Skyrmion Lattices Far from Equilibrium

  • Andreas Bauer
  • Alfonso Chacon
  • Marco Halder
  • Christian PfleidererEmail author
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 192)


Magnetic skyrmions are spin whirls with non-trivial topology that are remarkably robust. We review current research on skyrmion lattices far from equilibrium in cubic chiral magnets obtained under field cooling, providing access to generic properties of skyrmions in the non-thermal limit as well as concepts of spintronics applications.


  1. 1.
    T.H.R. Skyrme, A non-linear field theory. Proc. R. Soc. Lond. A 260, 127 (1961). Scholar
  2. 2.
    T.H.R. Skyrme, A unified field theory of mesons and baryons. Nucl. Phys. 31, 556 (1962). Scholar
  3. 3.
    G.S. Adkins, C.R. Nappi, E. Witten, Static properties of nucleons in the Skyrme model. Nucl. Phys. B 228, 552 (1983). Scholar
  4. 4.
    S.L. Sondhi, A. Karlhede, S.A. Kivelson, E.H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies. Phys. Rev. B 47, 16419 (1993). Scholar
  5. 5.
    T.-L. Ho, Spinor Bose condensates in optical traps. Phys. Rev. Lett. 81, 742 (1998). Scholar
  6. 6.
    U.A. Khawaja, H. Stoof, Skyrmions in a ferromagnetic Bose–Einstein condensate. Nature 411, 918 (2001). Scholar
  7. 7.
    J. Fukuda, S. Žumer, Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal. Nat. Commun. 2, 246 (2011).
  8. 8.
    S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.  Rosch, A. Neubauer, R. Georgii, P. Böni, Skyrmion lattice in a chiral magnet. Science 323, 915 (2009). Scholar
  9. 9.
    X.Z. Yu, Y. Onose, N. Kanazawa, J.H. Park, J.H. Han, Y. Matsui, N. Nagaosa, Y. Tokura, Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901 (2010). Scholar
  10. 10.
    S. Heinze, K.V. Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, S. Blügel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions. Nat. Phys. 7, 713 (2011). Scholar
  11. 11.
    M. Finazzi, M. Savoini, A.R. Khorsand, A. Tsukamoto, A. Itoh, L. Duò, A. Kirilyuk, T. Rasing, M. Ezawa, Laser-induced magnetic nanostructures with tunable topological properties. Phys. Rev. Lett. 110, 177205 (2013).
  12. 12.
    N. Romming, C. Hanneken, M. Menzel, J.E. Bickel, B. Wolter, K.V. Bergmann, A. Kubetzka, R. Wiesendanger, Writing and deleting single magnetic skyrmions. Science 341, 636 (2013). Scholar
  13. 13.
    J. Sampaio, V. Cros, S. Rohart, A. Thiaville, A. Fert, Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. Nat. Nano. 8, 839 (2013). Scholar
  14. 14.
    F. Büttner, C. Moutafis, M. Schneider, B. Krüger, C.M. Günther, J. Geilhufe, C.V. Korff Schmising, J. Mohanty, B. Pfau, S. Schaffert, A. Bisig, M. Foerster, T. Schulz, C.A.F. Vaz, J.H. Franken, H.J.M. Swagten, M. Kläui, S. Eisebitt, Dynamics and inertia of skyrmionic spin structures. Nat. Phys. 11, 225 (2015). Scholar
  15. 15.
    W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M.B. Jungfleisch, F.Y. Fradin, J.E. Pearson, Y. Tserkovnyak, K.L. Wang, O. Heinonen, S.G.E. te Velthuis, A. Hoffmann, Blowing magnetic skyrmion bubbles. Science 349, 283 (2015). Scholar
  16. 16.
    I. Kézsmárki, S. Bordács, P. Milde, E. Neuber, L.M. Eng, J.S. White, H.M. Rønnow, C.D. Dewhurst, M. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, A. Loidl, Néel-type skyrmion lattice with confined orientation in the polar magnetic semiconductor GaV\(_{4}\)S\(_{8}\). Nat. Mater. 14, 1116 (2015). Scholar
  17. 17.
    Y. Zhou, E. Iacocca, A.A. Awad, R.K. Dumas, F.C. Zhang, H.B. Braun, J. Åkerman, Dynamically stabilized magnetic skyrmions. Nat. Commun. 6, 8193 (2015).
  18. 18.
    O. Boulle, J. Vogel, H. Yang, S.  Pizzini, D.  de Souza Chaves, A.  Locatelli, T.O. Menteş, L.D. B.-P. A. Sala, O. Klein, M. Belmeguenai, Y. Roussign, A. Stashkevich, S.M. Chérif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I.M. Miron, G. Gaudin, Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures. Nat. Nano. 11, 449 (2016). Scholar
  19. 19.
    C. Moreau-Luchaire, C. Moutas, N. Reyren, J. Sampaio, C.A.F. Vaz, N.V. Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M. George, M. Weigand, J. Raabe, V. Cros, A. Fert, Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature. Nat. Nano. 11, 444 (2016). Scholar
  20. 20.
    P. Milde, D. Köhler, J. Seidel, L.M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. Mühlbauer, C. Pfleiderer, S. Buhrandt, C. Schütte, A. Rosch, Unwinding of a skyrmion lattice by magnetic monopoles. Science 340, 1076 (2013). Scholar
  21. 21.
    J. Li, A. Tan, K.W. Moon, A. Doran, M.A. Marcus, A.T. Young, E. Arenholz, S. Ma, R.F. Yang, C. Hwang, Z.Q. Qiu, Tailoring the topology of an artificial magnetic skyrmion. Nat. Commun. 5, 4704 (2014).
  22. 22.
    J. Wild, T.N.G. Meier, S. Pöllath, M. Kronseder, A. Bauer, A. Chacon, M. Halder, M. Schowalter, A. Rosenauer, J. Zweck, J. Müller, A. Rosch, C. Pfleiderer, C. H. Back, Entropy-limited topological protection of skyrmions. Sci. Adv. 3, e1701704 (2017). Scholar
  23. 23.
    A. Fert, V. Cros, J. Sampaio, Skyrmions on the track. Nat. Nano. 8, 152 (2013). Scholar
  24. 24.
    J. Iwasaki, M. Mochizuki, N. Nagaosa, Current-induced skyrmion dynamics in constricted geometries. Nat. Nano. 8, 742 (2013). Scholar
  25. 25.
    N. Nagaosa, Y. Tokura, Topological properties and dynamics of magnetic skyrmions. Nat. Nano. 8, 899 (2013). Scholar
  26. 26.
    Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota, S. Seki, S. Ishiwata, M. Kawasaki, Y. Onose, Y. Tokura, Microwave magnetoelectric effect via skyrmion resonance modes in a helimagnetic multiferroic. Nat. Commun. 4, 2391 (2013).
  27. 27.
    L. Sun, R.X. Cao, B.F. Miao, Z. Feng, B. You, D. Wu, W. Zhang, A. Hu, H.F. Ding, Creating an artificial two-dimensional skyrmion crystal by nanopatterning. Phys. Rev. Lett. 110, 167201 (2013).
  28. 28.
    X. Yu, J.P. DeGrave, Y. Hara, T. Hara, S. Jin, Y. Tokura, Observation of the magnetic skyrmion lattice in a MnSi nanowire by Lorentz TEM. Nano Lett. 13, 3755 (2013). Scholar
  29. 29.
    J. Hagemeister, N. Romming, K. von Bergmann, E.Y. Vedmedenko, R. Wiesendanger, Tailoring the topology of an artificial magnetic skyrmion. Nat. Commun. 6, 8455 (2015).
  30. 30.
    S.-Z. Lin, C.D. Batista, C. Reichhardt, A. Saxena, AC current generation in chiral magnetic insulators and skyrmion motion induced by the spin seebeck effect. Phys. Rev. Lett. 112, 187203 (2014).
  31. 31.
    X Zhang, G.P. Zhao, H. Fangohr, J.P. Liu, W.X. Xia, J. Xia, F.J. Morvan, Skyrmion-skyrmion and skyrmion-edge repulsions in skyrmion-based racetrack memory. Sci. Rep. 5, 7643 (2015).
  32. 32.
    P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, R. Wiesendanger, Electric-field-driven switching of individual magnetic skyrmions. Nat. Nano. 12, 123 (2016). Scholar
  33. 33.
    W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M.B. Jungfleisch, J.E. Pearson, X. Cheng, O. Heinonen, K.L. Wang, Y. Zhou, A. Hoffmann, S.G.E. te Velthuis, Direct observation of the skyrmion Hall effect. Nat. Phys. 13, 162 (2016). Scholar
  34. 34.
    R. Wiesendanger, Nanoscale magnetic skyrmions in metallic films and multilayers: a new twist for spintronics. Nat. Rev. Mater. 1, 16044 (2016).
  35. 35.
    S. Woo, K. Litzius, B. Krüger, 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, and G.S.D. Beach, Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater. 15, 501 (2016). Scholar
  36. 36.
    J. Müller, Magnetic skyrmions on a two-lane racetrack. New J. Phys. 19, 025002 (2017). Scholar
  37. 37.
    P. Huang, M. Cantoni, A. Kruchkov, R. Jayaraman, A. Magrez, F. Carbone, H.M. Rønnow, In situ electric field skyrmion creation in magnetoelectric Cu\(_2\)OSeO\(_3\) (2017). arXiv:1710.09200
  38. 38.
    T. Adams, S. Mühlbauer, C. Pfleiderer, F. Jonietz, A. Bauer, A. Neubauer, R. Georgii, P. Böni, U. Keiderling, K. Everschor, M. Garst, A. Rosch, Long-range crystalline nature of the skyrmion lattice in MnSi. Phys. Rev. Lett. 107, 217206 (2011).
  39. 39.
    S.L. Zhang, A. Bauer, D.M. Burn, P. Milde, E. Neuber, L.M. Eng, H. Berger, C. Pfleiderer, G. van der Laan, T. Hesjedal, Multidomain skyrmion lattice state in Cu\(_2\)OSeO\(_3\). Nano Lett. 16, 3285 (2016). Scholar
  40. 40.
    P. Milde, E. Neuber, A. Bauer, C. Pfleiderer, H. Berger, L.M. Eng, Heuristic description of magnetoelectricity of Cu\(_2\)OSeO\(_3\). Nano Lett. 16, 5612 (2016). Scholar
  41. 41.
    A. Bauer, C. Pfleiderer, Generic aspects of skyrmion lattices in chiral magnets, in Topological Structures in Ferroic Materials: Domain Walls, Vortices and Skyrmions (Springer International Publishing, 2016) p. 1. Scholar
  42. 42.
    A.N. Bogdanov, D.A. Yablonskii, Thermodynamically stable "vortices" in magnetically ordered crystals. The mixed state of magnets. Sov. Phys. JETP 95, 178 (1989),
  43. 43.
    A. Bogdanov, A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crystals, J. Magn. Magn. Mater. 138, 255 (1994). Scholar
  44. 44.
    A. Chacon, A. Bauer, T. Adams, F. Rucker, G. Brandl, R. Georgii, M. Garst, C. Pfleiderer, Uniaxial pressure dependence of magnetic order in MnSi. Phys. Rev. Lett. 115, 267202 (2015).
  45. 45.
    Y. Nii, T. Nakajima, A. Kikkawa, Y. Yamasaki, K. Ohishi, J. Suzuki, Y. Taguchi, T. Arima, Y. Tokura, Y. Iwasa, Uniaxial stress control of skyrmion phase. Nat. Commun. 6, 8539 (2015).
  46. 46.
    W. Münzer, A. Neubauer, T. Adams, S. Mühlbauer, C. Franz, F. Jonietz, R. Georgii, P. Böni, B. Pedersen, M. Schmidt, A. Rosch, C. Pfleiderer, Skyrmion lattice in the doped semiconductor Fe\(_{1-x}\)Co\(_x\)Si. Phys. Rev. B 81, 041203 (R) (2010).
  47. 47.
    C. Pfleiderer, T. Adams, A. Bauer, W. Biberacher, B. Binz, F. Birkelbach, P. Böni, C. Franz, R. Georgii, M. Janoschek, F. Jonietz, T. Keller, R. Ritz, S. Mühlbauer, W. Münzer, A. Neubauer, B. Pedersen, A. Rosch, Skyrmion lattices in metallic and semiconducting B20 transition metal compounds. J. Phys.: Condens. Matter 22, 164207 (2010). Scholar
  48. 48.
    T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl, B. Pedersen, H. Berger, P. Lemmens, C. Pfleiderer, Long-wavelength helimagnetic order and skyrmion lattice phase in Cu\(_2\)OSeO\(_3\). Phys. Rev. Lett. 108, 237204 (2012).
  49. 49.
    S. Seki, J.-H. Kim, D.S. Inosov, R. Georgii, B. Keimer, S. Ishiwata, Y. Tokura, Formation and rotation of skyrmion crystal in the chiral-lattice insulator Cu\(_2\)OSeO\(_3\). Phys. Rev. B 85, 220406 (R) (2012).
  50. 50.
    E. Moskvin, S. Grigoriev, V. Dyadkin, H. Eckerlebe, M. Baenitz, M. Schmidt, H. Wilhelm, Complex chiral modulations in FeGe close to magnetic ordering. Phys. Rev. Lett. 110, 077207 (2013).
  51. 51.
    Y. Tokunaga, X.Z. Yu, J.S. White, H.M. Rønnow, D. Morikawa, Y. Taguchi, Y. Tokura, A new class of chiral materials hosting magnetic skyrmions beyond room temperature. Nat. Commun. 6, 7638 (2015).
  52. 52.
    M.C. Langner, S. Roy, S.K. Mishra, J.C.T. Lee, X.W. Shi, M.A. Hossain, Y.-D. Chuang, S. Seki, Y. Tokura, S.D. Kevan, R.W. Schoenlein, Coupled skyrmion sublattices in Cu\(_2\)OSeO\(_3\). Phys. Rev. Lett. 112, 167202 (2014).
  53. 53.
    S.L. Zhang, A. Bauer, H. Berger, C. Pfleiderer, G. van der Laan, T. Hesjedal, Multidomain skyrmion lattice state in Cu\(_2\)OSeO\(_3\). Appl. Phys. Lett. 109, 192406 (2016). Scholar
  54. 54.
    X.Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W.Z. Zhang, S. Ishiwata, Y. Matsui, Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe. Nat. Mater. 10, 106 (2011). Scholar
  55. 55.
    S. Seki, X.Z. Yu, S. Ishiwata, Y. Tokura, Observation of skyrmions in a multiferroic material. Science 336, 198 (2012). Scholar
  56. 56.
    J. Rajeswari, P. Huang, G.F. Mancini, Y. Murooka, T. Latychevskaia, D. McGrouther, M. Cantoni, E. Baldini, J.S. White, A. Magrez, T. Giamarchi, H.M. Rønnow, F. Carbone, Filming the formation and fluctuation of skyrmion domains by cryo-Lorentz transmission electron microscopy. Proc. Natl. Acad. Sci. USA 112, 14212 (2015). Scholar
  57. 57.
    S. Pöllath, J. Wild, L. Heinen, T.N.G. Meier, M. Kronseder, L. Tutsch, A. Bauer, H. Berger, C. Pfleiderer, J Zweck, A. Rosch, C.H. Back, Dynamical defects in rotating magnetic skyrmion lattices. Phys. Rev. Lett. 118, 207205 (2017).
  58. 58.
    H.S. Park, X. Yu, S. Aizawa, T. Tanigaki, T. Akashi, Y. Takahashi, T. Matsuda, N. Kanazawa, Y. Onose, D. Shindo, A. Tonomura, Y. Tokura, Observation of the magnetic flux and three-dimensional structure of skyrmion lattices by electron holography. Nat. Nano. 9, 337 (2014). Scholar
  59. 59.
    M. Mochizuki, Spin-wave modes and their intense excitation effects in skyrmion crystals. Phys. Rev. Lett. 108, 017601 (2012).
  60. 60.
    Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, Y. Tokura, Observation of magnetic excitations of skyrmion crystal in a helimagnetic insulator Cu\(_2\)OSeO\(_3\). Phys. Rev. Lett. 109, 037603 (2012).
  61. 61.
    T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, A. Rosch, C. Pfleiderer, D. Grundler, Universal helimagnon and skyrmion excitations in metallic, semiconducting and insulating chiral magnets. Nat. Mater. 14, 478 (2015). Scholar
  62. 62.
    I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. Berger, M. Garst, C. Pfleiderer, D. Grundler, Linearly polarized GHz magnetization dynamics of spin helix modes in the ferrimagnetic insulator Cu\(_2\)OSeO\(_3\). Sci. Rep. 7, 7037 (2017).
  63. 63.
    N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, Y. Tokura, Large topological Hall effect in a short-period helimagnet MnGe. Phys. Rev. Lett. 106, 156603 (2011).
  64. 64.
    Y. Shiomi, N. Kanazawa, K. Shibata, Y. Onose, Y. Tokura, Topological Nernst effect in a three-dimensional skyrmion-lattice phase. Phys. Rev. B 88, 064409 (2013).
  65. 65.
    T. Tanigaki, K. Shibata, N. Kanazawa, X.Z. Yu, S. Aizawa, Y. Onose, H.S. Park, D. Shindo, Y. Tokura, Real-space observation of short-period cubic lattice of skyrmions in MnGe. Nano Lett. 15, 5438 (2015). Scholar
  66. 66.
    N. Kanazawa, Y. Nii, X.-X. Zhang, A.S. Mishchenko, G.D. Filippis, F. Kagawa, Y. Iwasa, N. Nagaosa, and Y. Tokura, Critical phenomena of emergent magnetic monopoles in a chiral magnet. Nat. Commun. 7, 11622 (2016). Scholar
  67. 67.
    E. Ruff, S. Widmann, P. Lunkenheimer, V. Tsurkan, S. Bordács, I. Kézsmárki, A. Loidl, Multiferroicity and skyrmions carrying electric polarization in GaV\(_4\)S\(_8\). Sci. Adv. 1, e1500916 (2015). Scholar
  68. 68.
    T. Kurumaji, T. Nakajima, V. Ukleev, A. Feoktystov, T. h. Arima, K. Kakurai, Y. Tokura, Néel-type skyrmion lattice in tetragonal polar magnet VOSe\(_2\)O\(_5\). Phys. Rev. Lett. 119, 237201 (2017)Google Scholar
  69. 69.
    A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P.G. Niklowitz, P. Böni, Topological Hall effect in the \(A\) phase of MnSi. Phys. Rev. Lett. 102, 186602 (2009).
  70. 70.
    C. Franz, F. Freimuth, A. Bauer, R. Ritz, C. Schnarr, C. Duvinage, T. Adams, S. Blügel, A. Rosch, Y.  Mokrousov, C. Pfleiderer, Real-space and reciprocal-space Berry phases in the Hall effect of Mn\(_{1-x}\)Fe\(_x\)Si. Phys. Rev. Lett. 112, 186601 (2014).
  71. 71.
    N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald, N.P. Ong, Anomalous Hall effect. Rev. Mod. Phys. 82, 1539 (2010). Scholar
  72. 72.
    J.C. Slonczewski, Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1 (1996). Scholar
  73. 73.
    L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353 (1996). Scholar
  74. 74.
    M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, P. Wyder, Excitation of a magnetic multilayer by an electric current. Phys. Rev. Lett. 80, 4281 (1998). Scholar
  75. 75.
    E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, R.A. Buhrman, Current-induced switching of domains in magnetic multilayer devices. Science 285, 867 (1999). Scholar
  76. 76.
    J. Grollier, P. Boulenc, V. Cros, A. Hamzić, A. Vaurès, A. Fert, G. Faini, Switching a spin valve back and forth by current-induced domain wall motion. Appl. Phys. Lett. 83, 509 (2003). Scholar
  77. 77.
    S.S.P. Parkin, M. Hayashi, L. Thomas, Magnetic domain-wall racetrack memory. Science 320, 190 (2008). Scholar
  78. 78.
    S.S.P. Parkin, S.-H. Yang, Memory on the racetrack. Nat. Nano. 10, 221 (2015). Scholar
  79. 79.
    F. Jonietz, S. Mühlbauer, C. Pfleiderer, A. Neubauer, W. Münzer, A. Bauer, T. Adams, R. Georgii, P. Böni, R.A. Duine, K. Everschor, M. Garst, A. Rosch, Spin transfer torques in MnSi at ultralow current densities. Science 330, 1648 (2010). Scholar
  80. 80.
    K. Everschor, M. Garst, R.A. Duine, A. Rosch, Current-induced rotational torques in the skyrmion lattice phase of chiral magnets. Phys. Rev. B 84, 064401 (2011).
  81. 81.
    X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. Onose, Y. Tokura, Skyrmion flow near room temperature in an ultralow current density. Nat. Commun. 3, 988 (2012).
  82. 82.
    J. Iwasaki, M. Mochizuki, N. Nagaosa, Universal current-velocity relation of skyrmion motion in chiral magnets. Nat. Commun. 4, 1463 (2013).
  83. 83.
    T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor, M. Garst, A. Rosch, Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys. 8, 301 (2012). Scholar
  84. 84.
    M. Mochizuki, X.Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, J. Zang, M. Mostovoy, Y. Tokura, N. Nagaosa, Thermally driven ratchet motion of a skyrmion microcrystal and topological magnon Hall effect. Nat. Mater. 13, 241 (2014). Scholar
  85. 85.
    S. Seki, S. Ishiwata, Y. Tokura, Magnetoelectric nature of skyrmions in a chiral magnetic insulator Cu\(_2\)OSeO\(_3\). Phys. Rev. B 86, 060403 (2012).
  86. 86.
    J.S. White, K. Prša, P. Huang, A.A. Omrani, I. Živković, M. Bartkowiak, H. Berger, A. Magrez, J.L. Gavilano, G. Nagy, J. Zang, H.M. Rønnow, Electric-field-induced skyrmion distortion and giant lattice rotation in the magnetoelectric insulator Cu\(_2\)OSeO\(_3\). Phys. Rev. Lett. 113, 107203 (2014).
  87. 87.
    Y. Okamura, F. Kagawa, S. Seki, Y. Tokura, Transition to and from the skyrmion lattice phase by electric fields in a magnetoelectric compound. Nat. Commun. 7, 12669 (2016). Scholar
  88. 88.
    A.J. Kruchkov, J.S. White, M. Bartkowiak, I. Zivcovic, A. Magrez, H.M. Rønnow, Direct control of the skyrmion phase stability by electric field in a magnetoelectric insulator (2017). arXiv:1703.06081
  89. 89.
    A. Bauer, M. Garst, C. Pfleiderer, History dependence of the magnetic properties of single-crystal Fe\(_{1-x}\)Co\(_x\)Si. Phys. Rev. B 93, 235144 (2016).
  90. 90.
    A. Bauer, A. Chacon, M. Halder, J. Kindervater, S. Mühlbauer, A. Heinemann, C. Pfleiderer, Topological protection of super-cooled skyrmion lattice order in Fe\(_{1-x}\)Co\(_x\)Si (2018)Google Scholar
  91. 91.
    L.J. Bannenberg, A.J.E. Lefering, K. Kakurai, Y. Onose, Y. Endoh, Y. Tokura, C. Pappas, Magnetic relaxation phenomena in the chiral magnet Fe\(_{1-x}\)Co\(_x\)Si: An ac susceptibility study. Phys. Rev. B 94, 134433 (2016).
  92. 92.
    A. Bauer, A. Neubauer, C. Franz, W. Münzer, M. Garst, C. Pfleiderer, Quantum phase transitions in single-crystal Mn\(_{1-x}\)Fe\(_x\)Si and Mn\(_{1-x}\)Co\(_x\)Si: Crystal growth, magnetization, ac susceptibility, and specific heat. Phys. Rev. B 82, 064404 (2010).
  93. 93.
    J. Kindervater, T. Adams, A. Bauer, F. Haslbeck, A. Chacon, S. Mühlbauer, F. Jonietz, A. Neubauer, U. Gasser, G. Nagy, N. Martin, W. Häußler, R. Georgii, M. Garst, C. Pfleiderer, Helical and skyrmion lattice order in Mn\(_{1-x}\)Fe\(_x\)Si and Mn\(_{1-x}\)Co\(_x\)Si (2018)Google Scholar
  94. 94.
    L.J. Bannenberg, K. Kakurai, F. Qian, E. Lelièvre-Berna, C.D. Dewhurst, Y. Onose, Y. Endoh, Y. Tokura, C. Pappas, Extended skyrmion lattice scattering and long-time memory in the chiral magnet Fe\(_{1-x}\)Co\(_x\)Si. Phys. Rev. B 94, 104406 (2016).
  95. 95.
    T. Adams, S. Mühlbauer, A. Neubauer, W. Münzer, F. Jonietz, R. Georgii, B. Pedersen, P. Böni, A. Rosch, C. Pfleiderer, Skyrmion lattice domains in Fe\(_{1-x}\)Co\(_x\)Si. J. Phys.: Conf. Ser. 200, 032001 (2010). Scholar
  96. 96.
    K. Makino, J.D. Reim, D. Higashi, D. Okuyama, T.J. Sato, Y. Nambu, E.P. Gilbert, N. Booth, S. Seki, Y. Tokura, Thermal stability and irreversibility of skyrmion-lattice phases in Cu\(_2\)OSeO\(_3\). Phys. Rev. B 95, 134412 (2017).
  97. 97.
    L.J. Bannenberg, F. Qian, R.M. Dalgliesh, N. Martin, G. Chaboussant, M. Schmidt, D.L. Schlagel, T.A. Lograsso, H. Wilhelm, C. Pappas, Reorientations, relaxations, metastabilities and domains of skyrmion lattices. Phys. Rev. B 96, 184416 (2017).
  98. 98.
    A. Bauer, C. Pfleiderer, Magnetic phase diagram of MnSi inferred from magnetization and ac susceptibility. Phys. Rev. B 85, 214418 (2012).
  99. 99.
    A. Bauer, M. Garst, C. Pfleiderer, Specific heat of the skyrmion lattice phase and field-induced tricritical point in MnSi. Phys. Rev. Lett. 110, 177207 (2013).
  100. 100.
    I. Levatić, V. Šurija, H. Berger, I. Živković, Dissipation processes in the insulating skyrmion compound Cu\(_2\)OSeO\(_3\). Phys. Rev. B 90, 224412 (2014).
  101. 101.
    F. Qian, H. Wilhelm, A. Aqeel, T.T.M. Palstra, A.J.E. Lefering, E.H. Brück, C. Pappas, Phase diagram and magnetic relaxation phenomena in Cu\(_2\)OSeO\(_3\). Phys. Rev. B 94, 064418 (2016).
  102. 102.
    K. Karube, J.S. White, N. Reynolds, J.L. Gavilano, H. Oike, A. Kikkawa, F. Kagawa, Y. Tokunaga, H.M. Rønnow, Y. Tokura, and Y. Taguchi, Robust metastable skyrmions and their triangular-square lattice structural transition in a high-temperature chiral magnet. Nat. Mater. 15, 1237 (2015). Scholar
  103. 103.
    K. Karube, J.S. White, D. Morikawa, M. Bartkowiak, A. Kikkawa, Y. Tokunaga, T. Arima, H.M. Rønnow, Y. Tokura, Y. Taguchi, Skyrmion formation in a bulk chiral magnet at zero magnetic field and above room temperature. Phys. Rev. Mater. 1, 074405 (2017).
  104. 104.
    H. Oike, A. Kikkawa, N. Kanazawa, Y. Taguchi, M. Kawasaki, Y. Tokura, F. Kagawa, Interplay between topological and thermodynamic stability in a metastable magnetic skyrmion lattice. Nat. Phys. 12, 62 (2016). Scholar
  105. 105.
    F.N. Rybakov, A.B. Borisov, S. Blügel, N.S. Kiselev, New type of stable particlelike states in chiral magnets. Phys. Rev. Lett. 115, 117201 (2015).
  106. 106.
    R. Ritz, M. Halder, C. Franz, A. Bauer, M. Wagner, R. Bamler, A. Rosch, C. Pfleiderer, Giant generic topological Hall resistivity of MnSi under pressure. Phys. Rev. B 87, 134424 (2013).
  107. 107.
    C. Pfleiderer, P. Böni, T. Keller, U.K. Rößler, A. Rosch, Non-fermi liquid metal without quantum criticality. Science 316, 1871 (2007). Scholar
  108. 108.
    R. Ritz, M. Halder, M. Wagner, C. Franz, A. Bauer, C. Pfleiderer, Formation of a topological non-Fermi liquid in MnSi. Nature 497, 231 (2013). Scholar
  109. 109.
    C. Pfleiderer, G.J. McMullan, S.R. Julian, G.G. Lonzarich, Magnetic quantum phase transition in MnSi under hydrostatic pressure. Phys. Rev. B 55, 8330 (1997). Scholar
  110. 110.
    C. Thessieu, C. Pfleiderer, A.N. Stepanov, J. Flouquet, Field dependence of the magnetic quantum phase transition in MnSi. J. Phys.: Condens. Matter 9, 6677 (1997). Scholar
  111. 111.
    C. Pfleiderer, S.R. Julian, G.G. Lonzarich, Non-Fermi-liquid nature of the normal state of itinerant-electron ferromagnets. Nature 414, 427 (2001). Scholar
  112. 112.
    N. Doiron-Leyraud, I.R. Walker, L. Taillefer, M.J. Steiner, S.R. Julian, G.G. Lonzarich, Fermi-liquid breakdown in the paramagnetic phase of a pure metal. Nature 425, 595 (2003). Scholar
  113. 113.
    C. Pfleiderer, D. Reznik, L. Pintschovius, H.V. Löhneysen, M. Garst, A. Rosch, Partial order in the non-Fermi-liquid phase of MnSi. Nature 427, 227 (2004). Scholar
  114. 114.
    Y.J. Uemura, T. Goko, I.M. Gat-Malureanu, J.P. Carlo, P.L. Russo, A.T. Savici, A. Aczel, G.J. MacDougall, J.A. Rodriguez, G.M. Luke, S.R. Dunsiger, A. McCollam, J. Arai, C. Pfleiderer, P. Böni, K. Yoshimura, E. Baggio-Saitovitch, M.B. Fontes, J. Larrea, Y.V. Sushko, J. Sereni, Phase separation and suppression of critical dynamics at quantum phase transitions of MnSi and (Sr\(_{1-x}\)Ca\(_x\))RuO\(_3\). Nat. Phys. 3, 29 (2007). Scholar
  115. 115.
    T. Nakajima, H. Oike, A. Kikkawa, E.P. Gilbert, N. Booth, K. Kakurai, Y. Taguchi, Y. Tokura, F. Kagawa, T.-H. Arima, Skyrmion lattice structural transition in MnSi. Sci. Adv. 3, e1602562 (2017). Scholar
  116. 116.
    N. Kanazawa, M. Kubota, A. Tsukazaki, Y. Kozuka, K.S. Takahashi, M. Kawasaki, M. Ichikawa, F. Kagawa, Y. Tokura, Discretized topological Hall effect emerging from skyrmions in constricted geometry. Phys. Rev. B 91, 041122 (2015).
  117. 117.
    A. Dussaux, P. Schoenherr, K. Koumpouras, J. Chico, K. Chang, L. Lorenzelli, N. Kanazawa, Y. Tokura, M. Garst, A. Bergman, C.L. Degen, D. Meier, Local dynamics of topological magnetic defects in the itinerant helimagnet FeGe. Nat. Commun. 7, 12430 (2016). Scholar
  118. 118.
    A. Bauer, A. Chacon, M. Wagner, M. Halder, R. Georgii, A. Rosch, C. Pfleiderer, M. Garst, Symmetry breaking, slow relaxation dynamics, and topological defects at the field-induced helix reorientation in MnSi. Phys. Rev. B 95, 024429 (2017).
  119. 119.
    J. Müller, J. Rajeswari, P. Huang, Y. Murooka, H.M. Rønnow, F. Carbone, A. Rosch, magnetic skyrmions and skyrmion clusters in the Helical phase of Cu\(_2\)OSeO\(_3\). Phys. Rev. Lett. 119, 137201 (2017).

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Andreas Bauer
    • 1
  • Alfonso Chacon
    • 1
  • Marco Halder
    • 1
  • Christian Pfleiderer
    • 1
    Email author
  1. 1.Physik-DepartmentTechnische Universität MünchenGarchingGermany

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