## Abstract

The pair density wave (PDW) is an extraordinary superconducting state in which Cooper pairs carry non-zero momentum^{1,2}. Evidence for the existence of intrinsic PDW order in high-temperature (high-*T*_{c}) cuprate superconductors^{3,4} and kagome superconductors^{5} 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 SrTiO_{3}(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} superconductors.

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## 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 availability

The code used to analyse the data reported in this study is available from the corresponding authors on reasonable request.

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## 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

### Authors and Affiliations

### 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.

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## 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 nm^{2}, *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 2^{nd}-UC Fe(Te,Se) film. **b**, Line profile taken along the orange curve in **a**. The thickness of the 1-UC FeTe_{1-x}Se_{x} 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 nm^{2}, *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 nm^{2}, *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 **q**_{1,x(y)} = **q**_{2,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** ≈ 0.28**Q**_{Fe} 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 \({\phi }_{{{\bf{Q}}}_{{\rm{t}}}}^{g}({{\bf{r}}}_{{\rm{DW}}})\) (**r**_{DW} represents the area of the domain wall) at the domain wall D1 for a series of tentative **Q**_{t}. 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 nm^{2}, *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** = (±**Q**_{Fe},0), (0,±**Q**_{Fe}). The modulation wavevector **Q** ≈ ±0.28**Q**_{Fe} 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 nm^{2}, *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)**Q**_{Fe} direction in **c**, **e**, and **g**. Peaks at **Q** ≈ 0.28**Q**_{Fe} appear for all curves. The profiles are normalized and shifted vertically for comparison. Orange crosses in **c**, **e**, and **g** are at **q** = (±**Q**_{Fe},0), (0,±**Q**_{Fe}). 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 2**Q** *ρ*(**r**) modulation at 2 mV at the domain wall shown in Fig. 4 (D3).

*ρ*(**q**, *V*) exhibits maxima at around ±2**Q** (marked by dashed blue circles). The amplitude of the modulation is restricted to the domain wall region and the phase of the 2**Q** modulation is nearly uniform in the domain wall region, indicating the existence of the 2**Q** 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 0.5**Q**_{Fe} (**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 2**Q** (**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 0.5**Q**_{Fe} (**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 2**Q** (**d**). The spatial variations of the phase and amplitude show many differences between the atomic lattice and 2**Q** CDW, indicating that the 2**Q** 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**).

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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). https://doi.org/10.1038/s41586-023-06072-x

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DOI: https://doi.org/10.1038/s41586-023-06072-x

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