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Statics and dynamics of skyrmions interacting with disorder and nanostructures

C. Reichhardt*, C. J. O. Reichhardt, and M. V. Milošević

C. Reichhardt* and C. J. O. Reichhardt

  • Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

M. V. Milošević

  • NANOlab Center of Excellence, Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

  • *reichhardt@lanl.gov
  • cjrx@lanl.gov
  • milorad.milosevic@uantwerpen.be

Rev. Mod. Phys. 94, 035005 – Published 20 September, 2022

DOI: https://doi.org/10.1103/RevModPhys.94.035005

Abstract

Magnetic skyrmions are topologically stable nanoscale particlelike objects that were discovered in 2009. Since that time, intense research interest in the field has led to the identification of numerous compounds that support skyrmions over a range of conditions spanning from cryogenic to room temperatures. Skyrmions can be set into motion under various types of driving, and the combination of their size, stability, and dynamics makes them ideal candidates for numerous applications. At the same time, skyrmions represent a new class of system in which the energy scales of the skyrmion-skyrmion interactions, sample disorder, temperature, and drive can compete. A growing body of work indicates that the static and dynamic states of skyrmions can be influenced strongly by pinning or disorder in the sample; thus, an understanding of such effects is essential for the eventual use of skyrmions in applications. The current state of knowledge regarding individual skyrmions and skyrmion assemblies interacting with quenched disorder or pinning is reviewed. The microscopic mechanisms for skyrmion pinning, including the repulsive and attractive interactions that can arise from impurities, grain boundaries, or nanostructures, are outlined. This is followed by descriptions of depinning phenomena, sliding states over disorder, the effect of pinning on the skyrmion Hall angle, the competition between thermal and pinning effects, the control of skyrmion motion using ordered potential landscapes such as one- or two-dimensional periodic asymmetric substrates, the creation of skyrmion diodes, and skyrmion ratchet effects. Highlighted are the distinctions arising from internal modes and the strong gyrotropic or Magnus forces that cause the dynamical states of skyrmions to differ from those of other systems with pinning, such as vortices in type-II superconductors, charge density waves, or colloidal particles. Throughout this review future directions and open questions related to the pinning and dynamics in skyrmion systems are also discussed.

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References (501)

  1. Ackerman, P. J., R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, 2014, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505.
  2. Akhtar, W., et al., 2019, “Current-Induced Nucleation and Dynamics of Skyrmions in a Co-Based Heusler Alloy,” Phys. Rev. Applied 11, 034066.
  3. Akosa, C. A., O. A. Tretiakov, G. Tatara, and A. Manchon, 2018, “Theory of the Topological Spin Hall Effect in Antiferromagnetic Skyrmions: Impact on Current-Induced Motion,” Phys. Rev. Lett. 121, 097204.
  4. Al Khawaja, U., and H. Stoof, 2001, “Skyrmions in a ferromagnetic Bose-Einstein condensate,” Nature (London) 411, 918.
  5. Anderson, P. W., and Y. B. Kim, 1964, “Hard superconductivity: Theory of the motion of Abrikosov flux lines,” Rev. Mod. Phys. 36, 39–43.
  6. Arjana, I. G., I. L. Fernandes, J. Chico, and S. Lounis, 2020, “Sub-nanoscale atom-by-atom crafting of skyrmion-defect interaction profiles,” Sci. Rep. 10, 14655.
  7. Auslaender, O. M., L. Luan, E. W. J. Straver, J. E. Hoffman, N. C. Koshnick, E. Zeldov, D. A. Bonn, R. Liang, W. N. Hardy, and K. A. Moler, 2009, “Mechanics of individual isolated vortices in a cuprate superconductor,” Nat. Phys. 5, 35–39.
  8. Avci, S., Z. L. Xiao, J. Hua, A. Imre, R. Divan, J. Pearson, U. Welp, W. K. Kwok, and G. W. Crabtree, 2010, “Matching effect and dynamic phases of vortex matter in Bi2Sr2CaCu2O8 nanoribbon with a periodic array of holes,” Appl. Phys. Lett. 97, 042511.
  9. Avraham, N., et al., 2001, “ ‘Inverse’ melting of a vortex lattice,” Nature (London) 411, 451–454.
  10. Back, C., et al., 2020, “The 2020 skyrmionics roadmap,” J. Phys. D 53, 363001.
  11. Baert, M., V. V. Metlushko, R. Jonckheere, V. V. Moshchalkov, and Y. Bruynseraede, 1995, “Composite Flux-Line Lattices Stabilized in Superconducting Films by a Regular Array of Artificial Defects,” Phys. Rev. Lett. 74, 3269–3272.
  12. Bag, B., G. Shaw, S. S. Banerjee, S. Majumdar, A. K. Sood, and A. K. Grover, 2017, “Negative velocity fluctuations and non-equilibrium fluctuation relation for a driven high critical current vortex state,” Sci. Rep. 7, 5531.
  13. Bak, P., 1982, “Commensurate phases, incommensurate phases and the Devil’s staircase,” Rep. Prog. Phys. 45, 587–629.
  14. Bak, P., C. Tang, and K. Wiesenfeld, 1988, “Self-organized criticality,” Phys. Rev. A 38, 364–374.
  15. Balents, L., M. C. Marchetti, and L. Radzihovsky, 1998, “Nonequilibrium steady states of driven periodic media,” Phys. Rev. B 57, 7705–7739.
  16. Banerjee, S. S., et al., 2000, “Peak effect, plateau effect, and fishtail anomaly: The reentrant amorphization of vortex matter in 2H-NbSe2,” Phys. Rev. B 62, 11838–11845.
  17. Barker, J., and O. A. Tretiakov, 2016, “Static and Dynamical Properties of Antiferromagnetic Skyrmions in the Presence of Applied Current and Temperature,” Phys. Rev. Lett. 116, 147203.
  18. Barkhausen, H., 1919, “Two phenomena discovered with the help of the new amplifiers,” Z. Phys. 20, 401.
  19. Baumard, J., J. Cayssol, F. S. Bergeret, and A. Buzdin, 2019, “Generation of a superconducting vortex via Néel skyrmions,” Phys. Rev. B 99, 014511.
  20. Beg, M., et al., 2017, “Dynamics of skyrmionic states in confined helimagnetic nanostructures,” Phys. Rev. B 95, 014433.
  21. Benassi, A., A. Vanossi, and E. Tosatti, 2011, “Nanofriction in cold ion traps,” Nat. Commun. 2, 236.
  22. Benz, S. P., M. S. Rzchowski, M. Tinkham, and C. J. Lobb, 1990, “Fractional Giant Shapiro Steps and Spatially Correlated Phase Motion in 2D Josephson Arrays,” Phys. Rev. Lett. 64, 693–696.
  23. Berdiyorov, G. R., M. V. Milošević, and F. M. Peeters, 2006, “Novel Commensurability Effects in Superconducting Films with Antidot Arrays,” Phys. Rev. Lett. 96, 207001.
  24. Berger, L., 1970, “Side-jump mechanism for the Hall effect of ferromagnets,” Phys. Rev. B 2, 4559–4566.
  25. Bertotti, G., G. Durin, and A. Magni, 1994, “Scaling aspects of domain wall dynamics and Barkhausen effect in ferromagnetic materials,” J. Appl. Phys. 75, 5490.
  26. Bhattacharya, S., and M. J. Higgins, 1993, “Dynamics of a Disordered Flux Line Lattice,” Phys. Rev. Lett. 70, 2617–2620.
  27. Bhatti, S., and S. N. Piramanayagam, 2019, “Effect of Dzyaloshinkii-Moriya interaction energy confinement on current-driven dynamics of skyrmions,” Phys. Status Solidi RRL 13, 1900090.
  28. Birch, M. T., et al., 2020, “Real-space imaging of confined magnetic skyrmion tubes,” Nat. Commun. 11, 1726.
  29. Blatter, G., M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, 1994, “Vortices in high-temperature superconductors,” Rev. Mod. Phys. 66, 1125–1388.
  30. Bloom, I., A. C. Marley, and M. B. Weissman, 1993, “Nonequilibrium Dynamics of Discrete Fluctuators in Charge-Density Waves in NbSe3,” Phys. Rev. Lett. 71, 4385–4388.
  31. Bogdanov, A. N., and C. Panagopoulos, 2020, “Physical foundations and basic properties of magnetic skyrmions,” Nat. Rev. Phys. 2, 492–498.
  32. Bogdanov, A. N., and D. A. Yablonskii, 1989, “Thermodynamically stable ‘vortices’ in magnetically ordered crystals. The mixed state of magnets,” Sov. Phys. JETP 68, 101–103, http://jetp.ras.ru/cgi-bin/dn/e_068_01_0101.pdf .
  33. Bohlein, T., and C. Bechinger, 2012, “Experimental Observation of Directional Locking and Dynamical Ordering of Colloidal Monolayers Driven across Quasiperiodic Substrates,” Phys. Rev. Lett. 109, 058301.
  34. Bohlein, T., J. Mikhael, and C. Bechinger, 2012, “Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces,” Nat. Mater. 11, 126–130.
  35. Bömerich, T., L. Heinen, and A. Rosch, 2020, “Skyrmion and tetarton lattices in twisted bilayer graphene,” Phys. Rev. B 102, 100408.
  36. Boulle, O., et al., 2016, “Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures,” Nat. Nanotechnol. 11, 449–454.
  37. Braun, H.-B., 2012, “Topological effects in nanomagnetism: From superparamagnetism to chiral quantum solitons,” Adv. Phys. 61, 1–116.
  38. Braun, O. M., and Y. S. Kivshar, 1998, “Nonlinear dynamics of the Frenkel-Kontorova model,” Phys. Rep. 306, 1–108.
  39. Brearton, R., L. A. Turnbull, J. A. T. Verezhak, G. Balakrishnan, P. D. Hatton, G. van der Laan, and T. Hesjedal, 2021, “Deriving the skyrmion Hall angle from skyrmion lattice dynamics,” Nat. Commun. 12, 2723.
  40. Brey, L., H. A. Fertig, R. Côté, and A. H. MacDonald, 1995, “Skyrme Crystal in a Two-Dimensional Electron Gas,” Phys. Rev. Lett. 75, 2562–2565.
  41. Brown, B. L., U. C. Täuber, and M. Pleimling, 2018, “Effect of the Magnus force on skyrmion relaxation dynamics,” Phys. Rev. B 97, 020405.
  42. Brunner, M., and C. Bechinger, 2002, “Phase Behavior of Colloidal Molecular Crystals on Triangular Light Lattices,” Phys. Rev. Lett. 88, 248302.
  43. Büchler, H. P., G. Blatter, and W. Zwerger, 2003, “Commensurate-Incommensurate Transition of Cold Atoms in an Optical Lattice,” Phys. Rev. Lett. 90, 130401.
  44. Büttner, F., et al., 2015, “Dynamics and inertia of skyrmionic spin structures,” Nat. Phys. 11, 225–228.
  45. Carlson, J. M., J. S. Langer, and B. E. Shaw, 1994, “Dynamics of earthquake faults,” Rev. Mod. Phys. 66, 657–670.
  46. Casiraghi, A., H. Corte-León, V. Vafaee, F. Garcia-Sanchez, G. Durin, M. Pasquale, G. Jakob, M. Kläui, and O. Kazakova, 2019, “Individual skyrmion manipulation by local magnetic field gradients,” Commun. Phys. 2, 145.
  47. Castell-Queralt, J., L. Gonzalez-Gomez, N. Del-Valle, A. Sanchez, and C. Navau, 2019, “Accelerating, guiding, and compressing skyrmions by defect rails,” Nanoscale 11, 12589–12594.
  48. Cha, M.-C., and H. A. Fertig, 1994, “Topological defects, orientational order, and depinning of the electron solid in a random potential,” Phys. Rev. B 50, 14368–14380.
  49. Cha, M.-C., and H. A. Fertig, 1995, “Disorder-Induced Phase Transitions in Two-Dimensional Crystals,” Phys. Rev. Lett. 74, 4867–4870.
  50. Cha, M.-C., and H. A. Fertig, 1998, “Peak Effect and the Transition from Elastic to Plastic Depinning,” Phys. Rev. Lett. 80, 3851–3854.
  51. Chai, Y., et al., 2021, “Probe of skyrmion phases and dynamics in MnSi via the magnetoelectric effect in a composite configuration,” Phys. Rev. B 104, L100413.
  52. Chen, Q.-H., and X. Hu, 2003, “Nonequilibrium Phase Transitions of Vortex Matter in Three-Dimensional Layered Superconductors,” Phys. Rev. Lett. 90, 117005.
  53. Chen, R., Y. Li, V. F. Pavlidis, and C. Moutafis, 2020, “Skyrmionic interconnect device,” Phys. Rev. Research 2, 043312.
  54. Chen, W., L. Liu, Y. Ji, and Y. Zheng, 2019, “Skyrmion ratchet effect driven by a biharmonic force,” Phys. Rev. B 99, 064431.
  55. Chen, W., L. Liu, and Y. Zheng, 2020, “Ultrafast Ratchet Dynamics of Skyrmions by Defect Engineering in Materials with Poor Conductivity under Gigahertz Magnetic Fields,” Phys. Rev. Applied 14, 064014.
  56. Choi, H. C., S.-Z. Lin, and J.-X. Zhu, 2016, “Density functional theory study of skyrmion pinning by atomic defects in MnSi,” Phys. Rev. B 93, 115112.
  57. Chudnovsky, E. M., and D. A. Garanin, 2018, “Skyrmion glass in a 2D Heisenberg ferromagnet with quenched disorder,” New J. Phys. 20, 033006.
  58. Civale, L., 1997, “Vortex pinning and creep in high-temperature superconductors with columnar defects,” Supercond. Sci. Technol. 10, A11–A28.
  59. Coey, J. M. D., 2010, Magnetism and Magnetic Materials (Cambridge University Press, Cambridge, England).
  60. Coppersmith, S. N., and P. B. Littlewood, 1986, “Interference Phenomena and Mode Locking in the Model of Deformable Sliding Charge-Density Waves,” Phys. Rev. Lett. 57, 1927–1930.
  61. Cote, P. J., and L. V. Meisel, 1991, “Self-Organized Criticality and the Barkhausen Effect,” Phys. Rev. Lett. 67, 1334–1337.
  62. Crabtree, G. W., and D. R. Nelson, 1997, “Vortex physics in high-temperature superconductors,” Phys. Today 50, No. 4, 38–45.
  63. Cubitt, R., et al., 1993, “Direct observation of magnetic flux lattice melting and decomposition in the high-Tc superconductor Bi2.15Sr1.95CaCu2O8+x,” Nature (London) 365, 407–411.
  64. Dahir, S. M., A. F. Volkov, and I. M. Eremin, 2019, “Interaction of Skyrmions and Pearl Vortices in Superconductor-Chiral Ferromagnet Heterostructures,” Phys. Rev. Lett. 122, 097001.
  65. D’Anna, G., P. L. Gammel, H. Safar, G. B. Alers, D. J. Bishop, J. Giapintzakis, and D. M. Ginsberg, 1995, “Vortex-Motion-Induced Voltage Noise in YBa2Cu3O7δ Single Crystals,” Phys. Rev. Lett. 75, 3521–3524.
  66. Danneau, R., A. Ayari, D. Rideau, H. Requardt, J. E. Lorenzo, L. Ortega, P. Monceau, R. Currat, and G. Grübel, 2002, “Motional Ordering of a Charge-Density Wave in the Sliding State,” Phys. Rev. Lett. 89, 106404.
  67. Das, S., et al., 2019, “Observation of room-temperature polar skyrmions,” Nature (London) 568, 368–372.
  68. Davis, T. J., D. Janoschka, P. Dreher, B. Frank, F.-J. M. zu Heringdorf, and H. Giessen, 2020, “Ultrafast vector imaging of plasmonic skyrmion dynamics with deep subwavelength resolution,” Science 368, eaba6415.
  69. Denisov, K. S., I. V. Rozhansky, N. S. Averkiev, and E. Lähderanta, 2017, “A nontrivial crossover in topological Hall effect regimes,” Sci. Rep. 7, 17204.
  70. Denisov, K. S., I. V. Rozhansky, N. S. Averkiev, and E. Lähderanta, 2018, “General theory of the topological Hall effect in systems with chiral spin textures,” Phys. Rev. B 98, 195439.
  71. Desplat, L., J.-V. Kim, and R. L. Stamps, 2019, “Paths to annihilation of first- and second-order (anti)skyrmions via (anti)meron nucleation on the frustrated square lattice,” Phys. Rev. B 99, 174409.
  72. Desplat, L., D. Suess, J.-V. Kim, and R. L. Stamps, 2018, “Thermal stability of metastable magnetic skyrmions: Entropic narrowing and significance of internal eigenmodes,” Phys. Rev. B 98, 134407.
  73. Deutschländer, S., T. Horn, H. Löwen, G. Maret, and P. Keim, 2013, “Two-Dimensional Melting under Quenched Disorder,” Phys. Rev. Lett. 111, 098301.
  74. Díaz, S. A., C. Reichhardt, D. P. Arovas, A. Saxena, and C. J. O. Reichhardt, 2018, “Avalanches and Criticality in Driven Magnetic Skyrmions,” Phys. Rev. Lett. 120, 117203.
  75. Díaz, S. A., C. J. O. Reichhardt, D. P. Arovas, A. Saxena, and C. Reichhardt, 2017, “Fluctuations and noise signatures of driven magnetic skyrmions,” Phys. Rev. B 96, 085106.
  76. Ding, J., X. Yang, and T. Zhu, 2015, “Manipulating current induced motion of magnetic skyrmions in the magnetic nanotrack,” J. Phys. D 48, 115004.
  77. Di Scala, N., E. Olive, Y. Lansac, Y. Fily, and J. C. Soret, 2012, “The elastic depinning transition of vortex lattices in two dimensions,” New J. Phys. 14, 123027.
  78. Dobramysl, U., M. Pleimling, and U. C. Täuber, 2014, “Pinning time statistics for vortex lines in disordered environments,” Phys. Rev. E 90, 062108.
  79. Dobrovolskiy, O. V., and M. Huth, 2015, “Dual cut-off direct current-tunable microwave low-pass filter on superconducting Nb microstrips with asymmetric nanogrooves,” Appl. Phys. Lett. 106, 142601.
  80. Dohi, T., S. DuttaGupta, S. Fukami, and H. Ohno, 2019, “Formation and current-induced motion of synthetic antiferromagnetic skyrmion bubbles,” Nat. Commun. 10, 5153.
  81. Dreyfus, R., Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato, 2015, “Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres,” Phys. Rev. E 91, 012302.
  82. Du, C. H., Y. R. Lee, C. Y. Lo, H. H. Lin, S. L. Chang, M. T. Tang, Y. P. Stetsko, and J. J. Lee, 2006, “Direct measurement of spatial distortions of charge density waves in K0.3MoO3,” Appl. Phys. Lett. 88, 241916.
  83. Du, H., et al., 2018, “Interaction of Individual Skyrmions in a Nanostructured Cubic Chiral Magnet,” Phys. Rev. Lett. 120, 197203.
  84. Durán, C. A., P. L. Gammel, R. Wolfe, V. J. Fratello, D. J. Bishop, J. P. Rice, and D. M. Ginsberg, 1992, “Real-time imaging of the magnetic flux distribution in superconducting YBa2Cu3O7δ,” Nature (London) 357, 474–477.
  85. Duzgun, A., C. Nisoli, C. J. O. Reichhardt, and C. Reichhardt, 2020, “Commensurate states and pattern switching via liquid crystal skyrmions trapped in a square lattice,” Soft Matter 16, 3338–3343.
  86. Duzgun, A., J. V. Selinger, and A. Saxena, 2018, “Comparing skyrmions and merons in chiral liquid crystals and magnets,” Phys. Rev. E 97, 062706.
  87. Eichhorn, R., P. Reimann, and P. Hänggi, 2002, “Brownian Motion Exhibiting Absolute Negative Mobility,” Phys. Rev. Lett. 88, 190601.
  88. Ertaş, D, and M. Kardar, 1996, “Anisotropic scaling in threshold critical dynamics of driven directed lines,” Phys. Rev. B 53, 3520–3542.
  89. Everschor-Sitte, K., J. Masell, R. M. Reeve, and M. Kläui, 2018, “Perspective: Magnetic skyrmions—Overview of recent progress in an active research field,” J. Appl. Phys. 124, 240901.
  90. Everschor-Sitte, K., and M. Sitte, 2014, “Real-space Berry phases: Skyrmion soccer (invited),” J. Appl. Phys. 115, 172602.
  91. Fangohr, H., S. J. Cox, and P. A. J. de Groot, 2001, “Vortex dynamics in two-dimensional systems at high driving forces,” Phys. Rev. B 64, 064505.
  92. Fassbender, J., J. Grenzer, O. Roshchupkina, Y. Choi, J. S. Jiang, and S. D. Bader, 2009, “The effect of ion irradiation and annealing on exchange spring magnets,” J. Appl. Phys. 105, 023902.
  93. Feigel’man, M. V., V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, 1989, “Theory of Collective Flux Creep,” Phys. Rev. Lett. 63, 2303–2306.
  94. Feilhauer, J., S. Saha, J. Tobik, M. Zelent, L. J. Heyderman, and M. Mruczkiewicz, 2020, “Controlled motion of skyrmions in a magnetic antidot lattice,” Phys. Rev. B 102, 184425.
  95. Fernandes, I. L., J. Bouaziz, S. Blügel, and S. Lounis, 2018, “Universality of defect-skyrmion interaction profiles,” Nat. Commun. 9, 4395.
  96. Fernandes, I. L., M. Bouhassoune, and S. Lounis, 2020, “Defect-implantation for the all-electrical detection of non-collinear spin-textures,” Nat. Commun. 11, 1602.
  97. Fernandes, I. L., J. Chico, and S. Lounis, 2020, “Impurity-dependent gyrotropic motion, deflection and pinning of current-driven ultrasmall skyrmions in PdFe/Ir(111) surface,” J. Phys. Condens. Matter 32, 425802.
  98. Fert, A., V. Cros, and J. Sampaio, 2013, “Skyrmions on the track,” Nat. Nanotechnol. 8, 152–156.
  99. Fert, A., N. Reyren, and V. Cros, 2017, “Magnetic skyrmions: Advances in physics and potential applications,” Nat. Rev. Mater. 2, 17031.
  100. Fidler, J., and T. Schrefl, 2000, “Micromagnetic modelling—The current state of the art,” J. Phys. D 33, R135.
  101. Fily, Y., E. Olive, N. Di Scala, and J. C. Soret, 2010, “Critical behavior of plastic depinning of vortex lattices in two dimensions: Molecular dynamics simulations,” Phys. Rev. B 82, 134519.
  102. Finocchio, G., F. Büttner, R. Tomasello, M. Carpentieri, and M. Kläui, 2016, “Magnetic skyrmions: From fundamental to applications,” J. Phys. D 49, 423001.
  103. Fiory, A. T., 1971, “Quantum Interference Effects of a Moving Vortex Lattice in Al Films,” Phys. Rev. Lett. 27, 501–503.
  104. Fisher, D. S., 1985, “Sliding charge-density waves as a dynamic critical phenomenon,” Phys. Rev. B 31, 1396–1427.
  105. Fisher, D. S., 1998, “Collective transport in random media: From superconductors to earthquakes,” Phys. Rep. 301, 113–150.
  106. Fisher, D. S., M. P. A. Fisher, and D. A. Huse, 1991, “Thermal fluctuations, quenched disorder, phase transitions, and transport in type-II superconductors,” Phys. Rev. B 43, 130–159.
  107. Foster, D., C. Kind, P. J. Ackerman, J.-S. B. Tai, M. R. Dennis, and I. I. Smalyukh, 2019, “Two-dimensional skyrmion bags in liquid crystals and ferromagnets,” Nat. Phys. 15, 655.
  108. Franken, J. H., H. J. M. Swagten, and B. Koopmans, 2012, “Shift registers based on magnetic domain wall ratchets with perpendicular anisotropy,” Nat. Nanotechnol. 7, 499–503.
  109. Fujishiro, Y., et al., 2019, “Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets,” Nat. Commun. 10, 1059.
  110. Ganguli, S. C., H. Singh, G. Saraswat, R. Ganguly, V. Bagwe, P. Shirage, A. Thamizhavel, and P. Rayhaudhuri, 2015, “Disordering of the vortex lattice through successive destruction of positional and orientational order in a weakly pinned Co0.0075NbSe2 single crystal,” Sci. Rep. 5, 10613.
  111. Gao, S., et al., 2020, “Fractional antiferromagnetic skyrmion lattice induced by anisotropic couplings,” Nature (London) 586, 37.
  112. Garst, M., J. Waizner, and D. Grundler, 2017, “Collective spin excitations of helices and magnetic skyrmions: Review and perspectives of magnonics in non-centrosymmetric magnets,” J. Phys. D 50, 293002.
  113. Giamarchi, T., and P. Le Doussal, 1995, “Elastic theory of flux lattices in the presence of weak disorder,” Phys. Rev. B 52, 1242–1270.
  114. Giamarchi, T., and P. Le Doussal, 1996, “Moving Glass Phase of Driven Lattices,” Phys. Rev. Lett. 76, 3408–3411.
  115. Gilbert, D. A., B. B. Maranville, A. L. Balk, B. J. Kirby, P. Fischer, D. T. Pierce, J. Unguris, J. A. Borchers, and K. Liu, 2015, “Realization of ground-state artificial skyrmion lattices at room temperature,” Nat. Commun. 6, 8462.
  116. Gilbert, D. A., et al., 2019, “Precipitating ordered skyrmion lattices from helical spaghetti and granular powders,” Phys. Rev. Mater. 3, 014408.
  117. Giller, D., et al., 1997, “Disorder-Induced Transition to Entangled Vortex Solid in Nd-Ce-Cu-O Crystal,” Phys. Rev. Lett. 79, 2542–2545.
  118. Goa, P. E., H. Hauglin, M. Baziljevich, E. Il’yashenko, P. L. Gammel, and T. H. Johansen, 2001, “Real-time magneto-optical imaging of vortices in superconducting NbSe2,” Supercond. Sci. Technol. 14, 729–731.
  119. Göbel, B., J. Henk, and I. Mertig, 2019, “Forming individual magnetic biskyrmions by merging two skyrmions in a centrosymmetric nanodisk,” Sci. Rep. 9, 9521.
  120. Göbel, B., and I. Mertig, 2021, “Skyrmion ratchet propagation: Utilizing the skyrmion Hall effect in ac racetrack storage devices,” Sci. Rep. 11, 3020.
  121. Göbel, B., I. Mertig, and O. A. Tretiakov, 2021, “Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles,” Phys. Rep. 895, 1.
  122. Gong, X., H. Y. Yuan, and X. R. Wang, 2020, “Current-driven skyrmion motion in granular films,” Phys. Rev. B 101, 064421.
  123. Gotcheva, V., A. T. J. Wang, and S. Teitel, 2004, “Lattice Gas Dynamics: Application to Driven Vortices in Two Dimensional Superconductors,” Phys. Rev. Lett. 92, 247005.
  124. Grigorenko, A. N., S. J. Bending, M. J. Van Bael, M. Lange, V. V. Moshchalkov, H. Fangohr, and P. A. J. de Groot, 2003, “Symmetry Locking and Commensurate Vortex Domain Formation in Periodic Pinning Arrays,” Phys. Rev. Lett. 90, 237001.
  125. Grollier, J., D. Querlioz, K. Y. Camsari, K. Everschor-Sitte, S. Fukami, and M. D. Stiles, 2020, “Neuromorphic spintronics,” Nat. Electron. 3, 360–370.
  126. Gross, I., et al., 2018, “Skyrmion morphology in ultrathin magnetic films,” Phys. Rev. Mater. 2, 024406.
  127. Grüner, G., A. Zawadowski, and P. M. Chaikin, 1981, “Nonlinear Conductivity and Noise due to Charge-Density-Wave Depinning in NbSe3,” Phys. Rev. Lett. 46, 511–515.
  128. Guillamón, I., R. Córdoba, J. Sesé, J. M. De Teresa, M. R. Ibarra, S. Vieira, and H. Suderow, 2014, “Enhancement of long-range correlations in a 2D vortex lattice by an incommensurate 1D disorder potential,” Nat. Phys. 10, 851–856.
  129. Güngördü, U., R. Nepal, O. A. Tretiakov, K. Belashchenko, and A. A. Kovalev, 2016, “Stability of skyrmion lattices and symmetries of quasi-two-dimensional chiral magnets,” Phys. Rev. B 93, 064428.
  130. Gutierrez, J., A. V. Silhanek, J. Van de Vondel, W. Gillijns, and V. V. Moshchalkov, 2009, “Transition from turbulent to nearly laminar vortex flow in superconductors with periodic pinning,” Phys. Rev. B 80, 140514.
  131. Haberkorn, N., B. Maiorov, I. O. Usov, M. Weigand, W. Hirata, S. Miyasaka, S. Tajima, N. Chikumoto, K. Tanabe, and L. Civale, 2012, “Influence of random point defects introduced by proton irradiation on critical current density and vortex dynamics of Ba(Fe0.925Co0.075)2As2 single crystals,” Phys. Rev. B 85, 014522.
  132. Hals, K. M. D., M. Schecter, and M. S. Rudner, 2016, “Composite Topological Excitations in Ferromagnet-Superconductor Heterostructures,” Phys. Rev. Lett. 117, 017001.
  133. Hänggi, P., and F. Marchesoni, 2009, “Artificial Brownian motors: Controlling transport on the nanoscale,” Rev. Mod. Phys. 81, 387–442.
  134. Hanneken, C., A. Kubetzka, K. von Bergmann, and R. Wiesendanger, 2016, “Pinning and movement of individual nanoscale magnetic skyrmions via defects,” New J. Phys. 18, 055009.
  135. Harada, K., O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura, and V. V. Moshchalkov, 1996, “Direct observation of vortex dynamics in superconducting films with regular arrays of defects,” Science 274, 1167–1170.
  136. Harris, J. M., N. P. Ong, R. Gagnon, and L. Taillefer, 1995, “Washboard Frequency of the Moving Vortex Lattice in YBa2Cu3O6.93 Detected by ac-dc Interference,” Phys. Rev. Lett. 74, 3684–3687.
  137. Heinze, S., K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, 2011, “Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions,” Nat. Phys. 7, 713–718.
  138. Henderson, W., E. Y. Andrei, M. J. Higgins, and S. Bhattacharya, 1996, “Metastability and Glassy Behavior of a Driven Flux-Line Lattice,” Phys. Rev. Lett. 77, 2077–2080.
  139. Herrero-Albillos, J., et al., 2018, “2D magnetic domain wall ratchet: The limit of submicromagnetic holes,” Mater. Des. 138, 111–118.
  140. Hess, H. F., R. B. Robinson, R. C. Dynes, J. M. Valles, and J. V. Waszczak, 1989, “Scanning-Tunneling-Microscope Observation of the Abrikosov Flux Lattice and the Density of States near and inside a Fluxoid,” Phys. Rev. Lett. 62, 214–216.
  141. Hirata, Y., et al., 2019, “Vanishing skyrmion Hall effect at the angular momentum compensation temperature of a ferrimagnet,” Nat. Nanotechnol. 14, 232–236.
  142. Hoffmann, M., B. Zimmermann, G. P. Müller, D. Schürhoff, N. S. Kiselev, C. Melcher, and S. Blügel, 2017, “Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions,” Nat. Commun. 8, 308.
  143. Hoshino, S., and N. Nagaosa, 2018, “Theory of the magnetic skyrmion glass,” Phys. Rev. B 97, 024413.
  144. Hrabec, A., J. Sampaio, M. Belmeguenai, I. Gross, R. Weil, S. M. Chérif, A. Stashkevich, V. Jacques, A. Thiaville, and S. Rohart, 2017, “Current-induced skyrmion generation and dynamics in symmetric bilayers,” Nat. Commun. 8, 15765.
  145. Hsu, P.-J., et al., 2018, “Inducing skyrmions in ultrathin Fe films by hydrogen exposure,” Nat. Commun. 9, 1571.
  146. Hu, J., and R. M. Westervelt, 1995, “Collective transport in two-dimensional magnetic bubble arrays,” Phys. Rev. B 51, 17279–17282.
  147. Huang, P., T. Schonenberger, M. Cantoni, L. Heinen, A. Magrez, A. Rosch, F. Carbone, and H. M. Rønnow, 2020, “Melting of a skyrmion lattice to a skyrmion liquid via a hexatic phase,” Nat. Nanotechnol. 15, 761.
  148. Hwa, T., P. Le Doussal, D. R. Nelson, and V. M. Vinokur, 1993, “Flux Pinning and Forced Vortex Entanglement by Splayed Columnar Defects,” Phys. Rev. Lett. 71, 3545–3548.
  149. Ikka, M., A. Takeuchi, and M. Mochizuki, 2018, “Resonance modes and microwave-driven translational motion of a skyrmion crystal under an inclined magnetic field,” Phys. Rev. B 98, 184428.
  150. Iwasaki, J., W. Koshibae, and N. Nagaosa, 2014, “Colossal spin transfer torque effect on skyrmion along the edge,” Nano Lett. 14, 4432–4437.
  151. Iwasaki, J., M. Mochizuki, and N. Nagaosa, 2013a, “Current-induced skyrmion dynamics in constricted geometries,” Nat. Nanotechnol. 8, 742–747.
  152. Iwasaki, J., M. Mochizuki, and N. Nagaosa, 2013b, “Universal current-velocity relation of skyrmion motion in chiral magnets,” Nat. Commun. 4, 1463.
  153. Jani, H., et al., 2021, “Antiferromagnetic half-skyrmions and bimerons at room temperature,” Nature (London) 590, 74.
  154. Jena, J., B. Göbel, T. Ma, V. Kumar, R. Saha, I. Mertig, C. Felser, and S. S. P. Parkin, 2020, “Elliptical Bloch skyrmion chiral twins in an antiskyrmion system,” Nat. Commun. 11, 1115.
  155. Jensen, H. J., A. Brass, and A. J. Berlinsky, 1988, “Lattice Deformations and Plastic Flow through Bottlenecks in a Two-Dimensional Model for Flux Pinning in Type-II Superconductors,” Phys. Rev. Lett. 60, 1676–1679.
  156. Jiang, W., G. Chen, K. Liu, J. Zang, S. G. E. te Velthuis, and A. Hoffmann, 2017, “Skyrmions in magnetic multilayers,” Phys. Rep. 704, 1–49.
  157. Jiang, W., et al., 2015, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286.
  158. Jiang, W., et al., 2017, “Direct observation of the skyrmion Hall effect,” Nat. Phys. 13, 162–169.
  159. Jin, Z., et al., 2020, “Dynamics of antiferromagnetic skyrmions in the absence or presence of pinning defects,” Phys. Rev. B 102, 054419.
  160. Jonietz, F., et al., 2010, “Spin transfer torques in MnSi at ultralow current densities,” Science 330, 1648–1651.
  161. Juge, R., et al., 2019, “Current-Driven Skyrmion Dynamics and Drive-Dependent Skyrmion Hall Effect in an Ultrathin Film,” Phys. Rev. Applied 12, 044007.
  162. Juge, R., et al., 2021, “Helium ions put magnetic skyrmions on the track,” Nano Lett. 21, 2989–2996.
  163. Juniper, M. P. N., A. V. Straube, R. Besseling, D. G. A. L. Aarts, and R. P. A. Dullens, 2015, “Microscopic dynamics of synchronization in driven colloids,” Nat. Commun. 6, 7187.
  164. Kafri, Y., D. R. Nelson, and A. Polkovnikov, 2007, “Unzipping vortices in type-II superconductors,” Phys. Rev. B 76, 144501.
  165. Kagawa, F., H. Oike, W. Koshibae, A. Kikkawa, Y. Okamura, Y. Taguchi, N. Nagaosa, and Y. Tokura, 2017, “Current-induced viscoelastic topological unwinding of metastable skyrmion strings,” Nat. Commun. 8, 1332.
  166. Kang, W., Y. Huang, C. Zheng, W. Lv, N. Lei, Y. Zhang, X. Zhang, Y. Zhou, and W. Zhao, 2016, “Voltage controlled magnetic skyrmion motion for racetrack memory,” Sci. Rep. 6, 23164.
  167. Kardar, M., 1998, “Nonequilibrium dynamics of interfaces and lines,” Phys. Rep. 301, 85–112.
  168. Karube, K., et al., 2016, “Robust metastable skyrmions and their triangular-square lattice structural transition in a high-temperature chiral magnet,” Nat. Mater. 15, 1237–1242.
  169. Karube, K., et al., 2018, “Disordered skyrmion phase stabilized by magnetic frustration in a chiral magnet,” Sci. Adv. 4, eaar7043.
  170. Kemmler, M., C. Gürlich, A. Sterck, H. Pöhler, M. Neuhaus, M. Siegel, R. Kleiner, and D. Koelle, 2006, “Commensurability Effects in Superconducting Nb Films with Quasiperiodic Pinning Arrays,” Phys. Rev. Lett. 97, 147003.
  171. Kent, N., et al., 2021, “Creation and observation of hopfions in magnetic multilayer systems,” Nat. Commun. 12, 1562.
  172. Kim, J.-V., and M.-W. Yoo, 2017, “Current-driven skyrmion dynamics in disordered films,” Appl. Phys. Lett. 110, 132404.
  173. Kindervater, J., et al., 2020, “Evolution of magnetocrystalline anisotropies in Mn1xFexSi and Mn1xCoxSi as inferred from small-angle neutron scattering and bulk properties,” Phys. Rev. B 101, 104406.
  174. Kirkpatrick, S., C. D. Gelatt, and M. P. Vecchi, 1983, “Optimization by simulated annealing,” Science 220, 671–680.
  175. Klein, T., I. Joumard, S. Blanchard, J. Marcus, R. Cubitt, T. Giamarchi, and P. Le Doussal, 2001, “A Bragg glass phase in the vortex lattice of a type II superconductor,” Nature (London) 413, 404–406.
  176. Klongcheongsan, T., T. J. Bullard, and U. C. Täuber, 2010, “Nonequilibrium steady states of driven magnetic flux lines in disordered type-II superconductors,” Supercond. Sci. Technol. 23, 025023.
  177. Kolesnikov, A. G., M. E. Stebliy, A. S. Samardak, and A. V. Ognev, 2018, “Skyrmionium—High velocity without the skyrmion Hall effect,” Sci. Rep. 8, 16966.
  178. Kolton, A. B., D. Domínguez, and N. Grønbech-Jensen, 1999, “Hall Noise and Transverse Freezing in Driven Vortex Lattices,” Phys. Rev. Lett. 83, 3061–3064.
  179. Kolton, A. B., R. Exartier, L. F. Cugliandolo, D. Domínguez, and N. Grønbech-Jensen, 2002, “Effective Temperature in Driven Vortex Lattices with Random Pinning,” Phys. Rev. Lett. 89, 227001.
  180. Kong, L., and J. Zang, 2013, “Dynamics of an Insulating Skyrmion under a Temperature Gradient,” Phys. Rev. Lett. 111, 067203.
  181. Korda, P. T., M. B. Taylor, and D. G. Grier, 2002, “Kinetically Locked-In Colloidal Transport in an Array of Optical Tweezers,” Phys. Rev. Lett. 89, 128301.
  182. Koshelev, A. E., and V. M. Vinokur, 1994, “Dynamic Melting of the Vortex Lattice,” Phys. Rev. Lett. 73, 3580–3583.
  183. Koshibae, W., and N. Nagaosa, 2018, “Theory of current-driven skyrmions in disordered magnets,” Sci. Rep. 8, 6328.
  184. Koshibae, W., and N. Nagaosa, 2019, “Dynamics of skyrmion in disordered chiral magnet of thin film form,” Sci. Rep. 9, 5111.
  185. Kosterlitz, J. M., and D. J. Thouless, 1973, “Ordering, metastability and phase-transitions in two-dimensional systems,” J. Phys. C 6, 1181–1203.
  186. Koushik, R., S. Kumar, K. R. Amin, M. Mondal, J. Jesudasan, A. Bid, P. Raychaudhuri, and A. Ghosh, 2013, “Correlated Conductance Fluctuations close to the Berezinskii-Kosterlitz-Thouless Transition in Ultrathin NbN Films,” Phys. Rev. Lett. 111, 197001.
  187. Kovalev, A. A., 2014, “Skyrmionic spin Seebeck effect via dissipative thermomagnonic torques,” Phys. Rev. B 89, 241101.
  188. Kovalev, A. A., and S. Sandhoefner, 2018, “Skyrmions and antiskyrmions in quasi-two-dimensional magnets,” Front. Phys. 6, 98.
  189. Kruchkov, A. J., J. S. White, M. Bartkowiak, I. Živković, A. Magrez, and H. M. Rønnow, 2018, “Direct electric field control of the skyrmion phase in a magnetoelectric insulator,” Sci. Rep. 8, 10466.
  190. Kumar, M., A. Laitinen, and P. Hakonen, 2018, “Unconventional fractional quantum Hall states and Wigner crystallization in suspended Corbino graphene,” Nat. Commun. 9, 2776.
  191. Lai, P., G. P. Zhao, H. Tang, N. Ran, S. Q. Wu, J. Xia, X. Zhang, and Y. Zhou, 2017, “An improved racetrack structure for transporting a skyrmion,” Sci. Rep. 7, 45330.
  192. Latimer, M. L., G. R. Berdiyorov, Z. L. Xiao, F. M. Peeters, and W. K. Kwok, 2013, “Realization of Artificial Ice Systems for Magnetic Vortices in a Superconducting MoGe Thin Film with Patterned Nanostructures,” Phys. Rev. Lett. 111, 067001.
  193. Lavergne, F. A., A. Curran, D. G. A. L. Aarts, and R. P. A. Dullens, 2018, “Dislocation-controlled formation and kinetics of grain boundary loops in two-dimensional crystals,” Proc. Natl. Acad. Sci. U.S.A. 115, 6922–6927.
  194. Lee, C. S., B. Jankó, I. Derényi, and A. L. Barabási, 1999, “Reducing vortex density in superconductors using the ‘ratchet effect,’ ” Nature (London) 400, 337–340.
  195. Legrand, W., D. Maccariello, N. Reyren, K. Garcia, C. Moutafis, C. Moreau-Luchaire, S. Collin, K. Bouzehouane, V. Cros, and A. Fert, 2017, “Room-temperature current-induced generation and motion of sub-100 nm skyrmions,” Nano Lett. 17, 2703–2712.
  196. Leliaert, J., M. Dvornik, J. Mulkers, J. De Clercq, M. V. Milošević, and B. Van Waeyenberge, 2018, “Fast micromagnetic simulations on GPU-recent advances made with mumax(3),” J. Phys. D 51, 123002.
  197. Leliaert, J., P. Gypens, M. V. Milošević, B. Van Waeyenberge, and J. Mulkers, 2019, “Coupling of the skyrmion velocity to its breathing mode in periodically notched nanotracks,” J. Phys. D 52, 024003.
  198. Leonov, A. O., and M. Mostovoy, 2015, “Multiply periodic states and isolated skyrmions in an anisotropic frustrated magnet,” Nat. Commun. 6, 8275.
  199. Leonov, A. O., and C. Pappas, 2019, “Skyrmion clusters and conical droplets in bulk helimagnets with cubic anisotropy,” Phys. Rev. B 99, 144410.
  200. Leroux, M., M. J. Stolt, S. Jin, D. V. Pete, C. Reichhardt, and B. Maiorov, 2018, “Skyrmion lattice topological Hall effect near room temperature,” Sci. Rep. 8, 15510.
  201. Le Thien, Q., D. McDermott, C. J. O. Reichhardt, and C. Reichhardt, 2017, “Enhanced pinning for vortices in hyperuniform pinning arrays and emergent hyperuniform vortex configurations with quenched disorder,” Phys. Rev. B 96, 094516.
  202. Levy, J., and M. S. Sherwin, 1991, “Poincaré Sections of Charge-Density-Wave Dynamics: Mode Locking,” Phys. Rev. Lett. 67, 2846–2849.
  203. Li, B., and A. A. Kovalev, 2020, “Magnon Landau Levels and Spin Responses in Antiferromagnets,” Phys. Rev. Lett. 125, 257201.
  204. Li, S., W. Kang, X. Zhang, T. Nie, Y. Zhou, K. L. Wang, and W. Zhao, 2021, “Magnetic skyrmions for unconventional computing,” Mater. Horiz. 8, 854–868.
  205. Li, S., J. Xia, X. Zhang, M. Ezawa, W. Kang, X. Liu, Y. Zhou, and W. Zhao, 2018, “Dynamics of a magnetic skyrmionum driven by spin waves,” Appl. Phys. Lett. 112, 142404.
  206. Li, Z.-A., F. Zheng, A. H. Tavabi, J. Caron, C. Jin, H. Du, A. Kovács, M. Tian, M. Farle, and R. E. Dunin-Borkowski, 2017, “Magnetic skyrmion formation at lattice defects and grain boundaries studied by quantitative off-axis electron holography,” Nano Lett. 17, 1395–1401.
  207. Liang, D., J. P. DeGrave, M. J. Stolt, Y. Tokura, and S. Jin, 2015, “Current-driven dynamics of skyrmions stabilized in MnSi nanowires revealed by topological Hall effect,” Nat. Commun. 6, 8217.
  208. Liang, X., G. Zhao, L. Shen, J. Xia, L. Zhao, X. Zhang, and Y. Zhou, 2019, “Dynamics of an antiferromagnetic skyrmion in a racetrack with a defect,” Phys. Rev. B 100, 144439.
  209. Libál, A., C. J. Olson Reichhardt, and C. Reichhardt, 2009, “Creating Artificial Ice States Using Vortices in Nanostructured Superconductors,” Phys. Rev. Lett. 102, 237004.
  210. Lin, N. S., T. W. Heitmann, K. Yu, B. L. T. Plourde, and V. R. Misko, 2011, “Rectification of vortex motion in a circular ratchet channel,” Phys. Rev. B 84, 144511.
  211. Lin, S.-Z., 2016, “Edge instability in a chiral stripe domain under an electric current and skyrmion generation,” Phys. Rev. B 94, 020402.
  212. Lin, S.-Z., and C. D. Batista, 2018, “Face Centered Cubic and Hexagonal Close Packed Skyrmion Crystals in Centrosymmetric Magnets,” Phys. Rev. Lett. 120, 077202.
  213. Lin, S.-Z., C. D. Batista, C. Reichhardt, and A. Saxena, 2014, “ac Current Generation in Chiral Magnetic Insulators and Skyrmion Motion Induced by the Spin Seebeck Effect,” Phys. Rev. Lett. 112, 187203.
  214. Lin, S.-Z., and S. Hayami, 2016, “Ginzburg-Landau theory for skyrmions in inversion-symmetric magnets with competing interactions,” Phys. Rev. B 93, 064430.
  215. Lin, S.-Z., C. Reichhardt, C. D. Batista, and A. Saxena, 2013a, “Driven Skyrmions and Dynamical Transitions in Chiral Magnets,” Phys. Rev. Lett. 110, 207202.
  216. Lin, S.-Z., C. Reichhardt, C. D. Batista, and A. Saxena, 2013b, “Particle model for skyrmions in metallic chiral magnets: Dynamics, pinning, and creep,” Phys. Rev. B 87, 214419.
  217. Lin, S.-Z., and A. Saxena, 2016, “Dynamics of Dirac strings and monopolelike excitations in chiral magnets under a current drive,” Phys. Rev. B 93, 060401.
  218. Lin, S.-Z., J.-X. Zhu, and A. Saxena, 2019, “Kelvin modes of a skyrmion line in chiral magnets and the associated magnon transport,” Phys. Rev. B 99, 140408.
  219. Litzius, K., et al., 2017, “Skyrmion Hall effect revealed by direct time-resolved x-ray microscopy,” Nat. Phys. 13, 170–175.
  220. Litzius, K., et al., 2020, “The role of temperature and drive current in skyrmion dynamics,” Nat. Electron. 3, 30–36.
  221. Liu, L., W. Chen, and Y. Zheng, 2020, “Current-Driven Skyrmion Motion beyond Linear Regime: Interplay between Skyrmion Transport and Deformation,” Phys. Rev. Applied 14, 024077.
  222. Liu, Y., W. Hou, X. Han, and J. Zang, 2020, “Three-Dimensional Dynamics of a Magnetic Hopfion Driven by Spin Transfer Torque,” Phys. Rev. Lett. 124, 127204.
  223. Liu, Y., N. Lei, C. Wang, X. Zhang, W. Kang, D. Zhu, Y. Zhou, X. Liu, Y. Zhang, and W. Zhao, 2019, “Voltage-Driven High-Speed Skyrmion Motion in a Skyrmion-Shift Device,” Phys. Rev. Applied 11, 014004.
  224. Liu, Y.-H., and Y.-Q. Li, 2013, “A mechanism to pin skyrmions in chiral magnets,” J. Phys. Condens. Matter 25, 076005.
  225. Lonsky, M., and A. Hoffmann, 2020, “Dynamic excitations of chiral magnetic textures,” APL Mater. 8, 100903.
  226. Loreto, R. P., X. Zhang, Y. Zhou, M. Ezawa, X. Liu, and C. I. L. de Araujo, 2019, “Manipulation of magnetic skyrmions in a locally modified synthetic antiferromagnetic racetrack,” J. Magn. Magn. Mater. 482, 155–159.
  227. Loudon, J. C., A. O. Leonov, A. N. Bogdanov, M. Ciomaga Hatnean, and G. Balakrishnan, 2018, “Direct observation of attractive skyrmions and skyrmion clusters in the cubic helimagnet Cu2OSeO3,” Phys. Rev. B 97, 134403.
  228. Luo, M.-B., and X. Hu, 2007, “Depinning and Creep Motion in Glass States of Flux Lines,” Phys. Rev. Lett. 98, 267002.
  229. Luo, S., M. Song, X. Li, Y. Zhang, J. Hong, X. Yang, X. Zou, N. Xu, and L. You, 2018, “Reconfigurable skyrmion logic gates,” Nano Lett. 18, 1180–1184.
  230. Luo, S., and L. You, 2021, “Skyrmion devices for memory and logic applications,” APL Mater. 9, 050901.
  231. Luo, Y., S.-Z. Lin, D. M. Fobes, Z. Liu, E. D. Bauer, J. B. Betts, A. Migliori, J. D. Thompson, M. Janoschek, and B. Maiorov, 2018, “Anisotropic magnetocrystalline coupling of the skyrmion lattice in MnSi,” Phys. Rev. B 97, 104423.
  232. Luo, Y., et al., 2020, “Skyrmion lattice creep at ultra-low current densities,” Commun. Mater. 1, 83.
  233. Ma, C., X. Zhang, J. Xia, M. Ezawa, W. Jiang, T. Ono, S. N. Piramanayagam, A. Morisako, Y. Zhou, and X. Liu, 2019, “Electric field-induced creation and directional motion of domain walls and skyrmion bubbles,” Nano Lett. 19, 353–361.
  234. Ma, F., C. Reichhardt, W. Gan, C. J. O. Reichhardt, and W. S. Lew, 2016, “Emergent geometric frustration of artificial magnetic skyrmion crystals,” Phys. Rev. B 94, 144405.
  235. Ma, X., C. J. Olson Reichhardt, and C. Reichhardt, 2017, “Reversible vector ratchets for skyrmion systems,” Phys. Rev. B 95, 104401.
  236. Ma, X., C. J. O. Reichhardt, and C. Reichhardt, 2020, “Braiding Majorana fermions and creating quantum logic gates with vortices on a periodic pinning structure,” Phys. Rev. B 101, 024514.
  237. Maccariello, D., W. Legrand, N. Reyren, K. Garcia, K. Bouzehouane, S. Collin, V. Cros, and A. Fert, 2018, “Electrical detection of single magnetic skyrmions in metallic multilayers at room temperature,” Nat. Nanotechnol. 13, 233–237.
  238. MacDonald, M. P., G. C. Spalding, and K. Dholakia, 2003, “Microfluidic sorting in an optical lattice,” Nature (London) 426, 421–424.
  239. MacKinnon, C. R., S. Lepadatu, T. Mercer, and P. R. Bissell, 2020, “Role of an additional interfacial spin-transfer torque for current-driven skyrmion dynamics in chiral magnetic layers,” Phys. Rev. B 102, 214408.
  240. Mankalale, M. G., Z. Zhao, J.-P. Wang, and S. S. Sapatnekar, 2019, “SkyLogic—A proposal for a skyrmion logic device,” IEEE Trans. Electron Devices 66, 1990–1996.
  241. Marchiori, E., P. J. Curran, J. Kim, N. Satchell, G. Burnell, and S. J. Bending, 2017, “Reconfigurable superconducting vortex pinning potential for magnetic disks in hybrid structures,” Sci. Rep. 7, 45182.
  242. Marconi, V. I., A. B. Kolton, J. A. Capitán, J. A. Cuesta, A. Pérez-Junquera, M. Vélez, J. I. Martín, and J. M. R. Parrondo, 2011, “Crossed-ratchet effects and domain wall geometrical pinning,” Phys. Rev. B 83, 214403.
  243. Marley, A. C., M. J. Higgins, and S. Bhattacharya, 1995, “Flux Flow Noise and Dynamical Transitions in a Flux Line Lattice,” Phys. Rev. Lett. 74, 3029–3032.
  244. Marrows, C. H., and K. Zeissler, 2021, “Perspective on skyrmion spintronics,” Appl. Phys. Lett. 119, 250502.
  245. Martín, J. I., M. Vélez, J. Nogués, and I. K. Schuller, 1997, “Flux Pinning in a Superconductor by an Array of Submicrometer Magnetic Dots,” Phys. Rev. Lett. 79, 1929–1932.
  246. Martinez, J. C., W. S. Lew, W. L. Gan, and M. B. A. Jalil, 2018, “Theory of current-induced skyrmion dynamics close to a boundary,” J. Magn. Magn. Mater. 465, 685–691.
  247. Martinoli, P., O. Daldini, C. Leemann, and E. Stocker, 1975, “ac quantum interference in superconducting films with periodically modulated thickness,” Solid State Commun. 17, 205–209.
  248. Mascot, E., J. Bedow, M. Graham, S. Rachel, and D. K. Morr, 2021, “Topological superconductivity in skyrmion lattices,” npj Quantum Mater. 6, 6.
  249. Masell, J., D. R. Rodrigues, B. F. McKeever, and K. Everschor-Sitte, 2020, “Spin-transfer torque driven motion, deformation, and instabilities of magnetic skyrmions at high currents,” Phys. Rev. B 101, 214428.
  250. Matsuda, T., K. Harada, H. Kasai, O. Kamimura, and A. Tonomura, 1996, “Observation of dynamic interaction of vortices with pinning centers by Lorentz microscopy,” Science 271, 1393–1395.
  251. Matsumoto, T., Y.-G. So, Y. Kohno, H. Sawada, Y. Ikuhara, and N. Shibata, 2016a, “Direct observation of Σ7 domain boundary core structure in magnetic skyrmion lattice,” Sci. Adv. 2, e1501280.
  252. Matsumoto, T., Y.-G. So, Y. Kohno, H. Sawada, R. Ishikawa, Y. Ikuhara, and N. Shibata, 2016b, “Jointed magnetic skyrmion lattices at a small-angle grain boundary directly visualized by advanced electron microscopy,” Sci. Rep. 6, 35880.
  253. Mehta, A. P., A. C. Mills, K. A. Dahmen, and J. P. Sethna, 2002, “Universal pulse shape scaling function and exponents: Critical test for avalanche models applied to Barkhausen noise,” Phys. Rev. E 65, 046139.
  254. Menezes, R. M., J. Mulkers, C. C. de Souza Silva, and M. V. Milošević, 2019, “Deflection of ferromagnetic and antiferromagnetic skyrmions at heterochiral interfaces,” Phys. Rev. B 99, 104409.
  255. Menezes, R. M., J. F. S. Neto, C. C. de Souza Silva, and M. V. Milošević, 2019, “Manipulation of magnetic skyrmions by superconducting vortices in ferromagnet-superconductor heterostructures,” Phys. Rev. B 100, 014431.
  256. Merithew, R. D., M. W. Rabin, M. B. Weissman, M. J. Higgins, and S. Bhattacharya, 1996, “Persistent Metastable States in Vortex Flow at the Peak Effect in NbSe2,” Phys. Rev. Lett. 77, 3197–3199.
  257. Migita, K., K. Yamada, and Y. Nakatani, 2020, “Controlling skyrmion motion in an angelfish-type racetrack memory by an ac magnetic field,” Appl. Phys. Express 13, 073003.
  258. Mikhael, J., J. Roth, L. Helden, and C. Bechinger, 2008, “Archimedean-like tiling on decagonal quasicrystalline surfaces,” Nature (London) 454, 501–504.
  259. Milde, P., et al., 2013, “Unwinding of a skyrmion lattice by magnetic monopoles,” Science 340, 1076–1080.
  260. Milošević, M. V., G. R. Berdiyorov, and F. M. Peeters, 2007, “Fluxonic cellular automata,” Appl. Phys. Lett. 91, 212501.
  261. Mirebeau, I., N. Martin, M. Deutsch, L. J. Bannenberg, C. Pappas, G. Chaboussant, R. Cubitt, C. Decorse, and A. O. Leonov, 2018, “Spin textures induced by quenched disorder in a reentrant spin glass: Vortices versus ‘frustrated’ skyrmions,” Phys. Rev. B 98, 014420.
  262. Mochizuki, M., X. Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, J. Zang, M. Mostovoy, Y. Tokura, and N. Nagaosa, 2014, “Thermally driven ratchet motion of a skyrmion microcrystal and topological magnon Hall effect,” Nat. Mater. 13, 241–246.
  263. Mohan, S., J. Sinha, S. S. Banerjee, A. K. Sood, S. Ramakrishnan, and A. K. Grover, 2009, “Large Low-Frequency Fluctuations in the Velocity of a Driven Vortex Lattice in a Single Crystal of 2H-NbSe2 Superconductor,” Phys. Rev. Lett. 103, 167001.
  264. Montoya, S. A., R. Tolley, I. Gilbert, S.-G. Je, M.-Y. Im, and E. E. Fullerton, 2018, “Spin-orbit torque induced dipole skyrmion motion at room temperature,” Phys. Rev. B 98, 104432.
  265. Moon, K., R. T. Scalettar, and G. T. Zimányi, 1996, “Dynamical Phases of Driven Vortex Systems,” Phys. Rev. Lett. 77, 2778–2781.
  266. Moreau-Luchaire, C., et al., 2016, “Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature,” Nat. Nanotechnol. 11, 444–448.
  267. Moretti, P., and M.-C. Miguel, 2009, “Irreversible flow of vortex matter: Polycrystal and amorphous phases,” Phys. Rev. B 80, 224513.
  268. Morin, A., N. Desreumaux, J.-B. Caussin, and D. Bartolo, 2017, “Distortion and destruction of colloidal flocks in disordered environments,” Nat. Phys. 13, 63–67.
  269. Mühlbauer, S., B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, 2009, “Skyrmion lattice in a chiral magnet,” Science 323, 915–919.
  270. Müller, J., 2017, “Magnetic skyrmions on a two-lane racetrack,” New J. Phys. 19, 025002.
  271. Müller, J., J. Rajeswari, P. Huang, Y. Murooka, H. M. Rønnow, F. Carbone, and A. Rosch, 2017, “Magnetic Skyrmions and Skyrmion Clusters in the Helical Phase of Cu2OSeO3,” Phys. Rev. Lett. 119, 137201.
  272. Müller, J., and A. Rosch, 2015, “Capturing of a magnetic skyrmion with a hole,” Phys. Rev. B 91, 054410.
  273. Nagaosa, N., and Y. Tokura, 2013, “Topological properties and dynamics of magnetic skyrmions,” Nat. Nanotechnol. 8, 899–911.
  274. Nakajima, H., A. Kotani, M. Mochizuki, K. Harada, and S. Mori, 2017, “Formation process of skyrmion lattice domain boundaries: The role of grain boundaries,” Appl. Phys. Lett. 111, 192401.
  275. Nakajima, T., H. Oike, A. Kikkawa, E. P. Gilbert, N. Booth, K. Kakurai, Y. Taguchi, Y. Tokura, F. Kagawa, and T. Arima, 2017, “Skyrmion lattice structural transition in MnSi,” Sci. Adv. 3, e1602562.
  276. Nattermann, T., and S. Scheidl, 2000, “Vortex-glass phases in type-II superconductors,” Adv. Phys. 49, 607–704.
  277. Navau, C., N. Del-Valle, and A. Sanchez, 2016, “Analytical trajectories of skyrmions in confined geometries: Skyrmionic racetracks and nano-oscillators,” Phys. Rev. B 94, 184104.
  278. Navau, C., N. Del-Valle, and A. Sanchez, 2018, “Interaction of isolated skyrmions with point and linear defects,” J. Magn. Magn. Mater. 465, 709–715.
  279. Nayak, A. K., V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U. K. Rößler, C. Felser, and S. S. P. Parkin, 2017, “Magnetic antiskyrmions above room temperature in tetragonal Heusler materials,” Nature (London) 548, 561–566.
  280. Nelson, D. R., 1983, “Reentrant melting in solid films with quenched random impurities,” Phys. Rev. B 27, 2902–2914.
  281. Nelson, D. R., 1988, “Vortex Entanglement in High-Tc Superconductors,” Phys. Rev. Lett. 60, 1973–1976.
  282. Nelson, D. R., and B. I. Halperin, 1979, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B 19, 2457–2484.
  283. Nepal, R., U. Güngördü, and A. A. Kovalev, 2018, “Magnetic skyrmion bubble motion driven by surface acoustic waves, Appl. Phys. Lett. 112, 112404.
  284. Neubauer, A., C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, 2009, “Topological Hall Effect in the A Phase of MnSi,” Phys. Rev. Lett. 102, 186602.
  285. Nishikawa, Y., K. Hukushima, and W. Krauth, 2019, “Solid-liquid transition of skyrmions in a two-dimensional chiral magnet,” Phys. Rev. B 99, 064435.
  286. Nozaki, T., Y. Jibiki, M. Goto, E. Tamura, T. Nozaki, H. Kubota, A. Fukushima, S. Yuasa, and Y. Suzuki, 2019, “Brownian motion of skyrmion bubbles and its control by voltage applications,” Appl. Phys. Lett. 114, 012402.
  287. Nych, A., J. Fukuda, U. Ognysta, S. Žumer, and I. Muševič, 2017, “Spontaneous formation and dynamics of half-skyrmions in a chiral liquid-crystal film,” Nat. Phys. 13, 1215.
  288. Ogawa, N., W. Koshibae, A. J. Beekman, N. Nagaosa, M. Kubota, M. Kawasaki, and Y. Tokura, 2015, “Photodrive of magnetic bubbles via magnetoelastic waves,” Proc. Natl. Acad. Sci. U.S.A. 112, 8977–8981.
  289. O’Hern, C. S., L. E. Silbert, A. J. Liu, and S. R. Nagel, 2003, “Jamming at zero temperature and zero applied stress: The epitome of disorder,” Phys. Rev. E 68, 011306.
  290. Okuma, S., H. Imaizumi, D. Shimamoto, and N. Kokubo, 2011, “Quantum melting and lattice orientation of driven vortex matter,” Phys. Rev. B 83, 064520.
  291. Okuma, S., J. Inoue, and N. Kokubo, 2007, “Suppression of broadband noise at mode locking in driven vortex matter,” Phys. Rev. B 76, 172503.
  292. Okuyama, D., et al., 2019, “Deformation of the moving magnetic skyrmion lattice in MnSi under electric current flow,” Commun. Phys. 2, 79.
  293. Olive, E., and J. C. Soret, 2006, “Chaotic Dynamics of Superconductor Vortices in the Plastic Phase,” Phys. Rev. Lett. 96, 027002.
  294. Olson, C. J., C. Reichhardt, and F. Nori, 1998a, “Fractal Networks, Braiding Channels, and Voltage Noise in Intermittently Flowing Rivers of Quantized Magnetic Flux,” Phys. Rev. Lett. 80, 2197–2200.
  295. Olson, C. J., C. Reichhardt, and F. Nori, 1998b, “Nonequilibrium Dynamic Phase Diagram for Vortex Lattices,” Phys. Rev. Lett. 81, 3757–3760.
  296. Olson, C. J., C. Reichhardt, R. T. Scalettar, G. T. Zimányi, and N. Grønbech-Jensen, 2003, “Metastability and transient effects in vortex matter near a decoupling transition,” Phys. Rev. B 67, 184523.
  297. Olson, C. J., G. T. Zimányi, A. B. Kolton, and N. Grønbech-Jensen, 2000, “Static and Dynamic Coupling Transitions of Vortex Lattices in Disordered Anisotropic Superconductors,” Phys. Rev. Lett. 85, 5416–5419.
  298. Olszewski, M. W., M. R. Eskildsen, C. Reichhardt, and C. J. O. Reichhardt, 2018, “Structural transitions in vortex systems with anisotropic interactions,” New J. Phys. 20, 023005.
  299. Onose, Y., Y. Okamura, S. Seki, S. Ishiwata, and Y. Tokura, 2012, “Observation of Magnetic Excitations of Skyrmion Crystal in a Helimagnetic Insulator Cu2OSeO3,” Phys. Rev. Lett. 109, 037603.
  300. Ortiz-Ambriz, A., and P. Tierno, 2016, “Engineering of frustration in colloidal artificial ices realized on microfeatured grooved lattices,” Nat. Commun. 7, 10575.
  301. Palermo, X., et al., 2020, “Tailored Flux Pinning in Superconductor-Ferromagnet Multilayers with Engineered Magnetic Domain Morphology from Stripes to Skyrmions,” Phys. Rev. Applied 13, 014043.
  302. Paltiel, Y., E. Zeldov, Y. N. Myasoedov, H. Shtrikman, S. Bhattacharya, M. J. Higgins, Z. L. Xiao, E. Y. Andrei, P. L. Gammel, and D. J. Bishop, 2000, “Dynamic instabilities and memory effects in vortex matter,” Nature (London) 403, 398–401.
  303. Pardo, F., F. de la Cruz, P. L. Gammel, E. Bucher, and D. J. Bishop, 1998, “Observation of smectic and moving-Bragg-glass phases in flowing vortex lattices,” Nature (London) 396, 348–350.
  304. Park, H. S., et al., 2014, “Observation of the magnetic flux and three-dimensional structure of skyrmion lattices by electron holography,” Nat. Nanotechnol. 9, 337–342.
  305. Pasquini, G., D. Pérez Daroca, C. Chiliotte, G. S. Lozano, and V. Bekeris, 2008, “Ordered, Disordered, and Coexistent Stable Vortex Lattices in NbSe2 Single Crystals,” Phys. Rev. Lett. 100, 247003.
  306. Pathak, S. A., and R. Hertel, 2021, “Geometrically constrained skyrmions,” Magnetochemistry 7, 26.
  307. Peng, L., R. Takagi, W. Koshibae, K. Shibata, K. Nakajima, T. Arima, N. Nagaosa, S. Seki, X. Yu, and Y. Tokura, 2020, “Controlled transformation of skyrmions and antiskyrmions in a non-centrosymmetric magnet,” Nat. Nanotechnol. 15, 181.
  308. Perković, O, K. Dahmen, and J. P. Sethna, 1995, “Avalanches, Barkhausen Noise, and Plain Old Criticality,” Phys. Rev. Lett. 75, 4528–4531.
  309. Pertsinidis, A., and X. S. Ling, 2008, “Statics and Dynamics of 2D Colloidal Crystals in a Random Pinning Potential,” Phys. Rev. Lett. 100, 028303.
  310. Petrović, A. P., et al., 2021, “Skyrmion-(Anti)Vortex Coupling in a Chiral Magnet-Superconductor Heterostructure,” Phys. Rev. Lett. 126, 117205.
  311. Pinna, D., F. Abreu Araujo, J.-V. Kim, V. Cros, D. Querlioz, P. Bessiere, J. Droulez, and J. Grollier, 2018, “Skyrmion Gas Manipulation for Probabilistic Computing,” Phys. Rev. Applied 9, 064018.
  312. Pinsolle, E., N. Kirova, V. L. R. Jacques, A. A. Sinchenko, and D. Le Bolloc’h, 2012, “Creep, Flow, and Phase Slippage Regimes: An Extensive View of the Sliding Charge-Density Wave Revealed by Coherent X-Ray Diffraction,” Phys. Rev. Lett. 109, 256402.
  313. Plettenberg, J., M. Stier, and M. Thorwart, 2020, “Steering of the Skyrmion Hall Angle by Gate Voltages,” Phys. Rev. Lett. 124, 207202.
  314. Pöllath, S., et al., 2017, “Dynamical Defects in Rotating Magnetic Skyrmion Lattices,” Phys. Rev. Lett. 118, 207205.
  315. Prychynenko, D., M. Sitte, K. Litzius, B. Krüger, G. Bourianoff, M. Kläui, J. Sinova, and K. Everschor-Sitte, 2018, “Magnetic Skyrmion as a Nonlinear Resistive Element: A Potential Building Block for Reservoir Computing,” Phys. Rev. Applied 9, 014034.
  316. Psaroudaki, C., and D. Loss, 2018, “Skyrmions Driven by Intrinsic Magnons,” Phys. Rev. Lett. 120, 237203.
  317. Puertas, A. M., and T. Voigtmann, 2014, “Microrheology of colloidal systems,” J. Phys. Condens. Matter 26, 243101.
  318. Purnama, I., W. L. Gan, D. W. Wong, and W. S. Lew, 2015, “Guided current-induced skyrmion motion in 1D potential well,” Sci. Rep. 5, 10620.
  319. Purnama, I., C. S. Murapaka, W. S. Lew, and T. Ono, 2014, “Remote driving of multiple magnetic domain walls due to topological interaction,” Appl. Phys. Lett. 104, 092414.
  320. Rajeswari, J., et al., 2015, “Filming the formation and fluctuation of skyrmion domains by cryo-Lorentz transmission electron microscopy,” Proc. Natl. Acad. Sci. U.S.A. 112, 14212–14217.
  321. Raju, M., A. Yagil, A. Soumyanarayanan, A. K. C. Tan, A. Almoalem, F. Ma, O. M. Auslaender, and C. Panagopoulos, 2019, “The evolution of skyrmions in Ir/Fe/Co/Pt multilayers and their topological Hall signature,” Nat. Commun. 10, 696.
  322. Ralph, D. C., and M. D. Stiles, 2008, “Spin transfer torques,” J. Magn. Magn. Mater. 320, 1190–1216.
  323. Reichhardt, C., and F. Nori, 1999, “Phase Locking, Devil’s Staircases, Farey Trees, and Arnold Tongues in Driven Vortex Lattices with Periodic Pinning,” Phys. Rev. Lett. 82, 414–417.
  324. Reichhardt, C., C. J. Olson, N. Grønbech-Jensen, and F. Nori, 2001, “Moving Wigner Glasses and Smectics: Dynamics of Disordered Wigner Crystals,” Phys. Rev. Lett. 86, 4354–4357.
  325. Reichhardt, C., C. J. Olson, and F. Nori, 1997, “Dynamic Phases of Vortices in Superconductors with Periodic Pinning,” Phys. Rev. Lett. 78, 2648–2651.
  326. Reichhardt, C., C. J. Olson, and F. Nori, 1998, “Commensurate and incommensurate vortex states in superconductors with periodic pinning arrays,” Phys. Rev. B 57, 7937–7943.
  327. Reichhardt, C., and C. J. Olson Reichhardt, 2015, “Shapiro steps for skyrmion motion on a washboard potential with longitudinal and transverse ac drives,” Phys. Rev. B 92, 224432.
  328. Reichhardt, C., and C. J. Olson Reichhardt, 2016, “Magnus-induced dynamics of driven skyrmions on a quasi-one-dimensional periodic substrate,” Phys. Rev. B 94, 094413.
  329. Reichhardt, C., D. Ray, and C. J. Olson Reichhardt, 2015, “Magnus-induced ratchet effects for skyrmions interacting with asymmetric substrates,” New J. Phys. 17, 073034.
  330. Reichhardt, C., D. Ray, and C. J. O. Reichhardt, 2015a, “Collective Transport Properties of Driven Skyrmions with Random Disorder,” Phys. Rev. Lett. 114, 217202.
  331. Reichhardt, C., D. Ray, and C. J. O. Reichhardt, 2015b, “Quantized transport for a skyrmion moving on a two-dimensional periodic substrate,” Phys. Rev. B 91, 104426.
  332. Reichhardt, C., D. Ray, and C. J. O. Reichhardt, 2018, “Nonequilibrium phases and segregation for skyrmions on periodic pinning arrays,” Phys. Rev. B 98, 134418.
  333. Reichhardt, C., and C. J. O. Reichhardt, 2016, “Noise fluctuations and drive dependence of the skyrmion Hall effect in disordered systems,” New J. Phys. 18, 095005.
  334. Reichhardt, C., and C. J. O. Reichhardt, 2017a, “Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: A review,” Rep. Prog. Phys. 80, 026501.
  335. Reichhardt, C., and C. J. O. Reichhardt, 2017b, “Shapiro spikes and negative mobility for skyrmion motion on quasi-one-dimensional periodic substrates,” Phys. Rev. B 95, 014412.
  336. Reichhardt, C., and C. J. O. Reichhardt, 2019a, “Nonlinear transport, dynamic ordering, and clustering for driven skyrmions on random pinning,” Phys. Rev. B 99, 104418.
  337. Reichhardt, C., and C. J. O. Reichhardt, 2019b, “Thermal creep and the skyrmion Hall angle in driven skyrmion crystals,” J. Phys. Condens. Matter 31, 07LT01.
  338. Reijnders, J. W., and R. A. Duine, 2004, “Pinning of Vortices in a Bose-Einstein Condensate by an Optical Lattice,” Phys. Rev. Lett. 93, 060401.
  339. Reimann, P., 2002, “Brownian motors: Noisy transport far from equilibrium,” Phys. Rep. 361, 57–265.
  340. Rex, S., I. V. Gornyi, and A. D. Mirlin, 2019, “Majorana bound states in magnetic skyrmions imposed onto a superconductor,” Phys. Rev. B 100, 064504.
  341. Risbud, S. R., and G. Drazer, 2014, “Directional locking in deterministic lateral-displacement microfluidic separation systems,” Phys. Rev. E 90, 012302.
  342. Ritzmann, U., L. Desplat, B. Dupé, R. E. Camley, and J.-V. Kim, 2020, “Asymmetric skyrmion-antiskyrmion production in ultrathin ferromagnetic films,” Phys. Rev. B 102, 174409.
  343. Ritzmann, U., S. von Malottki, J.-V. Kim, S. Heinze, J. Sinova, and B. Dupé, 2018, “Trochoidal motion and pair generation in skyrmion and antiskyrmion dynamics under spin-orbit torques,” Nat. Electron. 1, 451–457.
  344. Romming, N., C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, 2013, “Writing and deleting single magnetic skyrmions,” Science 341, 636–639.
  345. Ros, A., R. Eichhorn, J. Regtmeier, T. T. Duong, P. Reimann, and D. Anselmetti, 2005, “Brownian motion—Absolute negative particle mobility,” Nature (London) 436, 928.
  346. Rößler, U. K., A. N. Bogdanov, and C. Pfleiderer, 2006, “Spontaneous skyrmion ground states in magnetic metals,” Nature (London) 442, 797–801.
  347. Rousselet, J., L. Salome, A. Ajdari, and J. Prost, 1994, “Directional motion of Brownian particles induced by a periodic asymmetric potential,” Nature (London) 370, 446–448.
  348. Rózsa, L., A. Deák, E. Simon, R. Yanes, L. Udvardi, L. Szunyogh, and U. Nowak, 2016, “Skyrmions with Attractive Interactions in an Ultrathin Magnetic Film,” Phys. Rev. Lett. 117, 157205.
  349. Rybakov, F. N., A. B. Borisov, S. Blügel, and N. S. Kiselev, 2015, “New Type of Stable Particlelike States in Chiral Magnets,” Phys. Rev. Lett. 115, 117201.
  350. Rybakov, F. N., A. B. Borisov, S. Blügel, and N. S. Kiselev, 2016, “New spiral state and skyrmion lattice in 3D model of chiral magnets,” New J. Phys. 18, 045002.
  351. Rybakov, F. N., and N. S. Kiselev, 2019, “Chiral magnetic skyrmions with arbitrary topological charge,” Phys. Rev. B 99, 064437.
  352. Sadr-Lahijany, M. R., P. Ray, and H. E. Stanley, 1997, “Dispersity-Driven Melting Transition in Two-Dimensional Solids,” Phys. Rev. Lett. 79, 3206–3209.
  353. Safar, H., P. L. Gammel, D. A. Huse, D. J. Bishop, J. P. Rice, and D. M. Ginsberg, 1992, “Experimental Evidence for a First-Order Vortex-Lattice-Melting Transition in Untwinned, Single Crystal YBa2Cu3O7,” Phys. Rev. Lett. 69, 824–827.
  354. Saha, S., et al., 2019, “Formation of Néel-type skyrmions in an antidot lattice with perpendicular magnetic anisotropy,” Phys. Rev. B 100, 144435.
  355. Salger, T., S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, 2009, “Directed transport of atoms in a Hamiltonian quantum ratchet,” Science 326, 1241–1243.
  356. Salimath, A., A. Abbout, A. Brataas, and A. Manchon, 2019, “Current-driven skyrmion depinning in magnetic granular films,” Phys. Rev. B 99, 104416.
  357. Salomaa, M. M., and G. E. Volovik, 1987, “Quantized vortices in superfluid He3,” Rev. Mod. Phys. 59, 533–613.
  358. Sampaio, J., V. Cros, S. Rohart, A. Thiaville, and A. Fert, 2013, “Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures,” Nat. Nanotechnol. 8, 839–844.
  359. Sato, T., A. Kikkawa, Y. Taguchi, Y. Tokura, and F. Kagawa, 2020, “Mode locking phenomena of the current-induced skyrmion-lattice motion in microfabricated MnSi,” Phys. Rev. B 102, 180411.
  360. Sato, T., W. Koshibae, A. Kikkawa, T. Yokouchi, H. Oike, Y. Taguchi, N. Nagaosa, Y. Tokura, and F. Kagawa, 2019, “Slow steady flow of a skyrmion lattice in a confined geometry probed by narrow-band resistance noise,” Phys. Rev. B 100, 094410.
  361. Schulz, T., R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, 2012, “Emergent electrodynamics of skyrmions in a chiral magnet,” Nat. Phys. 8, 301–304.
  362. Schütte, C., J. Iwasaki, A. Rosch, and N. Nagaosa, 2014, “Inertia, diffusion, and dynamics of a driven skyrmion,” Phys. Rev. B 90, 174434.
  363. Schütte, C., and A. Rosch, 2014, “Dynamics and energetics of emergent magnetic monopoles in chiral magnets,” Phys. Rev. B 90, 174432.
  364. Seki, S., M. Garst, J. Waizner, R. Takagi, N. D. Khanh, Y. Okamura, K. Kondou, F. Kagawa, Y. Otani, and Y. Tokura, 2020, “Propagation dynamics of spin excitations along skyrmion strings,” Nat. Commun. 11, 256.
  365. Seki, S., X. Z. Yu, S. Ishiwata, and Y. Tokura, 2012, “Observation of skyrmions in a multiferroic material,” Science 336, 198–201.
  366. Sengupta, A., S. Sengupta, and G. I. Menon, 2010, “Driven disordered polymorphic solids: Phases and phase transitions, dynamical coexistence and peak effect anomalies,” Phys. Rev. B 81, 144521.
  367. Sethna, J. P., K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, 1993, “Hysteresis and Hierarchies: Dynamics of Disorder-Driven First-Order Phase Transformations,” Phys. Rev. Lett. 70, 3347–3350.
  368. Sethna, J. P., K. A. Dahmen, and C. R. Myers, 2001, “Crackling noise,” Nature (London) 410, 242–250.
  369. Shapiro, S., 1963, “Josephson Currents in Superconducting Tunneling: The Effect of Microwaves and Other Observations,” Phys. Rev. Lett. 11, 80–82.
  370. Shaw, G., P. Mandal, S. S. Banerjee, A. Niazi, A. K. Rastogi, A. K. Sood, S. Ramakrishnan, and A. K. Grover, 2012, “Critical behavior at depinning of driven disordered vortex matter in 2H-NbS2,” Phys. Rev. B 85, 174517.
  371. Shen, L. C., J. Xia, G. P. Zhao, X. C. Zhang, M. Ezawa, O. A. Tretiakov, X. X. Liu, and Y. Zhou, 2018, “Dynamics of the antiferromagnetic skyrmion induced by a magnetic anisotropy gradient,” Phys. Rev. B 98, 134448.
  372. Shen, M., Y. Zhang, J. Ou-Yang, X. Yang, and L. You, 2018, “Motion of a skyrmionium driven by spin wave,” Appl. Phys. Lett. 112, 062403.
  373. Shibata, K., T. Tanigaki, T. Akashi, H. Shinada, K. Harada, K. Niitsu, D. Shindo, N. Kanazawa, Y. Tokura, and T. Arima, 2018, “Current-driven motion of domain boundaries between skyrmion lattice and helical magnetic structure,” Nano Lett. 18, 929–933.
  374. Shklovskij, V. A., V. V. Sosedkin, and O. V. Dobrovolskiy, 2014, “Vortex ratchet reversal in an asymmetric washboard pinning potential subject to combined dc and ac stimuli,” J. Phys. Condens. Matter 26, 025703.
  375. Silva, R. L., L. D. Secchin, W. A. Moura-Melo, A. R. Pereira, and R. L. Stamps, 2014, “Emergence of skyrmion lattices and bimerons in chiral magnetic thin films with nonmagnetic impurities,” Phys. Rev. B 89, 054434.
  376. Singh, A., et al., 2019, “Scaling of domain cascades in stripe and skyrmion phases,” Nat. Commun. 10, 1988.
  377. Skyrme, T. H., 1961, “A non-linear field theory,” Proc. R. Soc. A 260, 127.
  378. Skyrme, T. H. R., 1962, “A unified field theory of mesons and baryons,” Nucl. Phys. 31, 556.
  379. Sondhi, S. L., A. Karlhede, S. A. Kivelson, and E. H. Rezayi, 1993, “Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies,” Phys. Rev. B 47, 16419–16426.
  380. Song, K. M., et al., 2020, “Skyrmion-based artificial synapses for neuromorphic computing,” Nat. Electron. 3, 148–155.
  381. Soumyanarayanan, A., et al., 2017, “Tunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayers,” Nat. Mater. 16, 898–904.
  382. Stier, M., R. Strobel, S. Krause, W. Häusler, and M. Thorwart, 2021, “Role of impurity clusters for the current-driven motion of magnetic skyrmions,” Phys. Rev. B 103, 054420.
  383. Stosic, D., T. B. Ludermir, and M. V. Milošević, 2017, “Pinning of magnetic skyrmions in a monolayer Co film on Pt(111): Theoretical characterization and exemplified utilization,” Phys. Rev. B 96, 214403.
  384. Strandburg, K. J., 1988, “Two-dimensional melting,” Rev. Mod. Phys. 60, 161–207.
  385. Straver, E. W. J., J. E. Hoffman, O. M. Auslaender, D. Rugar, and K. A. Moler, 2008, “Controlled manipulation of individual vortices in a superconductor,” Appl. Phys. Lett. 93, 172514.
  386. Suess, D., C. Vogler, F. Bruckner, P. Heistracher, and C. Abert, 2018, “A repulsive skyrmion chain as a guiding track for a racetrack memory,” AIP Adv. 8, 115301.
  387. Suess, D., C. Vogler, F. Bruckner, P. Heistracher, F. Slanovc, and C. Abert, 2019, “Spin torque efficiency and analytic error rate estimates of skyrmion racetrack memory,” Sci. Rep. 9, 4827.
  388. Sun, L., R. X. Cao, B. F. Miao, Z. Feng, B. You, D. Wu, W. Zhang, A. Hu, and H. F. Ding, 2013, “Creating an Artificial Two-Dimensional Skyrmion Crystal by Nanopatterning,” Phys. Rev. Lett. 110, 167201.
  389. Sun, L., H. Z. Wu, B. F. Miao, D. Wu, and H. F. Ding, 2018, “Tuning the stability and the skyrmion Hall effect in magnetic skyrmions by adjusting their exchange strengths with magnetic disks,” J. Magn. Magn. Mater. 455, 39.
  390. Takagi, R., Y. Yamasaki, T. Yokouchi, V. Ukleev, Y. Yokoyama, H. Nakao, T. Arima, Y. Tokura, and S. Seki, 2020, “Particle-size dependent structural transformation of skyrmion lattice,” Nat. Commun. 11, 5685.
  391. Takagi, R., X. Z. Yu, J. S. White, K. Shibata, Y. Kaneko, G. Tatara, H. M. Rønnow, Y. Tokura, and S. Seki, 2018, “Low-Field Bi-Skyrmion Formation in a Noncentrosymmetric Chimney Ladder Ferromagnet,” Phys. Rev. Lett. 120, 037203.
  392. Tatara, G., H. Kohno, and J. Shibata, 2008, “Microscopic approach to current-driven domain wall dynamics,” Phys. Rep. 468, 213–301.
  393. Thiele, A. A., 1973, “Steady-State Motion of Magnetic Domains,” Phys. Rev. Lett. 30, 230–233.
  394. Thorneywork, A. L., J. L. Abbott, D. G. A. L. Aarts, and R. P. A. Dullens, 2017, “Two-Dimensional Melting of Colloidal Hard Spheres,” Phys. Rev. Lett. 118, 158001.
  395. Tierno, P., 2012, “Depinning and Collective Dynamics of Magnetically Driven Colloidal Monolayers,” Phys. Rev. Lett. 109, 198304.
  396. Toft-Petersen, R., A. B. Abrahamsen, S. Balog, L. Porcar, and M. Laver, 2018, “Decomposing the Bragg glass and the peak effect in a type-II superconductor,” Nat. Commun. 9, 901.
  397. Togawa, Y., R. Abiru, K. Iwaya, H. Kitano, and A. Maeda, 2000, “Direct Observation of the Washboard Noise of a Driven Vortex Lattice in a High-Temperature Superconductor, Bi2Sr2CaCu2Oy,” Phys. Rev. Lett. 85, 3716–3719.
  398. Tokunaga, Y., X. Z. Yu, J. S. White, H. M. Rønnow, D. Morikawa, Y. Taguchi, and Y. Tokura, 2015, “A new class of chiral materials hosting magnetic skyrmions beyond room temperature,” Nat. Commun. 6, 7638.
  399. Tokura, Y., and N. Kanazawa, 2021, “Magnetic skyrmion materials,” Chem. Rev. 121, 2857–2897.
  400. Tolley, R., S. A. Montoya, and E. E. Fullerton, 2018, “Room-temperature observation and current control of skyrmions in Pt/Co/Os/Pt thin films,” Phys. Rev. Mater. 2, 044404.
  401. Tomasello, R., E. Martinez, R. Zivieri, L. Torres, M. Carpentieri, and G. Finocchio, 2014, “A strategy for the design of skyrmion racetrack memories,” Sci. Rep. 4, 6784.
  402. Tomasello, R., et al., 2018, “Micromagnetic understanding of the skyrmion Hall angle current dependence in perpendicularly magnetized ferromagnets,” Phys. Rev. B 98, 224418.
  403. Tong, Q., F. Liu, J. Xiao, and W. Yao, 2018, “Skyrmions in the moiré of van der Waals 2D magnets,” Nano Lett. 18, 7194–7199.
  404. Torquato, S., 2016, “Hyperuniformity and its generalizations,” Phys. Rev. E 94, 022122.
  405. Toscano, D., S. A. Leonel, P. Z. Coura, and F. Sato, 2019, “Building traps for skyrmions by the incorporation of magnetic defects into nanomagnets: Pinning and scattering traps by magnetic properties engineering,” J. Magn. Magn. Mater. 480, 171–185.
  406. Travesset, A., R. A. White, and K. A. Dahmen, 2002, “Crackling noise, power spectra, and disorder-induced critical scaling,” Phys. Rev. B 66, 024430.
  407. Troncoso, R. E., and A. S. Núñez, 2014, “Thermally assisted current-driven skyrmion motion,” Phys. Rev. B 89, 224403.
  408. Tsesses, S., E. Ostrovsky, K. Cohen, B. Gjonaj, N. H. Lindner, and G. Bartal, 2018, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361, 993–996.
  409. Tsoi, M., R. E. Fontana, and S. S. P. Parkin, 2003, “Magnetic domain wall motion triggered by an electric current,” Appl. Phys. Lett. 83, 2617–2619.
  410. Tung, S., V. Schweikhard, and E. A. Cornell, 2006, “Observation of Vortex Pinning in Bose-Einstein Condensates,” Phys. Rev. Lett. 97, 240402.
  411. Vakili, H., et al., 2021, “Skyrmionics—Computing and memory technologies based on topological excitations in magnets,” J. Appl. Phys. 130, 070908.
  412. Van Look, L., E. Rosseel, M. J. Van Bael, K. Temst, V. V. Moshchalkov, and Y. Bruynseraede, 1999, “Shapiro steps in a superconducting film with an antidot lattice,” Phys. Rev. B 60, R6998–R7000.
  413. Vanossi, A., N. Manini, M. Urbakh, S. Zapperi, and E. Tosatti, 2013, “Colloquium: Modeling friction: From nanoscale to mesoscale,” Rev. Mod. Phys. 85, 529–552.
  414. Villegas, J. E., M. I. Montero, C.-P. Li, and I. K. Schuller, 2006, “Correlation Length of Quasiperiodic Vortex Lattices,” Phys. Rev. Lett. 97, 027002.
  415. Villegas, J. E., S. Savel’ev, F. Nori, E. M. Gonzalez, J. V. Anguita, R. García, and J. L. Vicent, 2003, “A superconducting reversible rectifier that controls the motion of magnetic flux quanta,” Science 302, 1188–1191.
  416. Vizarim, N. P., C. Reichhardt, C. J. O. Reichhardt, and P. A. Venegas, 2020, “Skyrmion dynamics and topological sorting on periodic obstacle arrays,” New J. Phys. 22, 053025.
  417. Vizarim, N. P., C. J. O. Reichhardt, P. A. Venegas, and C. Reichhardt, 2020, “Skyrmion pinball and directed motion on obstacle arrays,” J. Phys. Commun. 4, 085001.
  418. Vlasko-Vlasov, V. K., L. A. Dorosinskii, A. A. Polyanskii, V. I. Nikitenko, U. Welp, B. W. Veal, and G. W. Crabtree, 1994, “Study of the Influence of Individual Twin Boundaries on the Magnetic Flux Penetration in YBa2Cu3O7δ,” Phys. Rev. Lett. 72, 3246–3249.
  419. Wang, C., D. Xiao, X. Chen, Y. Zhou, and Y. Liu, 2017, “Manipulating and trapping skyrmions by magnetic field gradients,” New J. Phys. 19, 083008.
  420. Wang, L., et al., 2018, “Ferroelectrically tunable magnetic skyrmions in ultrathin oxide heterostructures,” Nat. Mater. 17, 1087–1094.
  421. Wang, L., et al., 2019, “Construction of a room-temperature Pt/Co/Ta multilayer film with ultrahigh-density skyrmions for memory application,” ACS Appl. Mater. Interfaces 11, 12098–12104.
  422. Wang, W., M. Beg, B. Zhang, W. Kuch, and H. Fangohr, 2015, “Driving magnetic skyrmions with microwave fields,” Phys. Rev. B 92, 020403(R).
  423. Wang, W., et al., 2016, “A centrosymmetric hexagonal magnet with superstable biskyrmion magnetic nanodomains in a wide temperature range of 100–340 K,” Adv. Mater. 28, 6887.
  424. Wang, X. S., A. Qaiumzadeh, and A. Brataas, 2019, “Current-Driven Dynamics of Magnetic Hopfions,” Phys. Rev. Lett. 123, 147203.
  425. Wang, X.-G., L. Chotorlishvili, V. K. Dugaev, A. Ernst, I. V. Maznichenko, N. Arnold, C. Jia, J. Berakdar, I. Mertig, and J. Barnaś, 2020, “The optical tweezer of skyrmions,” npj Comput. Mater. 6, 140.
  426. Wang, Y. J., et al., 2020, “Polar meron lattice in strained oxide ferroelectrics,” Nat. Mater. 19, 881.
  427. Wang, Y.-L., X. Ma, J. Xu, Z.-L. Xiao, A. Snezhko, R. Divan, L. E. Ocola, J. E. Pearson, B. Jánko, and W.-K. Kwok, 2018, “Switchable geometric frustration in an artificial-spin-ice-superconductor heterosystem,” Nat. Nanotechnol. 13, 560.
  428. Wang, Z., et al., 2020, “Thermal generation, manipulation and thermoelectric detection of skyrmions,” Nat. Electron. 3, 672.
  429. Wei, Q.-H., C. Bechinger, D. Rudhardt, and P. Leiderer, 1998, “Experimental Study of Laser-Induced Melting in Two-Dimensional Colloids,” Phys. Rev. Lett. 81, 2606–2609.
  430. Weiss, J. A., A. E. Larsen, and D. G. Grier, 1998, “Interactions, dynamics, and elasticity in charge-stabilized colloidal crystals,” J. Chem. Phys. 109, 8659–8666.
  431. Weissman, M. B., 1988, “1/f noise and other slow, nonexponential kinetics in condensed matter,” Rev. Mod. Phys. 60, 537–571.
  432. White, J. S., et al., 2014, “Electric-Field-Induced Skyrmion Distortion and Giant Lattice Rotation in the Magnetoelectric Insulator Cu2OSeO3,” Phys. Rev. Lett. 113, 107203.
  433. Wiesendanger, R., 2016, “Nanoscale magnetic skyrmions in metallic films and multilayers: A new twist for spintronics,” Nat. Rev. Mater. 1, 16044.
  434. Williams, F. I. B., et al., 1991, “Conduction Threshold and Pinning Frequency of Magnetically Induced Wigner Solid,” Phys. Rev. Lett. 66, 3285–3288.
  435. Wolf, D., S. Schneider, U. K. Rößler, A. Kovács, M. Schmidt, R. E. Dunin-Borkowski, B. Büchner, B. Rellinghaus, and A. Lubk, 2022, “Unveiling the three-dimensional magnetic texture of skyrmion tubes,” Nat. Nanotechnol. 17, 250–255.
  436. Woo, S., et al., 2016, “Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets,” Nat. Mater. 15, 501–506.
  437. Woo, S., et al., 2018, “Current-driven dynamics and inhibition of the skyrmion Hall effect of ferrimagnetic skyrmions in GdFeCo films,” Nat. Commun. 9, 959.
  438. Xia, J., X. Zhang, M. Ezawa, Q. Shao, X. Liu, and Y. Zhou, 2020, “Dynamics of an elliptical ferromagnetic skyrmion driven by the spin-orbit torque,” Appl. Phys. Lett. 116, 022407.
  439. Xia, J., X. Zhang, K.-Y. Mak, M. Ezawa, O. A. Tretiakov, Y. Zhou, G. Zhao, and X. Liu, 2021, “Current-induced dynamics of skyrmion tubes in synthetic antiferromagnetic multilayers,” Phys. Rev. B 103, 174408.
  440. Xiao, K., Y. Roichman, and D. G. Grier, 2011, “Two-dimensional optical thermal ratchets based on Fibonacci spirals,” Phys. Rev. E 84, 011131.
  441. Xiao, Z. L., E. Y. Andrei, and M. J. Higgins, 1999, “Flow Induced Organization and Memory of a Vortex Lattice,” Phys. Rev. Lett. 83, 1664–1667.
  442. Xing, X., J. Åkerman, and Y. Zhou, 2020, “Enhanced skyrmion motion via strip domain wall,” Phys. Rev. B 101, 214432.
  443. Xu, X. B., H. Fangohr, Z. H. Wang, M. Gu, S. L. Liu, D. Q. Shi, and S. X. Dou, 2011, “Vortex dynamics for low-κ type-II superconductors,” Phys. Rev. B 84, 014515.
  444. Yang, G., P. Stano, J. Klinovaja, and D. Loss, 2016, “Majorana bound states in magnetic skyrmions,” Phys. Rev. B 93, 224505.
  445. Yang, H., A. Thiaville, S. Rohart, A. Fert, and M. Chshiev, 2015, “Anatomy of Dzyaloshinskii-Moriya Interaction at Co/Pt Interfaces,” Phys. Rev. Lett. 115, 267210.
  446. Yang, S., K.-W. Moon, C. Kim, D.-H. Kim, J. Shin, J. Hong, S. K. Kim, and C. Hwang, 2021, “Control of the half-skyrmion Hall effect and its application to adder-subtractor,” Adv. Quantum Technol. 4, 2000060.
  447. Yi, S. D., S. Onoda, N. Nagaosa, and J. H. Han, 2009, “Skyrmions and anomalous Hall effect in a Dzyaloshinskii-Moriya spiral magnet,” Phys. Rev. B 80, 054416.
  448. Yokouchi, T., S. Sugimoto, B. Rana, S. Seki, N. Ogawa, S. Kasai, and Y. Otani, 2020, “Creation of magnetic skyrmions by surface acoustic waves,” Nat. Nanotechnol. 15, 361–366.
  449. Yokouchi, T., et al., 2018, “Current-induced dynamics of skyrmion strings,” Sci. Adv. 4, eaat1115.
  450. Young, A. P., 1979, “Melting and the vector Coulomb gas in two dimensions,” Phys. Rev. B 19, 1855–1866.
  451. Yu, X., 2021, “Magnetic imaging of various topological spin textures and their dynamics,” J. Magn. Magn. Mater. 539, 168332.
  452. Yu, X., D. Morikawa, T. Yokouchi, K. Shibata, N. Kanazawa, F. Kagawa, T. Arima, and Y. Tokura, 2018, “Aggregation and collapse dynamics of skyrmions in a non-equilibrium state,” Nat. Phys. 14, 832.
  453. Yu, X., et al., 2017, “Current-induced nucleation and annihilation of magnetic skyrmions at room temperature in a chiral magnet,” Adv. Mater. 29, 1606178.
  454. Yu, X., et al., 2020, “Real-space observation of topological defects in extended skyrmion-strings,” Nano Lett. 20, 7313–7320.
  455. Yu, X. Z., N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, 2011, “Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe,” Nat. Mater. 10, 106–109.
  456. Yu, X. Z., N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. Onose, and Y. Tokura, 2012, “Skyrmion flow near room temperature in an ultralow current density,” Nat. Commun. 3, 988.
  457. Yu, X. Z., W. Koshibae, Y. Tokunaga, K. Shibata, Y. Taguchi, N. Nagaosa, and Y. Tokura, 2018, “Transformation between meron and skyrmion topological spin textures in a chiral magnet,” Nature (London) 564, 95–98.
  458. Yu, X. Z., D. Morikawa, K. Nakajima, K. Shibata, N. Kanazawa, T. Arima, N. Nagaosa, and Y. Tokura, 2020, “Motion tracking of 80-nm-size skyrmions upon directional current injections,” Sci. Adv. 6, eaaz9744.
  459. Yu, X. Z., Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, 2010, “Real-space observation of a two-dimensional skyrmion crystal,” Nature (London) 465, 901–904.
  460. Yu, X. Z., Y. Tokunaga, Y. Kaneko, W. Z. Zhang, K. Kimoto, Y. Matsui, Y. Taguchi, and Y. Tokura, 2014, “Biskyrmion states and their current-driven motion in a layered manganite,” Nat. Commun. 5, 3198.
  461. Yuan, H. Y., X. S. Wang, M.-H. Yung, and X. R. Wang, 2019, “Wiggling skyrmion propagation under parametric pumping,” Phys. Rev. B 99, 014428.
  462. Zahn, K., R. Lenke, and G. Maret, 1999, “Two-Stage Melting of Paramagnetic Colloidal Crystals in Two Dimensions,” Phys. Rev. Lett. 82, 2721–2724.
  463. Zang, J., M. Mostovoy, J. H. Han, and N. Nagaosa, 2011, “Dynamics of Skyrmion Crystals in Metallic Thin Films,” Phys. Rev. Lett. 107, 136804.
  464. Zapperi, S., C. Castellano, F. Colaiori, and G. Durin, 2005, “Signature of effective mass in crackling-noise asymmetry,” Nat. Phys. 1, 46–49.
  465. Zapperi, S., P. Cizeau, G. Durin, and H. E. Stanley, 1998, “Dynamics of a ferromagnetic domain wall: Avalanches, depinning transition, and the Barkhausen effect,” Phys. Rev. B 58, 6353–6366.
  466. Zázvorka, J., F. Dittrich, Y. Ge, N. Kerber, K. Raab, T. Winkler, K. Litzius, M. Veis, P. Virnau, and M. Kläui, 2020, “Skyrmion lattice phases in thin film multilayer,” Adv. Funct. Mater. 30, 2004037.
  467. Zázvorka, J., et al., 2019, “Thermal skyrmion diffusion used in a reshuffler device,” Nat. Nanotechnol. 14, 658–661.
  468. Zeissler, K., et al., 2017, “Pinning and hysteresis in the field dependent diameter evolution of skyrmions in Pt/Co/Ir superlattice stacks,” Sci. Rep. 7, 15125.
  469. Zeissler, K., et al., 2018, “Discrete Hall resistivity contribution from Néel skyrmions in multilayer nanodiscs,” Nat. Nanotechnol. 13, 1161–1166.
  470. Zeissler, K., et al., 2020, “Diameter-independent skyrmion Hall angle observed in chiral magnetic multilayers,” Nat. Commun. 11, 428.
  471. Zeldov, E., D. Majer, M. Konczykowski, V. B. Geshkenbein, V. M. Vinokur, and H. Shtrikman, 1995, “Thermodynamic observation of first-order vortex-lattice melting transition in Bi2Sr2CaCu2O8,” Nature (London) 375, 373–376.
  472. Zhang, S., F. Kronast, G. van der Laan, and T. Hesjedal, 2018, “Real-space observation of skyrmionium in a ferromagnet-magnetic topological insulator heterostructure,” Nano Lett. 18, 1057.
  473. Zhang, S., A. K. Petford-Long, and C. Phatak, 2016, “Creation of artificial skyrmions and antiskyrmions by anisotropy engineering,” Sci. Rep. 6, 31248.
  474. Zhang, S., G. van der Laan, J. Müller, L. Heinen, M. Garst, A. Bauer, H. Berger, C. Pfleiderer, and T. Hesjedal, 2018, “Reciprocal space tomography of 3D skyrmion lattice order in a chiral magnet,” Proc. Natl. Acad. Sci. U.S.A. 115, 6386–6391.
  475. Zhang, S., J. Zhang, Y. Wen, E. M. Chudnovsky, and X. Zhang, 2018, “Determination of chirality and density control of Néel-type skyrmions with in-plane magnetic field,” Commun. Phys. 1, 36.
  476. Zhang, S., et al., 2020, “Direct imaging of an inhomogeneous electric current distribution using the trajectory of magnetic half-skyrmions,” Sci. Adv. 6, eaay1876.
  477. Zhang, S. L., A. Bauer, H. Berger, C. Pfleiderer, G. van der Laan, and T. Hesjedal, 2016, “Imaging and manipulation of skyrmion lattice domains in Cu2OSeO3, Appl. Phys. Lett. 109, 192406.
  478. Zhang, S. L., W. W. Wang, D. M. Burn, H. Peng, H. Berger, A. Bauer, C. Pfleiderer, G. van der Laan, and T. Hesjedal, 2018, “Manipulation of skyrmion motion by magnetic field gradients,” Nat. Commun. 9, 2115.
  479. Zhang, X., M. Ezawa, D. Xiao, G. P. Zhao, Y. Liu, and Y. Zhou, 2015, “All-magnetic control of skyrmions in nanowires by a spin wave,” Nanotechnology 26, 225701.
  480. Zhang, X., M. Ezawa, and Y. Zhou, 2015, “Magnetic skyrmion logic gates: Conversion, duplication and merging of skyrmions,” Sci. Rep. 5, 9400.
  481. Zhang, X., M. Ezawa, and Y. Zhou, 2016, “Thermally stable magnetic skyrmions in multilayer synthetic antiferromagnetic racetracks,” Phys. Rev. B 94, 064406.
  482. Zhang, X., J. Müller, J. Xia, M. Garst, X. Liu, and Y. Zhou, 2017, “Motion of skyrmions in nanowires driven by magnonic momentum-transfer forces,” New J. Phys. 19, 065001.
  483. Zhang, X., J. Xia, M. Ezawa, O. A. Tretiakov, H. T. Diep, G. Zhao, X. Liu, and Y. Zhou, 2021, “A frustrated bimeronium: Static structure and dynamics,” Appl. Phys. Lett. 118, 052411.
  484. Zhang, X., J. Xia, L. Shen, M. Ezawa, O. A. Tretiakov, G. Zhao, X. Liu, and Y. Zhou, 2020, “Static and dynamic properties of bimerons in a frustrated ferromagnetic monolayer,” Phys. Rev. B 101, 144435.
  485. Zhang, X., J. Xia, G. P. Zhao, X. Liu, and Y. Zhou, 2017, “Magnetic skyrmion transport in a nanotrack with spatially varying damping and non-adiabatic torque,” IEEE Trans. Magn. 53, 1500206.
  486. Zhang, X., J. Xia, Y. Zhou, X. Liu, H. Zhang, and M. Ezawa, 2017, “Skyrmion dynamics in a frustrated ferromagnetic film and current-induced helicity locking-unlocking transition,” Nat. Commun. 8, 1717.
  487. Zhang, X., J. Xia, Y. Zhou, D. Wang, X. Liu, W. Zhao, and M. Ezawa, 2016, “Control and manipulation of a magnetic skyrmionium in nanostructures,” Phys. Rev. B 94, 094420.
  488. Zhang, X., Y. Zhou, and M. Ezawa, 2016a “Antiferromagnetic skyrmion: Stability, creation and manipulation,” Sci. Rep. 6, 24795.
  489. Zhang, X., Y. Zhou, and M. Ezawa, 2016b, “Magnetic bilayer-skyrmions without skyrmion Hall effect,” Nat. Commun. 7, 10293.
  490. Zhang, X., Y. Zhou, M. Ezawa, G. P. Zhao, and W. Zhao, 2015, “Magnetic skyrmion transistor: Skyrmion motion in a voltage-gated nanotrack,” Sci. Rep. 5, 11369.
  491. Zhang, X., Y. Zhou, K. M. Song, T.-E. Park, J. Xia, M. Ezawa, X. Liu, W. Zhao, G. Zhao, and S. Woo, 2020, “Skyrmion-electronics: Writing, deleting, reading and processing magnetic skyrmions toward spintronic applications,” J. Phys. Condens. Matter 32, 143001.
  492. Zhang, X.-X., A. S. Mishchenko, G. De Filippis, and N. Nagaosa, 2016, “Electric transport in three-dimensional skyrmion/monopole crystal,” Phys. Rev. B 94, 174428.
  493. Zhao, H. J., V. R. Misko, and F. M. Peeters, 2013, “Dynamics of self-organized driven particles with competing range interaction,” Phys. Rev. E 88, 022914.
  494. Zhao, H. J., W. Wu, Wei Zhou, Z. X. Shi, V. R. Misko, and F. M. Peeters, 2016, “Reentrant dynamics of driven pancake vortices in layered superconductors,” Phys. Rev. B 94, 024514.
  495. Zhao, L., et al., 2020, “Topology-Dependent Brownian Gyromotion of a Single Skyrmion,” Phys. Rev. Lett. 125, 027206.
  496. Zheng, F., et al., 2018, “Experimental observation of chiral magnetic bobbers in B20-type FeGe,” Nat. Nanotechnol. 13, 451–455.
  497. Zhou, H., H. Polshyn, T. Taniguchi, K. Watanabe, and A. F. Young, 2020, “Solids of quantum Hall skyrmions in graphene,” Nat. Phys. 16, 154.
  498. Zhou, L., R. Qin, Y.-Q. Zheng, and Y. Wang, 2019, “Skyrmion Hall effect with spatially modulated Dzyaloshinskii-Moriya interaction,” Front. Phys. 14, 53602.
  499. Zhou, Y., R. Mansell, and S. van Dijken, 2019, “Driven gyrotropic skyrmion motion through steps in magnetic anisotropy,” Sci. Rep. 9, 6525.
  500. Zou, J., S. Zhang, and Y. Tserkovnyak, 2020, “Topological Transport of Deconfined Hedgehogs in Magnets,” Phys. Rev. Lett. 125, 267201.
  501. Zvezdin, A. K., V. I. Belotelov, and K. A. Zvezdin, 2008, “Gyroscopic force acting on the magnetic vortex in a weak ferromagnet,” JETP Lett. 87, 381–384.

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