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Sign reversal of the Josephson inductance magnetochiral anisotropy and 0–π-like transitions in supercurrent diodes


The recent discovery of the intrinsic supercurrent diode effect, and its prompt observation in a rich variety of systems, has shown that non-reciprocal supercurrents naturally emerge when both space-inversion and time-inversion symmetries are broken. In Josephson junctions, non-reciprocal supercurrent can be conveniently described in terms of spin-split Andreev states. Here we demonstrate a sign reversal of the Josephson inductance magnetochiral anisotropy, a manifestation of the supercurrent diode effect. The asymmetry of the Josephson inductance as a function of the supercurrent allows us to probe the current–phase relation near equilibrium, and to probe jumps in the junction ground state. Using a minimal theoretical model, we can then link the sign reversal of the inductance magnetochiral anisotropy to the so-called 0−π-like transition, a predicted but still elusive feature of multichannel junctions. Our results demonstrate the potential of inductance measurements as sensitive probes of the fundamental properties of unconventional Josephson junctions.

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Fig. 1: Multichannel Rashba Josephson junctions: experiment and theory.
Fig. 2: Reversal of the inductance MCA at the 0–π-like transition.
Fig. 3: The d.c. SDE.

Data availability

The data that support the findings of this study are available at the online depository EPUB of the University of Regensburg, with the identifier Source data are provided with this paper.

Code availability

The computer codes that support the theoretical results, the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


  1. Ando, F. et al. Observation of superconducting diode effect. Nature 584, 373–376 (2020).

  2. Baumgartner, C. et al. Supercurrent rectification and magnetochiral effects in symmetric Josephson junctions. Nat. Nanotechnol. 17, 39–44 (2022).

  3. Baumgartner, C. et al. Effect of Rashba and Dresselhaus spin–orbit coupling on supercurrent rectification and magnetochiral anisotropy of ballistic Josephson junctions. J. Phys. Condens. Matter 34, 154005 (2022).

    Article  CAS  Google Scholar 

  4. Wu, H. et al. The field-free Josephson diode in a van der Waals heterostructure. Nature 604, 653–656 (2022).

  5. Jeon, K.-R. et al. Zero-field polarity-reversible Josephson supercurrent diodes enabled by a proximity-magnetized Pt barrier. Nat. Mater. 21, 1008–1013 (2022).

  6. Pal, B. et al. Josephson diode effect from Cooper pair momentum in a topological semimetal. Nat. Phys. (2022).

  7. Bauriedl, L. et al. Supercurrent diode effect and magnetochiral anisotropy in few-layer NbSe2. Nat. Commun. 13, 4266 (2022).

    Article  CAS  Google Scholar 

  8. Turini, B. et al. Josephson diode effect in high-mobility InSb nanoflags. Nano Lett. 22, 8502–8508 (2022).

    Article  CAS  Google Scholar 

  9. Gupta, M. et al. Gate-tunable superconducting diode effect in a three-terminal Josephson device. Nat. Commun. 14, 3078 (2023).

  10. Zhang, B. et al. Evidence of φ0-Josephson junction from skewed diffraction patterns in Sn-InSb nanowires. Preprint at arXiv (2022).

  11. Mazur, G. P. et al. The gate-tunable Josephson diode. Preprint at arXiv (2022).

  12. Diez-Merida, J. et al. Symmetry-broken Josephson junctions and superconducting diodes in magic-angle twisted bilayer graphene. Nat. Commun. 14, 2396 (2023).

  13. Lin, J.-X. et al. Zero-field superconducting diode effect in small-twist-angle trilayer graphene. Nat. Phys. 18, 1221–1227 (2022).

  14. Scammell, H. D., Li, J. I. A. & Scheurer, M. S. Theory of zero-field superconducting diode effect in twisted trilayer graphene. 2D Mater. 9, 025027 (2022).

    Article  Google Scholar 

  15. Lu, B., Ikegaya, S., Burset, P., Tanaka, Y. & Nagaosa, N. Josephson diode effect on the surface of topological insulators. Preprint at arXiv (2022).

  16. Fu, P.-H., Xu, Y., Lee, C. H., Ang, Y. S. & Liu, J.-F. Gate-tunable high-efficiency topological Josephson diode. Preprint at arXiv (2022).

  17. Daido, A., Ikeda, Y. & Yanase, Y. Intrinsic superconducting diode effect. Phys. Rev. Lett. 128, 037001 (2022).

    Article  CAS  Google Scholar 

  18. Yuan, N. F. Q. & Fu, L. Supercurrent diode effect and finite-momentum superconductors. Proc. Natl Acad. Sci. USA 119, e2119548119 (2022).

    Article  CAS  Google Scholar 

  19. He, J. J., Tanaka, Y. & Nagaosa, N. A phenomenological theory of superconductor diodes. New J. Phys. 24, 053014 (2022).

    Article  Google Scholar 

  20. Ilić, S. & Bergeret, F. S. Theory of the supercurrent diode effect in Rashba superconductors with arbitrary disorder. Phys. Rev. Lett. 128, 177001 (2022).

    Article  Google Scholar 

  21. Legg, H. F., Loss, D. & Klinovaja, J. Superconducting diode effect due to magnetochiral anisotropy in topological insulators and rashba nanowires. Phys. Rev. B 106, 104501 (2022).

    Article  CAS  Google Scholar 

  22. Kochan, D., Costa, A., Zhumagulov, I. and Žutić, I. Phenomenological theory of the supercurrent diode effect: the Lifshitz invariant. Preprint at arXiv (2023).

  23. Andreev, A. F. Electron spectrum of the intermediate state of superconductors. Zh. Eksp. Teor. Fiz. 49, 655 (1966). J. Exp. Theor. Phys. 22, 455–458 (1966).

  24. Davydova, M., Prembabu, S. & and Fu, L. Universal Josephson diode effect. Sci. Adv. 8, eabo0309 (2022).

  25. Grein, R., Eschrig, M., Metalidis, G. & Schön, G. Spin-dependent Cooper pair phase and pure spin supercurrents in strongly polarized ferromagnets. Phys. Rev. Lett. 102, 227005 (2009).

  26. Bezuglyi, E. V., Rozhavsky, A. S., Vagner, I. D. & Wyder, P. Combined effect of Zeeman splitting and spin-orbit interaction on the Josephson current in a superconductor–two-dimensional electron gas–superconductor structure. Phys. Rev. B 66, 052508 (2002).

    Article  Google Scholar 

  27. Krive, I. V., Gorelik, L. Y., Shekhter, R. I. & Jonson, M. Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction. Low. Temp. Phys. 30, 398–404 (2004).

    Article  CAS  Google Scholar 

  28. Buzdin, A. Direct coupling between magnetism and superconducting current in the Josephson φ0 junction. Phys. Rev. Lett. 101, 107005 (2008).

    Article  CAS  Google Scholar 

  29. Reynoso, A. A., Usaj, G., Balseiro, C. A., Feinberg, D. & Avignon, M. Anomalous Josephson current in junctions with spin polarizing quantum point contacts. Phys. Rev. Lett. 101, 107001 (2008).

    Article  CAS  Google Scholar 

  30. Zazunov, A., Egger, R., Jonckheere, T. & Martin, T. Anomalous Josephson current through a spin-orbit coupled quantum dot. Phys. Rev. Lett. 103, 147004 (2009).

    Article  CAS  Google Scholar 

  31. Liu, J.-F. & Chan, K. S. Relation between symmetry breaking and the anomalous Josephson effect. Phys. Rev. B 82, 125305 (2010).

    Article  Google Scholar 

  32. Liu, J.-F. & Chan, K. S. Anomalous Josephson current through a ferromagnetic trilayer junction. Phys. Rev. B 82, 184533 (2010).

    Article  Google Scholar 

  33. Liu, J.-F., Chan, K. S. & Wang, J. Anomalous Josephson current through a ferromagnet-semiconductor hybrid structure. J. Phys. Soc. Jpn 80, 124708 (2011).

    Article  Google Scholar 

  34. Reynoso, A. A., Usaj, G., Balseiro, C. A., Feinberg, D. & Avignon, M. Spin-orbit-induced chirality of Andreev states in Josephson junctions. Phys. Rev. B 86, 214519 (2012).

    Article  Google Scholar 

  35. Yokoyama, T., Eto, M. & Nazarov, Y. V. Josephson current through semiconductor nanowire with spin–orbit interaction in magnetic field. J. Phys. Soc. Jpn 82, 054703 (2013).

    Article  Google Scholar 

  36. Brunetti, A., Zazunov, A., Kundu, A. & Egger, R. Anomalous Josephson current, incipient time-reversal symmetry breaking, and Majorana bound states in interacting multilevel dots. Phys. Rev. B 88, 144515 (2013).

    Article  Google Scholar 

  37. Yokoyama, T., Eto, M. & Nazarov, Y. V. Anomalous Josephson effect induced by spin-orbit interaction and Zeeman effect in semiconductor nanowires. Phys. Rev. B 89, 195407 (2014).

    Article  Google Scholar 

  38. Shen, K., Vignale, G. & Raimondi, R. Microscopic theory of the inverse Edelstein effect. Phys. Rev. Lett. 112, 096601 (2014).

    Article  Google Scholar 

  39. Konschelle, F., Tokatly, I. V. & Bergeret, F. S. Theory of the spin-galvanic effect and the anomalous phase shift φ0 in superconductors and Josephson junctions with intrinsic spin-orbit coupling. Phys. Rev. B 92, 125443 (2015).

    Article  Google Scholar 

  40. Szombati, D. B. et al. Josephson φ0-junction in nanowire quantum dots. Nat. Phys. 12, 568–572 (2016).

    Article  CAS  Google Scholar 

  41. Assouline, A. et al. Spin-orbit induced phase-shift in Bi2Se3 Josephson junctions. Nat. Commun. 10, 126 (2019).

    Article  Google Scholar 

  42. Mayer, W. et al. Gate controlled anomalous phase shift in Al/InAs Josephson junctions. Nat. Commun. 11, 212 (2020).

    Article  CAS  Google Scholar 

  43. Strambini, E. et al. A Josephson phase battery. Nat. Nanotechnol. 15, 656–660 (2020).

    Article  CAS  Google Scholar 

  44. Baumgartner, C. et al. Josephson inductance as a probe for highly ballistic semiconductor-superconductor weak links. Phys. Rev. Lett. 126, 037001 (2021).

    Article  CAS  Google Scholar 

  45. De Gennes, P. G. Superconductivity of Metals and Alloys (Addison Wesley, 1989).

  46. Li, C. et al. Zeeman-effect-induced 0−π transitions in ballistic Dirac semimetal Josephson junctions. Phys. Rev. Lett. 123, 026802 (2019).

    Article  CAS  Google Scholar 

  47. Hart, S. et al. Controlled finite momentum pairing and spatially varying order parameter in proximitized HgTe quantum wells. Nat. Phys. 13, 87–93 (2017).

  48. Chen, A. Q. et al. Finite momentum Cooper pairing in three-dimensional topological insulator Josephson junctions. Nat. Commun. 9, 3478 (2018).

    Article  Google Scholar 

  49. Ke, C. T. et al. Ballistic superconductivity and tunable π–junctions in InSb quantum wells. Nat. Commun. 10, 3764 (2019).

    Article  Google Scholar 

  50. Whiticar, A. M. et al. Zeeman-driven parity transitions in an Andreev quantum dot. Phys. Rev. B 103, 245308 (2021).

    Article  CAS  Google Scholar 

  51. Shin, J. et al. Magnetic proximity-induced superconducting diode effect and infinite magnetoresistance in a van der Waals heterostructure. Phys. Rev. Res. 5, L022064 (2023).

  52. Hou, Y. et al. Ubiquitous superconducting diode effect in superconductor thin films. Preprint at arXiv (2022).

  53. Suri, D. et al. Non-reciprocity of vortex-limited critical current in conventional superconducting micro-bridges. Appl. Phys. Lett. 121, 102601 (2022).

    Article  CAS  Google Scholar 

  54. Sundaresh, A., Vayrynen, J. I., Lyanda-Geller, Y. & Rokhinson, L. P. Diamagnetic mechanism of critical current non-reciprocity in multilayered superconductors. Nat. Commun. 14, 1628 (2023).

  55. Legg, H. F., Laubscher, K., Loss, D. & Klinovaja, J. Parity protected superconducting diode effect in topological Josephson junctions. Preprint at arXiv (2023).

  56. Frattini, N. E. et al. 3-wave mixing Josephson dipole element. Appl. Phys. Lett. 110, 222603 (2017).

    Article  Google Scholar 

  57. Leroux, C. et al. Nonreciprocal devices based on voltage-tunable junctions. Preprint at arXiv (2022).

  58. Roudsari, A. F. et al. Three-wave mixing traveling-wave parametric amplifier with periodic variation of the circuit parameters. Appl. Phys. Lett. 122, 052601 (2023).

    Article  Google Scholar 

  59. Banerjee, A. et al. Phase asymmetry of Andreev spectra from Cooper-pair momentum. Preprint at arXiv (2023).

  60. Lotfizadeh, N. et al. Superconducting diode effect sign change in epitaxial Al-InAs Josepshon junctions. Preprint at arXiv (2023).

  61. Žutić, I. & Valls, O. T. Tunneling spectroscopy for ferromagnet/superconductor junctions. Phys. Rev. B 61, 1555–1566 (2000).

    Article  Google Scholar 

  62. Blonder, G. E., Tinkham, M. & Klapwijk, T. M. Transition from metallic to tunneling regimes in superconducting microconstrictions: excess current, charge imbalance, and supercurrent conversion. Phys. Rev. B 25, 4515–4532 (1982).

    Article  CAS  Google Scholar 

  63. Dartiailh, M. C. et al. Phase signature of topological transition in Josephson junctions. Phys. Rev. Lett. 126, 036802 (2021).

    Article  CAS  Google Scholar 

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Work at Regensburg University was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 314695032 (SFB 1277 (subprojects B05, B07 and B08)) and project no. 454646522, research grant ‘Spin and magnetic properties of superconducting tunnel junctions’ (A.C. and J.F.). D.K. acknowledges partial support from the project IM-2021-26 (SUPERSPIN) funded by the Slovak Academy of Sciences via the programme IMPULZ 2021.

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Authors and Affiliations



C.B. and J.B. fabricated the devices and performed the measurements. A.C. performed the numerical simulations. A.C., D.K. and J.F. formulated the theoretical model. S.R. developed and optimized the measurement method. T.L., S.G. and G.C.G. designed the heterostructure, conducted molecular-beam epitaxy growth and performed the initial characterization of the hybrid superconductor/semiconductor wafer. C.B. and N.P. analysed the data. N.P. and C.S. conceived the experiment. C.S. and M.J.M. supervised research activities at Regensburg and Purdue, respectively. N.P., A.C. and D.K. wrote the manuscript. All authors contributed to discussions and to the refinement of the manuscript.

Corresponding author

Correspondence to N. Paradiso.

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Nature Nanotechnology thanks Mathias Scheurer, Elia Strambini and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–5 and Discussion.

Source data

Source Data Fig. 1

Results of calculations based on our analytical model.

Source Data Fig. 2

Experimental data (Fig. 2a–c) and calculations (Fig. 2d,e).

Source Data Fig. 3

Experimental data (Fig. 3a,b) and calculations (Fig. 3c–e).

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Costa, A., Baumgartner, C., Reinhardt, S. et al. Sign reversal of the Josephson inductance magnetochiral anisotropy and 0–π-like transitions in supercurrent diodes. Nat. Nanotechnol. (2023).

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