- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Evolution of a demagnetized spin state toward equilibrium

I. A. Yastremsky1,* and B. A. Ivanov2,3,†

  • *Contact author: yastremsky@knu.ua
  • †Contact author: bor.a.ivanov@gmail.com
PDF

Phys. Rev. B 113, 224422 – Published 11 June, 2026

DOI: https://doi.org/10.1103/9b7b-pyvc

Abstract

The temporal evolution of the nonequilibrium nearly homogeneous ferromagnetic state is analyzed within the general phenomenological spin-dynamics theory proposed by Baryakhtar, which employs the concepts of both local (relativistic) and nonlocal (exchange) spin relaxation. We demonstrate that strongly inhomogeneous spin dynamics can emerge on nanometer length scales. The mechanism responsible for this inhomogeneous dynamics is related to instabilities of the demagnetized spin state. These instabilities qualitatively modify the relaxation process toward equilibrium and may result in a dramatic underestimation of the characteristic time and amplitude of the spin dynamics. This finding can be exploited for the development of a new type of source for coherent spin-wave generation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (52)

  1. E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Ultrafast spin dynamics in ferromagnetic nickel, Phys. Rev. Lett. 76, 4250 (1996).
  2. I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, Th Rasing, and A. V. Kimel, Transient ferromagneticlike state mediating ultrafast reversal of antiferromagnetically coupled spins, Nature (London) 472, 205 (2011).
  3. J. H. Mentink, J. Hellsvik, D. V. Afanasiev, B. A. Ivanov, A. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson, and Th Rasing, Ultrafast spin dynamics in multisublattice magnets, Phys. Rev. Lett. 108, 057202 (2012).
  4. A. Kirilyuk, A. V. Kimel, and T. Rasing, Laser-induced magnetization dynamics and reversal in ferrimagnetic alloys, Rep. Prog. Phys. 76, 026501 (2013).
  5. C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and T. F. Heinz, Ultrafast dynamics in van der Waals heterostructures, Nat. Nanotechnol. 13, 994 (2018).
  6. D. Afanasiev, J. R. Hortensius, B. A. Ivanov, A. Sasani, E. Bousquet, Y. M. Blanter, R. V. Mikhaylovskiy, A. V. Kimel, and A. D. Caviglia, Ultrafast control of magnetic interactions via light-driven phonons, Nat. Mater. 20, 607 (2021).
  7. K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Z. Hagström, S. S. Phani, K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Deinert, et al., Inertial spin dynamics in ferromagnets, Nat. Phys. 17, 245 (2021).
  8. Q. Remy, J. Hohlfeld, M. Vergès, Y. L. Guen, J. Gorchon, G. Malinowski, S. Mangin, and M. Hehn, Accelerating ultrafast magnetization reversal by nonlocal spin transfer, Nat. Commun. 14, 445 (2023).
  9. M. Hennecke, C. von Korff Schmising, K. Yao, E. Jal, B. Vodungbo, V. Chardonnet, K. Légaré, F. Capotondi, D. Naumenko, E. Pedersoli, I. Lopez-Quintas, et al., Ultrafast opto-magnetic effects in the extreme ultraviolet spectral range, Commun. Phys. 7, 191 (2024).
  10. N. Wu, S. Zhang, D. Chen, Y. Wang, and S. Meng, Three-stage ultrafast demagnetization dynamics in a monolayer ferromagnet, Nat. Commun. 15, 2804 (2024).
  11. X. Lu, Z. Lin, H. Pi, T. Zhang, G. Li, Y. Gong, Y. Yan, X. Ruan, Y. Li, H. Zhang, et al., Ultrafast magnetization enhancement via the dynamic spin-filter effect of type-II Weyl nodes in a kagome ferromagnet, Nat. Commun. 15, 2410 (2024).
  12. T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Directional control of spin-wave emission by spatially shaped light, Nat. Photonics 6, 662 (2012).
  13. Y. Au, M. Dvornik, T. Davison, E. Ahmad, P. S. Keatley, A. Vansteenkiste, B. Van Waeyenberge, and V. V. Kruglyak, Direct excitation of propagating spin waves by focused ultrashort optical pulses, Phys. Rev. Lett. 110, 097201 (2013).
  14. J. R. Hortensius, D. Afanasiev, M. Matthiesen, R. Leenders, R. Citro, A. V. Kimel, R. V. Mikhaylovskiy, B. A. Ivanov, and A. D. Caviglia, Coherent spin-wave transport in an antiferromagnet, Nat. Phys. 17, 1001 (2021).
  15. C. E. Graves, A. Reid, T. Wang, B. Wu, S. de Jong, K. Vahaplar, I. Radu, D. P. Bernstein, M. Messerschmidt, L. Müller, et al., Nanoscale spin reversal by nonlocal angular momentum transfer following ultrafast laser excitation in ferrimagnetic GdFeCo, Nat. Mater. 12, 293 (2013).
  16. V. I. Sugakov, Formation of new phase inclusions in the system of quasiequilibrium magnons of high density, Phys. Rev. B 94, 014407 (2016).
  17. V. I. Sugakov, Formation of phases with inverted spin orientation in magnetic crystals at rapid cooling, Phys. Rev. B 108, 064412 (2023).
  18. Nara Rubiano da Silva, Marcel Möller, Armin Feist, Henning Ulrichs, Claus Ropers, and Sascha Schäfer, Nanoscale mapping of ultrafast magnetization dynamics with femtosecond Lorentz microscopy, Phys. Rev. X 8, 031052 (2018).
  19. W. Zhang, P. Maldonado, Z. Jin, T. S. Seifert, J. Arabski, G. Schmerber, E. Beaurepaire, M. Bonn, T. Kampfrath, P. M. Oppeneer, and D. Turchinovich, Ultrafast terahertz magnetometry, Nat. Commun. 11, 4247 (2020).
  20. S. Zayko, O. Kfir, M. Heigl, M. Lohmann, M. Sivis, M. Albrecht, and C. Ropers, Ultrafast high-harmonic nanoscopy of magnetization dynamics, Nat. Commun. 12, 6337 (2021).
  21. T. T. Gareev, N. E. Khokhlov, L. Körber, and A. V. Kimel, Strongly inhomogeneous spin dynamics induced by ultrashort laser pulses with a gradient intensity profile, Phys. Rev. Lett. 135, 156701 (2025).
  22. H-T. Chang, S. Zayko, T. Schmidt, O. Kfir, M. Sivis, J. H. Mentink, M. Albrecht, and C. Ropers, Sub-wavelength extreme ultraviolet microscopy reveals domain-wall stability during ultrafast demagnetization, Nat. Mater. (to be published).
  23. A. I. Akhiezer, V. G. Bar'yakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, Amsterdam, 1968).
  24. F. J. Dyson, Thermodynamic behavior of an ideal ferromagnet, Phys. Rev. 102, 1230 (1956).
  25. U. Atxitia and O. Chubykalo-Fesenko, Ultrafast magnetization dynamics rates within the Landau-Lifshitz-Bloch model, Phys. Rev. B 84, 144414 (2011).
  26. I. A. Yastremsky, Peter M. Oppeneer, and B. A. Ivanov, Theory of fast time evolution of nonequilibrium spin states in magnetic heterostructures, Phys. Rev. B 90, 024409 (2014).
  27. M. O. A. Ellis, R. F. L. Evans, T. A. Ostler, J. Barker, U. Atxitia, O. Chubykalo-Fesenko, and R. W. Chantrell, The Landau–Lifshitz equation in atomistic models, Low Temp. Phys. 41, 9 (2015).
  28. L. D. Landau and E. M. Lifshitz, To the theory of magnetic permeability of ferromagnetic bodies, In: L. D. Landau, The collection of works (Nauka, Moscow 1969), Vol. 1, p. 128 [in Russian].
  29. T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40, 3443 (2004).
  30. D. A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B 55, 3050 (1997).
  31. D. A. Garanin and O. Chubykalo-Fesenko, Thermal fluctuations and longitudinal relaxation of single-domain magnetic particles at elevated temperatures, Phys. Rev. B 70, 212409 (2004).
  32. R. F. L. Evans, D. Hinzke, U. Atxitia, U. Nowak, R. W. Chantrell, and O. Chubykalo-Fesenko, Stochastic form of the Landau-Lifshitz-Bloch equation, Phys. Rev. B 85, 014433 (2012).
  33. I. A. Yastremsky, Longitudinal Evolution of the Magnetization in Nanostructures, NATO Science for Peace and Security Series B: Physics and Biophysics (Springer, Dordrecht 2024).
  34. Y. Tserkovnyak, A. Brataas, Gerrit E. W. Bauer, and B. I. Halperin, Nonlocal magnetization dynamics in ferromagnetic heterostructures, Rev. Mod. Phys. 77, 1375 (2005).
  35. I. A. Yastremsky, Size dependence of the relaxation rate for nonequilibrium redistributions of the magnetization in Ni–Fe heterestuctures: Exchange vs. Relativistic damping scenarios, J. Magn. Magn. Mater. 382, 349 (2015).
  36. T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. E. Moussaoui, L. L. Guyader, E. Mengotti, L. J. Heyderman, et al., Ultrafast heating as a sufficient stimulus for magnetization reversal in a ferrimagnet, Nat. Commun. 3, 666 (2012).
  37. V. G. Bar'yakhtar, B. A. Ivanov, and K. A. Safaryan, On the phenomenological description of the damping of the domain walls in ferrite-garnets, Solid State Comm. 72, 1117 (1989).
  38. V. G. Bar'yakhtar, B. A. Ivanov, T. K. Soboleva, and A. L. Sukstanskii, Theory of dynamical-soliton relaxation in ferromagnets, Zh. Eksp. Theor. Fiz. 91, 1454 (1986) [Sov. Phys. JETP 64, 857 (1986)].
  39. E. G. Galkina, B. A. Ivanov, and V. A. Stephanovich, Phenomenological theory of Bloch point relaxation, J. Magn. Magn. Mater. 118, 373 (1993).
  40. V. G. Baryakhtar, B. A. Ivanov, A. L. Sukstanskii, and E. Yu Melikhov, Soliton relaxation in magnets, Phys. Rev. B 56, 619 (1997).
  41. W. Wang, M. Dvornik, M.-A. Bisotti, D. Chernyshenko, M. Beg, M. Albert, A. Vansteenkiste, B. V. Waeyenberge, A. N. Kuchko, V. V. Kruglyak, and H. Fangohr, Phenomenological description of the nonlocal magnetization relaxation in magnonics, spintronics, and domain-wall dynamics, Phys. Rev. B 92, 054430 (2015).
  42. Ivan A. Yastremsky, Jürgen Fassbender, Boris A. Ivanov, and Denys Makarov, Enhanced longitudinal relaxation of magnetic solitons in ultrathin films, Phys. Rev. Appl. 17, L061002 (2022).
  43. Ivan A. Yastremsky, Magnetization length in the Bloch point at finite temperatures, J. Magn. Magn. Mater. 616, 172834 (2025).
  44. V. G. Bar'yakhtar, Phenomenological description of relaxation processes in magnetic materials, Zh. Eksp. Theor. Fiz. 87, 1501 (1984) [Sov. Phys. JETP 60, 863 (1984)].
  45. V. G. Bar'yakhtar, Spontaneous ground state degeneracy and relaxation processes in a ferromagnet, Fiz. Tverd. Tela 29, 1317 (1987) [Sov. Phys. Solid State 29, 754 (1987)].
  46. V. G. Bar'yakhtar, Crystal symmetry and the structure of the relaxation terms in the dynamic equations of motion for magnetization, Physica B 159, 20 (1989).
  47. P. Collet and J. P. Eckmann, Instabilities and Fronts in Extended Systems (Princeton University Press, Princeton, NJ, 1990).
  48. U. Ebert and W. van Saarloos, Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts, Physica D 146, 1 (2000).
  49. Agus L. Soenjaya, Mixed finite element methods for the Landau-Lifshitz-Baryakhtar and the regularised Landau-Lifshitz-Bloch equations in micromagnetics, J. Sci. Comput. 103, 65 (2025).
  50. H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J. Silva, Mode- and size-dependent Landau-Lifshitz damping in magnetic nanostructures: Evidence for nonlocal damping, Phys. Rev. Lett. 110, 117201 (2013).
  51. K. Binder, Time-dependent Ginzburg-Landau theory of nonequilibrium relaxation, Phys. Rev. B 8, 3423 (1973).
  52. I. Yastremsky and B. Ivanov, Evolution of a demagnetized spin state toward equilibrium, Zenodo, 2026, https://doi.org/10.5281/zenodo.20548719.

Outline

Information

Phys. Rev. B 113, 224422– Published 11 June, 2026
  • Received 15 February 2026
  • Revised 10 May 2026
  • Accepted 20 May 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation