- 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

Fokker-Planck approach to thermal fluctuations in antiferromagnetic systems

E. Martello1, G. A. Falci1,2, E. Paladino1,2, and F. M. D. Pellegrino1,2

PDF

Phys. Rev. B 113, 104413 – Published 9 March, 2026

DOI: https://doi.org/10.1103/qrcv-bcdn

Abstract

We develop a Fokker-Planck approach to describe the dynamics of staggered magnetization and thermal fluctuations in a two-dimensional antiferromagnetic system with uniaxial anisotropy. Beginning with a classical model for the antiferromagnetic system, we incorporate a Landau-Lifshitz-Gilbert equation augmented by Langevin fields to account for thermal fluctuations, and we derive the Fokker-Planck equation governing the probability distribution function of the spin configuration. Employing the mean-field approximation, we derive the equations of motion for the spin polarization and the two-time spin-spin correlation functions. The methodology is applied to the study of spin-wave dynamics and to the formulation of a phenomenological model for resistance fluctuations in two-dimensional antiferromagnetic semiconductors.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (52)

  1. S. Du, P. Tang, J. Li, Z. Lin, Y. Xu, W. Duan, and A. Rubio, Berry curvature engineering by gating two-dimensional antiferromagnets, Phys. Rev. Res. 2, 022025(R) (2020).
  2. X.-X. Zhang, L. Li, D. Weber, J. Goldberger, K. F. Mak, and J. Shan, Gate-tunable spin waves in antiferromagnetic atomic bilayers, Nat. Mater. 19, 838 (2020).
  3. J. Wang, H. Zeng, W. Duan, and H. Huang, Intrinsic nonlinear hall detection of the néel vector for two-dimensional antiferromagnetic spintronics, Phys. Rev. Lett. 131, 056401 (2023).
  4. A. Pancholi, A. Kumar, and S. Roychowdhury, Journey of the 2D intrinsic antiferromagnetic topological insulators in the (MnBi2Te4)(Bi2Te3)n homologous series, Small 21, 2411464 (2025).
  5. C. Liu, Y. Wang, H. Li, Y. Wu, Y. Li, J. Li, K. He, Y. Xu, J. Zhang, and Y. Wang, Robust axion insulator and Chern insulator phases in a two-dimensional antiferromagnetic topological insulator, Nat. Mater. 19, 522 (2020).
  6. S. Ghosh, F. Kargar, A. Mohammadzadeh, S. Rumyantsev, and A. A. Balandin, Low-frequency electronic noise in quasi-2D van der Waals antiferromagnetic semiconductor FePS3—signatures of phase transitions, Adv. Electron. Mater. 7, 2100408 (2021).
  7. S. Liang, T. Xie, N. A. Blumenschein, T. Zhou, T. Ersevim, Z. Song, J. Liang, M. A. Susner, B. S. Conner, S.-J. Gong, et al., Small-voltage multiferroic control of two-dimensional magnetic insulators, Nat. Electron. 6, 199 (2023).
  8. T.-Y. Wu, K.-H. Lin, J.-Y. Li, C.-N. Kuo, C.-S. Lue, and C.-Y. Chen, Delaminated MnPS3 with multiple layers coupled with Si featuring ultrahigh detectivity and environmental stability for UV photodetection, ACS Appl. Mater. Interfaces 15, 54643 (2023).
  9. S. Ghosh, Z. E. Nataj, F. Kargar, and A. A. Balandin, Electronic noise spectroscopy of quasi-two-dimensional antiferromagnetic semiconductors, ACS Appl. Mater. Interfaces 16, 20920 (2024).
  10. V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Antiferromagnetic spintronics, Rev. Mod. Phys. 90, 015005 (2018).
  11. E. V. Gomonay and V. M. Loktev, Spintronics of antiferromagnetic systems (review article), Low Temp. Phys. 40, 17 (2014).
  12. C. C. Mayorga-Martinez, Z. Sofer, D. Sedmidubsky, S. Huber, A. Y. S. Eng, and M. Pumera, Layered metal thiophosphite materials: magnetic, electrochemical, and electronic properties, ACS Appl. Mater. Interfaces 9, 12563 (2017).
  13. F. Wang, T. A. Shifa, P. Yu, P. He, Y. Liu, F. Wang, Z. Wang, X. Zhan, X. Lou, F. Xia, et al., New frontiers on van der Waals layered metal phosphorous trichalcogenides, Adv. Funct. Mater. 28, 1802151 (2018).
  14. R. Samal, G. Sanyal, B. Chakraborty, and C. S. Rout, Two-dimensional transition metal phosphorous trichalcogenides (MPX3): A review on emerging trends, current state and future perspectives, J. Mater. Chem. A 9, 2560 (2021).
  15. T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetization field, Phys. Rev. 100, 1243 (1955).
  16. T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40, 3443 (2004).
  17. A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas, Gilbert damping phenomenology for two-sublattice magnets, Phys. Rev. B 98, 184402 (2018).
  18. W. F. Brown, Thermal fluctuations of a single-domain particle, Phys. Rev. 130, 1677 (1963).
  19. Q. Liu, H. Y. Yuan, K. Xia, and Z. Yuan, Mode-dependent damping in metallic antiferromagnets due to intersublattice spin pumping, Phys. Rev. Mater. 1, 061401(R) (2017).
  20. D. A. Garanin, Fokker-Planck and Landau-Lifshitz-Bloch equations for classical ferromagnets, Phys. Rev. B 55, 3050 (1997).
  21. L. Chotorlishvili, Z. Toklikishvili, V. K. Dugaev, J. Barnaś, S. Trimper, and J. Berakdar, Fokker-Planck approach to the theory of the magnon-driven spin Seebeck effect, Phys. Rev. B 88, 144429 (2013).
  22. U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Landau-Lifshitz-Bloch equation for ferrimagnetic materials, Phys. Rev. B 86, 104414 (2012).
  23. S. M. Rezende, A. Azevedo, and R. L. Rodríguez-Suárez, Introduction to antiferromagnetic magnons, J. Appl. Phys. 126, 151101 (2019).
  24. B. L. Chittari, Y. Park, D. Lee, M. Han, A. H. MacDonald, E. Hwang, and J. Jung, Electronic and magnetic properties of single-layer MPX3 metal phosphorous trichalcogenides, Phys. Rev. B 94, 184428 (2016).
  25. T. Olsen, Antiferromagnetism in two-dimensional materials: progress and computational challenges, 2D Mater. 11, 033005 (2024).
  26. Z. Y. Chen, Z. R. Yan, M. H. Qin, and J.-M. Liu, Landau-Lifshitz-Bloch equation for domain wall motion in antiferromagnets, Phys. Rev. B 99, 214436 (2019).
  27. F. M. D. Pellegrino, G. G. N. Angilella, and R. Pucci, Effect of impurities in high-symmetry lattice positions on the local density of states and conductivity of graphene, Phys. Rev. B 80, 094203 (2009).
  28. A. Kamra and W. Belzig, Spin pumping and shot noise in ferrimagnets: Bridging ferro- and antiferromagnets, Phys. Rev. Lett. 119, 197201 (2017).
  29. J. L. Garcia-Palacios, Introduction to the theory of stochastic processes and Brownian motion problems, .
  30. S. V. Titov, W. T. Coffey, M. Zarifakis, Y. P. Kalmykov, and A. S. Titov, Compact formulation of the statistical moment method for the solution of the Fokker–Planck equation for two coupled macrospins, J. Magn. Magn. Mater. 539, 168365 (2021).
  31. H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7, 284 (1940).
  32. J. E. Moyal, Quantum mechanics as a statistical theory, Math. Proc. Cambridge Philos. Soc. 45, 99 (1949).
  33. K. K. Likharev, Statistical Mechanics: Lecture Notes (Academic Commons, Stony Brook, 2019).
  34. S. Kogan, Electronic Noise and Fluctuations in Solids (Cambridge University Press, Cambridge, UK, 2008).
  35. P. Langevin, Sur la theorie du magnetisme, J. Phys. Thor. Appl. 4, 678 (1905).
  36. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Frontiers in Physics (Addison-Wesley, Reading, MA, 1992).
  37. R. S. Gekht, V. A. Ignatchenko, Y. L. Raikher, and M. I. Shliomis, Magnetic resonance of an isotropic superparamagnet, Sov. Phys. JETP 43, 677 (1976) [Zh. Eksp. Teor. Fiz. 70, 1290 (1976)].
  38. J. E. Nitschke, D. L. Esteras, M. Gutnikov, K. Schiller, S. Mañas-Valero, E. Coronado, M. Stupar, G. Zamborlini, S. Ponzoni, J. J. Baldoví, et al., Valence band electronic structure of the van der Waals antiferromagnet FePS3, Mater. Today Electron. 6, 100061 (2023).
  39. H. Yamada and S. Takada, Magnetoresistance of antiferromagnetic metals due to s−d interaction, J. Phys. Soc. Jpn. 34, 51 (1973).
  40. J. Hu and T. F. Rosenbaum, Classical and quantum routes to linear magnetoresistance, Nat. Mater. 7, 697 (2008).
  41. G. Long, H. Henck, M. Gibertini, D. Dumcenco, Z. Wang, T. Taniguchi, K. Watanabe, E. Giannini, and A. F. Morpurgo, Persistence of magnetism in atomically thin MnPS3 crystals, Nano Lett. 20, 2452 (2020).
  42. M. Huang, J. C. Green, J. Zhou, V. Williams, S. Li, H. Lu, D. Djugba, H. Wang, B. Flebus, N. Ni, and C. R. Du, Layer-dependent magnetism and spin fluctuations in atomically thin van der waals magnet CrPS4, Nano Lett. 23, 8099 (2023).
  43. L. Esaki, P. J. Stiles, and S. v. Molnar, Magnetointernal field emission in junctions of magnetic insulators, Phys. Rev. Lett. 19, 852 (1967).
  44. M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer, Berlin, 2008).
  45. N. Wiener, Generalized harmonic analysis, Acta Math. 55, 117 (1930).
  46. A. Khintchine, Korrelationstheorie der stationären stochastischen prozesse, Math. Ann. 109, 604 (1934).
  47. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  48. E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, 1/f noise: Implications for solid-state quantum information, Rev. Mod. Phys. 86, 361 (2014).
  49. F. M. D. Pellegrino, G. Falci, and E. Paladino, Second spectrum of charge carrier density fluctuations in graphene due to trapping/detrapping processes, Appl. Phys. Lett. 122, 253102 (2023).
  50. F. M. D. Pellegrino, G. Falci, and E. Paladino, Charge carrier density noise in graphene: Effect of localized/delocalized traps, J. Stat. Mech. (2019) 094015.
  51. F. M. D. Pellegrino, G. Falci, and E. Paladino, 1/f critical current noise in short ballistic graphene Josephson junctions, Commun. Phys. 3, 6 (2020).
  52. H. T. Hardner, M. B. Weissman, M. B. Salamon, and S. S. P. Parkin, Fluctuation-dissipation relation for giant magnetoresistive 1/f noise, Phys. Rev. B 48, 16156 (1993).

Outline

Information

Phys. Rev. B 113, 104413– Published 9 March, 2026
  • Received 24 November 2025
  • Revised 26 January 2026
  • Accepted 17 February 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