Magnetic anisotropy
Encyclopedia
Magnetic anisotropy is the direction dependence
of a material's magnetic properties. In the absence of an applied magnetic field
, a magnetically isotropic
material has no preferential direction for its magnetic moment
while a magnetically anisotropic material will align its moment with one of the easy axes. An easy axis is an energetically favorable direction of spontaneous magnetization
that is determined by the sources of magnetic anisotropy listed below. The two opposite directions along an easy axis are usually equivalent, and the actual direction of magnetization can be either of them (see spontaneous symmetry breaking
).
Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is superparamagnetic.
is
and the volume of the particle is 
, the magnetization is 
, where 
is the saturation magnetization and 
are direction cosines (components of a unit vector) so 
. The energy associated with magnetic anisotropy can depend on the direction cosines in various ways, the most common of which are discussed below.
direction, the anisotropy energy
can be expressed as one of the forms:
where
is the volume, 
the anisotropy constant, and 
the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol 
is often used to indicate the anisotropy constant, instead of 
. In the widely used Stoner–Wohlfarth model, the anisotropy is uniaxial.
in the energy). The coordinates can be chosen so the energy has the form
If
the easy axis is the 
direction, the intermediate axis is the 
direction and the hard axis is the 
direction.
If
the easy axes are the 
and 
axes. If 
there are four easy axes characterized by 
.
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
of a material's magnetic properties. In the absence of an applied magnetic field
Magnetic 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;...
, a magnetically isotropic
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
material has no preferential direction for its magnetic moment
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...
while a magnetically anisotropic material will align its moment with one of the easy axes. An easy axis is an energetically favorable direction of spontaneous magnetization
Spontaneous magnetization
Spontaneous magnetization is the term used to describe the appearance of an ordered spin state at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or .-Overview:...
that is determined by the sources of magnetic anisotropy listed below. The two opposite directions along an easy axis are usually equivalent, and the actual direction of magnetization can be either of them (see spontaneous symmetry breaking
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....
).
Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is superparamagnetic.
Sources of magnetic anisotropy
There are different sources of magnetic anisotropy:- Magnetocrystalline anisotropyMagnetocrystalline anisotropyMagnetocrystalline 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...
: the atomic structure of a crystalCrystalA crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
introduces preferential directions for the magnetisation. - Shape anisotropy: when a particle is not perfectly spherical, the demagnetizing fieldDemagnetizing fieldThe demagnetizing field, also called the stray field, is the magnetic field generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement...
will not be equal for all directions, creating one or more easy axes. - Magnetoelastic anisotropy: tension may alter magnetic behaviour, leading to magnetic anisotropy.
- Exchange anisotropyExchange biasExchange bias or exchange anisotropy occurs in bilayers of magnetic materials where the hard magnetization behavior of an antiferromagnetic thin film causes a shift in the soft magnetization curve of a ferromagnetic film...
: a relatively new type that occurs when antiferromagnetic and ferromagnetic materials interact.
Anisotropy energy of a single-domain magnet
Suppose that a ferromagnet is single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the magnetic momentMagnetic 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...
is
and the volume of the particle is , the magnetization is , where is the saturation magnetization and are direction cosines (components of a unit vector) so . The energy associated with magnetic anisotropy can depend on the direction cosines in various ways, the most common of which are discussed below.Uniaxial
A magnetic particle with uniaxial anisotropy has one easy axis. If the easy axis is in the direction, the anisotropy energyEnergy
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...
can be expressed as one of the forms:
where
is the volume, the anisotropy constant, and the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol is often used to indicate the anisotropy constant, instead of . In the widely used Stoner–Wohlfarth model, the anisotropy is uniaxial.Triaxial
A magnetic particle with triaxial anisotropy still has a single easy axis, but it also has a hard axis (direction of maximum energy) and an intermediate axis (direction associated with a saddle pointSaddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
in the energy). The coordinates can be chosen so the energy has the form
If
the easy axis is the direction, the intermediate axis is the direction and the hard axis is the direction.Cubic
A magnetic particle with cubic anisotropy has three or four easy axes, depending on the anisotropy parameters. The energy has the formIf
the easy axes are the and axes. If there are four easy axes characterized by .