Skip to main content

EDITORIAL article

Front. Phys., 23 August 2023
Sec. Condensed Matter Physics
This article is part of the Research Topic Nucleation and Stability of Exotic Solitons in Condensed Matter View all 6 articles

Editorial: Nucleation and stability of exotic solitons in condensed matter

  • 1Faculty of Physics, ITMO University, St. Petersburg, Russia
  • 2School of Science and Technology, Örebro University, Örebro, Sweden
  • 3Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
  • 4Institute of Physics, Martin Luther University Halle-Wittenberg, Halle, Germany
  • 5Department of Physics and Electrical Engineering, Linnaeus University, Kalmar, Sweden
  • 6Science Institute, University of Iceland, Reykjavík, Iceland

Topological solitons in condensed matter are of particular interest for fundamental theory due to a deep connection between topology and physics manifested in these systems. At the same time, they are praised as the basis for new technologies of data storage, information processing, machine learning and neuromorphic computing. The most well-studied magnetic solitons are quasi-two-dimensional skyrmions and bubble domains. However, in recent years, attention has shifted to other two-dimensional and even three-dimensional localized topological structures appearing not only in magnetic materials, but also in liquid crystals, ferroelectrics and multiferroics, which expands our knowledge about topological effects in physics and possible scope of topological soliton applications [1].

Although skyrmions and related objects owe their stability to topology, the topological protection is not strict in real systems due to discrete nature of condensed matter, e.g., magnetic moments localized on atomic lattices. Instead, topological solitons can be nucleated and annihilated by overcoming finite energy barriers. Such over-the-barrier transitions can be induced spontaneously by thermal fluctuations leading to a finite lifetime of the states. Successful implementation of topological solitons in technology requires their lifetime to be sufficiently long, many orders of magnitude longer than characteristic times of the microscopic dynamics. This hierarchy of the timescales makes it challenging to study the thermal stability of the solitons. Consistently with the Néel-Brown theory of thermally activated magnetization reversal [2, 3], experimental observations [4] and numerical simulations [5] of magnetic skyrmions identified Arrhenius dependency of their nucleation/annihilation rates k on temperature T:

k=k0eΔE/kBT.(1)

However, both the energy barrier ΔE and, remarkably, the pre-exponential factor k0 which is often taken to be a phenomenological constant, turned out to be highly sensitive to various control parameters such as an external magnetic field [4]. The physics of the thermal stability of magnetic skyrmions was revealed using a statistical approach based on the rate theory for magnetic degrees of freedom [69]. Calculations of minimum energy paths (MEPs) connecting the skyrmion state with topologically trivial background state have uncovered skyrmion collapse mechanisms [1012], some of which were confirmed experimentally [13]. The rate theory has made it possible to identify, in a definite way, both the energy barrier and the Arrhenius pre-exponential factor. In particular, the unexpected variations of the pre-exponential factor were explained by large entropy difference between the skyrmion state and the transition state–the bottleneck for the skyrmion collapse [11, 1416]. Overall, recent developments of theoretical and computational methods for the rate theory [1720] have made it possible to establish a coherent picture about thermal stability of magnetic skyrmions. At the same time, the theoretical framework is quite general and can be applied to solitons beyond magnetic skyrmions.

Two-dimensional magnetic films with Dzyaloshinskii–Moriya interaction (DMI) can host, together with axisymmetric skyrmions, other locally stable configurations even with the same topological charge. Among them are so-called tailed skyrmions Kuchkin et al.. They have an elongated shape and can exist in a narrow range of fields near the transition from spin spirals to a uniform ferromagnetic state. “Growing a tail” is an additional mechanism for obtaining new solitons. There is a continuous transition (homotopy) between such structures and usual skyrmions. The homotopies can be efficiently found by calculating MEPs using the geodesic nudged elastic band method [17]. The discovery of tailed skyrmions extends the range of already known soliton solutions.

Additional possibilities appear in multilayer systems due to controlled modification of the interlayer exchange coupling (IEC). For example, synthetic antiferromagnet can be obtained by establishing an antiferromagnetic (AFM) IEC between the ferromagnetic (FM) layers through a non-magnetic spacer. DMI-stabilized ferromagnetic skyrmions in each layer can couple with each other in these systems thus forming composite AFM skyrmions [21]. AFM skyrmions can also be created intrinsically in AFM materials. Ab initio calculations predict that this can be done, for example, by depositing a row-wise AFM Cr layer on the PdFeIr(111) structure hosting FM skyrmions [22]. In this case, only exchange interactions may be required to form a complex AF structure. Aldarawsheh et al. investigate this system using the Heisenberg model, which includes basic magnetic interactions necessary to form AFM skyrmions on a triangular lattice. Interestingly, deposited Cr layer does not introduce additional DMI interaction but leads to long-range exchange interaction involving several neighbor shells.

In three-dimensional magnets, the possibility of forming even more exotic topological structures can be realized. In cubic magnets with competing magnetic interactions, hopfions can be stabilized even in the absence of DMI. However, theoretical estimates show that they are stable only at low temperatures of a few kelvins [23, 24]. In the presence of DMI and an external magnetic field, hopfions embedded in a conical magnetic structure form heliknotons. Their experimental observation, however, is a challenging task. Additional challenge lies in the interpretation of experimental data, which is far from being unambiguous, especially in three-dimensional systems. Therefore, it is necessary to use mutually complementary techniques and explore different interpretation options. Savchenko et al. show this using magnetic bubbles with alternating chirality in domain walls as an example. There, mathematical modeling of the system response, obtained in the framework of different experimental methods in combination with various theoretical approaches to the study of dynamics and stability assessment, is very useful. Kuchkin et al. discuss the stability of heliknotons and conditions of their detection based on micromagnetic modeling, rate theory, and stochastic spin dynamics simulations.

In three-dimensional samples with chiral interactions, delocalized states are possible in addition to localized topological structures. Leonov and Pappas carried out a systematic study of the states of an inclined spiral arising due to competition of cubic and exchange anisotropies inherent to cubic helimagnets. Field-controlled reorientation of metastable skyrmion lattices caused by competing anisotropies, may be responsible for some features in the experimental phase diagrams of Cu2OSeO3.

Author contributions

VU: Writing–original draft, Writing–review and editing. DT: Writing–original draft, Writing–review and editing. BG: Writing–original draft, Writing–review and editing. PB: Writing–original draft, Writing–review and editing.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References

1. Göbel B, Mertig I, Tretiakov OA. Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles. Phys Rep (2021) 895:1–28. doi:10.1016/j.physrep.2020.10.001

CrossRef Full Text | Google Scholar

2. Brown WF. Thermal fluctuations of a single-domain particle. Phys Rev (1963) 130:1677–86. doi:10.1103/PhysRev.130.1677

CrossRef Full Text | Google Scholar

3. Brown WF. Thermal fluctuation of fine ferromagnetic particles. IEEE Trans Magn (1979) 15:1196–208. doi:10.1109/TMAG.1979.1060329

CrossRef Full Text | Google Scholar

4. Wild J, Meier TNG, Pöllath S, Kronseder M, Bauer A, Chacon A, et al. Entropy-limited topological protection of skyrmions. Sci Adv (2017) 3:e1701704. doi:10.1126/sciadv.1701704

PubMed Abstract | CrossRef Full Text | Google Scholar

5. Hagemeister J, Romming N, von Bergmann K, Vedmedenko EY, Wiesendanger R. Stability of single skyrmionic bits. Nat Commun (2015) 6:8455. doi:10.1038/ncomms9455

PubMed Abstract | CrossRef Full Text | Google Scholar

6. Bessarab PF, Uzdin VM, Jónsson H. Harmonic transition-state theory of thermal spin transitions. Phys Rev B (2012) 85:184409. doi:10.1103/PhysRevB.85.184409

CrossRef Full Text | Google Scholar

7. Fiedler G, Fidler J, Lee J, Schrefl T, Stamps RL, Braun HB, et al. Direct calculation of the attempt frequency of magnetic structures using the finite element method. J Appl Phys (2012) 111:093917. doi:10.1063/1.4712033

CrossRef Full Text | Google Scholar

8. Vogler C, Bruckner F, Bergmair B, Huber T, Suess D, Dellago C. Simulating rare switching events of magnetic nanostructures with forward flux sampling. Phys Rev B (2013) 88:134409. doi:10.1103/PhysRevB.88.134409

CrossRef Full Text | Google Scholar

9. Lobanov IS, Potkina MN, Uzdin VM. Stability and lifetimes of magnetic states of nano- and microstructures (brief review). JETP Lett (2021) 113:801–13. doi:10.1134/S0021364021120109

CrossRef Full Text | Google Scholar

10. Lobanov IS, Jónsson H, Uzdin VM. Mechanism and activation energy of magnetic skyrmion annihilation obtained from minimum energy path calculations. Phys Rev B (2016) 94:174418. doi:10.1103/PhysRevB.94.174418

CrossRef Full Text | Google Scholar

11. Bessarab PF, Müller GP, Lobanov IS, Rybakov FN, Kiselev NS, Jónsson H, et al. Lifetime of racetrack skyrmions. Sci Rep (2018) 8:3433. doi:10.1038/s41598-018-21623-3

PubMed Abstract | CrossRef Full Text | Google Scholar

12. Desplat L, Kim JV, Stamps RL. Paths to annihilation of first- and second-order (anti)skyrmions via (anti)meron nucleation on the frustrated square lattice. Phys Rev B (2019) 99:174409. doi:10.1103/PhysRevB.99.174409

CrossRef Full Text | Google Scholar

13. Muckel F, von Malottki S, Holl C, Pestka B, Pratzer M, Bessarab PF, et al. Experimental identification of two distinct skyrmion collapse mechanisms. Nat Phys (2021) 17:395–402. doi:10.1038/s41567-020-01101-2

CrossRef Full Text | Google Scholar

14. Desplat L, Suess D, Kim JV, Stamps RL. Thermal stability of metastable magnetic skyrmions: Entropic narrowing and significance of internal eigenmodes. Phys Rev B (2018) 98:134407. doi:10.1103/PhysRevB.98.134407

CrossRef Full Text | Google Scholar

15. von Malottki S, Bessarab PF, Haldar S, Delin A, Heinze S. Skyrmion lifetime in ultrathin films. Phys Rev B (2019) 99:060409. doi:10.1103/PhysRevB.99.060409

CrossRef Full Text | Google Scholar

16. Varentcova AS, von Malottki S, Potkina MN, Kwiatkowski G, Heinze S, Bessarab PF. Toward room-temperature nanoscale skyrmions in ultrathin films. NPJ Comput Mater (2020) 6:193. doi:10.1038/s41524-020-00453-w

CrossRef Full Text | Google Scholar

17. Bessarab PF, Uzdin VM, Jónsson H. Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation. Comput Phys Commun (2015) 196:335–47. doi:10.1016/j.cpc.2015.07.001

CrossRef Full Text | Google Scholar

18. Desplat L, Vogler C, Kim JV, Stamps RL, Suess D. Path sampling for lifetimes of metastable magnetic skyrmions and direct comparison with Kramers’ method. Phys Rev B (2020) 101:060403. doi:10.1103/PhysRevB.101.060403

CrossRef Full Text | Google Scholar

19. Lobanov IS, Uzdin VM. The lifetime of micron scale topological chiral magnetic states with atomic resolution. Comput Phys Commun (2021) 269:108136. doi:10.1016/j.cpc.2021.108136

CrossRef Full Text | Google Scholar

20. Potkina MN, Lobanov IS, Jónsson H, Uzdin VM. Stability of magnetic skyrmions: Systematic calculations of the effect of size from nanometer scale to microns. Phys Rev B (2023) 107:184414. doi:10.1103/PhysRevB.107.184414

CrossRef Full Text | Google Scholar

21. Voronin KV, Lobanov IS, Uzdin VM. Activation energy and mechanisms for skyrmion collapse in synthetic antiferromagnets. JETP Lett (2022) 116:240–5. doi:10.1134/S0021364022601361

CrossRef Full Text | Google Scholar

22. Aldarawsheh A, Fernandes IL, Brinker S, Sallermann M, Abusaa M, Blügel S, et al. Emergence of zero-field non-synthetic single and interchained antiferromagnetic skyrmions in thin films. Nat Commun (2022) 13:7369. doi:10.1038/s41467-022-35102-x

PubMed Abstract | CrossRef Full Text | Google Scholar

23. Sallermann M, Jónsson H, Blügel S. Stability of hopfions in bulk magnets with competing exchange interactions. Phys Rev B (2023) 107:104404. doi:10.1103/PhysRevB.107.104404

CrossRef Full Text | Google Scholar

24. Lobanov IS, Uzdin VM. Lifetime, collapse, and escape paths for hopfions in bulk magnets with competing exchange interactions. Phys Rev B (2023) 107:104405. doi:10.1103/PhysRevB.107.104405

CrossRef Full Text | Google Scholar

Keywords: soliton, condensed matter physics, skyrmion, thermal stability, magnetic material

Citation: Uzdin VM, Thonig D, Göbel B and Bessarab PF (2023) Editorial: Nucleation and stability of exotic solitons in condensed matter. Front. Phys. 11:1275990. doi: 10.3389/fphy.2023.1275990

Received: 11 August 2023; Accepted: 15 August 2023;
Published: 23 August 2023.

Edited and reviewed by:

Matthias Eschrig, University of Greifswald, Germany

Copyright © 2023 Uzdin, Thonig, Göbel and Bessarab. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Pavel F. Bessarab, pavel.bessarab@lnu.se

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.