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Micromagnetism

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Handbook of Magnetism and Magnetic Materials

Abstract

Computational micromagnetics is widely used for the design and development of magnetic devices. The theoretical background of these simulations is the continuum theory of micromagnetism. It treats magnetization processes on a significant length scale which is small enough to resolve magnetic domain walls and large enough to replace atomic spins by a continuous function of position. The continuous expression for the micromagnetic energy terms is either derived from their atomistic counterpart or result from symmetry arguments. The equilibrium conditions for the magnetization and the equation of motion are introduced. The focus of the discussion lies on the basic building blocks of micromagnetic solvers. Numerical examples illustrate the micromagnetic concepts. An open-source simulation environment was used to address the ground state of thin film magnetic element, initial magnetization curves, stress-driven switching of magnetic storage elements, the grain size dependence of coercivity of permanent magnets, and damped oscillations in magnetization dynamics.

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Acknowledgements

The authors thank the Austrian Science Fund (FWF) under grant No. F4112 SFB ViCoM and grant No. P31140-N32 for financial support. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged.

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Authors and Affiliations

  1. University of Vienna Research Platform MMM Mathematics - Magnetism - Materials, University of Vienna, and Wolfgang Pauli Institute, Wien, Austria

    Lukas Exl

  2. University of Vienna Research Platform MMM Mathematics - Magnetism - Materials, and Physics of Functional Materials, Faculty of Physics, University of Vienna, Wien, Austria

    Dieter Suess

  3. Christian Doppler Laboratory for Magnet design through physics informed machine learning, Department of Integrated Sensor Systems, Danube University Krems, Wiener Neustadt, Austria

    Thomas Schrefl

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  1. Lukas Exl
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  2. Dieter Suess
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  3. Thomas Schrefl
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Correspondence to Lukas Exl .

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Editors and Affiliations

  1. School of Physics, Trinity College, Dublin, Ireland

    Michael Coey

  2. Max Planck Institute of Microstructure Physics, Halle (Saale), Germany

    Stuart Parkin

7 Appendix

7 Appendix

The intrinsic material properties listed in Table 1 are taken from [77]. The exchange lengths and the wall parameter are calculated as follows: \(l_{\text{ex}}=\sqrt {A/(\mu _0M^2_{\text{s}})}\), \(\delta _0 = \sqrt {A/|K_{1}|}\).

Table 1 Intrinsic magnetic properties and characteristic lengths of selected magnetic materials
Full size table

The examples given in Figs. 3 to 7 were computed using the micromagnetic simulation environment FIDIMAG [43]. FIDIMAG solves finite difference micromagnetic problems using a Python interface. The reader is encouraged to run computer experiments for further exploration of micromagnetism. In the following we illustrate the use of the Python interface for simulating the switching dynamics of a magnetic nano-element (see Fig. 7). The function relax_system computes the initial magnetic state. The function apply_field computes the response of the magnetization under the influence of a time varying external field.

 import numpy as np  from fidimag . micro import Sim  from fidimag .common import CuboidMesh  from fidimag . micro import UniformExchange ,  Demag  from fidimag . micro import TimeZeeman  mu0  =  4 * np . pi * 1e −7  A     =   1.0 e−11  Ms    =   1./mu0  def relax_system (mesh) :      sim  =  Sim(mesh ,  name = ’ relax ’ )      sim . driver . set_tols ( r t o l=1e −10,   atol =1 e −10)      sim . driver . alpha  =   0.5      sim . driver .gamma  =  2.211 e5      sim .Ms   =   Ms      sim . do_precession  =  False      sim . set_m ((0.577350269 , 0.577350269 , 0.577350269) )      sim . add( UniformExchange (A = A) )      sim . add(Demag() )      sim . relax ()      np . save ( ’m0. npy ’ , sim . spin )  def  apply_field (mesh) :      sim  =  Sim(mesh ,  name = ’dyn ’ )      sim . driver . set_tols ( r t o l=1e −10,   atol =1 e −10)      sim . driver . alpha  =  0.02      sim . driver .gamma  =  2.211 e5      sim .Ms   =   Ms      sim . set_m (np . load ( ’m0. npy ’ ) )      sim . add( UniformExchange (A = A) )      sim . add(Demag() )      sigma  =   0.1 e−9      def gaussian_fun ( t ) :          return np . exp (−0.5   *   (( t −3* sigma )   /   sigma ) **2)       mT   =  0.001 / mu0      zeeman  =  TimeZeeman ([−100   *   mT ,   −100   *   mT ,   −100   *   mT ] ,   time_fun = gaussian_fun ,   name = ’H’ )      sim . add(zeeman ,   save_field= True)      sim . relax ( dt=1.e −12, max_steps =10000)   if  __name__  ==   ’__main__ ’ :      mesh  =  CuboidMesh(nx=50, ny=10, nz=1, dx=2, dy=2, dz=2,  unit_length=1e −9)      relax_system (mesh)      apply_field (mesh)

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Exl, L., Suess, D., Schrefl, T. (2021). Micromagnetism. In: Coey, M., Parkin, S. (eds) Handbook of Magnetism and Magnetic Materials. Springer, Cham. https://doi.org/10.1007/978-3-030-63101-7_7-1

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