Anisotropy energy

Figure 2.3: Energy density due to uniaxial anisotropy as a function of the angle $\theta$ from a magnetic moment $\mu$. The maximum energy has been normalised to zero for clarity.

Anisotropy is a dependence of energy level on some direction. If the magnetic moments in a material have a bias towards one particular direction (the easy axis) then the material is said to have uniaxial anisotropy, like cobalt. If the bias is towards many particular directions, then the material has multiple easy axes and it possesses cubic anisotropy (see figure 2.4). Cubic crystals such as iron and nickel have this property (Aharoni, 2000, p86). Uniaxial and cubic anisotropy are forms of magnetocrystalline anisotropy as their properties in this respect arise from the crystalline structure of the material.

Figure 2.4: Normalised cubic anisotropy energy surfaces $w_c(\theta,\phi)$ for (left) iron and (right) nickel. The different shapes of the surfaces are a reflection of the sign of $K_1$ (O'Handley, 1999) -- iron has a positive $K_1$, nickel a negative $K_1$ (see appendix C)

The anisotropy energy in transition metal magnets arises from spin-orbit coupling. The typical fourth-order approximation of the parameterisation of uniaxial anisotropy (expressed as an energy density) is (Aharoni, 2000):

$$\mathcal{E}_{\mathrm{uni}}^{i} = -K_1 \cos^2(\theta_i) - K_2 \cos^4(\theta_i)$$ (2.5)

$$= K_1S^2_z + K_2S^4_z$$ (2.6)

$\mathcal{E}_{\mathrm{uni}}^{i}$The uniaxial anisotropy energy of a magnetic moment $\mu$$_i$ $K_1$The primary anisotropy constant of a material procured through experiment measurements, expressed as a temperature-dependent energy density $K_2$The secondary anisotropy constant of a material procured through experimental measurements, expressed as a temperature-dependent energy density

where $\theta_i$ is the angle between $\ensuremath{\mathbf{S}}_i$ and the easy axis (being here the component of $\ensuremath{\mathbf{S}}$ in the direction of the crystallographic axis, $z$). $K_1$ and $K_2$ are temperature-dependent energy densities derived from experiment, and can exist with either a positive or negative sign. When $K_1 > 0$ the axis is easy, when $K_1 < 0$ the axis becomes hard (which yields an easy plane).

Since constant terms can be neglected, an equivalent parameterisation is:

$$\mathcal{E}_{\mathrm{uni}}^{i} = K_1 \sin^2(\theta_i) + K_2 \sin^4(\theta_i)$$ (2.7)

The typical parameterisation of cubic anisotropy is not straightforward trigonometrically (O'Handley, 1999):

$$\mathcal{E}_{\mathrm{cub}}^{i} = K_1(S^2_xS^2_y + S^2_yS^2_z + S^2_zS^2_x) + K_2(S^2_xS^2_yS^2_z)$$ (2.8)

$\mathcal{E}_{\mathrm{cub}}^{i}$The cubic anisotropy energy of a magnetic moment $\mu$$_i$

A positive sign for $K_1$ yields easy axes along the body edges (100). Conversely, a negative sign for $K_1$ indicates that the easy axes exist across the diagonals (111) (Blundell, 2001).

The energy for a system of magnetic moments is given by:

$$\mathcal{E}_{\mathrm{an}} = \sum_i \mathcal{E}_{\mathrm{an}}^{i}$$ (2.9)

where $\mathcal{E}_{\mathrm{an}}$ is either $\mathcal{E}_{\mathrm{uni}}$ or $\mathcal{E}_{\mathrm{cub}}$.

It is worth noting that in some materials which are considered isotropic (i.e. K$_1$ = K$_2$ = 0) from a crystalline perspective, such as permalloy, the contribution to the total energy from the anisotropy is zero.

There are other types of anisotropy than magnetocrystalline. Magnetostriction is an anisotropy caused by the expansion or contraction of a ferromagnet along the direction of the magnetisation (Aharoni, 2000, p87). The so-called shape anisotropy (Paine et al., 1955) (also known as ``configurational stability'' (Ha et al., 2003)) is the direction in which the magnetisation will prefer to lie on account of the physical geometry of the sample. This becomes more and more influential the smaller one's sample becomes. This is one of the properties we investigate in this report.