Micromagnetism
Encyclopedia
Micromagnetics deals with the interactions between magnetic moment
s on sub-micrometre
length scales. These are governed by several competing energy
terms. Dipolar energy is the energy which causes magnet
s to align north
to south pole
. Exchange
energy will attempt to make the magnetic moments in the immediately surrounding space lie parallel to one another (if the material is ferromagnetic
) or antiparallel
to one another (if antiferromagnetic
). Anisotropy
energy is low when the magnetic moments are aligned along a particular crystal direction. Zeeman energy
is at its lowest when magnetic moments lie parallel to an external magnetic field.
Since the most efficient magnetic alignment (also known as a configuration) is the one in which the energy is lowest, the sum of these four energy terms will attempt to become as small as possible at the expense of the others, yielding complex physical interactions.
The competition of these interactions under different conditions is responsible for the overall behavior of a magnetic system.
published a paper on antiparallel domain wall
structures. Until comparatively recently computation micromagnetics has been prohibitively expensive in terms of computational power, but smaller problems are now solveable on a modern desktop PC
.
is used to solve time-dependent micromagnetic problems, where
is the magnetic moment per unit volume, 
is the effective magnetic field
,
is the Gilbert phenomenological damping parameter and 
is the electron gyromagnetic ratio. Furthermore, 
is the magnitude of the magnetization vector 
Equation (1) can be shown to be equivalent to the more complicated form
Originally, in 1935, Landau and Lifshitz used this expression without the denominator
, which arose from Gilbert's modification in 1955.
, then we get the famous, damping-free, Landau-Lifshitz equation (LLE)
depends on the relevant interactions, namely, (i), on the exchange interaction; (ii), on the so-called anisotropy interaction; (iii), on the magnetic dipole-dipole interaction; and, (iv), on the external field (the so-called "Zeeman field").
The answer is somewhat involved: let the energies corresponding to (i) and (ii) be given by
and
Here we use the decomposition
of the magnetization vector into its magnitude MS and the direction vector 
while A is the so-called exchange constant. V is the magnetic volume.
Then we have:
Here the third term on the r.h.s. is the internal field produced at the position
by the dipole-dipole interaction, whereas the fourth term is the external field, also called "Zeeman field". Usually the first and the third term play the dominating role, usually a competing one, in this complicated sum. In particular: due to the third term the effective field is a nonlocal function of the magnetization, i.e. although the Landau-Lifshitz-Gilbert equation looks relatively harmless, one is actually dealing with a complicated nonlinear set of integro-differential equations.
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
.
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
s on sub-micrometre
Micrometre
A micrometer , is by definition 1×10-6 of a meter .In plain English, it means one-millionth of a meter . Its unit symbol in the International System of Units is μm...
length scales. These are governed by several competing energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
terms. Dipolar energy is the energy which causes magnet
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
s to align north
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...
to south pole
South Pole
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...
. Exchange
Exchange interaction
In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap...
energy will attempt to make the magnetic moments in the immediately surrounding space lie parallel to one another (if the material is ferromagnetic
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
) or antiparallel
Antiparallel (mathematics)
-Definitions:Given two lines m_1 \, and m_2 \,, lines l_1 \, and l_2 \, are anti-parallel with respect to m_1 \, and m_2 \, if \angle 1 = \angle 2 \,....
to one another (if antiferromagnetic
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...
). Anisotropy
Magnetocrystalline anisotropy
Magnetocrystalline anisotropy is the dependence of the internal energy of a ferromagnet on the direction of its magnetization. As a result, certain crystallographic directions are preferred directions, or easy axes, for the magnetization. It is a special case of magnetic anisotropy...
energy is low when the magnetic moments are aligned along a particular crystal direction. Zeeman energy
Zeeman Energy
Zeeman energy is the potential energy of a magnetised body in an external magnetic field. It is named after the dutch physicist Pieter Zeeman, and closely related to the Zeeman effect...
is at its lowest when magnetic moments lie parallel to an external magnetic field.
Since the most efficient magnetic alignment (also known as a configuration) is the one in which the energy is lowest, the sum of these four energy terms will attempt to become as small as possible at the expense of the others, yielding complex physical interactions.
The competition of these interactions under different conditions is responsible for the overall behavior of a magnetic system.
History
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.William Fuller Brown, Jr.
William Fuller Brown, Jr. was an American physicist who developed the theory of micromagnetics, a continuum theory of ferromagnetism that has had numerous applications in physics and engineering...
published a paper on antiparallel domain wall
Domain wall
A domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.-Magnetism:In magnetism,...
structures. Until comparatively recently computation micromagnetics has been prohibitively expensive in terms of computational power, but smaller problems are now solveable on a modern desktop PC
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...
.
Landau-Lifshitz-Gilbert equation
Generally, a form of the Landau-Lifshitz-Gilbert equation:is used to solve time-dependent micromagnetic problems, where
is the magnetic moment per unit volume, is the effective magnetic fieldMagnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
,
is the Gilbert phenomenological damping parameter and is the electron gyromagnetic ratio. Furthermore, is the magnitude of the magnetization vector Equation (1) can be shown to be equivalent to the more complicated form
Originally, in 1935, Landau and Lifshitz used this expression without the denominator
, which arose from Gilbert's modification in 1955.Landau-Lifshitz equation
If in (1) we put the Gilbert damping parameter, then we get the famous, damping-free, Landau-Lifshitz equation (LLE)The "effective field"
An essential merit of the micromagnetic theory concerns the answer on the question, how the effective field depends on the relevant interactions, namely, (i), on the exchange interaction; (ii), on the so-called anisotropy interaction; (iii), on the magnetic dipole-dipole interaction; and, (iv), on the external field (the so-called "Zeeman field").The answer is somewhat involved: let the energies corresponding to (i) and (ii) be given by
and
Here we use the decomposition
of the magnetization vector into its magnitude MS and the direction vector while A is the so-called exchange constant. V is the magnetic volume.Then we have:
Here the third term on the r.h.s. is the internal field produced at the position
by the dipole-dipole interaction, whereas the fourth term is the external field, also called "Zeeman field". Usually the first and the third term play the dominating role, usually a competing one, in this complicated sum. In particular: due to the third term the effective field is a nonlocal function of the magnetization, i.e. although the Landau-Lifshitz-Gilbert equation looks relatively harmless, one is actually dealing with a complicated nonlinear set of integro-differential equations.Applications
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 vortexVortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a 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 numbers are thought to be used as information carriers, to apply the most recent, and already studied, propositions in information technology
Information technology
Information technology is the acquisition, processing, storage and dissemination of vocal, pictorial, textual and numerical information by a microelectronics-based combination of computing and telecommunications...
.
External links
- µMAG -- Micromagnetic Modeling Activity Group.
- Magnetization dynamics applet.
- OOMMF - The Object-Oriented Micromagnetic Framework - a popular free micromagnetic simulation tool using finite differenceFinite differenceA finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
lattice discretisations of space and FFT. - Magpar - a parallelizableParallel computingParallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...
, finite element based, free micromagnetic simulation package. - Nmag - a parallelizableParallel computingParallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...
, finite element based, free micromagnetic simulator that is scriptable in PythonPython (programming language)Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
. - FEMME -- Finite element based micromagnetic package, commercial.
- http://llgmicro.home.mindspring.com/LLGMicromagnetics -- Finite differenceFinite differenceA finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
based micromagnetic package, commercial]. - http://www.magoasis.com/Magsimus Deluxe -- Finite differenceFinite differenceA finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
, MultiphysicsMultiphysicsMultiphysics treats simulations that involve multiple physical models or multiple simultaneous physical phenomena. For example, combining chemical kinetics and fluid mechanics or combining finite elements with molecular dynamics...
based micromagnetic package, commercial].