# Superconductivity with broken time-reversal symmetry inside a superconducting s-wave state

## Abstract

In general, magnetism and superconductivity are antagonistic to each other. However, there are several families of superconductors in which superconductivity coexists with magnetism, and a few examples are known where the superconductivity itself induces spontaneous magnetism. The best known of these compounds are Sr2RuO4 and some non-centrosymmetric superconductors. Here, we report the finding of a narrow dome of an $$s+is^{\prime}$$ superconducting phase with apparent broken time-reversal symmetry (BTRS) inside the broad s-wave superconducting region of the centrosymmetric multiband superconductor Ba1 − xKxFe2As2 (0.7 x 0.85). We observe spontaneous magnetic fields inside this dome using the muon spin relaxation (μSR) technique. Furthermore, our detailed specific heat study reveals that the BTRS dome appears very close to a change in the topology of the Fermi surface. With this, we experimentally demonstrate the likely emergence of a novel quantum state due to topological changes of the electronic system.

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## Data availability

The data represented in Figs. 25 are available as Source Data. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The code for calculating spontaneous fields within the Ginsburg–Landau model is available with the Supplementary Information files attached to this paper.

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## Acknowledgements

This work was supported by DFG (GR 4667/1, GRK 1621 and SFB 1143 project C02). The work of M. A. Silaev was supported by the Academy of Finland (project no. 297439). The work of I.E. was carried out with support from the Ministry of Science and Higher Education of the Russian Federation in the framework of Increase Competitiveness Program of NUST MISiS grant no. К2-2020-001. This work was partly performed at the Swiss Muon Source (SμS), PSI, Villigen. We acknowledge fruitful discussions with A. Amato, S. Borisenko, E. Babaev, A. Charnukha, O. Dolgov, C. Hicks, C. Meingast and A. de Visser. We especially acknowledge D. Efremov for support with DFT calculations. We are grateful for technical assistance from C. Baines.

## Author information

Authors

### Contributions

V.G. designed the study, initiated and supervised the project and wrote the paper, performed the μSR, specific heat, magnetic susceptibility and X-ray diffraction measurements. R.S. performed the μSR experiments. K.K. and C.H.L. prepared Ba1 − xKxFe2As2 single crystals in the doping range 0.65 x 0.85. I.M. and S.A. prepared Ba1 − xKxFe2As2 single crystals with x ≈ 0.98. P.C. performed and W.S. supervised TEM measurements. K. Nenkov performed specific heat and magnetization measurements. B.B, R.H. and K. Nielsch supervised the research at IFW. S.-L.D. provided interpretation of the experimental data. V.L.V. and M.A.S. performed calculation of the spontaneous magnetic fields in the BTRS state. P.A.V. and I.E. analysed specific heat in the BTRS state. H.L. performed μSR experiments and supervised the research at PSI. H.-H.K. initiated the project and supervised the research. All authors discussed the results and implications and commented on the manuscript.

### Corresponding authors

Correspondence to V. Grinenko or M. A. Silaev or H.-H. Klauss.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Thomas A. Maier and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Normal state depolarization rate.

(Left axis) Doping dependence of the normal state depolarization rate Λ0 extrapolated to zero temperature as shown in Fig. 3 of the main text obtained assuming temperature independent σ. The error-bars in the absolute value of the depolarization rate is caused by uncertainty in the background contribution. (Right axis) Doping dependence of the static susceptibility measured at T = 20 K given in Fig. 2a of the main text. Dashed lines are guides to the eyes. Inset: Doping dependence of the temperature independent depolarization rate σ.

### Extended Data Fig. 2 The structure of the spontaneous fields and currents in an s + is superconductor.

The structure of the spontaneous magnetic field (left panels) and spontaneous currents (right panels) produced by the spherically-symmetric inhomogeneity in anisotropic s + is and sis states. In the left panels the red/blue arrows show clockwise/counter-clockwise parts of the magnetic field distribution. The clockwise and counter-clockwise field is generated by the supercurrents with jz > 0 (red arrows) and jz < 0 (blue arrows) shown in the right panels. Notice that the magnetic field and current directions are opposite in s + is and sis states. The length-scale in the axes is given in units of the zero-temperature coherence length ħvF/Tc. The x, y and z axes are chosen along the a, b and c crystallographic directions, respectively.

### Extended Data Fig. 3 Anisotropy of the internal fields in an s + is superconductor.

Two orthogonal components of the normalized internal spontaneous fields Bint/Bc2 depending on a strength of the variation of the SC coupling constant Δη2 due to an inhomogeneity. Inset: Δη2 dependence of the anisotropy ratio of the internal fields γint. In the main text ΔλaΔη2 with a ~ 1 (see section 6 in the Supplementary information for details).

### Extended Data Fig. 4 Electrical resistivity.

(a) Temperature dependence of the resistivity of the Ba1−xKxFe2As2 single crystals with different doping levels. Inset: Zoom into the temperature range close Tc. (b) The resistivity versus squared temperature. Dashed curves are the fits using ρ(T) = ρ0 + AFLT2. Inset: Doping dependence of the AFL coefficient. The data for x = 1 are taken from Ref. 46.

### Extended Data Fig. 5 Impurity effect on the phase diagram.

Doping dependence of the residual resistivity ρ0 of the Ba1−xKxFe2As2 single crystals (green triangles, right axis) on top of the phase diagram with the region close to the BTRS dome.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–12 and discussion.

### Supplementary Software 1

The code for calculating spontaneous fields within the Ginsburg–Landau model. Software_1 file contains the code implementing relaxation minimization of the free energy functional defined on the triangular grid which is constructed in Software_2 file. The parameters of the Ginzburg–Landau model like temperature, pairing constants and effective masses are passed to the constructor of the main class SPlusIS_XZ. Parameter ‘kappa’ defines the ratio between the London penetration depth and the coherence length (the default value None corresponds to the infinite penetration depth), parameter ‘spis’ distinguishes between the s + is and s + id cases. The method ‘solve’ of this class minimizes the free energy functional.

### Supplementary Software 2

The code for calculating spontaneous fields within the Ginsburg–Landau model. Software_2 file contains a class that describes a triangular grid and vector operations like gradient, divergence and curl.

## Source data

### Source Data Fig. 2

Susceptibility data and raw SQUID data from Fig. 2a Electronic specific heat from Fig. 2b Tc, Electronic specific heat/T, Sommerfeld coefficient versus K-doping from Fig. 2c and 2d.

### Source Data Fig. 3

Zero-field muSR time spectra from Fig. 3a, 3b, and 3c muon spin depolarization rate at different temperatures and magnetic susceptibility at Bab = 0.5 mT from Figs. 3d, Fig. 3e, Fig. 3f, Fig. 3g and Fig. 3h.

### Source Data Fig. 4

Muon spin depolarization rate at different temperatures for Pμc, and for Pμa from Fig. 4a and Fig. 4b.

### Source Data Fig. 5

Tc defined from specific heat and magnetization data, TBTRS temperature, electronic specific heat, experimental Sommerfeld coefficient, calculated DOS, calculated Sommerfeld coefficient $${\gamma }_{{\rm{DFT}}}/{\gamma }_{{\rm{DFT}},\min }$$ and calculated mean field $${T}_{{\rm{c}}}/{T}_{{\rm{c}},\min }$$ versus K-doping.

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