400
X. S. Wang and X. R. Wang
80. X. Zhang, Y. Zhou, M. Ezawa, G.P. Zhao, W. Zhao, Magnetic skyrmion transistor: skyrmion
motion in a voltage-gated nanotrack. Sci. Rep. 5(1), 11369 (2015)
81. M.A. Castro, S. Allende, Skyrmion core size dependence as a function of the perpendicular
anisotropy and radius in magnetic nanodots. J. Magn. Magn. Mater. 417, 344–348 (2016)
82. N. Vidal-Silva, A. Riveros, J. Escrig, Stability of Néel skyrmions in ultra-thin nanodots considering Dzyaloshinskii-Moriya and dipolar interactions. J. Magn. Magn. Mater. 443, 116–123
(2017)
83. A.O. Leonov, T.L. Monchesky, N. Romming, A. Kubetzka, A.N. Bogdanov, R. Wiesendanger,
The properties of isolated chiral skyrmions in thin magnetic films. New J. Phys. 18(6), 065003
(2016)
84. X.S. Wang, H.Y. Yuan, X.R. Wang, A theory on skyrmion size. Commun. Phys. 1(1), 31
(2018)
85. H.-B. Braun, Fluctuations and instabilities of ferromagnetic domain-wall pairs in an external
magnetic field. Phys. Rev. B 50, 16485–16500 (1994)
86. A. Siemens, Y. Zhang, J. Hagemeister, E.Y. Vedmedenko, R. Wiesendanger, Minimal radius
of magnetic skyrmions: statics and dynamics. New J. Phys. 18(4), 045021 (2016)
87. A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, B. Van Waeyenberge,
The design and verification of MuMax3. AIP Adv. 4(10), 107133 (2014)
88. D. Haifeng, R. Che, L. Kong, X. Zhao, C. Jin, C. Wang, J. Yang, W. Ning, R. Li, C. Jin, X.
Chen, J. Zang, Y. Zhang, M. Tian, Edge-mediated skyrmion chain and its collective dynamics
in a confined geometry. Nat. Communications. 6(1), 8504 (2015)
89. I.E. Dzyaloshinskii, The theory of helicoidal structures in antiferromagnets. II. metals. J. Exp.
Heoretic. Phys. 20(1), 223 (1965)
90. H.Y. Yuan, X.R. Wang, Skyrmion creation and manipulation by nano-second current pulses.
Sci. Rep. 6(1), 22638 (2016)
91. M. Stone, Magnus force on skyrmions in ferromagnets and quantum hall systems. Phys. Rev.
B 53, 16573–16578 (1996)
92. J. Iwasaki, M. Mochizuki, N. Nagaosa, Current-induced skyrmion dynamics in constricted
geometries. Nat. Nanotechnol. 8(10), 742–747 (2013)
93. W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. Benjamin 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(2), 162–169 (2017)
94. M.-W. Yoo, V. Cros, J.-V. Kim, Current-driven skyrmion expulsion from magnetic nanostrips.
Phys. Rev. B 95, 184423 (2017)
95. N.R. Cooper, Propagating magnetic vortex rings in ferromagnets. Phys. Rev. Lett. 82, 1554–
1557 (1999)
96. G.H. Derrick, Comments on nonlinear wave equations as models for elementary particles. J.
Math. Phys. 5(9), 1252–1254 (1964)
97. L.D. Faddeev, Some comments on the many-dimensional solitons. Lett. Math. Phys. 1(4),
289–293 (1976)
98. J. Hietarinta, P. Salo, Faddeev-Hopf knots: dynamics of linked un-knots. Phys. Lett. B 451(1),
60–67 (1999)
99. H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math.
Ann. 104(1), 637–665 (1931)
100. L. Faddeev, A.J. Niemi, Partially dual variables in SU(2) Yang-Mills theory. Phys. Rev. Lett.
82, 1624–1627 (1999)
101. L. Faddeev, A.J. Niemi, Stable knot-like structures in classical field theory. Nature. 387(6628),
58–61 (1997)
102. E. Babaev, Dual neutral variables and knot solitons in triplet superconductors. Phys. Rev. Lett.
88, 177002 (2002)
103. E. Babaev, L.D. Faddeev, A.J. Niemi, Hidden symmetry and knot solitons in a charged twocondensate bose system. Phys. Rev. B 65, 100512 (2002)
104. Y. Kawaguchi, M. Nitta, M. Ueda, Knots in a spinor Bose-Einstein condensate. Phys. Rev.
Lett. 100, 180403 (2008)
X. S. Wang and X. R. Wang
80. X. Zhang, Y. Zhou, M. Ezawa, G.P. Zhao, W. Zhao, Magnetic skyrmion transistor: skyrmion
motion in a voltage-gated nanotrack. Sci. Rep. 5(1), 11369 (2015)
81. M.A. Castro, S. Allende, Skyrmion core size dependence as a function of the perpendicular
anisotropy and radius in magnetic nanodots. J. Magn. Magn. Mater. 417, 344–348 (2016)
82. N. Vidal-Silva, A. Riveros, J. Escrig, Stability of Néel skyrmions in ultra-thin nanodots considering Dzyaloshinskii-Moriya and dipolar interactions. J. Magn. Magn. Mater. 443, 116–123
(2017)
83. A.O. Leonov, T.L. Monchesky, N. Romming, A. Kubetzka, A.N. Bogdanov, R. Wiesendanger,
The properties of isolated chiral skyrmions in thin magnetic films. New J. Phys. 18(6), 065003
(2016)
84. X.S. Wang, H.Y. Yuan, X.R. Wang, A theory on skyrmion size. Commun. Phys. 1(1), 31
(2018)
85. H.-B. Braun, Fluctuations and instabilities of ferromagnetic domain-wall pairs in an external
magnetic field. Phys. Rev. B 50, 16485–16500 (1994)
86. A. Siemens, Y. Zhang, J. Hagemeister, E.Y. Vedmedenko, R. Wiesendanger, Minimal radius
of magnetic skyrmions: statics and dynamics. New J. Phys. 18(4), 045021 (2016)
87. A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, B. Van Waeyenberge,
The design and verification of MuMax3. AIP Adv. 4(10), 107133 (2014)
88. D. Haifeng, R. Che, L. Kong, X. Zhao, C. Jin, C. Wang, J. Yang, W. Ning, R. Li, C. Jin, X.
Chen, J. Zang, Y. Zhang, M. Tian, Edge-mediated skyrmion chain and its collective dynamics
in a confined geometry. Nat. Communications. 6(1), 8504 (2015)
89. I.E. Dzyaloshinskii, The theory of helicoidal structures in antiferromagnets. II. metals. J. Exp.
Heoretic. Phys. 20(1), 223 (1965)
90. H.Y. Yuan, X.R. Wang, Skyrmion creation and manipulation by nano-second current pulses.
Sci. Rep. 6(1), 22638 (2016)
91. M. Stone, Magnus force on skyrmions in ferromagnets and quantum hall systems. Phys. Rev.
B 53, 16573–16578 (1996)
92. J. Iwasaki, M. Mochizuki, N. Nagaosa, Current-induced skyrmion dynamics in constricted
geometries. Nat. Nanotechnol. 8(10), 742–747 (2013)
93. W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. Benjamin 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(2), 162–169 (2017)
94. M.-W. Yoo, V. Cros, J.-V. Kim, Current-driven skyrmion expulsion from magnetic nanostrips.
Phys. Rev. B 95, 184423 (2017)
95. N.R. Cooper, Propagating magnetic vortex rings in ferromagnets. Phys. Rev. Lett. 82, 1554–
1557 (1999)
96. G.H. Derrick, Comments on nonlinear wave equations as models for elementary particles. J.
Math. Phys. 5(9), 1252–1254 (1964)
97. L.D. Faddeev, Some comments on the many-dimensional solitons. Lett. Math. Phys. 1(4),
289–293 (1976)
98. J. Hietarinta, P. Salo, Faddeev-Hopf knots: dynamics of linked un-knots. Phys. Lett. B 451(1),
60–67 (1999)
99. H. Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math.
Ann. 104(1), 637–665 (1931)
100. L. Faddeev, A.J. Niemi, Partially dual variables in SU(2) Yang-Mills theory. Phys. Rev. Lett.
82, 1624–1627 (1999)
101. L. Faddeev, A.J. Niemi, Stable knot-like structures in classical field theory. Nature. 387(6628),
58–61 (1997)
102. E. Babaev, Dual neutral variables and knot solitons in triplet superconductors. Phys. Rev. Lett.
88, 177002 (2002)
103. E. Babaev, L.D. Faddeev, A.J. Niemi, Hidden symmetry and knot solitons in a charged twocondensate bose system. Phys. Rev. B 65, 100512 (2002)
104. Y. Kawaguchi, M. Nitta, M. Ueda, Knots in a spinor Bose-Einstein condensate. Phys. Rev.
Lett. 100, 180403 (2008)