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  • Letter
  • Open Access

Nonlinear spin-Seebeck diode in f-wave magnets, third-order spin-Nernst effects in g-wave magnets, and spin-Nernst effects in i-wave altermagnets

Motohiko Ezawa

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Phys. Rev. B 113, L241301 – Published 15 June, 2026

DOI: https://doi.org/10.1103/fnwg-lfzg

Abstract

We analyze thermally induced spin transport in X-wave magnets with X=p,d,f,g, and i. A prominent feature of d-wave altermagnets is that spin current is generated by applying a temperature gradient, which is known as the spin-Nernst effect. We show in f-wave magnets that spin current is generated proportional to the square of the temperature gradient, which we call the nonlinear spin-Seebeck current. It can be used as a spin current diode. In addition, we show in g-wave altermagnets that spin current is generated in the third order of the temperature gradient. We also show in i-wave altermagnets that spin current is generated perpendicular to the temperature gradient, which is the spin-Nernst current. We have derived analytic formulas for these spin currents. On the other hand, we show in p-wave magnets that spin current is not generated by temperature gradient.

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References (64)

  1. M. I. Dyakonov and V. I. Perel, Possibility of orienting electron spins with current, Zh. Eksp. Teor. Fiz. Pis. Red. 13, 657 (1971) [Sov. Phys. JETP Lett. 13, 467 (1971)].
  2. M. I. Dyakonov and V. I. Perel, Current-induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459 (1971).
  3. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects Rev. Mod. Phys. 87, 1213 (2015).
  4. K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Observation of the spin Seebeck effect, Nature (London) 455, 778 (2008).
  5. G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin caloritronics Nat. Mater. 11, 391 (2012).
  6. S.-g. Cheng, Y. Xing, Q.-f. Sun, and X. C. Xie, Spin Nernst effect and Nernst effect in two-dimensional electron systems, Phys. Rev. B 78, 045302 (2008).
  7. S. Meyer, Y.-T. Chen, S. Wimmer, M. Althammer, T. Wimmer, R. Schlitz, S. Geprags, H. Huebl, D. Kodderitzsch, H. Ebert, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Observation of the spin Nernst effect, Nat. Mater. 16, 977 (2017).
  8. Y. Gao, S. A. Yang, and Q. Niu, Field induced positional shift of Bloch electrons and its dynamical implications, Phys. Rev. Lett. 112, 166601 (2014).
  9. I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials, Phys. Rev. Lett. 115, 216806 (2015).
  10. T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu, Y. Kaneko, Y. Tokura, N. Nagaosa, and Y. Iwasa, Bulk rectification effect in a polar semiconductor, Nat. Phys. 13, 578 (2017).
  11. H. Liu, J. Zhao, Y.-X. Huang, W. Wu, X.-L. Sheng, C. Xiao, and S. A. Yang, Intrinsic second-order anomalous Hall effect and its application in compensated antiferromagnets, Phys. Rev. Lett. 127, 277202 (2021).
  12. Y. Michishita and N. Nagaosa, Dissipation and geometry in nonlinear quantum transports of multiband electronic systems, Phys. Rev. B 106, 125114 (2022).
  13. H. Watanabe and Y. Yanase, Nonlinear electric transport in odd-parity magnetic multipole systems: Application to Mn-based compounds, Phys. Rev. Res. 2, 043081 (2020).
  14. C. Wang, Y. Gao, and D. Xiao, Intrinsic nonlinear Hall effect in antiferromagnetic tetragonal CuMnAs, Phys. Rev. Lett. 127, 277201 (2021).
  15. R. Oiwa and H. Kusunose, Systematic analysis method for nonlinear response tensors, J. Phys. Soc. Jpn. 91, 014701 (2022).
  16. A. Gao, Y.-F. Liu, J.-X. Qiu, B. Ghosh, T. V. Trevisan, Y. Onishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen, et al., Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure, Science 381, 181 (2023).
  17. N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, et al., Quantum-metric-induced nonlinear transport in a topological antiferromagnet, Nature (London) 621, 487 (2023).
  18. K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agarwal, Intrinsic nonlinear conductivities induced by the quantum metric, Phys. Rev. B 108, L201405 (2023).
  19. D. Kaplan, T. Holder, and B. Yan, Unification of nonlinear anomalous Hall effect and nonreciprocal magnetoresistance in metals by the quantum geometry, Phys. Rev. Lett. 132, 026301 (2024).
  20. Y. D. Wang, Z. F. Zhang, Z.-G. Zhu, and G. Su, Intrinsic nonlinear Ohmic current, Phys. Rev. B 109, 085419 (2024).
  21. L. Xiang, B. Wang, Y. Wei, Z. Qiao, and J. Wang, Linear displacement current solely driven by the quantum metric, Phys. Rev. B 109, 115121 (2024).
  22. Z.-H. Gong, Z. Z. Du, H.-P. Sun, H.-Z. Lu, and X. C. Xie, Nonlinear transport theory at the order of quantum metric, arXiv:2410.04995.
  23. M. Ezawa, Intrinsic nonlinear conductivity induced by quantum geometry in altermagnets and measurement of the in-plane Neel vector, Phys. Rev. B 110, L241405 (2024).
  24. Y. Fang, J. Cano, S. Ali, and A. Ghorashi, Quantum geometry induced nonlinear transport in altermagnets, Phys. Rev. Lett. 133, 106701 (2024).
  25. M. Ezawa, Purely electrical detection of the spin-splitting vector in p-wave magnets based on linear and nonlinear conductivities, Phys. Rev. B 112, 125412 (2025).
  26. X.-Q. Yu, Z.-G. Zhu, J.-S. You, T. Low, and G. Su, Topological nonlinear anomalous Nernst effect in strained transition metal dichalcogenides, Phys. Rev. B 99, 201410(R) (2019).
  27. D. B. Karki and M. N. Kiselev, Nonlinear Seebeck effect of SU(N) Kondo impurity, Phys. Rev. B 100, 125426 (2019).
  28. C. Zeng, S. Nandy, A. Taraphder, and S. Tewari, Nonlinear Nernst effect in bilayer WTe2, Phys. Rev. B 100, 245102 (2019).
  29. G. Marchegiani, A. Braggio, and F. Giazotto, Nonlinear thermoelectricity with electron-hole symmetric systems, Phys. Rev. Lett. 124, 106801 (2020).
  30. H. Arisawa, Y. Fujimoto, T. Kikkawa, and E. Saitoh, Observation of nonlinear thermoelectric effect in MoGe/Y3Fe5O12 Nat. Commun. 15, 6912 (2024).
  31. Y. Hirata, T. Kikkawa, H. Arisawa, and E. Saitoh, Nonlinear Seebeck effect in Ni81Fe19|Pt at room temperature, Appl. Phys. Lett. 126, 252408 (2025).
  32. H. Varshney and A. Agarwal, Intrinsic nonlinear Nernst and Seebeck effect, New J. Phys. 27, 083506 (2025).
  33. H. Varshney and A. Agarwal, Asymmetric scattering drives large nonlinear Nernst and Seebeck effects, Phys. Rev. B 113, 195417 (2026).
  34. Y.-F. Zhang, Z.-F. Zhang, H. Jiang, Z.-G. Zhu, and G. Su, Fundamental relations as the leading order in nonlinear thermoelectric responses with time-reversal symmetry, arXiv:2601.19625.
  35. K. Hamamoto, M. Ezawa, K. W. Kim, T. Morimoto, and N. Nagaosa, Nonlinear spin current generation in noncentrosymmetric spin-orbit coupled systems, Phys. Rev. B 95, 224430 (2017).
  36. M. Kameda, D. Hirobe, S. Daimon, Y. Shiomi, S. Takahashi, and E. Saitoh, Microscopic formulation of nonlinear spin current induced by spin pumping, J. Magn. Magn. Mater. 476, 459 (2019).
  37. S. Hayami, M. Yatsushiro, and H. Kusunose, Nonlinear spin Hall effect in PT-symmetric collinear magnets, Phys. Rev. B 106, 024405 (2022).
  38. S. Hayami, Linear and nonlinear spin-current generation in polar collinear antiferromagnets without relativistic spin-orbit coupling, Phys. Rev. B 109, 214431 (2024).
  39. M. Ezawa, Third-order and fifth-order nonlinear spin-current generation in g-wave and i-wave altermagnets and perfectly nonreciprocal spin current in f-wave magnets, Phys. Rev. B 111, 125420 (2025).
  40. L. Smejkal, J. Sinova, and T. Jungwirth, Beyond conventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022).
  41. L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging research landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022).
  42. S. Hayami, Y. Yanagi, and H. Kusunose, Momentum-dependent spin splitting by collinear antiferromagnetic ordering, J. Phys. Soc. Jpn. 88, 123702 (2019).
  43. A. B. Hellenes, T. Jungwirth, R. Jaeschke-Ubiergo, A. Chakraborty, J. Sinova, and L. Šmejkal, P-wave magnets, arXiv:2309.01607.
  44. T. Jungwirth, R. M. Fernandes, E. Fradkin, A. H. MacDonald, J. Sinova, and L. Smejkal, Altermagnetism: An unconventional spin-ordered phase of matter, arXiv:2411.00717.
  45. M. Ezawa, Quantum geometry and X-wave magnets with X=p, d, f, g, i, Appl. Phys. Express 19, 030101 (2026).
  46. M. Ezawa, Almost half-quantized planar Hall effects in X-wave magnets with X=p, d, f, g, i, Phys. Rev. B 112, 235307 (2025).
  47. M. Ezawa, Tunneling magnetoresistance in a junction made of X-wave magnets with X=p,d,f,g,i, Phys. Rev. B 113, 155303 (2026).
  48. M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Motome, and H. Seo, Spin current generation in organic antiferromagnets, Nat. Commun. 10, 4305 (2019).
  49. R. Gonzalez-Hernandez, L. Šmejkal, K. Vborn, Y. Yahagi, J. Sinova, T. Jungwirth, and J. Železny, Efficient electrical spin splitter based on nonrelativistic collinear antiferromagnetism, Phys. Rev. Lett. 126, 127701 (2021).
  50. M. Naka, Y. Motome, and H. Seo, Perovskite as a spin current generator, Phys. Rev. B 103, 125114 (2021).
  51. A. Bose, N. J. Schreiber, R. Jain, D.-F. Shao, H. P. Nair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsymbal, D. G. Schlom, and D. C. Ralph, Tilted spin current generated by the collinear antiferromagnet ruthenium dioxide, Nat. Electron. 5, 267 (2022).
  52. M. Naka, Y. Motome, and H. Seo, Altermagnetic perovskites, npj Spintron. 3, 1 (2025).
  53. See Supplemental Material at http://link.aps.org/supplemental/10.1103/fnwg-lfzg for the derivation of the solution of the Boltzmann equation up to any order of temperature gradient, the Fermi surface symmetry and tight-binding models, and the leading order of Seebeck currents and spin currents for various X-wave magnets both in two and three dimensions.
  54. C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, Inc, Hoboken, New Jersey, 2004).
  55. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, Philadelphia, 1976).
  56. C. Jacoboni and L. Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55, 645 (1983).
  57. C. Herring, Theory of the thermoelectric power of semiconductors, Phys. Rev. 96, 1163 (1954).
  58. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, Oxford Classic Texts in the Physical Sciences (Oxford University Press, Oxford, 2001).
  59. H. Smith and H. H. Jensen, Transport Phenomena (Clarendon Press, Oxford, 1989).
  60. S. Hayami, Y. Yanagi, and H. Kusunose, Spontaneous antisymmetric spin splitting in noncollinear antiferromagnets without spin-orbit coupling, Phys. Rev. B 101, 220403(R) (2020).
  61. S. Hayami, Y. Yanagi, and H. Kusunose, Bottom-up design of spin-split and reshaped electronic band structures in antiferromagnets without spin-orbit coupling: Procedure on the basis of augmented multipoles, Phys. Rev. B 102, 144441 (2020).
  62. S. K. Das and B. Roy, From local spin nematicity to altermagnets: Footprints of band topology, Phys. Rev. B 111, L201102 (2025).
  63. Y. Liu, J. Yu, and C.-C. Liu, Twisted magnetic Van der Waals bilayers: An ideal platform for altermagnetism, Phys. Rev. Lett. 133, 206702 (2024).
  64. I. Mazin, R. Gonzalez-Hernandez, and L. Smejkal, Induced monolayer altermagnetism in MnP(S,Se)3 and FeSe, arXiv:2309.02355.

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Phys. Rev. B 113, L241301– Published 15 June, 2026
  • Received 21 February 2026
  • Revised 7 May 2026
  • Accepted 29 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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