * Hefei National Laboratory, Hefei, China Jian Wang * CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of

Sciences, Beijing, China Jian Wang * Beijing Academy of Quantum Information Sciences, Beijing, China Jian Wang Authors * Yanzhao Liu View author publications You can also search for this

author inPubMed Google Scholar * Tianheng Wei View author publications You can also search for this author inPubMed Google Scholar * Guanyang He View author publications You can also search

for this author inPubMed Google Scholar * Yi Zhang View author publications You can also search for this author inPubMed Google Scholar * Ziqiang Wang View author publications You can also

search for this author inPubMed Google Scholar * Jian Wang View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS The order of the first two

authors was determined arbitrarily. J.W. conceived and supervised the research. T.W., Y.L. and G.H. grew the samples. Y.L., T.W. and G.H. carried out the STM and STS experiments. T.W., Y.L.

and J.W. analysed the experimental data. Z.W. and Y.Z. proposed the theoretical model. Y.L., T.W., Z.W. and J.W. wrote the manuscript. CORRESPONDING AUTHORS Correspondence to Ziqiang Wang or

Jian Wang. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. PEER REVIEW PEER REVIEW INFORMATION _Nature_ thanks the anonymous reviewers for their

contribution to the peer review of this work. ADDITIONAL INFORMATION PUBLISHER’S NOTE Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional

affiliations. EXTENDED DATA FIGURES AND TABLES EXTENDED DATA FIG. 1 MORE INFORMATION ABOUT THE 1-UC FE(TE,SE)/STO. A, An STM topography of 1-2 UC Fe(Te,Se)/STO at large-scale (200×200 nm2,

_V_s = 1 V, _I_s = 0.5 nA) with terraces of the STO substrate (lighter color means the higher height). The cyan area at the edge of the lower STO terrace shows the 2nd-UC Fe(Te,Se) film. B,

Line profile taken along the orange curve in A. The thickness of the 1-UC FeTe1-xSex film is approximately 0.59 nm, corresponding to the composition x ≈ 0.5. C, An STM image of 1-UC

Fe(Te,Se)/STO (10×10 nm2, _V_s = 0.1 V, _I_s = 0.5 nA). D, Averaged tunneling spectrum (averaged over ~ 100 spectra) taken along the light grey arrow in C, which shows two pairs of coherence

peaks at _Δ_1 ≈ 11 meV and _Δ_2 ≈ 19 meV. E, Waterfall-plot of d_I_/d_V_ curves taken along the light grey arrow in C. EXTENDED DATA FIG. 2 CHARACTERIZATION OF DOMAIN WALLS. A, B, STM

images (16×16 nm2, _V_s = 0.1 V, _I_s = 0.5 nA) of two domain walls labeled as D1 (A) and D4 (B). C, D, d_I_/d_V_ maps of two domain walls. The lattice vectors Q_i,x_(_y_) of two regions

marked by white spots in A and B satisfy Q1_,x_(_y_) = Q2_,x_(_y_). d_I_/d_V_ maps shown in C and D are measured at the red dashed box in A and B, respectively. EXTENDED DATA FIG. 3

NON-DISPERSIVE PDW AND ABSENCE OF PDW IN HIGH ENERGY D_I_/D_V_ MAPS. A, The STM topography of the domain wall D3 shown in Fig. 4. B, The bias dependence of the wavevector of the LDOS

modulation at the domain wall. C,D, d_I_/d_V_ map (C) taken at 80 mV over the same FOV of A and corresponding magnitude of the Fourier transform (D). There is no spatial modulation in the

d_I_/d_V_ map (C) and no FFT peak at around Q ≈ 0.28QFe at the energy much higher than the superconducting gap (D). E,F, Spatial variation of the amplitude (E) and phase (F) of the LDOS

modulation at bias voltage from 0 to 14 mV. The averaging length scales in E and F are denoted by dashed circles. The LDOS modulations mainly exist in the energies within the superconducting

gap, which is also observed at domain wall D2 (Supplementary Fig. 13). Dashed lines in A,C,E and F mark the edges of the domain wall. EXTENDED DATA FIG. 4 DETERMINATION OF MODULATION

WAVEVECTOR Q AT DOMAIN WALL D1. A, The STM topography of the domain wall D1. B, Zero-bias conductance (ZBC) map _g_(R, _V_ = 0 mV) taken at the same area in A. C, The magnitude of the

Fourier transform _g_(Q, _V_ = 0 mV) of the ZBC map in B. D–F, Spatial distribution of the modulation _g_Q(R) phase \({\phi }_{{{\bf{Q}}}_{{\rm{t}}}}^{g}({\bf{r}})\). The averaging length

scales in D–F are denoted by dashed circles. The domain wall area is marked by black dashed lines in A,B and D–F. G–I, Statistical histogram of \({\phi

}_{{{\bf{Q}}}_{{\rm{t}}}}^{g}({{\bf{r}}}_{{\rm{DW}}})\) (RDW represents the area of the domain wall) at the domain wall D1 for a series of tentative Qt. Only the points inside the domain

wall area in D–F are counted. Black dashed lines in G–I are located at the half maximum of the peak. The determined wavevector Q corresponds to the minimum of the full width at half maximum

(FWHM) of the peak in the distribution histogram \({\phi }_{{{\bf{Q}}}_{{\rm{t}}}}^{g}({{\bf{r}}}_{{\rm{DW}}})\), as shown in H. EXTENDED DATA FIG. 5 SUPERCONDUCTING GAP ENERGY MODULATIONS

AT THE DOMAIN WALL LABELED AS D8 IN ANOTHER 1-UC FE(TE,SE)/STO SAMPLE (S2). A, The STM topography of the domain wall. (4.8×4.8 nm2, _V_s = 0.04 V, _I_s = 2.5 nA). B, Spatial distribution of

_Δ_1(R) measured in the same area in A. White dashed lines in B are guides to the eye. C, The magnitude of the Fourier transform of B. Orange crosses are at Q = (±QFe,0), (0,±QFe). The

modulation wavevector Q ≈ ±0.28QFe is marked by dashed blue circles. Black dashed lines in A and B mark the edges of the domain wall. EXTENDED DATA FIG. 6 SPATIAL MODULATIONS OF THE LDOS,

SUPERCONDUCTING GAP ENERGY AND COHERENCE PEAK HEIGHT AT ONE DOMAIN WALL LABELED AS D9 IN ANOTHER 1-UC FE(TE,SE) FILM (S2). A, The STM topography of the domain wall. (4.3×5.4 nm2, _V_s = 0.04

 V, _I_s = 2.5 nA). B, d_I_/d_V_ map _g_(R, _V_ = 3 mV) taken at the same area in A. C, The magnitude of the Fourier transform of B. D, Spatial distribution of _Δ_1(R) measured in the same

area in A. E, The magnitude of the Fourier transform of D. F, Spatial distribution of superconducting coherence peak height measured in the same area in A. G, The magnitude of the Fourier

transform of F. H, Line profiles along the (0,0) to (1,0)QFe direction in C, E, and G. Peaks at Q ≈ 0.28QFe appear for all curves. The profiles are normalized and shifted vertically for

comparison. Orange crosses in C, E, and G are at Q = (±QFe,0), (0,±QFe). Black dashed lines in A, B, D and F mark the edges of the domain wall. Orange dashed lines in B, D and F are guides

to the eye. EXTENDED DATA FIG. 7 _Ρ_(R, _V_) MAP (A), CORRESPONDING FOURIER TRANSFORM _Ρ_(Q, _V_) (B), SPATIAL VARIATION OF THE AMPLITUDE \(|{A}_{2{\BF{Q}}}^{\RHO (V)}({\BF{R}})|\) (C) AND

THE PHASE \({\PHI }_{2{\BF{Q}}}^{\RHO (V)}({\BF{R}})\) (D) OF THE 2Q _Ρ_(R) MODULATION AT 2 MV AT THE DOMAIN WALL SHOWN IN FIG. 4 (D3). _ρ_(Q, _V_) exhibits maxima at around ±2Q (marked by

dashed blue circles). The amplitude of the modulation is restricted to the domain wall region and the phase of the 2Q modulation is nearly uniform in the domain wall region, indicating the

existence of the 2Q CDW. The averaging length scales in C and D are denoted by dashed circles. The edge of the domain wall is marked by the dashed lines in A, C and D. EXTENDED DATA FIG. 8

COMPARISON BETWEEN THE ATOMIC LATTICE AND 2Q CDW STATE AT THE DOMAIN WALL SHOWN IN FIG. 4 (D3). A,B, Spatial variation of the phase \({\phi

}_{0.5{{\bf{Q}}}_{{\rm{Fe}}}}^{T}\left({\bf{r}}\right)\) of the STM topology _T_(R) modulation at 0.5QFe (A) and the phase \({\phi }_{2{\bf{Q}}}^{\rho (2{\rm{mV}})}\left({\bf{r}}\right)\) of

the _ρ_(R, _V_ = 2 mV) (charge density) modulation at 2Q (B). Topological defects in A and B are marked by black dots. C,D, Spatial variation of the amplitude

\(\left|{A}_{0.5{{\bf{Q}}}_{{\rm{Fe}}}}^{T}\left({\bf{r}}\right)\right|\) of the STM topology _T_(R) modulation at 0.5QFe (C) and the amplitude \(\left|{A}_{2{\bf{Q}}}^{\rho

(2{\rm{mV}})}\left({\bf{r}}\right)\right|\) of the _ρ_(R, _V_ = 2 mV) modulation at 2Q (D). The spatial variations of the phase and amplitude show many differences between the atomic lattice

and 2Q CDW, indicating that the 2Q CDW is independent of the atomic lattice. The averaging length scales are denoted by dashed circles. The edge of the domain wall is marked by the dashed

lines. EXTENDED DATA FIG. 9 POSSIBLE THEORETICAL SCENARIOS FOR THE PDW STATE AT THE DOMAIN WALL. A, Schematic of the PDW state at the domain wall. \({\varDelta }_{{\rm{SC}}}^{\mathrm{1,2}}\)

and \({\varDelta }_{{\rm{SC}}}^{{\rm{DW}}}\) represent the zero-momentum superconductivity in the domains (1 and 2) and at the domain wall, respectively. B, Illustration of the finite

center of mass momentum, equal-spin pairing in the presence of only Rashba SOC. \({\rm{\langle }}{c}_{{\bf{k}},\uparrow }{c}_{-{\bf{k}}+{\bf{Q}},\uparrow }{\rm{\rangle }}\) and

\({\rm{\langle }}{c}_{-{\bf{k}},\downarrow }{c}_{{\bf{k}}-{\bf{Q}},\downarrow }{\rm{\rangle }}\) are equal-spin pairing order parameters where one spin sector carries momentum +Q and the

other spin sector –Q. C, Illustration of the finite center of mass momentum equal-spin pairing in the presence of both Rashba and Dresselhaus SOC. In this case, the triplet equal-spin

pairing order parameters \({\rm{\langle }}{c}_{{\bf{k}},\sigma }{c}_{-{\bf{k}}+{\bf{Q}},\sigma }{\rm{\rangle }}\) and \({\rm{\langle }}{c}_{-{\bf{k}},\sigma }{c}_{{\bf{k}}-{\bf{Q}},\sigma

}{\rm{\rangle }}\) can have both spin components (_σ_), i.e., each equal-spin pairing sector carries both nonzero momentum +Q and –Q, giving rise to the equal-spin pairing PDW state. D,F,

Calculated tunneling density of states _ρ__i_(_ω_) with _i_ the site index, for two examples with Q = 2π/12 (D) and Q = 2π/4 (F), where the insets show the zoom-in feature around the

coherent peaks. E,G, The spatial modulations of the superconducting gap extracted from the corresponding coherence peak positions for the Q = 2π/12 (E) and Q = 2π/4 (G). SUPPLEMENTARY

INFORMATION SUPPLEMENTARY INFORMATION This file contains Supplementary Text, Figs. 1–20 and References. SOURCE DATA SOURCE DATA FIG. 1 SOURCE DATA FIG. 3 SOURCE DATA FIG. 4 SOURCE DATA FIG.

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permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Liu, Y., Wei, T., He, G. _et al._ Pair density wave state in a monolayer high-_T_c iron-based superconductor. _Nature_ 618, 934–939 (2023).

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