Energetics and switching of quasi-uniform states in small ferromagnetic particles
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 2, p. 235-248

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.

DOI : https://doi.org/10.1051/m2an:2004011
Classification:  65L60,  78M10,  82D40
Keywords: micromagnetics, finite elements
@article{M2AN_2004__38_2_235_0,
     author = {Alouges, Fran\c cois and Conti, Sergio and DeSimone, Antonio and Pokern, Yvo},
     title = {Energetics and switching of quasi-uniform states in small ferromagnetic particles},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {2},
     year = {2004},
     pages = {235-248},
     doi = {10.1051/m2an:2004011},
     zbl = {1085.82015},
     mrnumber = {2069145},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_2_235_0}
}
Alouges, François; Conti, Sergio; DeSimone, Antonio; Pokern, Yvo. Energetics and switching of quasi-uniform states in small ferromagnetic particles. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 2, pp. 235-248. doi : 10.1051/m2an:2004011. http://www.numdam.org/item/M2AN_2004__38_2_235_0/

[1] A. Aharoni, Introduction to the theory of ferromagnetism. Oxford Ed., Clarendon Press (1996).

[2] A. Aharoni, Angular dependence of nucleation by curling in a prolate spheroid. J. Appl. Phys. 82 (1997) 1281-1287.

[3] F. Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | Zbl 0886.35010

[4] F. Alouges, Computation of demagnetizing field in micromagnetics with the infinite elements method. ESAIM: COCV 6 (2001) 629-647. | Numdam | Zbl 0992.78007

[5] A. Bagnérés-Viallix, P. Baras and J.B. Albertini, 2d and 3d calculations of micromagnetic wall structures using finite elements. IEEE Trans. Magn. 27 (1991) 3819-3822.

[6] G. Bertotti, Hysteresis in magnetism. Academic Press, San Diego (1998).

[7] E. Bonet, W. Wernsdorfer, B. Barbara, A. Benoît, D. Mailly and A. Thiaville, Three-dimensional magnetization reversal measurements in nanoparticles. Phys. Rev. Lett. 83 (1999) 4188-4191.

[8] W.F. Brown, Criterion for uniform micromagnetization. Phys. Rev. 105 (1957) 1479-1482.

[9] T. Chang, J.-G. Zhu and J.H. Judy, Method for investigating the reversal properties of isolated barium ferrite fine particles utilizing magnetic force microscopy (mfm). J. Appl. Phys. 73 (1993) 6716-6718.

[10] W. Chen, D.R. Fredkin and T.R. Koehler, A new finite element method in micromagnetics. IEEE Trans. Magn. 29 (1993) 2124-2128.

[11] Y.M. Chen, The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69-74. | Zbl 0685.58015

[12] A. Desimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles. Meccanica 30 (1995) 591-603. | Zbl 0836.73060

[13] D.R. Fredkin and T.R. Koehler, Finite element methods for micromagnetics. IEEE Trans. Magn. 28 (1992) 1239-1244.

[14] E.H. Frei, S. Shtrikman and D. Treves, Critical size and nucleation field of ideal ferromagnetic particles. Phys. Rev. 106 (1957) 446-454. | Zbl 0078.23307

[15] R. Hertel and H. Kronmüller, Finite element calculations on the single-domain limit of a ferromagnetic cube - a solution to μmag standard problem no. 3. J. Magn. Magn. Mat. 238 (2002) 185-199.

[16] A. Hubert and R. Schäfer, Magnetic domains. Springer, Berlin (1998).

[17] Y. Ishii, Magnetization curling in an infinite cylinder with a uniaxial magnetocrystalline anisotropy. J. Appl. Phys. 70 (1991) 3765-3769.

[18] R.D. Mcmichael, Standard problem number 3, problem specification and reported solutions1998).

[19] A.J. Newell and R.T. Merrill, The curling nucleation mode in a ferromagnetic cube. J. Appl. Phys. 84 (1998) 4394-4402.

[20] R. O'Barr, M. Lederman, S. Schultz, W. Xu, A. Scherer and R.J. Tonucci, Preparation and quantitative magnetic studies of single-domain nickel cylinders. J. Appl. Phys. 79 (1996) 5303-5305.

[21] W. Rave, K. Fabian and A. Hubert, Magnetic states of small cubic particles with uniaxial anisotropy. J. Magn. Magn. Mat. 190 (1998) 332-348.

[22] F. Rogier, S. Labbé and P.Y. Bertin, Schéma en temps et calcul du champ démagnétisant pour le micromagnétisme. NUMELEC'97, École Centrale de Lyon (1997).

[23] M.E. Schabes and H.N. Bertram, Magnetization processes in ferromagnetic cubes. J. Appl. Phys. 64 (1988) 1347-1357.

[24] E.C. Stoner and E.P. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys. Phil. Trans. R. Soc. London Ser. A 240 (1948) 599-642. | Zbl 0031.38003

[25] A. Thiaville, Coherent rotation of magnetization in three dimensions: a geometrical approach. Phys. Rev. B 61 (2000) 12221.

[26] L.A. Ying, Infinite elements method. Beijing University Press (1995). | Zbl 0611.65076