Advertisement

Topology and Magnetic Domain Walls

  • André ThiavilleEmail author
  • Jacques Miltat
Chapter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 192)

Abstract

This chapter presents the various connections between topology and magnetic domain walls. To begin with, we expose what topology tells us about magnetic domain walls, answering questions like: may domain walls be topological defects?, are there topologically different classes of domain walls? We then turn to dynamics, explaining the profound link between topology and the dynamics of magnetic textures, domain walls here. Experimental aspects are reviewed in the next sections, constructed according to material and sample types, with a special role played by bubble garnet films where a number of fundamental concepts were introduced. The authors try to provide a unified view of the vast literature on the subject, that spreads over four decades and different research thematics.

Notes

Acknowledgements

The authors would like to acknowledge Maurice Kléman, formerly at LPS, for being an inspiration and, by his pionneering works, a source for the content of this chapter. They thank Jean-Yves Chauleau for providing MFM images of domain walls in Ni-Fe nanostrips, borrowed from his Ph.D. thesis [85]. The critical reading of the manuscript by O. Fruchart is gratefully acknowledged.

References

  1. 1.
    O.V. Pylypovskyi, V.P. Kravchuk, D.D. Sheka, D. Makarov, O.G. Schmidt, Y. Gaididei, Phys. Rev. Lett. 114, 197204 (2015)ADSCrossRefGoogle Scholar
  2. 2.
    A. Hubert, R. Schäfer, Magnetic Domains (Springer, Berlin, 1998)Google Scholar
  3. 3.
    W.F. Brown Jr., Micromagnetics (Interscience Publishers, New York, 1963)zbMATHGoogle Scholar
  4. 4.
    G. Toulouse, M. Kléman, J. Phys. Lett. 37, L149 (1976)CrossRefGoogle Scholar
  5. 5.
    M. Kléman, Points, Lines and Walls (Wiley, Chichester, 1983)Google Scholar
  6. 6.
    N.D. Mermin, Rev. Mod. Phys. 51, 591 (1979)ADSCrossRefGoogle Scholar
  7. 7.
    A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002), http://www.math.cornell.edu/~hatcher
  8. 8.
    E. Feldtkeller, Z. angew. Phys. 19, 530 (1965). English translation: IEEE. Trans. Magn. 53, 0700308 (2017)Google Scholar
  9. 9.
    A.P. Malozemoff, J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979)Google Scholar
  10. 10.
    L. Michel, Rev. Mod. Phys. 52, 617 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    H.R. Trebin, Adv. Phys. 31, 195 (1982)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    V. Mineyev, G. Volovik, Phys. Rev. B 18, 3197 (1978)ADSCrossRefGoogle Scholar
  13. 13.
    H.J. Williams, M. Goertz, J. Appl. Phys. 23, 316 (1952)ADSCrossRefGoogle Scholar
  14. 14.
    R.W. DeBlois, C.D. Graham Jr., J. Appl. Phys. 29, 931 (1958)ADSCrossRefGoogle Scholar
  15. 15.
    E.E. Huber Jr., D. Smith, J.B. Goodenough, J. Appl. Phys. 29, 294 (1958)ADSCrossRefGoogle Scholar
  16. 16.
    L. Landau, E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935)Google Scholar
  17. 17.
    T.L. Gilbert, Phys. Rev. 100, 1243 (1955). (abstract only)Google Scholar
  18. 18.
    M.D. Stiles, J. Miltat, Spin Dynamics in Confined Magnetic Structures III (Springer, Berlin, 2006), pp. 225–308CrossRefGoogle Scholar
  19. 19.
    D.C. Ralph, M.D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    D.C. Ralph, M.D. Stiles, J. Magn. Magn. Mater. 321, 2508 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    A. Brataas, A.D. Kent, H. Ohno, Nat. Mater. 11, 372 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    P. Gambardella, I.M. Miron, Phil. Trans. R. Soc. A 369(13), 3175 (2011).  https://doi.org/10.1098/rsta.2010.0336ADSCrossRefGoogle Scholar
  23. 23.
    A. Thiaville, Y. Nakatani, J. Miltat, Y. Suzuki, Europhys. Lett. 69, 990 (2005)ADSCrossRefGoogle Scholar
  24. 24.
    N.L. Schryer, L.R. Walker, J. Appl. Phys. 45, 5406 (1974)ADSCrossRefGoogle Scholar
  25. 25.
    J.C. Slonczewski, Int. J. Magn. 2, 85 (1972)Google Scholar
  26. 26.
    M.J. Donahue, D.G. Porter, OOMMF user’s guide, version 1.0, Interagency report NISTIR 6376 (1999)Google Scholar
  27. 27.
    T. Fischbacher, M. Franchin, G. Bordignon, H. Fangohr, IEEE Trans. Magn. 43, 2896 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, B. Van Waeyenberge, AIP Adv. 4, 107133 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973)ADSCrossRefGoogle Scholar
  30. 30.
    A.A. Thiele, J. Appl. Phys. 45, 377 (1974)ADSCrossRefGoogle Scholar
  31. 31.
    L. Berger, Phys. Rev. B 33, 1572 (1986)ADSCrossRefGoogle Scholar
  32. 32.
    A. Thiaville, Y. Nakatani, Domain wall dynamics in nanowires and nanostrips, in Spin Dynamics in Confined Magnetic Structures III (Springer, Berlin, 2006), pp. 161–206Google Scholar
  33. 33.
    J.F. Dillon Jr., Magnetism, vol. III (Academic Press, New York, 1963), pp. 415–464Google Scholar
  34. 34.
    Y. Nakatani, A. Thiaville, J. Miltat, J. Magn. Magn. Mater. 290–291, 750 (2005)ADSCrossRefGoogle Scholar
  35. 35.
    E. Jué, A. Thiaville, S. Pizzini, J. Miltat, J. Sampaio, L.D. Buda-Prejbeanu, S. Rohart, J. Vogel, M. Bonfim, O. Boulle, S. Auffret, I.M. Miron, G. Gaudin, Phys. Rev. B 93, 014403 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    W. Döring, Z. Naturforschg. 3a, 373 (1948)Google Scholar
  37. 37.
    G.M. Wysin, F.G. Mertens, Equations of motion for vortices in 2-D easy-plane magnets, in Nonlinear Coherent Structures in Physics and Biology (Springer, Berlin, 1991)Google Scholar
  38. 38.
    G.M. Wysin, F.G. Mertens, A.R. Völkel, A.R. Bishop, Mass and momentum for vortices in two-dimensional easy-plane magnets, in Nonlinear Coherent Structures in Physics and Biology (Plenum, New York, 1994)CrossRefGoogle Scholar
  39. 39.
    F.G. Mertens, A.R. Bishop, Dynamics of vortices in two-dimensional magnets, in Nonlinear Science at the Dawn of the 21st Century (Springer, Berlin, 2000), pp. 137–170Google Scholar
  40. 40.
    W. Koshibae, N. Nagaosa, New J. Phys. 18, 045007 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    A.P. Malozemoff, J.C. DeLuca, Appl. Phys. Lett. 26, 719 (1975)ADSCrossRefGoogle Scholar
  42. 42.
    A.A. Thiele, J. Appl. Phys. 47, 2759 (1976)ADSCrossRefGoogle Scholar
  43. 43.
    J.C. Slonczewski, J. Magn. Magn. Mater. 12, 108 (1979)ADSCrossRefGoogle Scholar
  44. 44.
    S. Komineas, N. Papanicolaou, Phys. D 99, 81 (1996)MathSciNetCrossRefGoogle Scholar
  45. 45.
    A. Thiaville, Y. Nakatani, F. Piéchon, J. Miltat, T. Ono, Eur. Phys. J. B 60, 15 (2007)ADSCrossRefGoogle Scholar
  46. 46.
    V.S. Gornakov, L.M. Dedukh, V.I. Nikitenko, V.T. Synogach, Sov. Phys. JETP 63, 1225 (1986)Google Scholar
  47. 47.
    L.J. Schwee, H.R. Irons, W.E. Anderson, in AIP Conference Proceedings, vol. 18, p. 1383 (1974)Google Scholar
  48. 48.
    O. Bostanjoglo, T. Rosin, Phys. Stat. Sol. A 57, 561 (1980)Google Scholar
  49. 49.
    B.E. Argyle, B. Petek, M.E. Re, F. Suits, D.A. Herman, J. Appl. Phys. 64, 6595 (1988)ADSCrossRefGoogle Scholar
  50. 50.
    B.E. Argyle, E. Terrenzio, J.C. Slonczewski, Phys. Rev. Lett. 53, 190 (1984)ADSCrossRefGoogle Scholar
  51. 51.
    Y.P. Kabanov, L.M. Dedukh, V.I. Nikitenko, JETP Lett. 49, 637 (1989)ADSGoogle Scholar
  52. 52.
    V.S. Gornakov, V.I. Nikitenko, I.A. Prudnikov, JETP Lett. 50, 513 (1990)ADSGoogle Scholar
  53. 53.
    J.C. Slonczewski, J. Appl. Phys. 44, 1759 (1973)ADSCrossRefGoogle Scholar
  54. 54.
    J.C. Slonczewski, J. Appl. Phys. 45, 2705 (1974)ADSCrossRefGoogle Scholar
  55. 55.
    B.E. MacNeal, F.B. Humphrey, IEEE Trans. Magn. 15, 1272 (1979)ADSCrossRefGoogle Scholar
  56. 56.
    K. Patek, A. Thiaville, J. Miltat, Phys. Rev. B 49, 6678 (1994)ADSCrossRefGoogle Scholar
  57. 57.
    J.C. Slonczewski, A.P. Malozemoff, O. Voegeli, in AIP Conference Proceedings vol. 10, p. 458 (1973)Google Scholar
  58. 58.
    J.J. Fernandez de Castro, F.J. Friedlaender, D. Musselman, J.A. Nyenhuis, IEEE Trans. Magn. 23, 3388 (1987)ADSCrossRefGoogle Scholar
  59. 59.
    J.A. Nyenhuis, F.J. Friedlaender, H. Sato, IEEE Trans. Magn. 19, 1796 (1983)ADSCrossRefGoogle Scholar
  60. 60.
    C.A. Jones, M.V.C. Stroomer, O. Voegeli, F.J. Friedlaender, IEEE Trans. Magn. 15, 926 (1979)ADSCrossRefGoogle Scholar
  61. 61.
    J.C. Slonczewski, J. Appl. Phys. 50, 7850 (1979)ADSCrossRefGoogle Scholar
  62. 62.
    A.P. Malozemoff, J.C. Slonczewski, IEEE Trans. Magn. 11, 1091 (1975)ADSCrossRefGoogle Scholar
  63. 63.
    B.E. Argyle, S. Maekawa, P. Dekker, J.C. Slonczewski, in AIP Conference Proceedings, vol. 34, p. 131 (1976)Google Scholar
  64. 64.
    A.P. Malozemoff, S. Maekawa, J. Appl. Phys. 47, 3321 (1976)ADSCrossRefGoogle Scholar
  65. 65.
    S. Maekawa, P. Dekker, in AIP Conference Proceedings, vol. 34, p. 148 (1976)Google Scholar
  66. 66.
    F.B. Hagedorn, J. Appl. Phys. 45, 3129 (1974)ADSCrossRefGoogle Scholar
  67. 67.
    A. Thiaville, L. Arnaud, F. Boileau, G. Sauron, J. Miltat, IEEE Trans. Magn. 24, 1722 (1988)ADSCrossRefGoogle Scholar
  68. 68.
    A. Thiaville, J. Ben Youssef, Y. Nakatani, J. Miltat, J. Appl. Phys. 69, 6090 (1991)ADSCrossRefGoogle Scholar
  69. 69.
    S. Konishi, K. Matsuyama, I. Chida, S. Kubota, H. Kawahara, M. Ohbo, IEEE Trans. Magn. 20, 1129 (1984)ADSCrossRefGoogle Scholar
  70. 70.
    K. Mizuno, H. Matsutera, H. Kawahara, Y. Hidaka, IEEE Trans. Magn. 26, 2520 (1990)ADSCrossRefGoogle Scholar
  71. 71.
    M.V. Chetkin, V.B. Smirnov, A.F. Novikov, I.V. Parygina, A.K. Zvezdin, S.V. Gomonov, Sov. Phys. JETP 67, 2269 (1988)Google Scholar
  72. 72.
    A. Thiaville, J. Miltat, J. Magn. Magn. Mater. 104–107, 335 (1992)ADSCrossRefGoogle Scholar
  73. 73.
    A. Thiaville, J. Miltat, Europhys. Lett. 26, 1006 (1994)CrossRefGoogle Scholar
  74. 74.
    Y. Maruyama, T. Ikeda, R. Suzuki, IEEE Trans. J. Magn. Jpn. 4, 730 (1989)CrossRefGoogle Scholar
  75. 75.
    W.J. Tabor, A.H. Bobeck, G.P. Vella-Coleiro, A. Rosencwaig, Bell Syst. Tech. J. 51, 1427 (1972)CrossRefGoogle Scholar
  76. 76.
    A. Rosencwaig, W.J. Tabor, T.J. Nelson, Phys. Rev. Lett. 29, 946 (1972)ADSCrossRefGoogle Scholar
  77. 77.
    S. Konishi, IEEE Trans. Magn. 19, 1838 (1983)ADSCrossRefGoogle Scholar
  78. 78.
    J. Haisma, K.L.L. van Mierloo, W.F. Druyvesteyn, U. Enz, Appl. Phys. Lett. 27, 459 (1975)ADSCrossRefGoogle Scholar
  79. 79.
    C.Y. You, I.M. Sung, B.K. Joe, Appl. Phys. Lett. 89, 222513 (2006)ADSCrossRefGoogle Scholar
  80. 80.
    D. Navas, C. Nam, D. Velazquez, C.A. Ross, Phys. Rev. B 81, 224439 (2010)ADSCrossRefGoogle Scholar
  81. 81.
    J.Y. Chauleau, B.J. McMorran, R. Belkhou, N. Bergeard, T.O. Mentes, M.A. Niño, A. Locatelli, J. Unguris, S. Rohart, J. Miltat, A. Thiaville, Phys. Rev. B 84, 094416 (2011)ADSCrossRefGoogle Scholar
  82. 82.
    H. Kronmüller, M. Fähnle, Micromagnetization and the Microstructure of Ferromagnetic Solids (Cambridge University Press, Cambridge, 2003)Google Scholar
  83. 83.
    R.D. McMichael, M.J. Donahue, IEEE Trans. Magn. 33, 4167 (1997)ADSCrossRefGoogle Scholar
  84. 84.
    J.Y. Chauleau, R. Weil, A. Thiaville, J. Miltat, Phys. Rev. B 82, 214414 (2010)ADSCrossRefGoogle Scholar
  85. 85.
    J.Y. Chauleau, Textures magnétiques et parois de domaines: les objets du transfert de spin. Ph.D. thesis, Université Paris-Sud, 2010Google Scholar
  86. 86.
    O. Tchernyshyov, G.W. Chern, Phys. Rev. Lett. 95, 197204 (2005)ADSCrossRefGoogle Scholar
  87. 87.
    G.W. Chern, H. Youk, O. Tchernyshyov, J. Appl. Phys. 99, 08Q505 (2006)CrossRefGoogle Scholar
  88. 88.
    D.J. Clarke, O.A. Tretiakov, G.W. Chern, Y.B. Bazaliy, O. Tchernyshyov, Phys. Rev. B 78, 134412 (2008)ADSCrossRefGoogle Scholar
  89. 89.
    H.A.M. van den Berg, IBM, J. Res. Develop. 33, 540 (1989)Google Scholar
  90. 90.
    T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, T. Shinjo, Science 284, 468 (1999)ADSCrossRefGoogle Scholar
  91. 91.
    D. Atkinson, D.A. Allwood, G. Xiong, M.D. Cooke, C.C. Faulkner, R.P. Cowburn, Nature Mater. 2, 85 (2003)ADSCrossRefGoogle Scholar
  92. 92.
    J.Y. Lee, K.S. Lee, S. Choi, K.Y. Guslienko, S.K. Kim, Phys. Rev. B 76, 184408 (2007)ADSCrossRefGoogle Scholar
  93. 93.
    G.D. Beach, C. Nistor, C. Knutson, M. Tsoi, J.L. Erskine, Nat. Mater. 4, 741 (2005)ADSCrossRefGoogle Scholar
  94. 94.
    J. Yang, C. Nistor, G.S.D. Beach, J.L. Erskine, Phys. Rev. B 77, 014413 (2008)ADSCrossRefGoogle Scholar
  95. 95.
    M. Hayashi, L. Thomas, C. Rettner, R. Moriya, S.S.P. Parkin, Nat. Phys. 3, 21 (2007)CrossRefGoogle Scholar
  96. 96.
    S. Glathe, R. Mattheis, D.V. Berkov, Appl. Phys. Lett. 93, 072508 (2008)ADSCrossRefGoogle Scholar
  97. 97.
    Y. Nakatani, A. Thiaville, J. Miltat, Nat. Mater. 2, 521 (2003)ADSCrossRefGoogle Scholar
  98. 98.
    L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, S. Parkin, Nature 443, 197 (2006)ADSCrossRefGoogle Scholar
  99. 99.
    S. Zhang, Z. Li, Phys. Rev. Lett. 93, 127204 (2004)ADSCrossRefGoogle Scholar
  100. 100.
    L. Heyne, M. Kläui, D. Backes, T.A. Moore, S. Krzyk, U. Rüdiger, L.J. Heyderman, A.F. Rodríguez, F. Nolting, T.O. Mentes, M. Niño, A. Locatelli, K. Kirsch, R. Mattheis, Phys. Rev. Lett. 100, 066603 (2008)ADSCrossRefGoogle Scholar
  101. 101.
    M. Kläui, P.O. Jubert, R. Allenspach, A. Bischof, J.A.C. Bland, G. Faini, U. Rüdiger, C.A.F. Vaz, L. Vila, C. Vouille, Phys. Rev. Lett. 95, 026601 (2005)ADSCrossRefGoogle Scholar
  102. 102.
    M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang, S.S.P. Parkin, Phys. Rev. Lett. 97, 207205 (2006)ADSCrossRefGoogle Scholar
  103. 103.
    N. Vernier, D.A. Allwood, D. Atkinson, M.D. Cooke, R.P. Cowburn, Europhys. Lett. 65, 526 (2004)ADSCrossRefGoogle Scholar
  104. 104.
    S.S.P. Parkin, M. Hayashi, L. Thomas, Science 320, 190 (2008)ADSCrossRefGoogle Scholar
  105. 105.
    G.E. Volovik, J. Phys. C Solid State Phys. 20, L83 (1987)ADSCrossRefGoogle Scholar
  106. 106.
    S.E. Barnes, S. Maekawa, Concepts in Spin Electronics (Oxford University Press, Oxford, 2006)zbMATHGoogle Scholar
  107. 107.
    S.A. Yang, G.S.D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi, J.L. Erskine, Phys. Rev. Lett. 102, 067201 (2009)ADSCrossRefGoogle Scholar
  108. 108.
    S.S.L. Zhang, S. Zhang, Phys. Rev. B 82, 184423 (2010)ADSCrossRefGoogle Scholar
  109. 109.
    S. Jamet, N. Rougemaille, J.C. Toussaint, O. Fruchart, Magnetic Nano- and Microwires, chap. 25 (Elsevier, Amsterdam, 2015), pp. 783–811CrossRefGoogle Scholar
  110. 110.
    H. Forster, T. Schrefl, W. Scholz, D. Suess, V. Tsiantos, J. Fidler, J. Magn. Magn. Mater. 249, 181 (2002)ADSCrossRefGoogle Scholar
  111. 111.
    R. Hertel, J. Magn. Magn. Mater. 249, 251 (2002)ADSCrossRefGoogle Scholar
  112. 112.
    A. Hubert, J. Magn. Magn. Mater. 2, 25 (1976)ADSCrossRefGoogle Scholar
  113. 113.
    S. Da Col, S. Jamet, N. Rougemaille, A. Locatelli, T.O. Mentes, B. Santos Burgos, R. Afid, M. Darques, L. Cagnon, J.C. Toussaint, O. Fruchart, Phys. Rev. B 89, 180405(R) (2014)CrossRefGoogle Scholar
  114. 114.
    R. Hertel, J. Kirschner, Phys. B 343, 206 (2004)ADSCrossRefGoogle Scholar
  115. 115.
    R. Hertel, J. Kirschner, J. Magn. Magn. Mater. 278, L291 (2004)ADSCrossRefGoogle Scholar
  116. 116.
    C. Andreas, A. Kákay, R. Hertel, Phys. Rev. B 89, 134403 (2014)ADSCrossRefGoogle Scholar
  117. 117.
    R. Hertel, J. Phys. Condens. Matter 28, 483002 (2016)CrossRefGoogle Scholar
  118. 118.
    P. Landeros, A.S. Núñez, J. Appl. Phys. 108, 033917 (2010)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Physique des SolidesUniversité Paris-SudOrsayFrance

Personalised recommendations