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| '''Micromagnetics''' is a field of [[physics]] dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored (the [[Continuum mechanics|continuum approximation]]), yet small enough to resolve magnetic structures such as [[Domain wall (magnetism)|domain walls]] or vortices.
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| Micromagnetics can deal with static [[Mechanical equilibrium|equilibria]], by minimizing the magnetic energy, and with dynamic behavior, by solving the time-dependent dynamical equation.
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| ==History==
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| Micromagnetics as a field (''i.e.'', that which deals specifically with the behaviour of (ferro)magnetic materials at sub-micrometer length scales) was introduced in 1963 when [[William Fuller Brown, Jr.]] published a paper on antiparallel [[Domain wall (magnetism)|domain wall]] structures. Until comparatively recently computational micromagnetics has been prohibitively expensive in terms of computational power, but smaller problems are now solvable on a modern desktop [[personal computer|PC]].
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| == Static micromagnetics ==
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| The purpose of static micromagnetics is to solve for the spatial distribution of the magnetization '''M''' at equilibrium. In most cases, as the temperature is much lower than the [[Curie temperature]] of the material considered, the modulus |'''M'''| of the magnetization is assumed to be everywhere equal to the [[saturation magnetization]] ''M''<sub>s</sub>. The problem then consists in finding the spatial orientation of the magnetization, which is given by the ''reduced magnetization'' '''m''' = '''M'''/''M''<sub>s</sub>.
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| The static equilibria are found by minimizing the magnetic energy,
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| :<math>E = E_\text{exch} + E_\text{anis} + E_\text{Z} + E_\text{demag}</math>,
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| subject to the constraint |'''M'''|='''M'''/''M''<sub>s</sub>.
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| The contributions to this energy are the following:
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| === Exchange energy === | |
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| The exchange energy is a macroscopic continuous-medium description of the microscopic [[exchange interaction]]. It is written as:
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| :<math>E_\text{exch} = A \int_V \left((\nabla m_x)^2 + (\nabla m_y)^2 + (\nabla m_z)^2\right) \mathrm{d}V</math>
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| where ''A'' is the ''exchange constant''; ''m''<sub>x</sub>, ''m''<sub>y</sub> and ''m''<sub>z</sub> are the components of '''m''';
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| and the integral is performed over the volume of the sample.
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| The exchange energy tends to favor configurations where the magnetization varies only slowly across the sample. This energy is minimized when the magnetization is perfectly uniform.
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| === Anisotropy energy ===
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| {{Main|Magnetic anisotropy|Anisotropy energy}}
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| Magnetic anisotropy arises from the microscopic anisotropy of the material, usually related to the [[Rotational symmetry|symmetry axes]] of its [[crystal structure]]. It can be generally written as:
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| :<math>E_\text{anis} = \int_V F_\text{anis}(\mathbf{m}) \mathrm{d}V</math>
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| where ''F''<sub>anis</sub>, the anisotropy energy density, is a function of the orientation of the magnetization. Minimum-energy directions for ''F''<sub>anis</sub> are called ''easy axes''.
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| [[Time-reversal symmetry]] ensures that ''F''<sub>anis</sub> is an even function of '''m'''. The simplest such function is
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| :<math>F_\text{anis}(\mathbf{m}) = -K m_z^2</math>.
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| where ''K'' is called the ''anisotropy constant''. In this approximation, called ''uniaxial anisotropy'', the easy axis is the ''z'' direction.
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| The anisotropy energy favors magnetic configurations where the magnetization is everywhere aligned along an easy axis.
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| === Zeeman energy === | |
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| {{Main|Zeeman energy}}
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| The Zeeman energy is the interaction energy between the magnetization and any externally applied field. It's written as:
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| :<math>E_\text{Z} = -\mu_0 \int_V \mathbf{M}\cdot\mathbf{H}_\text{a} \mathrm{d}V</math>
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| where '''H'''<sub>a</sub> is the applied field and µ<sub>0</sub> is the [[vacuum permeability]].
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| The Zeeman energy favors alignment of the magnetization parallel to the applied field.
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| === Energy of the demagnetizing field ===
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| [[File:Micromagnetics by Zureks.svg|thumb|upright|Example of micromagnetic configuration. Compared to a uniform state, the flux closure structure lowers the energy of the demagnetizing field, at the expense of some exchange energy.]]
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| {{Main|Demagnetizing field}}
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| The demagnetizing field is the magnetic field created by the magnetic sample upon itself. The associated energy is:
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| :<math>E_\text{demag} = -\frac{\mu_0}{2} \int_V \mathbf{M}\cdot\mathbf{H}_\text{d} \mathrm{d}V</math>
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| where '''H'''<sub>d</sub> is the [[demagnetizing field]]. This field depends on the magnetic configuration itself, and it can be found by solving:
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| :<math>\nabla\cdot\mathbf{H}_\text{d} = -\nabla\cdot\mathbf{M}</math> | |
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| :<math>\nabla\times\mathbf{H}_\text{d} = 0</math>
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| where −∇·'''M''' is sometimes called ''magnetic charge density''. The solution of these equations (c.f. [[magnetostatics]]) is:
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| :<math>\mathbf{H}_\text{d} = -\frac{1}{4\pi} \int_V \nabla\cdot\mathbf{M} \frac{\mathbf{r}}{r^3} \mathrm{d}V</math>
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| where '''r''' is the vector going from the current integration point to the point where '''H'''<sub>d</sub> is being calculated.
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| It is worth noting that the magnetic charge density can be infinite at the edges of the sample, due to '''M''' changing discontinuously from a finite value inside to zero outside of the sample. This is usually dealt with by using suitable [[boundary condition]]s on the edge of the sample.
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| The energy of the demagnetizing field favors magnetic configurations that minimize magnetic charges. In particular, on the edges of the sample, the magnetization tends to run parallel to the surface. In most cases it is not possible to minimize this energy term at the same time as the others. The static equilibrium then is a compromise that minimizes the total magnetic energy, although it may not minimize individually any particular term.
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| {{Clear}}
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| == Dynamic micromagnetics ==
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| The purpose of dynamic micromagnetics is to predict the time evolution of the magnetic configuration of a sample subject to some non-steady conditions such as the application of a field pulse or an AC field. This is done by solving the [[Landau-Lifshitz-Gilbert equation]], which is a [[partial differential equation]] describing the evolution of the magnetization in term of the local ''effective field'' acting on it.
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| === Effective field ===
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| The '''effective field''' is the local field ''felt'' by the magnetization. It can be described informally as the derivative of the magnetic energy density with respect to the orientation of the magnetization, as in:
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| :<math>\mathbf{H}_\mathrm{eff} = - \frac{1}{\mu_0 M_s} \frac{\mathrm{d}^2 E}{\mathrm{d}\mathbf{m}\mathrm{d}V}</math>
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| where d''E''/d''V'' is the energy density. In [[Calculus of variations|variational]] terms, a change d'''m''' of the magnetization and the associated change d''E'' of the magnetic energy are related by:
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| :<math>\mathrm{d}E = -\mu_0 M_s \int_V (\mathrm{d}\mathbf{m})\cdot\mathbf{H}_\text{eff}\,\mathrm{d}V</math>
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| It should be noted that, since '''m''' is a unit vector, d'''m''' is always perpendicular to '''m'''. Then the above definition leaves unspecified the component of '''H'''<sub>eff</sub> that is parallel to '''m'''. This is usually not a problem, as this component has no effect on the magnetization dynamics.
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| From the expression of the different contributions to the magnetic energy, the effective field can be found to be:
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| :<math>\mathbf{H}_\mathrm{eff} = \frac{2A}{\mu_0 M_s} \nabla^2 \mathbf{m} - \frac{1}{\mu_0 M_s} \frac{\partial
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| F_\text{anis}}{\partial \mathbf{m}} + \mathbf{H}_\text{a} + \mathbf{H}_\text{d}</math>
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| === Landau-Lifshitz-Gilbert equation ===
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| [[File:Damped Magnetization Precession.jpg|thumb|upright|The terms of the Landau-Lifshitz-Gilbert equation: precession (red) and damping (blue). The trajectory of the magnetization (dotted spiral) is drawn under the simplifying assumption that the effective field '''H'''<sub>eff</sub> is constant.]]
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| {{Main|Landau-Lifshitz-Gilbert equation}}
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| This is the equation of motion of the magnetization. It describes a [[Larmor precession]] of the magnetization around the effective field, with an additional [[damping]] term arising from the coupling of the magnetic system to the environment. The equation can be written in the so called ''Gilbert form'' (or implicit form) as:
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| :<math>\frac{\partial \mathbf m}{\partial t} = - |\gamma| \mathbf{m} \times \mathbf{H}_\mathrm{eff} + \alpha \mathbf{m}\times\frac{\partial \mathbf{m}} {\partial t}</math>
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| where γ is the electron gyromagnetic ratio and α the Gilbert damping constant.
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| It can be shown that this is mathematically equivalent to the following ''Landau-Lifshitz'' (or explicit) form:
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| :<math>\frac{\partial\mathbf m}{\partial t} = - \frac{|\gamma|}{1+\alpha^2} \mathbf{m} \times \mathbf{H}_\mathrm{eff} - \frac{\alpha|\gamma|}{1+\alpha^2} \mathbf{m}\times(\mathbf{m}\times\mathbf{H}_\text{eff})</math>
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| {{Clear}}
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| ==Applications==
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| Apart from conventional magnetic domains and domain-walls, the theory also treats the statics and dynamics of topological line and point configurations, e.g. magnetic [[vortex]] and antivortex states;<ref>{{cite arXiv |last=Komineas |first=S. N. |authorlink= |eprint=0712.3684 |title=Dynamics of vortex-antivortex pairs in ferromagnets |class=cond-mat.mtrl-sci |year=2007 |version=1 |accessdate=14 November 2013}}</ref> or even 3d-Bloch points,<ref>{{cite journal|last=Thiaville|first=André|coauthors=García, José; Dittrich, Rok; Miltat, Jacques; Schrefl, Thomas|title=Micromagnetic study of Bloch-point-mediated vortex core reversal|journal=Physical Review B|date=March 2003|volume=67|issue=9|doi=10.1103/PhysRevB.67.094410}}</ref><ref name="Döring">{{cite journal|last=Döring|first=W.|title=Point Singularities in Micromagnetism|journal=Journal of Applied Physics|year=1968|volume=39|issue=2|pages=1006|doi=10.1063/1.1656144}}</ref> where, for example, the magnetization leads radially into all directions from the origin, or into topologically equivalent configurations. Thus in space, and also in time, nano- (and even pico-)scales are used.
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| The corresponding topological quantum numbers<ref name="Döring" /> are thought to be used as information carriers, to apply the most recent, and already studied, propositions in [[information technology]].
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| ==See also==
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| *[[Magnetism]]
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| ==Footnotes and references==
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| <references />
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| ==Further reading==
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| {{Refbegin}}
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| * {{cite book|first=William Fuller, Jr.|last=Brown|title=Micromagnetics|place=New York|publisher=Wiley|year=1963|isbn=0-88275-665-6}}
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| * {{cite journal|last=Gilbert|first=Thomas L.|title=A Phenomenological Theory of Damping in Ferromagnetic Materials|journal=IEEE Transactions on Magnetics|volume=40|issue=6|pages=3443–3449|year=2004|issn=0018-9464|doi=10.1109/TMAG.2004.836740|bibcode = 2004ITM....40.3443G }}
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| * {{cite journal|last=Kruzik Martin|first=Prohl Andreas |title=Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism|journal=SIAM Review|volume=48|issue=3|pages=439–483|year=2006|doi=10.1137/S0036144504446187 }}
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| {{Refend}}
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| ==External links==
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| * [http://www.ctcms.nist.gov/mumag/mumag.org.html µMAG -- Micromagnetic Modeling Activity Group].
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| [[Category:Dynamical systems]]
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| [[Category:Magnetic ordering]]
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| [[Category:Magnetostatics]]
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