Abstract
In superconductors possessing both time and inversion symmetries, the Zeeman effect of an external magnetic field can break the time-reversal symmetry, forming a conventional Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state characterized by Cooper pairings with finite momentum1,2. In superconductors lacking (local) inversion symmetry, the Zeeman effect may still act as the underlying mechanism of FFLO states by interacting with spin–orbit coupling (SOC). Specifically, the interplay between the Zeeman effect and Rashba SOC can lead to the formation of more accessible Rashba FFLO states that cover broader regions in the phase diagram3,4,5. However, when the Zeeman effect is suppressed because of spin locking in the presence of Ising-type SOC, the conventional FFLO scenarios are no longer effective. Instead, an unconventional FFLO state is formed by coupling the orbital effect of magnetic fields with SOC, providing an alternative mechanism in superconductors with broken inversion symmetries6,7,8. Here we report the discovery of such an orbital FFLO state in the multilayer Ising superconductor 2H-NbSe2. Transport measurements show that the translational and rotational symmetries are broken in the orbital FFLO state, providing the hallmark signatures of finite-momentum Cooper pairings. We establish the entire orbital FFLO phase diagram, consisting of a normal metal, a uniform Ising superconducting phase and a six-fold orbital FFLO state. This study highlights an alternative route to achieving finite-momentum superconductivity and provides a universal mechanism to preparing orbital FFLO states in similar materials with broken inversion symmetries.
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All relevant data shown are provided with this paper. Additional data that support the plots and other analyses in this work are available from the corresponding author upon request.
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Acknowledgements
We thank J. Zoestbergen for technical support. This publication is part of the project TOPCORE (with project number OCENW.GROOT.2019.048) of the research programme Open Competition ENW Groot, which is (partly) financed by the Dutch Research Council (NWO). P.W. acknowledges the research program ‘Materials for the Quantum Age’ (QuMat) for financial support. This program (registration number 024.005.006) is part of the Gravitation program financed by the Dutch Ministry of Education, Culture and Science (OCW). O.Z. acknowledges financial support from the CogniGron research center and the Ubbo Emmius Funds (University of Groningen). N.F.Q.Y. acknowledges the National Natural Science Foundation of China (grant number 12174021) for their financial support. The high-field measurement was supported by HFML-RU/NWO-I, a member of the European Magnetic Field Laboratory (EMFL). It is part of the research programme of the Netherlands Organisation for Scientific Research (NWO) funded by the National Roadmap for Large-Scale Research Facilities.
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P.W., O.Z. and J.Y. conceived the research. P.W., O.Z., X.P., L.Z. and M.L. fabricated the devices. P.W. carried out the magnetotransport and anisotropy measurements. P.W., O.Z., X.P., S.W. and U.Z. carried out the high-field magnetotransport measurements. N.F.Q.Y. constructed the theoretical model of the orbital FFLO states. P.W. and J.Y. analysed the data. P.W., N.F.Q.Y. and J.Y. interpreted the data and prepared the manuscript with inputs from N.E.H. and T.T.M.P. All authors commented on the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Absence of the upturn in upper critical fields in a downgraded device.
a, The temperature dependence of sheet resistances for two flakes showing RRR = 12.6 and 28 for 11 and 17 nm thick flakes, respectively. b, The Bc2,|| measured for the 11 and 17 nm thick flakes. The 2D Ginzburg-Landau fittings, that is, the solid black curves, yield thickness dfit = 8 and 12 nm, respectively. Overall, the reduced thicknesses obtained from the GL fittings are due to the protection from the Ising SOC, which becomes more robust in thinner flakes59,60,61. Close to Tc0, the 11 nm flake shows a steeper temperature dependence of Bc2, consistent with its reduced thickness. Nevertheless, for the 17 nm flake with a larger RRR, an upturn in the Bc2 can be observed at B = 0.36BP, indicating the orbital FFLO state, which eventually enhances Bc2 to exceed that measured in the 11 nm flake. As a larger RRR indicates better sample quality, the contrasting behaviour in the temperature dependence of Bc2 suggests that the absence of the orbital FFLO phase in the thin flake might be caused by the downgraded quality, which suppresses the finite-momentum pairing via scattering13,62.
Extended Data Fig. 2 Illustration of 2-axis rotation of a 2D sample in an external magnetic field with an installation canting angle γ.
A 2D sample is mounted on a 2-axis rotational stage. The 2D surface of the sample (yellow plane) makes a canting angle γ with respect to one of the rotation planes of the stage (grey plane). To simplify the discussion and isolate the effect of canting angle γ, we assume that the stage can make precise rotations so that, as shown in Fig. 3b, we can always align the sample plane precisely parallel to the external B field. When this exact parallelism is aligned at a given φ, due to the canting angle γ, further rotation along the stage axis 1 or 2 can cause a correlation between θ and φ, which are labelled as different θ(φ) values.
Extended Data Fig. 3 Procedure for subtracting the canting angle γ.
a, Magnetoresistance R||(φ) in an in-plane B field B|| (for the 17 nm device). b, Variation of θ0 (as defined in Extended Data Fig. 2) as a function of φ when the B field is adjusted to be parallel to the sample plane. The solid black curve is fitted using Eqn. 2 when θ = 0°, which yields a canting angle γ = 0.71°. c, The data are shown in Fig. 2f before correcting the effect of γ. The black line is the same fitting that is shown in panel b. d, After correcting for the canting angle, the magnetoresistance R(φ, θ) shows a two-fold anisotropy.
Extended Data Fig. 4 Measurement configuration of the 17 nm device.
a, Device orientation and the applied current direction. One crystalline direction of NbSe2, as indicated by the white dashed line, is defined as φ = 0. b, c, Transport measurements using two sets of electrode pairs on two sides of the Hall bar show a small shift (~5°). It is consistent with the small deviation of φ when changing the current direction in Fig. 3j.
Extended Data Fig. 5 Six-fold anisotropy of another multilayer NbSe2 flake.
The thickness of the flake is 22 nm. The anisotropy is measured at B|| = 8.9 T. a, The magnetoresistance R||(φ) in the coexisting state shows a six-fold anisotropy in the B|| field. b, The mapping of R(θ, φ) exhibits six-fold anisotropy when rotating θ close to 0.
Extended Data Fig. 6 Determination of the critical point where the upturn of Bc2(Jc) occurs.
a, An example of I-V measurements at different B|| fields at a fixed temperature. b, The critical current densities Jc were extracted from a. The critical current density is determined as the point where V/I is half the normal resistance RN at T = 10 K. c, The upturn in the B – Jc plot is determined by the kink in dB/dJc.
Extended Data Fig. 7 Critical current density as a function of temperature under zero magnetic field.
At B = 0 T, no upturn was observed in the temperature dependence of Jc, ruling out the two-gap scenario as the cause of the upturns.
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Wan, P., Zheliuk, O., Yuan, N.F.Q. et al. Orbital Fulde–Ferrell–Larkin–Ovchinnikov state in an Ising superconductor. Nature 619, 46–51 (2023). https://doi.org/10.1038/s41586-023-05967-z
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DOI: https://doi.org/10.1038/s41586-023-05967-z
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