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Micromagnetics is a field of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
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 approximation), yet small enough to resolve magnetic structures such as
domain walls A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with pol ...
or vortices. Micromagnetics can deal with static equilibria, by minimizing the magnetic energy, and with dynamic behavior, by solving the time-dependent dynamical equation.


History

Micromagnetics as a field (''i.e.'', that deals specifically with the behaviour of ferromagnetic materials at sub-micrometer length scales) was introduced in 1963 when William Fuller Brown Jr. published a paper on antiparallel 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 PC.


Static micromagnetics

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 In physics and materials science, the Curie temperature (''T''C), or Curie point, is the temperature above which certain materials lose their permanent magnetic properties, which can (in most cases) be replaced by induced magnetism. The Cur ...
of the material considered, the modulus , M, of the magnetization is assumed to be everywhere equal to the
saturation magnetization Seen in some magnetic materials, saturation is the state reached when an increase in applied external magnetic field ''H'' cannot increase the magnetization of the material further, so the total magnetic flux density ''B'' more or less levels off ...
''M''s. The problem then consists in finding the spatial orientation of the magnetization, which is given by the ''magnetization direction vector'' m = M/''M''s, also called ''reduced magnetization''. The static equilibria are found by minimizing the magnetic energy, :E = E_\text + E_\text + E_\text + E_\text+E_\text, subject to the constraint , M, =''M''s or , m, =1. The contributions to this energy are the following:


Exchange energy

The exchange energy is a phenomenological continuum description of the quantum-mechanical
exchange interaction In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
. It is written as: :E_\text = A \int_V \left((\nabla m_x)^2 + (\nabla m_y)^2 + (\nabla m_z)^2\right) \mathrmV where ''A'' is the ''exchange constant''; ''m''x, ''m''y and ''m''z are the components of m; and the integral is performed over the volume of the sample. 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.


Anisotropy energy

Magnetic anisotropy arises due to a combination of
crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
and spin-orbit interaction. It can be generally written as: :E_\text = \int_V F_\text(\mathbf) \mathrmV where ''F''anis, the anisotropy energy density, is a function of the orientation of the magnetization. Minimum-energy directions for ''F''anis are called ''easy axes''.
Time-reversal symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the futur ...
ensures that ''F''anis is an even function of m. The simplest such function is :F_\text(\mathbf) = -K m_z^2. where ''K'' is called the ''anisotropy constant''. In this approximation, called ''uniaxial anisotropy'', the easy axis is the ''z'' direction. The anisotropy energy favors magnetic configurations where the magnetization is everywhere aligned along an easy axis.


Zeeman energy

The Zeeman energy is the interaction energy between the magnetization and any externally applied field. It's written as: :E_\text = -\mu_0 \int_V \mathbf\cdot\mathbf_\text \mathrmV where Ha is the applied field and µ0 is the
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
. The Zeeman energy favors alignment of the magnetization parallel to the applied field.


Energy of the demagnetizing field

The demagnetizing field is the magnetic field created by the magnetic sample upon itself. The associated energy is: :E_\text = -\frac \int_V \mathbf\cdot\mathbf_\text \mathrmV where Hd is the
demagnetizing field The demagnetizing field, also called the stray field (outside the magnet), is the magnetic field (H-field) generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields ...
. This field depends on the magnetic configuration itself, and it can be found by solving: :\nabla\cdot\mathbf_\text = -\nabla\cdot\mathbf :\nabla\times\mathbf_\text = 0 where −∇·M is sometimes called ''magnetic charge density''. The solution of these equations (c.f.
magnetostatics Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge, charges are stationary. The magnetization need not be st ...
) is: :\mathbf_\text = -\frac \int_V \nabla\cdot\mathbf \frac \mathrmV where r is the vector going from the current integration point to the point where Hd is being calculated. 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 In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s on the edge of the sample. 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.


Magnetoelastic Energy

The magnetoelastic energy describes the energy storage due to elastic lattice distortions. It may be neglected if magnetoelastic coupled effects are neglected. There exists a preferred local distortion of the crystalline solid associated with the magnetization director m, . For a simple model, one can assume this strain to be isochoric and fully isotropic in the lateral direction, yielding the deviatoric ansatz \mathbf_0(\mathbf) = \frac E\, \left mathbf\otimes \mathbf - \frac\mathbf\right/math> where the material parameter ''E > 0'' is the magnetostrictive constant. Clearly, ''E'' is the strain induced by the magnetization in the direction m. With this ansatz at hand, we consider the elastic energy density to be a function of the elastic, stress-producing strains \mathbf_e := \mathbf -\mathbf_0. A quadratic form for the magnetoelastic energy is E_\text = \frac mathbf -\mathbf_0(\mathbf): \mathbb : mathbf -\mathbf_0(\mathbf) where \mathbb :=\lambda \mathbf\otimes \mathbf + 2\mu \mathbb is the fourth-order elasticity tensor. Here the elastic response is assumed to be isotropic (based on the two Lamé constants λ and μ). Taking into account the constant length of m, we obtain the invariant-based representation E_\text = \frac \mbox^2 mathbf + \mu \, \mbox mathbf^2 - 3\mu E \big\ . This energy term contributes to magnetostriction.


Dynamic micromagnetics

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 In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
describing the evolution of the magnetization in terms of the local ''effective field'' acting on it.


Effective field

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: :\mathbf_\mathrm = - \frac \frac where d''E''/d''V'' is the energy density. In variational terms, a change dm of the magnetization and the associated change d''E'' of the magnetic energy are related by: :\mathrmE = -\mu_0 M_s \int_V (\mathrm\mathbf)\cdot\mathbf_\text\,\mathrmV Since m is a unit vector, dm is always perpendicular to m. Then the above definition leaves unspecified the component of Heff that is parallel to m. This is usually not a problem, as this component has no effect on the magnetization dynamics. From the expression of the different contributions to the magnetic energy, the effective field can be found to be: :\mathbf_\mathrm = \frac \nabla^2 \mathbf - \frac \frac + \mathbf_\text + \mathbf_\text


Landau-Lifshitz-Gilbert equation

This is the equation of motion of the magnetization. It describes a
Larmor precession In physics, Larmor precession (named after Joseph Larmor) is the precession of the magnetic moment of an object about an external magnetic field. The phenomenon is conceptually similar to the precession of a tilted classical gyroscope in an extern ...
of the magnetization around the effective field, with an additional
damping Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
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: :\frac = - , \gamma, \mathbf \times \mathbf_\mathrm + \alpha \mathbf\times\frac where γ is the electron gyromagnetic ratio and α the Gilbert damping constant. It can be shown that this is mathematically equivalent to the following ''Landau-Lifshitz'' (or explicit) form: :\frac = - \frac \mathbf \times \mathbf_\mathrm - \frac \mathbf\times(\mathbf\times\mathbf_\text)


Applications

The interaction of micromagnetics with mechanics is also of interest in the context of industrial applications that deal with magneto-acoustic resonance such as in hypersound speakers, high frequency magnetostrictive transducers etc. FEM simulations taking into account the effect of magnetostriction into micromagnetics are of importance. Such simulations use models described above within a finite element framework. 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 In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
and antivortex states; or even 3d-Bloch points, 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. The corresponding topological quantum numbers are thought to be used as information carriers, to apply the most recent, and already studied, propositions in
information technology Information technology (IT) is the use of computers to create, process, store, retrieve, and exchange all kinds of data . and information. IT forms part of information and communications technology (ICT). An information technology system (I ...
. Another application that has emerged in the last decade is the application of micromagnetics towards neuronal stimulation. In this discipline, numerical methods such as finite-element analysis are used to analyze the electric/magnetic fields generated by the stimulation apparatus; then the results are validated or explored further using in-vivo or in-vitro neuronal stimulation. Several distinct set of neurons have been studied using this methodology including retinal neurons, cochlear neurons, vestibular neurons, and cortical neurons of embryonic rats.


See also

*
Magnetism Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
*
Magnetic nanoparticles Magnetic nanoparticles are a class of nanoparticle that can be manipulated using magnetic fields. Such particles commonly consist of two components, a magnetic material, often iron, nickel and cobalt, and a chemical component that has functionali ...


Footnotes and references


Further reading

* * * * * * * * * * {{Refend


External links


µMAG -- Micromagnetic Modeling Activity Group

OOMMF -- Micromagnetic Modeling Tool

MuMax -- GPU-accelerated Micromagnetic Modeling Tool
Dynamical systems Magnetic ordering Magnetostatics