Domain wall
Encyclopedia
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
.
In magnetism
, a domain wall is an interface separating magnetic domains. It is a transition between different magnetic moments
and usually undergoes an angular displacement
of 90° or 180°. Domain wall is a gradual reorientation of individual moments across a finite distance. The domain wall thickness depends on the anisotropy of the material, but on average spans across around 100–150 atoms.
The energy of a domain wall is simply the difference between the magnetic moments before and after the domain wall was created. This value is usually expressed as energy per unit wall area.
The width of the domain wall varies due to the two opposing energies that create it: the Magnetocrystalline anisotropy
energy and the exchange energy (
), both of which tend to be as low as possible so as to be in a more favorable energetic state. The anisotropy energy is lowest when the individual magnetic moments are aligned with the crystal lattice axes thus reducing the width of the domain wall. Whereas the exchange energy is reduced when the magnetic moments are aligned parallel to each other and thus makes the wall thicker, due to the repulsion between them. (Where anti-parallel alignment would bring them closer - working to reduce the wall thickness.) In the end an equilibrium is reached between the two and the domain wall's width is set as such.
An ideal domain wall would be fully independent of position, however, they are not ideal and so get stuck on inclusion sites within the medium, also known as Crystallographic defect
s. These include missing or different (foreign) atoms, oxides, insulators and even stresses within the crystal. This prevents the formation of domain walls and also inhibits their propagation through the medium. Thus a greater applied magnetic field is required to overcome these sites.
Note that the magnetic domain walls are exact solutions to classical nonlinear equations of magnets (Landau-Lifshitz equation, nonlinear Schrodinger equation
and so on).
then these groups transform to magnetic point group
s. It was shown that there are 125 of such groups. It was found that if magnetic point group
is pyroelectric and/or pyromagnetic then domain wall carries polarization
and/or magnetization
respectively . These criteria were derived from the conditions of the appearing of the uniform polarization
and/or magnetization
. After their application to any inhomogeneous region they predict existing of even parts in functions of distribution of order parameters. Identification of the remained odd parts of these functions, were formulated based on symmetry transformations which interrelate domain
s. The symmetry classification of the magnetic domain walls contains 64 magnetic point group
s.
Symmetry based predictions of the structure of the multiferroic domain walls have been proven using phenomenology
coupling via magnetization
and/or polarization
spatial derivatives (flexomagnetoelectric effect).
Non-magnetic inclusion
s in the volume of a ferromagnetic material, or dislocation
s in crystallographic structure, can cause "pinning" of the domain walls (see animation). Such pinning sites cause the domain wall to seat in a local energy minimum and external field is required to "unpin" the domain wall from its pinned position. The act of unpinning will cause sudden movement of the domain wall and sudden change of the volume of both neighbouring domains. This causes Barkhausen noise and in effect it is most likely to be the source of magnetic hysteresis.
was only observed in fiber lasers of positive dispersion, a phase-locked dark-bright vector soliton was obtained in fiber lasers of either positive or negative dispersion. Numerical simulations confirmed the experimental observations, and further showed that the observed vector solitons are the two types of phase-locked polarization domain-wall solitons theoretically predicted. Another novel type domain wall soliton is the vector dark domain wall consisting of stable localized structures separating the two orthogonal linear polarization eigenstates of the laser emission and dark structure is visible only when the total laser emission is measured.
, a domain wall is a theoretical 2-dimensional singularity
. A domain wall is meant to represent an object of codimension
one embedded into space (a defect in space localized in one spatial dimension). For example, D8-branes are domain walls in type II string theory
. In M-theory
, the existence of Horava-Witten domain wall
s, "ends of the world" that carry an E8
gauge theory
, is important for various relations between superstring theory
and M-theory
.
If domain walls exist, it seems plausible that they will violently emit gravitational wave
s if two such walls would collide. As the Laser Interferometer Gravitational-Wave Observatory
and future observatories of its kind will search for direct evidence of gravitational waves, this phenomenon would be included as well in such searches.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
which can have one of two distinct but similar meanings in magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, or string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry
Discrete symmetry
A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of...
is spontaneously broken
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
.
Magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
, a domain wall is an interface separating magnetic domains. It is a transition between different magnetic moments
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...
and usually undergoes an angular displacement
Angular displacement
Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis....
of 90° or 180°. Domain wall is a gradual reorientation of individual moments across a finite distance. The domain wall thickness depends on the anisotropy of the material, but on average spans across around 100–150 atoms.
The energy of a domain wall is simply the difference between the magnetic moments before and after the domain wall was created. This value is usually expressed as energy per unit wall area.
The width of the domain wall varies due to the two opposing energies that create it: the Magnetocrystalline 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 and the exchange energy (
), both of which tend to be as low as possible so as to be in a more favorable energetic state. The anisotropy energy is lowest when the individual magnetic moments are aligned with the crystal lattice axes thus reducing the width of the domain wall. Whereas the exchange energy is reduced when the magnetic moments are aligned parallel to each other and thus makes the wall thicker, due to the repulsion between them. (Where anti-parallel alignment would bring them closer - working to reduce the wall thickness.) In the end an equilibrium is reached between the two and the domain wall's width is set as such.An ideal domain wall would be fully independent of position, however, they are not ideal and so get stuck on inclusion sites within the medium, also known as Crystallographic defect
Crystallographic defect
Crystalline solids exhibit a periodic crystal structure. The positions of atoms or molecules occur on repeating fixed distances, determined by the unit cell parameters. However, the arrangement of atom or molecules in most crystalline materials is not perfect...
s. These include missing or different (foreign) atoms, oxides, insulators and even stresses within the crystal. This prevents the formation of domain walls and also inhibits their propagation through the medium. Thus a greater applied magnetic field is required to overcome these sites.
Note that the magnetic domain walls are exact solutions to classical nonlinear equations of magnets (Landau-Lifshitz equation, nonlinear Schrodinger equation
Nonlinear Schrödinger equation
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...
and so on).
Symmetry of multiferroic domain walls
Since domain walls can be considered as thin layers, their symmetry is described by one of the 528 magnetic layer groups . To determine the layer's physical properties continuum approximation is used which leads to point-like layer groups . If continuous translation operation is considering as identityIdentity
-Philosophical topics:* Identity , also called sameness, is whatever makes an entity definable and recognizable* Law of identity, principle of logic stating that an object is the same as itself...
then these groups transform to magnetic point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s. It was shown that there are 125 of such groups. It was found that if magnetic point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
is pyroelectric and/or pyromagnetic then domain wall carries polarization
Polarization
Polarization is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves, such as light, and gravitational waves exhibit polarization; acoustic waves in a gas or liquid do not have polarization because the direction of vibration and...
and/or magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
respectively . These criteria were derived from the conditions of the appearing of the uniform polarization
Polarization
Polarization is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves, such as light, and gravitational waves exhibit polarization; acoustic waves in a gas or liquid do not have polarization because the direction of vibration and...
and/or magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
. After their application to any inhomogeneous region they predict existing of even parts in functions of distribution of order parameters. Identification of the remained odd parts of these functions, were formulated based on symmetry transformations which interrelate domain
Domain
-General:*Territory , a non-sovereign geographic area which has come under the authority of another government*Public domain, a body of works and knowledge without proprietary interest...
s. The symmetry classification of the magnetic domain walls contains 64 magnetic point group
Point group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s.
Symmetry based predictions of the structure of the multiferroic domain walls have been proven using phenomenology
Phenomenology (science)
The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...
coupling via magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
and/or polarization
Polarization
Polarization is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves, such as light, and gravitational waves exhibit polarization; acoustic waves in a gas or liquid do not have polarization because the direction of vibration and...
spatial derivatives (flexomagnetoelectric effect).
Depinning of a domain wall
Inclusion
Inclusion may refer to:- Metallurgy :*Inclusion , a type of metal casting defect*Inclusions in Aluminium Alloys, solid particles in liquid aluminium alloy- Social inclusion of persons :...
s in the volume of a ferromagnetic material, or dislocation
Dislocation
In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. The presence of dislocations strongly influences many of the properties of materials...
s in crystallographic structure, can cause "pinning" of the domain walls (see animation). Such pinning sites cause the domain wall to seat in a local energy minimum and external field is required to "unpin" the domain wall from its pinned position. The act of unpinning will cause sudden movement of the domain wall and sudden change of the volume of both neighbouring domains. This causes Barkhausen noise and in effect it is most likely to be the source of magnetic hysteresis.
Optics
Recently, a phase-locked dark-dark vector solitonVector soliton
In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one polarization component, while vector solitons have two distinct...
was only observed in fiber lasers of positive dispersion, a phase-locked dark-bright vector soliton was obtained in fiber lasers of either positive or negative dispersion. Numerical simulations confirmed the experimental observations, and further showed that the observed vector solitons are the two types of phase-locked polarization domain-wall solitons theoretically predicted. Another novel type domain wall soliton is the vector dark domain wall consisting of stable localized structures separating the two orthogonal linear polarization eigenstates of the laser emission and dark structure is visible only when the total laser emission is measured.
String theory
In string theoryString theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, a domain wall is a theoretical 2-dimensional singularity
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
. A domain wall is meant to represent an object of codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
one embedded into space (a defect in space localized in one spatial dimension). For example, D8-branes are domain walls in type II string theory
Type II string theory
In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings. These account for two of the five consistent superstring theories in ten dimensions. Both theories have the maximal amount of supersymmetry — namely 32 supercharges...
. In M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
, the existence of Horava-Witten domain wall
Horava-Witten domain wall
In theoretical physics, a Hořava–Witten domain wall is a type of domain wall that behaves as a boundary of the eleven-dimensional spacetime in M-theory....
s, "ends of the world" that carry an E8
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...
gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, is important for various relations between superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
and M-theory
M-theory
In theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
.
If domain walls exist, it seems plausible that they will violently emit gravitational wave
Gravitational wave
In physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...
s if two such walls would collide. As the Laser Interferometer Gravitational-Wave Observatory
LIGO
LIGO, which stands for the Laser Interferometer Gravitational-Wave Observatory, is a large-scale physics experiment aiming to directly detect gravitational waves. Cofounded in 1992 by Kip Thorne and Ronald Drever of Caltech and Rainer Weiss of MIT, LIGO is a joint project between scientists at MIT,...
and future observatories of its kind will search for direct evidence of gravitational waves, this phenomenon would be included as well in such searches.