×

A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree \(\pm 1\) harmonic maps from \(\mathbb{R}^2\) to \(\mathbb{S}^2\). (English) Zbl 1466.78007

Summary: We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate the existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree \(\pm 1\) harmonic maps from the two-dimensional sphere to itself.

MSC:

78A30 Electro- and magnetostatics
82D40 Statistical mechanics of magnetic materials
35B40 Asymptotic behavior of solutions to PDEs
35C08 Soliton solutions
49J20 Existence theories for optimal control problems involving partial differential equations
74F15 Electromagnetic effects in solid mechanics
74K35 Thin films
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abanov, A.; Pokrovsky, VL, Skyrmion in a real magnetic film, Phys. Rev. B, 58, R8889-R8892 (1998) · doi:10.1103/PhysRevB.58.R8889
[2] Abramowitz, M.; Stegun, IA, Handbook of Mathematical Functions (1972), Gaithersburg: National Bureau of Standards, Gaithersburg · Zbl 0543.33001
[3] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000), Oxford: Oxford University Press, Oxford · Zbl 0957.49001
[4] Bahouri, H.; Chemin, J-Y; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (2011), Berlin: Springer, Berlin · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[5] Beckner, W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 2, 138, 213-242 (1993) · Zbl 0826.58042 · doi:10.2307/2946638
[6] Belavin, AA; Polyakov, AM, Metastable states of two-dimensional isotropic ferromagnets, JETP Lett., 22, 10, 245-248 (1975)
[7] Bernand-Mantel, A.; Camosi, L.; Wartelle, A.; Rougemaille, N.; Darques, M.; Ranno, L., The skyrmion-bubble transition in a ferromagnetic thin film, SciPost Phys., 4, 027 (2018) · doi:10.21468/SciPostPhys.4.5.027
[8] Bernand-Mantel, A.; Muratov, CB; Simon, TM, Unraveling the role of dipolar vs. Dzyaloshinskii-Moriya interaction in stabilizing compact magnetic skyrmions, Phys. Rev. B, 101, 045302 (2020) · doi:10.1103/PhysRevB.101.045416
[9] Bogdanov, A.; Hubert, A., Thermodynamically stable magnetic vortex states in magnetic crystals, J. Magn. Magn. Mater., 138, 255-269 (1994) · doi:10.1016/0304-8853(94)90046-9
[10] Bogdanov, AN; Kudinov, MV; Yablonskii, DA, Theory of magnetic vortices in easy-axis ferromagnets, Sov. Phys. Solid State, 31, 1707-1710 (1989)
[11] Bogdanov, AN; Yablosnkii, DA, Thermodynamically stable “vortices” in magnetically ordered cyrstals. The mixed state of magnets, Sov. Phys. JETP, 68, 101-103 (1989)
[12] Borwein, JM; Lewis, AS, Convex Analysis and Nonlinear Optimization: Theory and Examples (2006), New York: Springer, New York · Zbl 1116.90001 · doi:10.1007/978-0-387-31256-9
[13] Boulle, O.; Vogel, J.; Yang, H.; Pizzini, S.; de Souza Chaves, D.; Locatelli, A.; Menteş, TO; Sala, A.; Buda-Prejbeanu, LD; Klein, O.; Belmeguenai, M.; Roussigné, Y.; Stashkevich, A.; Chérif, SM; Aballe, L.; Foerster, M.; Chshiev, M.; Auffret, S.; Miron, IM; Gaudin, G., Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures, Nature Nanotechnol., 11, 449-455 (2016) · doi:10.1038/nnano.2015.315
[14] Bracewell, RN, The Fourier Transform and Its Applications (2000), New York: McGraw-Hill, New York · Zbl 0149.08301
[15] Braides, A.; Truskinovsky, L., Asymptotic expansions by \(\Gamma \)-convergence, Continuum Mech. Thermodyn., 20, 21-62 (2008) · Zbl 1160.74363 · doi:10.1007/s00161-008-0072-2
[16] Brezis, H.; Coron, J-M, Large solutions for harmonic maps in two dimensions, Commun. Math. Phys., 92, 203-215 (1983) · Zbl 0532.58006 · doi:10.1007/BF01210846
[17] Brezis, H.; Coron, J-M, Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal., 89, 21-56 (1985) · Zbl 0584.49024 · doi:10.1007/BF00281744
[18] Brezis, H.; Nirenberg, L., Degree theory and BMO; Part I: Compact manifolds without boundaries, Selecta Math. (N.S.), 1, 2, 197-263 (1995) · Zbl 0852.58010 · doi:10.1007/BF01671566
[19] Büttner, F.; Lemesh, I.; Beach, GSD, Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications, Sci. Rep., 8, 4464 (2018) · doi:10.1038/s41598-018-22242-8
[20] Chanillo, S.; Malchiodi, A., Asymptotic Morse theory for the equation \(\Delta v = 2 v_x \wedge v_y\), Commun. Anal. Geom., 13, 187-251 (2005) · Zbl 1175.35049 · doi:10.4310/CAG.2005.v13.n1.a6
[21] Chen, G.; Liu, Y.; Wei, J., Nondegeneracy of harmonic maps from \(\mathbb{R}^2\) to \(\mathbb{S}^2\), Discrete Contin. Dyn. Syst., 40, 3215-3233 (2020) · Zbl 1436.53045 · doi:10.3934/dcds.2019228
[22] Corless, RM; Gonnet, GH; Hare, DEG; Jeffrey, DJ; Knuth, DE, On the Lambert W function, Adv. Comput. Math., 5, 329-359 (1996) · Zbl 0863.65008 · doi:10.1007/BF02124750
[23] Dávila, J.; del Pino, M.; Wei, J., Singularity formation for the two-dimensional harmonic map flow into \(S^2\), Invent. Math., 219, 345-466 (2019) · Zbl 1445.35082 · doi:10.1007/s00222-019-00908-y
[24] Derrick, GH, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys., 5, 1252-1254 (1964) · doi:10.1063/1.1704233
[25] Di Fratta, G.; Robbins, JM; Slastikov, V.; Zarnescu, A., Landau-de Gennes corrections to the Oseen-Frank theory of nematic liquid crystals, Arch. Ration. Mech. Anal., 236, 1089-1125 (2020) · Zbl 1436.76002 · doi:10.1007/s00205-019-01488-0
[26] Di Fratta, G.; Slastikov, V.; Zarnescu, A., On a sharp Poincaré-type inequality on the 2-sphere and its application in micromagnetics, SIAM J. Math. Anal., 51, 3373-3387 (2019) · Zbl 1420.35012 · doi:10.1137/19M1238757
[27] Do Carmo, MP, Differential Geometry of Curves and Surfaces (1976), Upper Saddle River: Prentice Hall, Upper Saddle River · Zbl 0326.53001
[28] Döring, L.; Melcher, C., Compactness results for static and dynamic chiral skyrmions near the conformal limit, Calc. Var. Partial Differ. Equ., 56, 60 (2017) · Zbl 1375.49065 · doi:10.1007/s00526-017-1172-2
[29] Eells, J.; Lemaire, L., A report on harmonic maps, Bull. Lond. Math. Soc., 10, 1-68 (1978) · Zbl 0401.58003 · doi:10.1112/blms/10.1.1
[30] Eells, J.; Lemaire, L., Another report on harmonic maps, Bull. Lond. Math. Soc., 20, 385-524 (1988) · Zbl 0669.58009 · doi:10.1112/blms/20.5.385
[31] Esteban, MJ, A direct variational approach to Skyrme’s model for meson fields, Commun. Math. Phys., 105, 571-591 (1986) · Zbl 0621.58035 · doi:10.1007/BF01238934
[32] Esteban, MJ; Berestycki, H., A new setting for Skyrme’s problem, Progress in Nonlinear Differential Equations and Their Applications (1990), Berlin: Birkhäuser, Berlin · Zbl 0724.49007
[33] Esteban, MJ, Existence of 3D skyrmions, Commun. Math. Phys., 251, 209-210 (2004) · Zbl 1085.58501 · doi:10.1007/s00220-004-1139-y
[34] Fleming, W.; Rishel, R., An integral formula for total gradient variation, Arch. Math. (Basel), 11, 218-222 (1960) · Zbl 0094.26301 · doi:10.1007/BF01236935
[35] Freeden, W.; Schreiner, M., Spherical functions of Mathematical Geosciences: A scalar, vectorial and tensorial setup (2009), Berlin: Springer, Berlin · Zbl 1167.86002 · doi:10.1007/978-3-540-85112-7
[36] Grafakos, L., Classical Fourier Analysis (2014), Berlin: Springer, Berlin · Zbl 1304.42001
[37] Greco, C., On the existence of skyrmions in planar liquid crystals, Topol. Methods Nonlinear Anal., 54, 567-586 (2019) · Zbl 1439.82051
[38] Gustafson, S.; Kang, K.; Tsai, T-P, Schrödinger flow near harmonic maps, Commun. Pure. Appl. Math., 60, 0463-0499 (2007) · Zbl 1144.53085 · doi:10.1002/cpa.20143
[39] Hélein, F.; Wood, JC; Krupka, D.; Saunders, D., Harmonic maps, Handbook of Global Analysis, 417-492 (2008), Amsterdam: Elsevier, Amsterdam · Zbl 1236.58002 · doi:10.1016/B978-044452833-9.50009-7
[40] Hellman, F.; Hoffmann, A.; Tserkovnyak, Y.; Beach, GSD; Fullerton, EE; Leighton, C.; MacDonald, AH; Ralph, DC; Arena, DA; Dürr, HA; Fischer, P.; Grollier, J.; Heremans, JP; Jungwirth, T.; Kimel, AV; Koopmans, B.; Krivorotov, IN; May, SJ; Petford-Long, AK; Rondinelli, JM; Samarth, N.; Schuller, IK; Slavin, AN; Stiles, MD; Tchernyshyov, O.; Thiaville, A.; Zink, BL, Interface-induced phenomena in magnetism, Rev. Mod. Phys., 89, 025006 (2017) · doi:10.1103/RevModPhys.89.025006
[41] Hoffmann, M.; Zimmermann, B.; Müller, GP; Schürhoff, D.; Kiselev, NS; Melcher, C.; Bügel, S., Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions, Nat. Commun., 8, 308 (2017) · doi:10.1038/s41467-017-00313-0
[42] Hsu, P-J; Kubetzka, A.; Finco, A.; Romming, N.; von Bergmann, K.; Wiesendanger, R., Electric-field-driven switching of individual magnetic skyrmions, Nat. Nanotechnol., 12, 123-126 (2017) · doi:10.1038/nnano.2016.234
[43] Isobe, T., On the asymptotic analysis of H-systems, I: Asymptotic behaviour of large solutions, Adv. Differ. Equ., 6, 513-546 (2001) · Zbl 1142.35345
[44] Ivanov, BA; Stephanovich, VA; Zhmudskii, AA, Magnetic vortices: the microscopic analogs of magnetic bubbles, J. Magn. Magn. Mater., 88, 116-120 (1990) · doi:10.1016/S0304-8853(97)90021-4
[45] Jonietz, F.; Mulbauer, S.; Pfleiderer, C.; Neubauer, A.; Munzer, W.; Bauer, A.; Adams, T.; Georgii, R.; Boni, P.; Duine, RA; Everschor, K.; Garst, M.; Rosch, A., Spin transfer torques in MnSi at ultralow current densities, Science, 330, 1648 (2011) · doi:10.1126/science.1195709
[46] Jost, J., Riemannian Geometry and Geometric Analysis (2011), Berlin: Springer, Berlin · Zbl 1227.53001 · doi:10.1007/978-3-642-21298-7
[47] Kiselev, NS; Bogdanov, AN; Schäfer, R.; Rößler, UK, Chiral skyrmions in thin magnetic films: new objects for magnetic storage technologies?, J. Phys. D, 44, 392001 (2011) · doi:10.1088/0022-3727/44/39/392001
[48] Knüpfer, H.; Muratov, CB; Nolte, F., Magnetic domains in thin ferromagnetic films with strong perpendicular anisotropy, Arch. Ration. Mech. Anal., 232, 727-761 (2019) · Zbl 1412.78005 · doi:10.1007/s00205-018-1332-3
[49] Komineas, S.; Melcher, C.; Venakides, S., The profile of chiral skyrmions of small radius, Nonlinearity, 33, 3395-3408 (2020) · Zbl 1440.82011 · doi:10.1088/1361-6544/ab81eb
[50] Komineas, S., Melcher, C., Venakides, S.: Chiral skyrmions of large radius, 2019. arXiv preprint arXiv:1910.04818 · Zbl 1440.82011
[51] Kravchuk, VP; Rößler, UK; Volkov, OM; Sheka, DD; van den Brink, J.; Makarov, D.; Fuchs, H.; Fangohr, H.; Gaididei, Y., Topologically stable magnetization states on a spherical shell: curvature-stabilized skyrmions, Phys. Rev. B, 94, 144402 (2016) · doi:10.1103/PhysRevB.94.144402
[52] Lemaire, L., Applications harmoniques de surfaces riemanniennes, J. Differ. Geom., 13, 51-78 (1978) · Zbl 0388.58003 · doi:10.4310/jdg/1214434347
[53] Li, J.; Zhu, X., Existence of 2D skyrmions, Math. Z., 268, 305-315 (2011) · Zbl 1216.81173 · doi:10.1007/s00209-010-0672-y
[54] Li, X.; Melcher, C., Stability of axisymmetric chiral skyrmions, J. Funct. Anal., 275, 2817-2844 (2018) · Zbl 1400.49057 · doi:10.1016/j.jfa.2018.01.019
[55] Lieb, EH; Loss, M., Analysis (2010), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0966.26002
[56] Lin, F., Mapping problems, fundamental groups and defect measures, Acta Math. Sin., 15, 25-52 (1999) · Zbl 0926.49025 · doi:10.1007/s10114-999-0059-3
[57] Lin, F.; Yang, Y., Existence of energy minimizers as stable knotted solitons in the Faddeev model, Commun. Math. Phys., 249, 273-303 (2004) · Zbl 1065.81118 · doi:10.1007/s00220-004-1110-y
[58] Lin, F.; Yang, Y., Existence of two-dimensional skyrmions via the concentration-compactness method, Commun. Pure Appl. Math., 57, 1332-1351 (2004) · Zbl 1059.81184 · doi:10.1002/cpa.20038
[59] Lions, P-L, The concentration-compactness principle in the Calculus of Variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 109-145 (1984) · Zbl 0541.49009 · doi:10.1016/S0294-1449(16)30428-0
[60] Luckhaus, S., Zemas, K.: Stability estimates for the conformal group of \({\mathbb{S}}^{n-1}\) in dimension \(n \ge 3, 2019\). arXiv preprint arXiv: 1910.01862
[61] Manton, N.; Sutcliffe, P., Topological Solitons (2004), Cambridge: Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge · Zbl 1100.37044 · doi:10.1017/CBO9780511617034
[62] Mazet, E., La formule de la variation seconde de l’energie au voisinage d’une application harmonique, J. Differ. Geom., 8, 279-296 (1973) · Zbl 0301.53012 · doi:10.4310/jdg/1214431644
[63] Melcher, C., Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions, Indiana Univ. Math. J., 61, 1175-1200 (2012) · Zbl 1272.35116 · doi:10.1512/iumj.2012.61.4717
[64] Melcher, C., Chiral skyrmions in the plane, Proc. R. Soc. A, 470, 20140394 (2014) · Zbl 1371.81320 · doi:10.1098/rspa.2014.0394
[65] Melcher, C.; Sakellaris, ZN, Curvature stabilized skyrmions with angular momentum, Lett. Math. Phys., 109, 2291-2304 (2019) · Zbl 1426.49051 · doi:10.1007/s11005-019-01188-6
[66] Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, 1077-1092 (1971) · Zbl 0203.43701 · doi:10.1512/iumj.1971.20.20101
[67] Mühlbauer, S.; Binz, B.; Jonietz, F.; Pfleiderer, C.; Rosch, A.; Neubauer, A.; Georgii, R.; Böni, P., Skyrmion lattice in a chiral magnet, Science, 323, 915-919 (2009) · doi:10.1126/science.1166767
[68] Muratov, CB, A universal thin film model for ginzburg-landau energy with dipolar interaction, Calc. Var. Partial Differ. Equ., 58, 52 (2019) · Zbl 1412.35319 · doi:10.1007/s00526-019-1493-4
[69] Muratov, CB; Slastikov, VV, Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction, Proc. R. Soc. A, 473, 20160666 (2017) · Zbl 1404.82101 · doi:10.1098/rspa.2016.0666
[70] Nagaosa, N.; Tokura, Y., Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol., 8, 899-911 (2013) · doi:10.1038/nnano.2013.243
[71] Pinna, D.; Abreu Araujo, F.; Kim, J-V; Cros, V.; Querlioz, D.; Bessiere, P.; Droulez, J.; Grollier, J., Skyrmion gas manipulation for probabilistic computing, Phys. Rev. Appl., 9, 064018 (2018) · doi:10.1103/PhysRevApplied.9.064018
[72] Prychynenko, D.; Sitte, M.; Litzius, K.; Krüger, B.; Bourianoff, G.; Kläui, M.; Sinova, J.; Everschor-Sitte, K., Magnetic skyrmion as a nonlinear resistive element: a potential building block for reservoir computing, Phys. Rev. Appl., 9, 014034 (2018) · doi:10.1103/PhysRevApplied.9.014034
[73] Rho, M.; Zahed, I., The Multifaceted Skyrmion (2016), Singapore: World Scientific, Singapore · Zbl 1347.81017 · doi:10.1142/9710
[74] Romming, N.; Hanneken, C.; Menzel, M.; Bickel, JE; Wolter, B.; von Bergmann, K.; Kubetzka, A.; Wiesendanger, R., Writing and deleting single magnetic skyrmions, Science, 341, 636-639 (2013) · doi:10.1126/science.1240573
[75] Schoen, R.; Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom., 18, 253-268 (1983) · Zbl 0547.58020 · doi:10.4310/jdg/1214437663
[76] Skyrme, THR, A unified field theory of mesons and baryons, Nucl. Phys., 31, 556-569 (1962) · doi:10.1016/0029-5582(62)90775-7
[77] Smith, RT, The second variation formula for harmonic mappings, Proc. Am. Math. Soc., 47, 229-236 (1975) · Zbl 0303.58008 · doi:10.1090/S0002-9939-1975-0375386-2
[78] Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2008), Berlin: Springer, Berlin · Zbl 1284.49004
[79] Tomasello, R.; Martinez, E.; Zivieri, R.; Torres, LL; Carpentieri, M.; Finocchio, G., A strategy for the design of skyrmion racetrack memories, Sci. Rep., 4, 6784 (2014) · doi:10.1038/srep06784
[80] Wood, J.C.: Harmonic mappings between surfaces. PhD thesis, Warwick University 1974
[81] Yu, XZ; Onose, Y.; Kanazawa, N.; Park, JH; Han, JH; Matsui, Y.; Nagaosa, N.; Tokura, Y., Real-space observation of a two-dimensional skyrmion crystal, Nature, 465, 901-904 (2010) · doi:10.1038/nature09124
[82] Zhang, X.; Ezawa, M.; Zhou, Y., Magnetic skyrmion logic gates: conversion, duplication and merging of skyrmions, Sci. Rep., 5, 9400 (2015) · doi:10.1038/srep09400
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.