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Smectic pair-density-wave order in EuRbFe4As4


The pair density wave (PDW) is a superconducting state in which Cooper pairs carry centre-of-mass momentum in equilibrium, leading to the breaking of translational symmetry1,2,3,4. Experimental evidence for such a state exists in high magnetic field5,6,7,8 and in some materials that feature density-wave orders that explicitly break translational symmetry9,10,11,12,13. However, evidence for a zero-field PDW state that exists independent of other spatially ordered states has so far been elusive. Here we show that such a state exists in the iron pnictide superconductor EuRbFe4As4, a material that features co-existing superconductivity (superconducting transition temperature (Tc) ≈ 37 kelvin) and magnetism (magnetic transition temperature (Tm) ≈ 15 kelvin)14,15. Using spectroscopic imaging scanning tunnelling microscopy (SI-STM) measurements, we show that the superconducting gap at low temperature has long-range, unidirectional spatial modulations with an incommensurate period of about eight unit cells. Upon increasing the temperature above Tm, the modulated superconductor disappears, but a uniform superconducting gap survives to Tc. When an external magnetic field is applied, gap modulations disappear inside the vortex halo. The SI-STM and bulk measurements show the absence of other density-wave orders, indicating that the PDW state is a primary, zero-field superconducting state in this compound. Both four-fold rotational symmetry and translation symmetry are recovered above Tm, indicating that the PDW is a smectic order.

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Fig. 1: Characterization of Rb surface in EuRbFe4As4.
Fig. 2: Visualizing spatial modulations of the superconducting gap.
Fig. 3: Temperature dependence of PDW gap modulations.
Fig. 4: Magnetic-field dependence of the PDW gap modulations.

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Data availability

All data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The computer code used for data analysis is available upon reasonable request from the corresponding authors.


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We thank A. Kreisel, R. Fernandes, J. Schmalian, I. Mazin, S. A. Kivelson, Y. Higashi, Y. Yanase, H. Aoki and S.-i. Uchida for discussions. Work at Brookhaven is supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, US Department of Energy under contract no. DE-SC0012704. Raman spectroscopy measurements at Columbia University are supported by supported by the National Science Foundation MRSEC programme in the Center for Precision-Assembled Quantum Materials under award number DMR-2011738 and by the Air Force Office of Scientific Research via grant FA9550-21-1- 0378. The work at AIST was supported by the Grant-in-Aid for Scientific Research on Innovative Areas ‘Quantum Liquid Crystals’ (KAKENHI grant no. JP19H05823) from JSPS of Japan.

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Authors and Affiliations



K.F. and A.N.P. led the project. H.Z. carried out spectroscopic imaging STM experiments with contribution from R.B. at Brookhaven National Laboratory. S.I., A.I. and H.E. synthesized and characterized the samples. M.T., T.H. and X.Z. carried out Raman spectroscopy measurements. H.Z. carried out analysis with contribution from R.B. K.F., A.N.P. and H.Z. wrote the paper. The paper reflects the contributions and ideas of all authors.

Corresponding authors

Correspondence to Abhay N. Pasupathy or Kazuhiro Fujita.

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Extended data figures and tables

Extended Data Fig. 1 Temperature dependent magnetic susceptibility (χ) measurements of EuRbFe4As4 single crystal.

(a) Temperature dependent χ measurements for Zero field cooled (ZFC) (solid blue circles) and field cooled (FC) (open blue circles) procedures over temperature range from above superconducting transition (Tc) to lower than magnetism transition (Tm). (b–d) The zoom-in of χ(T) plots in a focusing on the temperature close to Tm for FC, ZFC and temperature close to Tc, respectively. The magnetic field applied was perpendicular to the FeAs plane and kept at 10 Oe during the measurement. Volume of the sample was measured to be 1.09 × 10−4 cm3.

Extended Data Fig. 2 Comparison of superconducting gap modulation below and above Tm.

a,b Topograph T(r) and its associated Fourier transforms (FT). c,d Superconducting gap (∆(r)) map obtained over identical area as a and its associated FT, respectively. The black double-head arrows in c indicate the PDW stripe pattern. e Line cut of  the differential conductance (dI/dV) spectra along the dashed line as denoted in a,c. The inset is the dI/dV intensities focusing on the coherence peak. gk Corresponding results obtained over identical area at 16 K, which is above Tm, with exactly the same setup conditions except the T. f,l Histograms of the gap size distributions from c and i, respectively. The gray curves represent the applied Gaussian fittings with Δfit (4.5 K) = 4.97 meV, Δfit (16 K) = 4.91 meV. Setup conditions: Topographs: Vset = 12 mV, Iset = 500 pA; Gap maps: Vset = 12 mV, Iset = 500 pA, Vexc = 1 mV.

Extended Data Fig. 3 Additional large field-of-view N(r, E~Δ0) maps on different sample surfaces and regions.

Spatial resolutions of the N(r, E~Δ0) maps are (in unit of pixels/nm): top row from left to right: 1.37, 1.60, 2.06 and 1.80, respectively; bottom row from left to right: 1.37, 1.32, 1.37 and 1.30, respectively. The spatial resolution required to discern the modulations with an 8aFe periodicity in the N(r) maps should be at least 0.93 pixels/nm. Dashed circles are vortices introduced by external magnetic fields. The last two rows are spatially averaged differential conductance (dI/dV) spectra taken over the corresponding regions as described in the first two rows. The blue dashed lines indicate the coherence peak positions. Setup conditions: N(r, E~Δ0) maps: Sample I, region A: Vset = 12 mV, Iset = 500 pA, Vexc = 0.3 mV, 0T; Sample I, region B: Vset = 6 mV, Iset = 60 pA, Vexc = 1 mV, 0T; Sample II, region A: Vset = 5 mV, Iset = 50 pA, Vexc = 1 mV, 0T; Sample II, region B: Vset = 5 mV, Iset = 50 pA, Vexc = 1 mV, 0T; Sample II, region C: Vset = 5 mV, Iset = 50 pA, Vexc = 1 mV, 0.1T; Sample II, region D: Vset = 5 mV, Iset = 50 pA, Vexc = 1 mV, 0.2T; Sample II, region E: Vset = 5 mV, Iset = 50 pA, Vexc = 1 mV, 0T; Sample II, region F: Vset = 5 mV, Iset = 30 pA, Vexc = 1 mV, 0.1T. Spectra: Sample I, region A: Vset = 12 mV, Iset = 500 pA, Vexc = 0.3 mV, 0T; Sample I, region B: Vset = 20 mV, Iset = 400 pA, Vexc = 0.2 mV, 0T; Sample II, region A: Vset = 30 mV, Iset = 600 pA, Vexc = 0.6 mV, 0T; Sample II, region B: Vset = 12 mV, Iset = 500 pA, Vexc = 0.3 mV, 0T; Sample II, region C: Vset = 15 mV, Iset = 350 pA, Vexc = 0.2 mV, 0.1T; Sample II, region D: Vset = 30 mV, Iset = 1500 pA, Vexc = 0.15 mV, 0.2T; Sample II, region E Vset = 30 mV, Iset = 1500 pA, Vexc = 0.15 mV, 0T; Sample II, region F: Vset = 30 mV, Iset = 1500 pA, Vexc = 0.2 mV, 0.1T.

Extended Data Fig. 4 Large field-of-view T(r) maps and their Fourier transforms on different sample surfaces and regions acquired at different setup conditions.

Spatial resolutions of the T(r) maps are (in unit of pixels/nm): first column from top to bottom: 2.26, 1.60, 1.80 and 1.68, respectively; third column from top to bottom: 2.06, 2.41, 1.53 and 1.37, respectively. The red squares denote the area where the corresponding N(r, E~Δ0) maps are taken as described in Fig. S3. The second and fourth columns represent their associated Fourier transforms of the topographs. Area near QPDW (black squares) are zoomed in for the visual purpose, and the green crosses locate the (¼, ¼) 2π/aFe. The spatial resolution required to discern the modulations with an 8aFe periodicity in the N(r) maps should be at least 0.93 pixels/nm. Setup conditions: Sample I, region A: Vset= 12 mV, Iset= 500 pA, Vexc= 0.3 mV, 0T; Sample I, region B: Vset= 6 mV, Iset= 60 pA, Vexc= 1 mV, 0T; Sample II, region A: Vset= 5 mV, Iset= 50 pA, Vexc= 1 mV, 0T; Sample II, region B(column 1): Vset= 50 mV, Iset= 50 pA, 0T; Sample II, region B(column 3): Vset= 5 mV, Iset= 50 pA, 0T; region C: Vset= 205 mV, Iset= 50 pA, 0T; Sample II, region E(top): Vset= 100 mV, Iset= 40 pA, 0T; Sample II, region E(bottom): Vset= 5 mV, Iset= 30 pA, Vexc= 1 mV, 0.1T.

Extended Data Fig. 5 Comparison of Fourier transforms of N(r,E) images below and above the magnetic transition temperature on sample II, region E.

The purple squares highlight the layers where strong QPDW peaks can be detected. Setup conditions: Vset= 12 mV, Iset= 500 pA, Vexc= 0.3 mV, 0T.

Extended Data Fig. 6 Raman spectroscopy measurements on Eu-1144.

The left panel shows Raman spectra at different temperatures with offsets for clarity. The right panel shows the wavenumbers of the observed modes as a function of temperature. No mode softening is seen around the magnetic transition temperature of 16 K, and no additional modes are observed below the magnetic transition temperature.

Extended Data Fig. 7 C4 rotational symmetry breaking below Tm.

a,b, Fourier transforms of N(r, E= 3.12 meV) maps at 16 K (above Tm) and 4.5 K (below Tm), respectively, acquired over identical regions on the sample 2 (same sample as in Fig. 3j,k). The quasiparticle interference (QPI) pattern at 4.5 K is obviously C2, while that at 16 K is almost C4. Both images have been 2-fold-symmetrized with respect to the Qy axis. STM setup condition: Vset= 12 mV, Iset= 500 pA, Vexc= 0.3 mV.

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Zhao, H., Blackwell, R., Thinel, M. et al. Smectic pair-density-wave order in EuRbFe4As4. Nature 618, 940–945 (2023).

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