Numerical Micromagnetics: Finite Difference Methods

Micromagnetics is based on the one hand on a continuum approximation of exchange interactions, including boundary conditions, on the other hand on Maxwell equations in the nonpropagative (static) limit for the evaluation of the demagnetizing field. The micromagnetic energy is most often restricted to the sum of the exchange, demagnetizing or self-magnetostatic, Zeeman, and anisotropy energies. When supplemented with a time evolution equation, including field induced magnetization precession, damping and possibly additional torque sources, micromagnetics allows for a precise description of magnetization distributions within finite bodies both in space and time. Analytical solutions are, however, rarely available. Numerical micromagnetics enables the exploration of complexity in small size magnetic bodies. Finite difference methods are here applied to numerical micromagnetics in two variants for the description of both exchange interactions/boundary conditions and demagnetizing field evaluation. Accuracy in the time domain is also discussed and a simple tool provided in order to monitor time integration accuracy. A specific example involving large angle precession, domain wall motion as well as vortex/antivortex creation and annihilation allows for a fine comparison between two discretization schemes with as a net result, the necessity for mesh sizes well below the exchange length in order to reach adequate convergence. Keywords: micromagnetics; finite differences; boundary conditions; Landau—Lifshitz—Gilbert; magnetization dynamics; approximation order

[1]  J. C. Sloncxewski,et al.  Current-driven excitation of magnetic multilayers , 2003 .

[2]  W. Press.  Numerical recipes in Fortran 77 : the art of scientific computing : volume 1 of fortran numerical recipes , 1996 .

[3]  R. Hertel.  Thickness dependence of magnetization structures in thin Permalloy rectangles , 2002, International Journal of Materials Research.

[4]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[5]  C. Kittel,et al.  Physical Theory of Ferromagnetic Domains , 1949 .

[6]  G. Rado,et al.  Spin-wave resonance in a ferromagnetic metal , 1959 .

[7]  A. Slavin,et al.  Boundary conditions for magnetization in magnetic nanoelements , 2005, cond-mat/0503745.

[8]  A. Hubert,et al.  Solving Micromagnetic Problems. Towards an Optimal Numerical Method , 1993 .

[9]  J. Kools,et al.  On the ferromagnetic interlayer coupling in exchange-biased spin-valve multilayers , 1995 .

[10]  Jean-Raymond Abrial,et al.  On B , 1998, B.

[11]  M. D. Stiles,et al.  Phenomenological theory of current-induced magnetization precession , 2004 .

[12]  Jacques Miltat,et al.  Néel lines in ferrimagnetic garnet epilayers with orthorhombic anisotropy and canted magnetization , 1987 .

[13]  Michael J. Donahue,et al.  Behavior of μMAG standard problem No. 2 in the small particle limit , 2000 .

[14]  Gonçalo Albuquerque,et al.  Spin transfer into an inhomogeneous magnetization distribution , 2001 .

[15]  A. Marty,et al.  A new technique for ferromagnetic resonance calculations , 2002 .

[16]  H. Bertram,et al.  Interlayer coupling and surface roughness in GMR spin valves , 1996 .

[17]  Dennis W. Hewett,et al.  Embedded curve boundary method for micromagnetic simulations , 2000 .

[18]  A. Hubert,et al.  Optimizing stray field computations in finite-element micromagnetics , 1994 .

[19]  M. Labrune,et al.  Wall structures in ferro/antiferromagnetic exchange-coupled bilayers: A numerical micromagnetic approach , 1995 .

[20]  Avner Friedman.  Exchange energy representations in computational micromagnetics , 1998 .

[21]  E Weinan,et al.  Accurate numerical methods for micromagnetics simulations with general geometries , 2003 .

[22]  A. E. Labonte,et al.  Two‐Dimensional Bloch‐Type Domain Walls in Ferromagnetic Films , 1969 .

[23]  Michael J. Donahue,et al.  Exchange energy formulations for 3D micromagnetics , 2004 .

[24]  H. Kronmüller,et al.  Micromagnetism and the Microstructure of Ferromagnetic Solids , 2003 .

[25]  Denis Donnelly,et al.  The fast Fourier transform for experimentalists. Part II. convolutions , 2005, Comput. Sci. Eng..

[26]  M. Donahue.  A VARIATIONAL APPROACH TO EXCHANGE ENERGY CALCULATIONS IN MICROMAGNETICS , 1998 .

[27]  T. Schrefl,et al.  Micromagnetic study of Bloch-point-mediated vortex core reversal , 2003 .

[28]  Roger W. Hockney,et al.  P3M3DP-the three-dimensional periodic particle-particle/particle-mesh program , 1984 .

[29]  W. Williams,et al.  A generalization of the demagnetizing tensor for nonuniform magnetization , 1993 .

[30]  Jian-Gang Zhu,et al.  Switching field variation in patterned submicron magnetic film elements , 1997 .

[31]  S. Shtrikman,et al.  On the Resolution of Brown's Paradox , 1960 .

[32]  K. Ramstöck,et al.  Corners and nucleation in Micromagnetics , 1998 .

[33]  T. Gilbert.  A phenomenological theory of damping in ferromagnetic materials , 2004, IEEE Transactions on Magnetics.

[34]  H. Bertram,et al.  Size effects of switching fields of thin permalloy particles , 1992, 1992. Digests of Intermag. International Magnetics Conference.

[35]  S. W. Yuan,et al.  Fast adaptive algorithms for micromagnetics , 1992 .

[36]  Amikam Aharoni,et al.  Introduction to the Theory of Magnetism , 1998 .

[37]  Denis Donnelly,et al.  The fast Fourier transform for experimentalists. Part I. Concepts , 2005, Comput. Sci. Eng..

[38]  M. Schabes,et al.  Magnetostatic interaction fields for a three-dimensional array of ferromagnetic cubes , 1987 .

[39]  A. Hubert,et al.  Magnetic ground state of a thin-film element , 2000 .

[40]  K. Fabian,et al.  Magnetic states of small cubic particles with uniaxial anisotropy , 1998 .

[41]  Nobuo Hayashi,et al.  Volume average demagnetizing tensor of rectangular prisms , 1998 .

[42]  M. Donahue,et al.  Generalization of a two-dimensional micromagnetic model to nonuniform thickness , 2001 .

[43]  R W Hockney,et al.  Computer Simulation Using Particles , 1966 .

[44]  M J Donahue,et al.  OOMMF User's Guide, Version 1.0 , 1999 .