User:2over0/Anisotropy energy

Anisotropy energy is a form of spontaneous symmetry breaking where energy minimization acquires a directional dependence, or anisotropy. The most common form of anisotropic energy is magnetocrystalline anisotropy. For each atom in a ferromagnetic, ferrimagnetic, or antiferromagnetic crystalline lattice, the direction of its magnetic moment is determined by its position in the lattice. In non-magnetic materials, these moments orient without preference to a particular direction in the absence of an applied field. The energy cost per atom to flip away from the "preferred" direction is the anisotropy energy. The direction along which the moments prefer to align is referred to as the easy axis.

One of the many projects currently researching this phenomena is directed by Stefan Krause and Roland Wiesendanger of the University of Hamburg. Using spin-polarized scanning tunnelling microscopes they are observing the effect upon the macrospin states of domains by passing a spin-polarized current through the atoms, and observing their alignment and how long they maintain their spin state.

Magnetocrystalline anisotropy
Magnetocrystalline anisotropy is the energy cost per atom to align its magnetization from one crystallographic direction to another. It is the most common source of magnetic anisotropy in bulk materials. In the vicinity of a particular atomic site, the potential is given by the direction, distance, and properties of neighboring atoms. This distorts the outlying atomic orbitals, which, mediated by spin-orbit coupling, can lead to preferred orientation in the electron dipole moments.

Face-centered and body-centered cubic systems
Take BCC iron for example, if it is cut perpendicularly with [001] direction, the domain will lie along $$[010], [0\bar{1}0], [001],$$ and $$[00\bar{1}]$$. When it is appled a field H in the [010] direction, the moment pointing in the same direction ($$\theta=0$$) will be more preferred due to less pontential energy, where $$E_p=\int_{\frac{\pi}{2}}^{\theta}mHsin{\theta}d\theta=-mHcos\theta$$ The moment in the opposite direction will be eliminated eventually. Compare to the field H applied in the [010], if it is applied in the direction of [110], then the moment will lie along [100] and [010] direction as a result, and if the H-field is large enough, the moment will take enough energy to rotate the moment direction to lie along . The magnetic field H in the direction will need more energy to rotate the moment direction than the field in the direction, so in BCC iron, is the easy axis, is the medium axis, and is the hard axis. Also, this magnetocrystalline anisotropy depends on the crystal structure, which means BCC and FCC will have different easy axes, and it can be explained by the equation, $$E=K_o+K_1(\alpha_1^2\alpha_2^2+\alpha_2^2\alpha_3^2+\alpha_3^2 \alpha_1^2)+K_2(\alpha_1^2\alpha_2^2\alpha_3^2)+...$$ Where $$K_1$$ is positive in BCC but is negative in FCC. In order to have the lowest potential energy, BCC has its easy axis, which is its magnetocrystalline anisotropy, in the direction; FCC has its easy axis in the direction.

HCP system
Different from cubic, HCP has the energy density equation in the form, $$E=K_o+K_1sin^2\theta+K_2sin^4\theta\,$$

Depending on the values $$K_{1}$$ and $$K_{2}$$, there are four different cases:


 * When $$K_{1}=K_{2}=0$$, the structure is isotropic.
 * When $$K_{1}>0$$ and $$K_{2}>0$$, the structure has easy axis anisotropy.
 * When $$K_{1}<0$$ and $$K_{2}<0$$, the structure has easy plane anisotropy.
 * When $$K_{1}$$ and $$K_{2}$$ have different signs, the structure has easy cone anisotropy.

For cobalt, $$K_1=4.1\times10^5J/m^3$$ $$K_2=1.5\times10^5J/m^3$$ Where $$K_1$$ and $$K_2$$ are both positive, so cobalt has the easy axis in the [0001] direction, and the lowest anisotropy energy is E=Ko when $$\theta$$ is 0 or 180, which means the moment points either up or down.

Sources of magnetic anisotropy
There are different sources of magnetic anisotropy :
 * Magnetocrystalline anisotropy: the atomic structure of a crystal introduces preferential directions for the magnetisation.
 * Shape anisotropy: when a particle is not perfectly spherical, the demagnetizing field will not be equal for all directions, creating one or more easy axes.
 * Stress anisotropy: tension may alter magnetic behaviour, leading to magnetic anisotropy.
 * Exchange anisotropy: a relatively new type that occurs when antiferromagnetic and ferromagnetic materials interact.

Anisotropy energy
Consider a magnetic particle with one easy axis and a (forced) magnetisation. The anisotropy energy can be expressed as $$E = K \sin^2(\theta)$$, with $$K$$ the anisotropy constant, and $$\theta$$ the angle between the easy axis and the particle's magnetisation. When shape anisotropy is explicitly considered, the symbol $$\mathcal{N}$$ is often used to indicate the anisotropy constant, instead of $$K$$.