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Solitons in Real Space: Domain Walls, Vortices, Hedgehogs, and Skyrmions

  • Hans-Benjamin BraunEmail author
Chapter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 192)

Abstract

Recent years have seen tremendous progress in the understanding of topological phenomena in magnetism, in particular at the nanoscale. In this overview, we consider smooth topological textures such as smooth domain walls, meron or vortices, and most importantly skyrmions. These structures derive their topological stability from the fact that they cannot be undone without violating the continuity of the magnetization field, similar to a knot in a rope. Owing to their topological stability, domain walls and skyrmions are prominent candidates in racetrack-type memories introduced by Parkin and co-workers. These smooth textures should be contrasted with singular topological point defects where the magnetization field is forced to vanish in a submanifold. Such point defects include Ising domain walls, vortices of easy-plane spins, and 3D Bloch points, ‘hedgehogs’, or ‘monopoles’. As domain walls, vortices, and skyrmions including their dynamical versions will be discussed in detail in later chapters by Thiaville and Miltat, Behncke and Meier, Chen, Bauer et al., and Åkerman, we give analytical arguments how domain walls emerge in quasi 1D nanowires, how magnetization reverses via nucleation, and why skyrmions exist in thin films. A variational ansatz for skyrmions that is derived from an exact \(2\pi \) domain wall profile provides an excellent approximation to numerical and experimental observations in films that include Dzyaloshinskii-Moriya interaction (DMI) and dipolar interactions. In systems of vanishing DMI, the two helical states of a skyrmion are degenerate, and switching between the two helicities occurs in a topologically allowed fashion. This mechanism is closely related to domain wall nucleation in nanowires. Finally we show that dynamical skyrmions may be regarded as 2D siblings of domain wall breathers, and can be described by the same variational ansatz inspired from \(2\pi \) domain walls as static skyrmions in thin films.

Notes

Acknowledgements

I gratefully acknowledge numerous helpful discussions with J. Åkerman, P. Böni, R.V. Hügli, B. Roessli, and Y. Zhou. This research has been supported by Science Foundation Ireland under 11/PI/1048.

References

  1. 1.
    F. Bloch, Zeit. f. Phys. 61, 206–219 (1930)ADSCrossRefGoogle Scholar
  2. 2.
    D. Landau, E. Lifshitz, Phys. Zeit. Sowj. 61, 153–169 (1935)Google Scholar
  3. 3.
    T.H.R. Skyrme, Proc. R. Soc. Lond. Ser. A 260, 127–138 (1961)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    T.H.R. Skyrme, Nucl. Phys. 31, 556–569 (1962)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. Böni, Science 323, 915–919 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    U.K. Rössler, A.N. Bogdanov, C. Pfleiderer, Nature 442, 797–801 (2006)ADSCrossRefGoogle Scholar
  7. 7.
    B. Roessli, P. Böni, W.E. Fischer, Y. Endoh, Phys. Rev. Lett. 88, 237204 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    A.N. Bogdanov, D.A. Yablonsky, Sov. Phys. JETP 95, 178–182 (1989)Google Scholar
  9. 9.
    W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. Jungfleisch, F. Fradin, J. Pearson, Y. Tserkovnyak, K. Wang, O. Heinonen, S. te Velthuis, A. Hoffmann, Science 349, 283–286 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    N. Nagaosa, Y. Tokura, Nat. Nanotechnol. 8, 899–911 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    H.B. Braun, Adv. Phys. 61, 1–116 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    S.-H. Yang, K.-S. Ryu, S. Parkin, Nat. Nanotechnol. 10, 221–226 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, C.A.F. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhuter, J.M. George, M. Weigand, J. Raabe, V. Cros, A. Fert, Nat. Nanotechnol. 11, 444–449 (2016)ADSCrossRefGoogle Scholar
  14. 14.
    S.S.P. Parkin, M. Hayashi, L. Thomas, Science 320, 190–194 (2008)ADSCrossRefGoogle Scholar
  15. 15.
    H.B. Braun, Phys. Rev. Lett. 71, 3557–3560 (1993)ADSCrossRefGoogle Scholar
  16. 16.
    A. Aharoni, J. Appl. Phys. 80, 3133–3134 (1996)ADSCrossRefGoogle Scholar
  17. 17.
    H.J. Richter, J. Phys. D 40, R149–R177 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    A. Malozemoff, J. Slonczewski, Magnetic Domain Walls in Bubble Materials. Advances in Materials and Device Research (Academic Press Inc, 1979)Google Scholar
  19. 19.
    H.B. Braun, Phys. Rev. B 50, 16485 (1994)ADSCrossRefGoogle Scholar
  20. 20.
    N. Romming, A. Kubetzka, C. Hanneken, K. von Bergmann, R. Wiesendanger, Phys. Rev. Lett. 114, 177203 (2015)ADSCrossRefGoogle Scholar
  21. 21.
    J. Sampaio, V. Cros, S. Rohart, A. Thiaville, A. Fert, Nat. Nanotechnol. 8, 839–844 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    S. Woo, K. Litzius, B. Krueger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R.M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Kläui, G.R.S.D. Beach, Nat. Mat. 15, 501–506 (2016)CrossRefGoogle Scholar
  23. 23.
    W.F. Brown, Micromagnetics (Interscience, 1963)Google Scholar
  24. 24.
    M. Nakahara, Geometry, Topology and Physics. Graduate Student Series in Physics (Hilger, Bristol, 1990)Google Scholar
  25. 25.
    N. Mermin, Rev. Mod. Phys. 51, 591–648 (1979)ADSCrossRefGoogle Scholar
  26. 26.
    T. Eggebrecht, M. Möller, J.G. Gatzmann, N. Rubiano da Silva, A. Feist, U. Martens, H. Ulrichs, M. Münzenberg, C. Ropers, S. Schäfer, Phys. Rev. Lett. 118, 097203 (2017)ADSCrossRefGoogle Scholar
  27. 27.
    A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, R. Wiesendanger, Science 298, 577–580 (2002)ADSCrossRefGoogle Scholar
  28. 28.
    T. Kamionka, M. Martens, K.W. Chou, M. Curcic, A. Drews, G. Schütz, T. Tyliszczak, H. Stoll, B. Van Waeyenberge, G. Meier, Phys. Rev. Lett. 105, 137204 (2010)ADSCrossRefGoogle Scholar
  29. 29.
    G. Chen, A.T. N’Diaye, S.P. Kang, H.Y. Kwon, C. Won, Y. Wu, Z.Q. Qiu, A.K. Schmid, Nat. Commun. 6, 6598 (2015)ADSCrossRefGoogle Scholar
  30. 30.
    G. Toulouse, M. Kleman, J. De Phys. Lett. 37, L149–L151 (1976)CrossRefGoogle Scholar
  31. 31.
    V.P. Mineev, Topologically stable defects and solitons in ordered media, in Classic Reviews in Physics, vol. 1 (Harwood Academic Publishers, 1998)Google Scholar
  32. 32.
    J. Rubinstein, J. Math. Phys. 11, 258–267 (1970)ADSCrossRefGoogle Scholar
  33. 33.
    D. Finkelstein, C. Misner, Ann. Phys. 6, 230–243 (1959)ADSCrossRefGoogle Scholar
  34. 34.
    U. Enz, Helv. Phys. Acta 37, 245–253 (1964)Google Scholar
  35. 35.
    R. Rajaraman, Solitons and Instantons an Introduction to Solitons and Instantons in Quantum Field Theory (North-Holland, Amsterdam a.o, 1982)zbMATHGoogle Scholar
  36. 36.
    H.B. Braun, J. Kulda, B. Roessli, D. Visser, K.W. Krämer, H.U. Güdel, P. Böni, Nat. Phys. 1, 159–163 (2005)CrossRefGoogle Scholar
  37. 37.
    M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blügel, R. Wiesendanger, Nature 447, 190–193 (2007)ADSCrossRefGoogle Scholar
  38. 38.
    N. Grisewood, J. Eves, T. Usher, H.B. Braun, J. Appl. Phys. 111, 07c706 (2012)CrossRefGoogle Scholar
  39. 39.
    X.Z. Yu, Y. Onose, N. Kanazawa, J.H. Park, J.H. Han, Y. Matsui, N. Nagaosa, Y. Tokura, Nature 465, 901–904 (2010)ADSCrossRefGoogle Scholar
  40. 40.
    O. Boulle, J. Vogel, H.X. Yang, S. Pizzini, D.D. Chaves, A. Locatelli, T.O. Mentes, A. Sala, L.D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussigne, A. Stashkevich, S.M. Cherif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I.M. Miron, G. Gaudin, Nat. Nanotechnol. 11, 449–455 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    A.A. Belavin, A.M. Polyakov, JETP Lett. 22, 245 (1975)ADSGoogle Scholar
  42. 42.
    O. Chubykalo-Fesenko, U. Nowak, R.W. Chantrell, D. Garanin, Phys. Rev. B 74, 094436 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    A. Thiaville, Y. Nakatani, J. Miltat, Y. Suzuki, Europhys. Lett. 69, 990 (2005)ADSCrossRefGoogle Scholar
  44. 44.
    M.E. Schabes, H.N. Bertram, J. Appl. Phys. 64, 1347–1357 (1988)ADSCrossRefGoogle Scholar
  45. 45.
    R.P. Cowburn, A.O. Adeyeye, M.E. Welland, Phys. Rev. Lett. 81, 5414–5417 (1998)ADSCrossRefGoogle Scholar
  46. 46.
    C. Stamm, F. Marty, A. Vaterlaus, V. Weich, S. Egger, U. Maier, U. Ramsperger, H. Fuhrmann, D. Pescia, Science 282, 449–451 (1998)ADSCrossRefGoogle Scholar
  47. 47.
    M. Hehn, K. Ounadjela, J.P. Bucher, F. Rousseaux, D. Decanini, B. Bartenlian, C. Chappert, Science 272, 1782–1785 (1996)ADSCrossRefGoogle Scholar
  48. 48.
    U. Ebels, A. Radulescu, Y. Henry, L. Piraux, K. Ounadjela, Phys. Rev. Lett. 84, 983–986 (2000)ADSCrossRefGoogle Scholar
  49. 49.
    E.Y. Vedmedenko, A. Kubetzka, K. von Bergmann, O. Pietzsch, M. Bode, J. Kirschner, H.P. Oepen, R. Wiesendanger, Phys. Rev. Lett. 92, 077207 (2004)Google Scholar
  50. 50.
    G. Woltersdorf, C.H. Back, Phys. Rev. Lett. 99, 227207 (2007)ADSCrossRefGoogle Scholar
  51. 51.
    S. Parkin, U.S. Patents 6834005, 6898132, 6920062, 7031178Google Scholar
  52. 52.
    M. Hayashi, L. Thomas, R. Moriya, C. Rettner, S.S.P. Parkin, Science 320, 209–211 (2008)ADSCrossRefGoogle Scholar
  53. 53.
    L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, S.S.P. Parkin, Nature 443, 197–200 (2006)ADSCrossRefGoogle Scholar
  54. 54.
    J.L. Eves, N. Grisewood, R.V. Hügli, H.B. Braun, J. Magn. Magn. Mater. 322, 1381–1384 (2010)ADSCrossRefGoogle Scholar
  55. 55.
    H.B. Braun, O. Brodbeck, Phys. Rev. Lett. 70, 3335–3338 (1993)ADSCrossRefGoogle Scholar
  56. 56.
    H.B. Braun, J. Appl. Phys. 85, 6172–6174 (1999)ADSCrossRefGoogle Scholar
  57. 57.
    G.D. Chaves-O’Flynn, A.D. Kent, D.L. Stein, Phys. Rev. B 79, 184421 (2009)ADSCrossRefGoogle Scholar
  58. 58.
    R. Kohn, V. Slastikov, Arch. Ration. Mech. Anal. 178 (2005)Google Scholar
  59. 59.
    H.B. Braun, J. Appl. Phys. 76, 6310–6315 (1994)ADSCrossRefGoogle Scholar
  60. 60.
    A. Kubetzka, O. Pietzsch, M. Bode, R. Wiesendanger, Phys. Rev. B 67, 020401 (2003)ADSCrossRefGoogle Scholar
  61. 61.
    H.B. Braun, D. Loss, Europhys. Lett. 31, 555 (1995)ADSCrossRefGoogle Scholar
  62. 62.
    H.B. Braun, D. Loss, Phys. Rev. B 53, 3237–3255 (1996)ADSCrossRefGoogle Scholar
  63. 63.
    Y. Yoshimura, K.-J. Kim, T. Taniguchi, T. Tono, K. Ueda, R. Hiramatsu, T. Moriyama, K. Yamada, Y. Nakatani, T. Ono, Nat. Phys. 12, 157–161 (2016)CrossRefGoogle Scholar
  64. 64.
    E.R. Lewis, D. Petit, A.V. Jausovec, L. O’Brien, D.E. Read, H.T. Zeng, R.P. Cowburn, Phys. Rev. Lett. 102, 057209 (2009)ADSCrossRefGoogle Scholar
  65. 65.
    A. Cavallin, F.D. Natterer, S. Ouazi, G. Moulas, A. Lehnert, S. Rusponi, H. Brune, Phys. Rev. B 90, 144427 (2014)ADSCrossRefGoogle Scholar
  66. 66.
    P. Milde, D. Koehler, J. Seidel, L.M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. Mühlbauer, C. Pfleiderer, S. Buhrandt, C. Schuette, A. Rosch, Science 340, 1076–1080 (2013)ADSCrossRefGoogle Scholar
  67. 67.
    S.M. Mohseni, S.R. Sani, J. Persson, T.N.A. Nguyen, S. Chung, Y. Pogoryelov, P.K. Muduli, E. Iacocca, A. Eklund, R.K. Dumas, S. Bonetti, A. Deac, M.A. Hoefer, J. Akerman, Science 339, 1295–1298 (2013)ADSCrossRefGoogle Scholar
  68. 68.
    M. Nagao, Y.-G. So, H. Yoshida, M. Isobe, T. Hara, K. Ishizuka, K. Kimoto, Nat. Nanotechnol. 8, 325–328 (2013)ADSCrossRefGoogle Scholar
  69. 69.
    X.Z. Yu, K. Shibata, W. Koshibae, Y. Tokunaga, Y. Kaneko, T. Nagai, K. Kimoto, Y. Taguchi, N. Nagaosa, Y. Tokura, Phys. Rev. B 93, 134417 (2016)ADSCrossRefGoogle Scholar
  70. 70.
    Y. Zhou, E. Iacocca, A.A. Awad, R.K. Dumas, F.C. Zhang, H.B. Braun, J. Åkerman, Nat. Commun. 6, 8193 (2015)ADSCrossRefGoogle Scholar
  71. 71.
    N. Romming, C. Hanneken, M. Menzel, J.E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, R. Wiesendanger, Science 341, 636–639 (2013)ADSCrossRefGoogle Scholar
  72. 72.
    T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. Pfleiderer, D. Grundler, Nat. Mat. 14, 478–483 (2015)CrossRefGoogle Scholar
  73. 73.
    S.A. Montoya, S. Couture, J.J. Chess, J.C.T. Lee, N. Kent, M.-Y. Im, S.D. Kevan, P. Fischer, B.J. McMorran, S. Roy, V. Lomakin, E.E. Fullerton, Phys. Rev. B 95, 224405 (2017)ADSCrossRefGoogle Scholar
  74. 74.
    F. Ma, Y. Zhou, H.B. Braun, W.S. Lew, Nano Lett. 15, 4029–4036 (2015)ADSCrossRefGoogle Scholar
  75. 75.
    Z.K. Wang, V.L. Zhang, H.S. Lim, S.C. Ng, M.H. Kuok, S. Jain, A.O. Adeyeye, Appl. Phys. Lett. 94, 083112 (2009)ADSCrossRefGoogle Scholar
  76. 76.
    A.D. Karenowska, J.F. Gregg, V.S. Tiberkevich, A.N. Slavin, A.V. Chumak, A.A. Serga, B. Hillebrands, Phys. Rev. Lett. 108, 015505 (2012)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of PhysicsUniversity College DublinDublinIreland
  2. 2.School of Theoretical PhysicsDublin Institute of Advanced StudiesDublinIreland

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