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Spin-resolved Josephson diode effect through strongly spin-polarized conical magnets

Danilo Nikolić1,*, Niklas L. Schulz1,†, Alexander I. Buzdin2,3, and Matthias Eschrig1,‡

  • *Contact author: danilo.nikolic@uni-greifswald.de
  • †Contact author: niklas.schulz@uni-greifswald.de
  • ‡Contact author: matthias.eschrig@uni-greifswald.de
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Phys. Rev. B 112, 224507 – Published 8 December, 2025

DOI: https://doi.org/10.1103/35ss-brd7

Abstract

We present a theoretical study of the spin-resolved Josephson diode effect in junctions comprising strongly spin-polarized conical magnets (FM) coupled to singlet superconductors (SC). The system is treated by making use of the Gor'kov and quasiclassical Green's function methods. Modeling the SC/FM interfaces as spin-dependent δ potentials, we apply our model to an SC/FM/SC junction and account for the Josephson current-phase relation (CPR). The nontrivial coupling between the spin bands in the conical magnet gives rise to a strong Josephson diode effect with an efficiency greater than 40%. The effect essentially depends on the quantum spin-geometric phase that enters the Josephson CPR in a very similar manner to the superconducting phase difference. The former is generated nonlocally by the intrinsically noncoplanar spin arrangement of the conical magnet, which breaks the time-reversal and inversion symmetries. Strong spin polarization and a helical pitch of the conical magnet comparable to the superconducting coherence length are essential for the effect. We perform a harmonic analysis of the Josephson CPR and interpret the effect in terms of coherent transfer of multiple equal-spin triplet Cooper pairs across the conical magnet.

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References (97)

  1. M. Eschrig, J. Kopu, A. Konstandin, J. C. Cuevas, M. Fogelström, and G. Schön, Singlet-triplet mixing in superconductor–ferromagnet hybrid devices, in Advances in Solid State Physics, edited by B. Kramer (Springer, Berlin, Heidelberg, 2004), pp. 533–545.
  2. A. I. Buzdin, Proximity effects in superconductor-ferromagnet heterostructures, Rev. Mod. Phys. 77, 935 (2005).
  3. F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures, Rev. Mod. Phys. 77, 1321 (2005).
  4. M. Eschrig, Spin-polarized supercurrents for spintronics, Phys. Today 64, 43 (2011).
  5. M. Eschrig, Spin-polarized supercurrents for spintronics: A review of current progress, Rep. Prog. Phys. 78, 104501 (2015).
  6. J. Linder and J. W. A. Robinson, Superconducting spintronics, Nat. Phys. 11, 307 (2015).
  7. N. O. Birge, Spin-triplet supercurrents in Josephson junctions containing strong ferromagnetic materials, Philos. Trans. R. Soc., A 376, 20150150 (2018).
  8. J. Linder and A. V. Balatsky, Odd-frequency superconductivity, Rev. Mod. Phys. 91, 045005 (2019).
  9. G. Yang, C. Ciccarelli, and J. W. A. Robinson, Boosting spintronics with superconductivity, APL Mater. 9, 050703 (2021).
  10. R. Cai, I. Žutić, and W. Han, Superconductor/ferromagnet heterostructures: A platform for superconducting spintronics and quantum computation, Adv. Quantum Technol. 6, 2200080 (2023).
  11. M. Eschrig, J. Kopu, J. C. Cuevas, and G. Schön, Theory of half-metal/superconductor heterostructures, Phys. Rev. Lett. 90, 137003 (2003).
  12. M. Eschrig and T. Löfwander, Triplet supercurrents in clean and disordered half-metallic ferromagnets, Nat. Phys. 4, 138 (2008).
  13. R. Grein, M. Eschrig, G. Metalidis, and G. Schön, Spin-dependent cooper pair phase and pure spin supercurrents in strongly polarized ferromagnets, Phys. Rev. Lett. 102, 227005 (2009).
  14. M. Eschrig, Scattering problem in nonequilibrium quasi-classical theory of metals and superconductors: General boundary conditions and applications, Phys. Rev. B 80, 134511 (2009).
  15. R. Grein, T. Löfwander, and M. Eschrig, Inverse proximity effect and influence of disorder on triplet supercurrents in strongly spin-polarized ferromagnets, Phys. Rev. B 88, 054502 (2013).
  16. M. Houzet and J. S. Meyer, Quasiclassical theory of disordered Rashba superconductors, Phys. Rev. B 92, 014509 (2015).
  17. I. V. Bobkova, A. M. Bobkov, and M. A. Silaev, Gauge theory of the long-range proximity effect and spontaneous currents in superconducting heterostructures with strong ferromagnets, Phys. Rev. B 96, 094506 (2017).
  18. J. A. Ouassou, A. Pal, M. Blamire, M. Eschrig, and J. Linder, Triplet Cooper pairs induced in diffusive s-wave superconductors interfaced with strongly spin-polarized magnetic insulators or half-metallic ferromagnets, Sci. Rep. 7, 1932 (2017).
  19. M. Eschrig, Theory of Andreev bound states in S-F-S junctions and S-F proximity devices, Philos. Trans. R. Soc. A 376, 20150149 (2018).
  20. T. Tokuyasu, J. A. Sauls, and D. Rainer, Proximity effect of a ferromagnetic insulator in contact with a superconductor, Phys. Rev. B 38, 8823 (1988).
  21. E. A. Demler, G. B. Arnold, and M. R. Beasley, Superconducting proximity effects in magnetic metals, Phys. Rev. B 55, 15174 (1997).
  22. F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Long-Range proximity effects in superconductor-ferromagnet structures, Phys. Rev. Lett. 86, 4096 (2001).
  23. M. Eschrig, T. Löfwander, T. Champel, J. C. Cuevas, J. Kopu, and G. Schön, Symmetries of pairing correlations in superconductor–ferromagnet nanostructures, J. Low Temp. Phys. 147, 457 (2007).
  24. Y. Tanaka and A. A. Golubov, Theory of the proximity effect in junctions with unconventional superconductors, Phys. Rev. Lett. 98, 037003 (2007).
  25. R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, and A. Gupta, A spin triplet supercurrent through the half-metallic ferromagnet CrO2, Nature (London) 439, 825 (2006).
  26. T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge, Observation of spin-triplet superconductivity in Co-based Josephson junctions, Phys. Rev. Lett. 104, 137002 (2010).
  27. M. S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, and J. Aarts, Long-range supercurrents through half-metallic ferromagnetic CrO2, Phys. Rev. B 82, 100501(R) (2010).
  28. J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Controlled injection of spin-triplet supercurrents into a strong ferromagnet, Science 329, 59 (2010).
  29. J. A. Glick, A. B. Gougam, B. M. Niedzielski, E. C. Gingrich, R. Loloee, W. P. Pratt, Jr., and N. O. Birge, Phase control in a spin-triplet SQUID, Sci. Adv. 4, eaat9457 (2018).
  30. R. Caruso, D. Massarotti, G. Campagnano, A. Pal, H. G. Ahmad, P. Lucignano1, M. Eschrig, M. G. Blamire, and F. Tafuri, Tuning of magnetic activity in spin-filter Josephson junctions towards spin-triplet transport, Phys. Rev. Lett. 122, 047002 (2019).
  31. V. Aguilar, D. Korucu, J. A. Glick, R. Loloee, W. P. Pratt, Jr., and N. O. Birge, Spin-polarized triplet supercurrent in Josephson junctions with perpendicular ferromagnetic layers, Phys. Rev. B 102, 024518 (2020).
  32. N. L. Schulz, D. Nikolić, and M. Eschrig, Quantum-geometric spin- and charge Josephson diode effects, Phys. Rev. B 112, 104514 (2025).
  33. N. L. Schulz, D. Nikolić, and M. Eschrig, Theory of quantum-geometric charge and spin Josephson diode effects in strongly spin-polarized hybrid structures with noncoplanar spin textures, Phys. Rev. B 112, 104515 (2025).
  34. L. N. Bulaevskii, A. I. Rusinov, and M. Kulić, Helical ordering of spins in a superconductor, J. Low Temp. Phys. 39, 255 (1980).
  35. H. Meng, A. V. Samokhvalov, and A. I. Buzdin, Nonuniform superconductivity and Josephson effect in a conical ferromagnet, Phys. Rev. B 99, 024503 (2019).
  36. A. F. Volkov, Ya V. Fominov, and K. B. Efetov, Long-range odd triplet superconductivity in superconductor-ferromagnet structures with Néel walls, Phys. Rev. B 72, 184504 (2005).
  37. I. Sosnin, H. Cho, V. T. Petrashov, and A. F. Volkov, Superconducting phase coherent electron transport in proximity conical ferromagnets, Phys. Rev. Lett. 96, 157002 (2006).
  38. A. Majeed and H. Singh, Modelling of triplet proximity effects in conically magnetized NbN/Ho/NbN Josephson junctions, Sci. Rep. 15, 21000 (2025).
  39. G. A. Bobkov, A. V. Kornev, A. M. Bobkov, and I. V. Bobkova, Proximity effect and p-wave superconductivity in s-wave superconductor/helimagnet heterostructures, Phys. Rev. B 112, 184515 (2025).
  40. A. F. Volkov, A. Anishchanka, and K. B. Efetov, Odd triplet superconductivity in a superconductor/ferromagnet system with a spiral magnetic structure, Phys. Rev. B 73, 104412 (2006).
  41. Ya V. Fominov, A. F. Volkov, and K. B. Efetov, Josephson effect due to the long-range odd-frequency triplet superconductivity in SFS junctions with Néel domain walls, Phys. Rev. B 75, 104509 (2007).
  42. T. Champel, T. Löfwander, and M. Eschrig, 0−π transitions in a superconductor/chiral ferromagnet/superconductor junction induced by a homogeneous cycloidal spiral, Phys. Rev. Lett. 100, 077003 (2008).
  43. J. D. S. Witt, J. W. A. Robinson, and M. G. Blamire, Josephson junctions incorporating a conical magnetic holmium interlayer, Phys. Rev. B 85, 184526 (2012).
  44. T. Champel and M. Eschrig, Effect of an inhomogeneous exchange field on the proximity effect in disordered superconductor-ferromagnet hybrid structures, Phys. Rev. B 72, 054523 (2005).
  45. A. Spuri, D. Nikolić, S. Chakraborty, M. Klang, H. Alpern, O. Millo, H. Steinberg, W. Belzig, E. Scheer, and A. Di Bernardo, Signature of long-ranged spin triplets across a two-dimensional superconductor/helimagnet van der Waals interface, Phys. Rev. Res. 6, L012046 (2024).
  46. A. A. Golubov, M. Yu Kupriyanov, and E. Il'ichev, The current-phase relation in Josephson junctions, Rev. Mod. Phys. 76, 411 (2004).
  47. V. B. Geshkenbein and A. I. Larkin, The Josephson effect in superconductors with heavy fermions, Pis'ma Zh Eksp. Teor. Fiz. 43, 306 (1986) [JETP Lett. 43, 395 (1986)].
  48. S. Yip, Josephson current-phase relationships with unconventional superconductors, Phys. Rev. B 52, 3087 (1995).
  49. M. Sigrist, Time-reversal symmetry breaking states in high-temperature superconductors, Prog. Theor. Phys. 99, 899 (1998).
  50. A. Buzdin, Direct coupling between magnetism and superconducting current in the Josephson φ0 junction, Phys. Rev. Lett. 101, 107005 (2008).
  51. M. Nadeem, M. S. Fuhrer, and X. Wang, The superconducting diode effect, Nat. Rev. Phys. 5, 558 (2023).
  52. F. Ando, Y. Miyasaka, T. Li, J. Ishizuka, T. Arakawa, Y. Shiota, T. Moriyama, Y. Yanase, and T. Ono, Observation of superconducting diode effect, Nature (London) 584, 373 (2020).
  53. C. Baumgartner, L. Fuchs, A. Costa, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, P. E. Faria Junior, D. Kochan, J. Fabian, N. Paradiso, and C. Strunk, Supercurrent rectification and magnetochiral effects in symmetric Josephson junctions, Nat. Nanotechnol. 17, 39 (2022).
  54. A. Costa, C. Baumgartner, S. Reinhardt, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, J. Fabian, D. Kochan, N. Paradiso, and C. Strunk, Sign reversal of the Josephson inductance magnetochiral anisotropy and 0–π-like transitions in supercurrent diodes, Nat. Nanotechnol. 18, 1266 (2023).
  55. A. Gutfreund, H. Matsuki, V. Plastovets, A. Noah, L. Gorzawski, N. Fridman, G. Yang, A. Buzdin, O. Millo, J. W. A. Robinson, and Y. Anahory, Direct observation of a superconducting vortex diode, Nat. Commun. 14, 1630 (2023).
  56. Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F. Ritter, D. Z. Haxell, M. Davydova, S. Ilić, O. Glezakou-Elbert, A. Varambally, F. S. Bergeret, A. Kamra, L. Fu, P. A. Lee, and J. S. Moodera, Ubiquitous superconducting diode effect in superconductor thin films, Phys. Rev. Lett. 131, 027001 (2023).
  57. E. Strambini, M. Spies, N. Ligato, S. Ilić, M. Rouco, C. Gonzalez-Orellana, M. Ilyn, C. Rogero, F. S. Bergeret, J. S. Moodera, P. Virtanen, T. T. Heikkilä, and F. Giazotto, Superconducting spintronic tunnel diode, Nat. Commun. 13, 2431 (2022).
  58. M. Trahms, L. Melischek, J. F. Steiner, B. Mahendru, I. Tamir, N. Bogdanoff, O. Peters, G. Reecht, C. B. Winkelmann, F. von Oppen, and K. J. Franke, Diode effect in Josephson junctions with a single magnetic atom, Nature (London) 615, 628 (2023).
  59. S. Reinhardt, T. Ascherl, A. Costa, J. Berger, S. Gronin, G. C. Gardner, T. Lindemann, M. J. Manfra, J. Fabian, D. Kochan, C. Strunk, and N. Paradiso, Link between supercurrent diode and anomalous Josephson effect revealed by gate-controlled interferometry, Nat. Commun. 15, 4413 (2024).
  60. S. Pal and C. Benjamin, Quantized Josephson phase battery, EPL 126, 57002 (2019).
  61. A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic superconducting diode effect, Phys. Rev. Lett. 128, 037001 (2022).
  62. Ya V. Fominov and D. S. Mikhailov, Asymmetric higher-harmonic SQUID as a Josephson diode, Phys. Rev. B 106, 134514 (2022).
  63. K. Halterman, M. Alidoust, R. Smith, and S. Starr, Supercurrent diode effect, spin torques, and robust zero-energy peak in planar half-metallic trilayers, Phys. Rev. B 105, 104508 (2022).
  64. J. J. He, Y. Tanaka, and N. Nagaosa, A phenomenological theory of superconductor diodes, New J. Phys. 24, 053014 (2022).
  65. S. Ilić and F. S. Bergeret, Theory of the supercurrent diode effect in Rashba superconductors with arbitrary disorder, Phys. Rev. Lett. 128, 177001 (2022).
  66. T. Karabassov, I. V. Bobkova, A. A. Golubov, and A. S. Vasenko, Hybrid helical state and superconducting diode effect in superconductor/ferromagnet/topological insulator heterostructures, Phys. Rev. B 106, 224509 (2022).
  67. A. A. Kopasov, A. G. Kutlin, and A. S. Mel'nikov, Geometry-controlled superconducting diode and anomalous Josephson effect triggered by the topological phase transition in curved proximitized nanowires, Phys. Rev. B 103, 144520 (2021).
  68. K. Misaki and N. Nagaosa, Theory of the nonreciprocal Josephson effect, Phys. Rev. B 103, 245302 (2021).
  69. Y. Tanaka, B. Lu, and N. Nagaosa, Theory of giant diode effect in d-wave superconductor junctions on the surface of a topological insulator, Phys. Rev. B 106, 214524 (2022).
  70. N. F. Q. Yuan and L. Fu, Supercurrent diode effect and finite-momentum superconductors, Proc. Natl. Acad. Sci. USA 119, e2119548119 (2022).
  71. Y. Zhang, Y. Gu, P. Li, J. Hu, and K. Jiang, General theory of Josephson diodes, Phys. Rev. X 12, 041013 (2022).
  72. B. Zinkl, K. Hamamoto, and M. Sigrist, Symmetry conditions for the superconducting diode effect in chiral superconductors, Phys. Rev. Res. 4, 033167 (2022).
  73. R. S. Souto, M. Leijnse, and C. Schrade, Josephson diode effect in supercurrent interferometers, Phys. Rev. Lett. 129, 267702 (2022).
  74. J. F. Steiner, L. Melischek, M. Trahms, K. J. Franke, and F. Von Oppen, Diode effects in current-biased Josephson junctions, Phys. Rev. Lett. 130, 177002 (2023).
  75. A. Costa, J. Fabian, and D. Kochan, Microscopic study of the Josephson supercurrent diode effect in Josephson junctions based on two-dimensional electron gas, Phys. Rev. B 108, 054522 (2023).
  76. A. A. Kopasov, Zh. Devizorova, H. Meng, S. V. Mironov, A. S. Mel'nikov, and A. I. Buzdin, Adiabatic phase pumping in S/F/S hybrids with noncoplanar magnetization, Phys. Rev. B 108, 224511 (2023).
  77. J. S. Meyer and M. Houzet, Josephson diode effect in a ballistic single-channel nanowire, Appl. Phys. Lett. 125, 022603 (2024).
  78. A. V. Putilov, S. V. Mironov, and A. I. Buzdin, Nonreciprocal electron transport in finite-size superconductor/ferromagnet bilayers with strong spin-orbit coupling, Phys. Rev. B 109, 014510 (2024).
  79. J. B. Tjernshaugen, M. Amundsen, and J. Linder, Superconducting phase diagram and spin diode effect via spin accumulation, Phys. Rev. B 109, 094516 (2024).
  80. J. Hasan, D. Shaffer, M. Khodas, and A. Levchenko, Supercurrent diode effect in helical superconductors, Phys. Rev. B 110, 024508 (2024).
  81. S. Patil, G. Tang, and W. Belzig, Spin-split Andreev bound states and diode effect in an Ising superconductor Josephson junction, Phys. Rev. B 111, L060502 (2025).
  82. M. Roig, P. Kotetes, and B. M. Andersen, Superconducting diodes from magnetization gradients, Phys. Rev. B 109, 144503 (2024).
  83. L. J. Kamra and L. Fu, Nonreciprocal Josephson current through a conical magnet, arXiv:2409.00223.
  84. L. P. Gor'kov, On the energy spectrum of superconductors, Zh Eksp. Teor. Fiz. 34, 735 (1958) [Sov. Phys.–JETP 7, 505 (1958)].
  85. R. Grein, T. Löfwander, G. Metalidis, and M. Eschrig, Theory of superconductor-ferromagnet point-contact spectra: The case of strong spin polarization, Phys. Rev. B 81, 094508 (2010).
  86. A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Zh Eksp. Teor. Fiz. 55, 2262 (1968) [Sov. Phys. JETP 28, 1200 (1969)].
  87. G. D. Eilenberger, Transformation of Gorkov's equation for type II superconductors into transport-like equations, Z. Phys. A: Hadrons Nucl. 214, 195 (1968).
  88. W. Belzig, F. K. Wilhelm, C. Bruder, G. Schoön, and A. D. Zaikin, Quasiclassical Green's function approach to mesoscopic superconductivity, Superlattices Microstruct. 25, 1251 (1999).
  89. K. Y. Bliokh and Y. P. Bliokh, Ann. Phys. (NY) 319, 13 (2005).
  90. T. Fujita, M. B. A. Jalil, S. G. Tan, and S. Murakami, Gauge fields in spintronics, J. Appl. Phys. 110, 121301 (2011).
  91. G. Tatara, Effective gauge field theory of spintronics, Phys. E (Amsterdam, Neth.) 106, 208 (2019).
  92. W. Serene and D. Rainer, The quasiclassical approach to superfluid He3, Phys. Rep. 101, 221 (1983).
  93. M. Eschrig, J. A. Sauls, and D. Rainer, Electromagnetic response of a vortex in layered superconductors, Phys. Rev. B 60, 10447 (1999).
  94. M. Eschrig, Distribution functions in nonequilibrium theory of superconductivity and Andreev spectroscopy in unconventional superconductors, Phys. Rev. B 61, 9061 (2000).
  95. N. Schopohl, Transformation of the Eilenberger equations of superconductivity to a scalar Riccati equation, arXiv:cond-mat/9804064.
  96. G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion, Phys. Rev. B 25, 4515 (1982).
  97. Y. Cao, Z. Huang, Y. Yin, H. Xie, B. Liu, W. Wang, C. Zhu, D. Mandrus, L. Wang, and W. Huang, Overview and advances in a layered chiral helimagnet Cr1/3NbS2, Mater. Today Adv. 7, 100080 (2020).

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Phys. Rev. B 112, 224507– Published 8 December, 2025
  • Received 7 August 2025
  • Accepted 29 October 2025
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