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 approximation), yet small enough to resolve magnetic structures such as domain walls or vortices.
Micromagnetics can deal with static equilibria, by minimizing the magnetic energy, and with dynamic behavior, by solving the time-dependent dynamical equation.
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.
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 Ms. The problem then consists in finding the spatial orientation of the magnetization, which is given by the magnetization direction vectorm = M/Ms, also called reduced magnetization.
The static equilibria are found by minimizing the magnetic energy,
subject to the constraint |M|=Ms or |m|=1.
The contributions to this energy are the following:
The exchange energy is a phenomenological continuum description of the quantum-mechanical exchange interaction. It is written as:
where A is the exchange constant; mx, my and mz 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.
Magnetic anisotropy arises due to a combination of crystal structure and spin-orbit interaction. It can be generally written as:
where Fanis, the anisotropy energy density, is a function of the orientation of the magnetization. Minimum-energy directions for Fanis are called easy axes.
Time-reversal symmetry ensures that Fanis is an even function of m. The simplest such function is
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.
The Zeeman energy is the interaction energy between the magnetization and any externally applied field. It's written as:
where Ha is the applied field and µ0 is the vacuum permeability.
The Zeeman energy favors alignment of the magnetization parallel to the applied field.
The demagnetizing field is the magnetic field created by the magnetic sample upon itself. The associated energy is:
where Hd is the demagnetizing field. This field depends on the magnetic configuration itself, and it can be found by solving:
where −∇·M is sometimes called magnetic charge density. The solution of these equations (c.f. magnetostatics) is:
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 conditions 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.
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
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 . A quadratic form for the magnetoelastic energy is
where 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
This energy term contributes to magnetostriction.
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 terms of the local effective field acting on it.
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:
where dE/dV is the energy density. In variational terms, a change dm of the magnetization and the associated change dE of the magnetic energy are related by:
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:
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:
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:
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 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 numbersare thought to be used as information carriers, to apply the most recent, and already studied, propositions in information technology.
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.
In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer.
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In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations.
Magnetostatics is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory.
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Effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. EMAs or EMTs are developed from averaging the multiple values of the constituents that directly make up the composite material. At the constituent level, the values of the materials vary and are inhomogeneous. Precise calculation of the many constituent values is nearly impossible. However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters including the effective permittivity and permeability of the materials as a whole. In this sense, effective medium approximations are descriptions of a medium based on the properties and the relative fractions of its components and are derived from calculations, and effective medium theory. There are two widely used formulae.
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In physics, magnetization dynamics is the branch of solid-state physics that describes the evolution of the magnetization of a material.
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