Two-dimensional magnetic vortices

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

In the proposed review, the structure of peculiar topological excitations of magnetically ordered media, the so-called two-dimensional magnetic vortices, is described as completely and in detail as possible. Magnetic vortices represent a distinct category of defects within the field of condensed matter physics. Accordingly, the structure of vortices in hydrodynamics and superfluids, as well as dislocations in solid-state physics, is presented at the beginning of the review. A specific section of the review is dedicated to elucidating the structural characteristics of plane vortices, instantons, spiral vortices, magnetic “targets,” vortex stripes, and their interactions employing analytical methods. A general solution of a two-dimensional isotropic ferromagnetic system is presented using methods of differential geometry. The discussion encompasses twodimensional vortices with anisotropic exchange interactions. A substantial portion of the review is devoted to helicoidal structures and vortices (skyrmions) in chiral magnets, encompassing their theoretical characterization based on a functional incorporating the DMI, as well as the outcomes of the early experiments on the detection of one-dimensional helical structures. A theoretical description of skyrmions and two-dimensional skyrmion lattices in bulk crystals is provided. It is observed that the DMI significantly alters the morphology of skyrmions with an arbitrary topological charge. Such structures can be represented as a “sack” with the shell comprised of kπ-skyrmions. The observed Archimedean spiral vortices are described, and a hexagonal lattice of Archimedean spiral is predicted to represent a new equilibrium phase.

Full Text

Restricted Access

About the authors

A. B. Borisov

Mikheev Institute of Metal Physics, Ural Branch, Russian Academy of Sciences

Author for correspondence.
Email: borisov@imp.uran.ru
Russian Federation, Ekaterinburg

References

  1. Ковалев А.С., Косевич А.М., Маслов К.В. Магнитный вихрь — топологический солитон в ферромагнетике с анизотропией типа легкая ось // Письма в ЖЭТФ. 1979. Т. 30. № 6. С. 321–324.
  2. Косевич А.М., Иванов Б.А., Ковалев А.С. Нелинейные волны намагниченности. Динамические и топологические солитоны. Киев: Наукова думка, 1983. 192 с.
  3. Kosevich A.M., Ivanov B.A., Kovalev A.S. Magnetic Solitons // Phys. Reports. 1990. V. 194. № 3–4. P. 117–238.
  4. Seidel J. Topological structures in ferroic materials: domain walls, vortices and skyrmions. Springer, Berlin. 2016. 249 p.
  5. Seki S. and Mochizuki M. Skyrmions in magnetic materials. Cham, Switzerland: Springer, 2016. 69 p.
  6. Jung Hoon Han. Skyrmions in condensed matter. Springer Tracts in Modern Physics. Springer. 2017. 278 p.
  7. Liu J.P., Zhang Z.D. and Zhao G.P. Skyrmions: Topological structures, properties, and applications. Boca Raton, London, New York: CRC Press, 2016. 481 p.
  8. Gupta S. and Saxena A. The role of topology in materials. Springer International Publishing AG, 2018. 307 p.
  9. Борисов А.Б., Киселев В.В. Двумерные и трехмерные топологические дефекты, солитоны и текстуры в магнетиках. М.: Физматлит, 2022. 455 с.
  10. Göbel B., Mertig I., Tretiakov O.A. Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles // Phys. Reports. 2021. V. 895. P. 1–28.
  11. Стишов С.М., Петрова А.Е. Геликоидальный зонный магнетик MnSi // УФН. 2011. Т. 181. № 12. С. 1157–1170.
  12. Изюмов Ю.А. Модулированные или длиннопериодические магнитные структуры кристаллов // УФН. 1984. Т. 144. № 3. С. 439–474.
  13. Fert A., Reyren N. and Cros V. Magnetic skyrmions: advances in physics and potential applications // Nat. Rev. Mater. 2017. V. 2. Article # 17031.
  14. Nagaosa N. and Tokura Y. Topological properties and dynamics of magnetic skyrmions // Nature Nanotech. 2013. V. 8. P. 899–911.
  15. Finocchio G., Büttner F., Tomasello R. et al. Magnetic skyrmions: from fundamental to applications // J. Phys. D: Applied Physics. 2016. V. 49. Article No. 423001.
  16. Bihlmayer G., Buhl P.M., Dupй B., Fernandes I.L., Freimuth F., Gayles J., Heinze S., Kiselev N.S., Lounis S., Mokrousov Yu., Blьgel S. Magnetic skyrmions: structure, stability, and transport phenomena // Scientific Highlight of the Month. 2018. No. 139, February.
  17. Звездин К.А., Екомасов Е.Г. Спиновые токи и нелинейная динамика вихревых спин–трансферных наноосцилляторов // ФММ. 2022. Т. 123. № 3. С. 219–239.
  18. Самардак А.С., Колесников А.Г., Давыденко А.В., Стеблий М.Е., Огнев А.В. Топологически нетривиальные спиновые текстуры в тонких магнитных пленках // ФММ. 2022. Т. 123. № 3. С. 260–283.
  19. Лаврентьев М.А., Шабат Б.В. Проблемы гидродинамики и их математические модели. М.: Наука, 1973. 416 с.
  20. Ламб Г. Гидродинамика. Т. 1, 2. М.: “ОГИЗ”, 1947. 929 с.
  21. Хирт Дж., Лоте И. Теория дислокаций. М.: Атомиздат, 1972. 599 с.
  22. Питаевский Л.П. Вихревые нити в неидеальном Бозе-газе // ЖЭТФ. 1961. Т. 40. С. 646–651.
  23. Gross E.P. Structure of a quantized vortex in boson systems // Il Nuovo Cimento. 1961. V. 20. № 3. P. 454–457.
  24. Абрикосов А.А. Сверхпроводники второго рода и вихревая решетка // УФН. 2004. Т. 174. № 11. С. 1234–1239.
  25. Ахиезер А.И., Барьяхтар В.Г., Пелетминский С.В. Спиновые волны. М.: Наука, 1967. 368 c.
  26. Derrick G.M. Comments on Nonlinear Wave Equations as Models for Elementary Particles // J. Math. Phys. 1964. V. 5. № 9. P. 1252–1254.
  27. Hobart R.H. On the Instability of a Class of Unitary Field Models // Proc. Phys. Soc. 1963. V. 82. № 2. P. 201–203.
  28. Борисов А.Б., Танкеев А.П., Шагалов А.Г. Новые типы двумерных вихреподобных состояний в магнетиках // ФТТ. 1989. Т. 31. № 5. С. 140–147.
  29. Khodenkov H.E. Nonstationary equations of motion for magnetic bubble domains // Phys. St. Sol. (a). 1981. V. 63. № 2. P. 461–473.
  30. Hudák O. On vortex configurations in two-dimensional sine-Gordon systems with applications to phase transitions of the Kosterlitz-Thouless type and to Josephson junctions // Phys. Lett. A. 1982. V. 89. № 5. P. 245–248.
  31. Сонин А.С. Введение в физику жидких кристаллов. М.: Наука, 1983. 320 с.
  32. Курик М.В., Лаврентович О.Д. Дефекты в жидких кристаллах: гомотопическая теория и экспериментальные исследования // УФН. 1988. Т. 154. № 3. С. 381–431.
  33. Березинский В.Л. Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии I. Классические системы // ЖЭТФ. 1970. V. 59. № 3. C. 907–920.
  34. Kosterlitz J.M., Thouless D.J. Ordering, metastability and phase transitions in two-dimensional systems // J. Phys. C: Solid State Phys. 1973. V. 6. P. 1181–1203.
  35. Белавин А.А., Поляков А.М. Метастабильные состояния двумерного изотропного ферромагнетика // Письма в ЖЭТФ. 1975. Т. 22. № 10. С. 503–506.
  36. Рожков С.С. Топология, многообразия и гомотопия: основные понятия и приложения к моделям π–поля // УФН. 1986. Т. 149. № 2. С. 259–273.
  37. Розенфельд Б.А., Сергеева Н.Д. Стереографическая проекция. Серия “Популярные лекции по математике”. Вып. 53. М.: Наука, 1973. 48 c.
  38. Gross D.J. Meron configurations in the two-dimensional O(3) -model // Nucl. Phys. B. 1978. V. 132. № 5. P. 439–456.
  39. Переломов А.М. Решения типа инстантонов в киральных моделях // УФН. 1981. Т. 134. № 4. С. 577–609.
  40. Perelomov A.M. Chiral models: geometrical aspects // Phys. Rep. 1987. V. 146. № 3. P. 135–213.
  41. Сергеев А.Г. Гармонические отображения. Лекц. курсы НОЦ, 10. М.: МИАН, 2008. С. 3–117.
  42. Борисов А.Б. Спиральные вихри в ферромагнетике // Письма в ЖЭТФ. 2001. Т. 73. № 5. С. 279–282.
  43. Борисов А.Б., Долгих Д.В. Вихревые полосы в двумерном ферромагнетике // ФММ. 2023. Т. 124. № 4. С. 375–381.
  44. Byrd P.F. and Friedman M.D. Handbook of Elliptic Integrals for Engineers and Scientists. Springer–Verlag, New–York, Heidelberg, Berlin. 1971. 373 p.
  45. Изменение области определения двухвихревой структуры. Видеофильм https://youtu.be/vgMpEnrZSIY.
  46. Двухвихревая структура. Видеофильм https://youtu.be/gh0IbYMpfIU.
  47. Борисов А.Б. Интегрирование двумерной модели Гейзенберга методами дифференциальной геометрии // ТМФ. 2023. Т. 216. № 2. С. 302–314.
  48. Gouva M.E., Wysin G.M., Bishop A.R., Mertens F.G. Vortices in the classical two-dimensional anisotropic Heisenberg model // Phys. Rev. B. 1989. V. 39. № 16. P. 11840–11849.
  49. Wysin G.M. Instability of in-plane vortices in two-dimensional easy-plane ferromagnets // Phys. Rev. B. 1994. V. 49. № 13. P. 8780–8789.
  50. Борисов А.Б., Зыков С.А., Микушина Н.А. Вихри и магнитные структуры типа "мишени" в двумерном ферромагнетике с анизотропным обменным взаимодействием // ФТТ. 2002. Т. 44. № 2. С. 313–320.
  51. Skyrme T.H.R. A non-linear field theory // Proc. R. Soc. Lond. A. V. 260. P. 127–138.
  52. Skyrme T.H.R. Particle states of a quantized meson field // Proc. R. Soc. Lond. A. 1961. V. 262. P. 237–245.
  53. Skyrme T.H.R. A unified field theory of mesons and baryons // Nucl. Phys. 1962. V. 31. P. 556–569.
  54. Sondhi S.L., Karlhede A., Kivelson S.A. and Rezayi E.H. Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies // Phys. Rev. B. 1993. V. 47. № 24. P. 16419–16426.
  55. Khawaja U.A., Stoof H. Skyrmions in a ferromagnetic Bose–Einstein condensate // Nature. 2001. V. 411. P. 918–920.
  56. Fukuda J., Žumer Sl. Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal // Nat. Commun. 2011. № 2. Article No. 246.
  57. Богданов А.Н., Яблонский Д.А. Термодинамические устойчивые “вихри” в магнитоупорядоченных кристаллах. Смешанное состояние магнетиков // ЖЭТФ. 1989. Т. 95. № 1. C. 178–182.
  58. Богданов А.Н., Кудинов М.В., Яблонский Д.А. К теории магнитных вихрей в легкоосных ферромагнетиках // ФТТ. 1989. Т. 31. С. 99–104.
  59. Ivanov B.A., Stephanovich V.A., Zhmudskii A.A. Magnetic vortices — The microscopic analogs of magnetic bubbles // J. Magn. Magn. Mater. 1990. V. 88. № 1–2. P. 116–120.
  60. Bogdanov A.N., Hubert A. Thermodynamically stable magnetic vortex states in magnetic crystals // JMMM. 1994. V. 138. № 3. P. 255–269.
  61. Богданов А.Н. Новые локализованные решения нелинейных полевых уравнений // Письма в ЖЭТФ. 1995. Т. 62. № 3. C. 231–235.
  62. Bogdanov A., Hubert A. Thermodynamically stable magnetic vortex states in magnetic crystals // J. Magn. Magn. Mater. 1994. V. 138. № 3. P. 255–269.
  63. Rößler U.K., Bogdanov A.N., Pfleiderer C. Spontaneous skyrmion ground states in magnetic metals // Nature. 2006. V. 442. P. 797–801.
  64. Bogdanov A., Hubert A. The stability of vortex-like structures in uniaxial ferromagnets // J. Magn. Magn. Mater. 1999. V. 195. № 1. P. 182–192.
  65. Muhlbauer S., Binz B., Jonietz F. et al. Skyrmion Lattice in a Chiral Magnet // Science. 2009. V. 323. P. 915–919.
  66. Yu X.Z., Onose Y., Kanazawa N. et al. Real–space observation of a two-dimensional skyrmion crystal // Nature 2010. V. 465. P. 901–904.
  67. Munzer W., Neubauer A., Adams T. et al. Skyrmion lattice in the doped semiconductor Fe1–xCoxSi // Phys. Rev. B. 2010. V. 81. Article No. 041203(R).
  68. Moriya T. Anisotropic Superexchange Interaction and Weak Ferromagnetism // Phys. Rev. 1960. V. 120. № 1. P. 91–98.
  69. Дзялошинский И.Е. Теория геликоидальных структур в антиферромагнетиках. III // ЖЭТФ. 1964. Т. 47. № 3. С. 992–1002.
  70. Bak P., Jensen M.H. Theory of helical magnetic structures and phase transitions in MnSi and FeGe // J. Phys. C: Solid State Phys. 1980. V. 13. № 31. P. L881–L885.
  71. Chizhikov V.A., Dmitrienko V.E. Frustrated magnetic helices in MnSi-type crystals // Phys. Rev. B. 2012. V. 85. № 1. Article No. 014421.
  72. Ishikawa Y., Tajima K., Bloch D., Roth M. Helical spin structure in manganese silicide MnSi // Solid State Commun. 1976. V. 19. № 6. P. 525–528.
  73. Maleyev S.V. Investigation of Spin Chirality by Polarized Neutrons // Phys. Rev. Lett. 1995. V. 75. № 25. P. 4682–4685.
  74. Grigoriev S.V., Maleyev S.V., Okorokov A.I., Chetverikov Yu.O. and Eckerlebe H. Field-induced reorientation of the spin helix in MnSi near T c// Phys. Rev. B. 2006. V. 73. № 22. Article No. 224440.
  75. Grigoriev S.V., Maleyev S.V., Dyadkin V.A., Menzel D., Schoenes J. and Eckerlebe H. Principal interactions in the magnetic system Fe1–xCoxSi: Magnetic structure and critical temperature by neutron diffraction and SQUID measurements // Phys. Rev. B. 2007. V. 76. № 9. Article No. 092407.
  76. Grigoriev S.V., Dyadkin V.A., Menzel D., Schoenes J., Chetverikov Yu.O., Okorokov A.I., Eckerlebe H. and Maleyev S.V. Magnetic structure of Fe1–xCoxSi in a magnetic field studied via small-angle polarized neutron diffraction // Phys. Rev. B. 2007. V. 76. № 22. Article No. 224424.
  77. Grigoriev S.V., Maleyev S.V., Okorokov A.I., Chetverikov Yu.O., Böni P., Georgii R., Lamago D., Eckerlebe H. and Pranzas K. Magnetic structure of MnSi under an applied field probed by polarized small-angle neutron scattering // Phys. Rev. B. 2006. V. 74. № 21. Article No. 214414.
  78. Grigoriev S.V., Chernyshov D., Dyadkin V.A., Dmitriev V., Maleyev S.V., Moskvin E.V., Menzel D., Schoenes J. and Eckerlebe H. Crystal Handedness and Spin Helix Chirality in Fe1–xCoxSi // Phys. Rev. Lett. 2009. V. 102. № 3. Article No. 037204.
  79. Uchida M., Onose Y., Matsui Y., Tokura Y. Real-space observation of helical spin order // Science. 2006. V. 311. P. 359–361.
  80. Siemens A., Rуzsa L., Vedmedenko E.Y. Controlled creation and stability of kπ–skyrmions on a discrete lattice // Phys. Rev. B. 2018. V. 97. № 17. Article No. 174436.
  81. Rybakov F.N., Kiselev Ni.S. Chiral magnetic skyrmions with arbitrary topological charge // Phys. Rev. B. 2019. V. 99. № 6. Article No. 064437.
  82. Видеофильмы “Skyrmionic sacks” https://www.youtube.com/@skyrmionicsacks4892.
  83. Uchida M., Nagaosa N., Tokura Y., Matsui Y. Topological spin textures in the helimagnet FeGe // Phys. Rev. B. 2008. V. 77. № 18. Article No. 184402.
  84. Борисов А.Б., Рыбаков Ф.Н. Спиральные структуры в геликоидальных магнетиках // Письма в ЖЭТФ. 2012. Т. 96. № 8. С. 572–575.
  85. Видеофильм “Archimedian spiral” https://www.youtube.com/watch?v=LVapNv850OA.
  86. Мамалуй Ю.А., Сирюк Ю.А. Устойчивые спиральные домены в пленках ферритов-гранатов // Изв. РАН. Сер. физическая. 2008. Т. 72. N 8. С. 1091–1093.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Fig. 1. Distribution of the normalized velocity of the drain (a), vortex (b), vortex source (c).

Download (7KB)
Indexing metadata
3. Fig. 2. (a) An edge dislocation is formed by the presence of an unfinished half-plane of atoms; (b) a screw dislocation is a complete shift of a section of the lattice.

Download (53KB)
Indexing metadata
4. Fig. 3. Schematic representation of a vortex in a type II superconductor. The vortex is parallel to the external magnetic field. The field lines outside the conductor and in the center of the vortex are indicated by straight arrows, and the eddy currents are indicated by closed circular arrows.

Download (4KB)
Indexing metadata
5. Fig. 4. Triangular lattice of vortices, if you look in the direction of the magnetic field. Each circle with an arrow symbolically depicts an eddy current, and the dot in the middle of the circle is the magnetic field line directed towards us. Every three adjacent vortices form a regular triangle.

Download (2KB)
Indexing metadata
6. Fig. 5. Distribution of the vector n in the plane for a plane vortex at Q = 1 (a, b, c) and Q = −1 (d).

Download (9KB)
Indexing metadata
7. Fig. 6. Distribution of the vector n in a two-vortex structure at Q1 = Q2 = 1.

Download (5KB)
Indexing metadata
8. Fig. 7. Stereographic projection of a sphere onto a plane.

Download (13KB)
Indexing metadata
9. Fig. 8. Distribution of magnetization in a two-dimensional vortex (intanton) with N = 1.

Download (17KB)
Indexing metadata
10. Fig. 9. The structure of the core (surface n3 = n3 (x, y)), corresponding to a single-turn spiral (N = 1, Q = 1, r0 = 1, k = 1/2). At the bottom are shown regions on the xOy plane with positive (white) and negative (black) values ​​of the magnetization component n3. The energy of the spiral vortex, as for the flat vortex, is proportional to InR / l.

Download (8KB)
Indexing metadata
11. Fig. 10. Structure of the magnetic “target” type.

Download (8KB)
Indexing metadata
12. Fig. 11. Graph of the function F2(θ) for A = −1, U = 1.

Download (1KB)
Indexing metadata
13. Fig. 12. Distribution of the vector field n in a circular strip at A = −1, U = 1.

Download (18KB)
Indexing metadata
14. Fig. 13. Domains of definition of a two-vortex structure at b = 0 (a), (b), (c).

Download (3KB)
Indexing metadata
15. Fig. 14. Two-vortex structure at b = 0 (a), (b).

Download (3KB)
Indexing metadata
16. Fig. 15. Graphs of the functions θ(r) for regular solutions. The solid line corresponds to a vortex in the absence of anisotropic exchange.

Download (1KB)
Indexing metadata
17. Fig. 16. Dependence of the parameter B(λ) on the anisotropic exchange constant.

Download (1KB)
Indexing metadata
18. Fig. 17. Crystal structure of the right (a) and left (b) modifications of MnSi.

Download (13KB)
Indexing metadata
19. Fig. 18. Schematic representation of various modulated states in chiral magnets. (a) Spin helix in the absence of a magnetic field with a wave vector k along the Oz axis; (b) the location of the helix in planes. Under the influence of a magnetic field, the helix (a) is transformed into a conical helix; (c) with inclined magnetization and a wave vector along the magnetic field or into a longitudinal helicoid (d).

Download (20KB)
Indexing metadata
20. Fig. 19. Topological spin textures: a — Neel-type skyrmion N = 1; b — Bloch-type skyrmion N = 1; c — anti-skyrmion N = −1.

Download (35KB)
Indexing metadata
21. Fig. 20. Topological spin textures in the Fe0.5Co0.5Si film [66]. Helical (a) and skyrmion (b) structures predicted by Monte Carlo simulation; (c) scheme of the skyrmion spin configuration. Experimentally observed in real space images of spin textures (d–f), according to TEM data: (d) helical structure in the absence of a magnetic field, (d) structure of a skyrmion crystal in the presence of a weak magnetic field (50 mT) directed normal to the plate, (e) enlarged view of an individual skyrmion. The color map and white arrows indicate the direction of magnetization at each point of the film.

Download (62KB)
Indexing metadata
22. Fig. 21. Effect of magnetic field and temperature on the change in spin texture in Fe0.5Co0.5Si: (a–g) — TEM images of the dependence of texture on magnetic field; (e–h) — fast Fourier analysis processing of TEM images (a–g); (i–m) — temperature dependence of spin texture in a magnetic field of 50 mT. The magnetic field is directed along the normal (001) to the film surface. The color wheel determines the direction of magnetization at each point.

Download (114KB)
Indexing metadata
23. Fig. 22. Experimental phase diagram of the magnetic structure in a thin Fe0.5Co0.5Si film in the variables “magnetic induction – temperature” (B – T).

Download (16KB)
Indexing metadata
24. Fig. 23. Plot of θ(r) for 1π- and 2π-skyrmions.

Download (3KB)
Indexing metadata
25. Fig. 24. Color map for Sz of the first kπ-skyrmions.

Download (20KB)
Indexing metadata
26. Fig. 25. Morphology of stable chiral skyrmions with topological charges N = −3, −2, ..., 2. The upper row of images (a) corresponds to zero magnetocrystalline anisotropy (u = 0) in an external magnetic field applied perpendicular to the plane, h = 0.65. The lower row of images (b) corresponds to the case of uniaxial anisotropy, u = 1.3 and zero external field, h = 0. The colors reflect the direction of n vectors according to the scheme: black and white denote spins up and down, respectively, and red-green-blue reflect the azimuthal angle relative to the Ox axis.

Download (35KB)
Indexing metadata
27. Fig. 26. Vortex-like stripe spin patterns.

Download (72KB)
Indexing metadata
28. Fig. 27. Calculated AS surrounded by different structures: a, b — AS in a labyrinth structure in the absence of a magnetic field; c — in a skyrmion lattice in the absence of a field; d — in a field h = 0.5.

Download (87KB)
Indexing metadata
29. Fig. 28. The structure of the RAS with five turns in the absence of a magnetic field (a) and three turns in a field h = 0.15 (b).

Download (24KB)
Indexing metadata