1.1.4.1 Uniaxial anisotropy

The most common anisotropy effect is connected to the existence of one only easy direction, and in literature it is referred to as uniaxial anisotropy. Thus, the anisotropy free energy density $f_$an$(\textbf{{m}})$ will be rotationally-symmetric with respect to the easy axis and will depend only on the relative orientation of $\textbf{{m}}$ with respect to this axis. We suppose, for sake of simplicity, that the easy direction coincides with the cartesian axis $z$. Therefore, we can write the expression of $f_$an$(\textbf{{m}})$ as an even function of $m_z=\cos \theta$, or equivalently using as independent variable $m_x^2+m_y^2=1-m_z^2=\sin\theta$. This expression, developed in series assumes the following form:

$$f_ (\textbf{{m}})=K_0+K_1\sin^2\theta+K_2\sin^4\theta+K_3\sin^6\theta+\ldots$$ (1.41)

where $K_1$, $K_2$, $K_3,\ldots$, are the anisotropy constants having the dimensions of energy per unit volume [J/m$^3$]. Here we will limit our analysis to the case in which the expansion (1.41) is truncated after the $\sin^2\theta$ term:

$$f_ (\textbf{{m}})=K_0+K_1\sin^2\theta \quad.$$ (1.42)

In the latter case, the anisotropic behavior depends on the sign of the constant $K_1$. When $K_1>0$, the anisotropy energy admits two minima at $\theta=0$ and $\theta=\pi$, that is when the magnetization lies along the positive or negative $z$ direction with no preferential orientation. This case is often referred to as easy axis anisotropy (see Fig. 1.3). Conversely, when $K_1<0$ the energy is minimized for $\theta=\pi/2$, meaning that any direction in $x-y$ plane corresponds to an easy direction. For this reason, this case is often referred to as easy plane anisotropy.

Figure: Uniaxial anisotropy energy density. (left) easy axis anisotropy ($K_1>0$). (right) easy plane anisotropy ($K_1<0$).

In the sequel, referring to uniaxial anisotropy, we will intend to use the following anisotropy free energy, derived from the integration over the whole body of the energy density Eq. (1.42):

$$G_ (\textbf{{m}})=\int_\Omega K_1 [1-(\mathbf{e}_ (\textbf{r})\cdot\textbf{{m}}(\textbf{r}))^2] dV \quad,$$ (1.43)

where $\mathbf{e}_$an$(\textbf{r})$ is the easy axis unit-vector at the location $\textbf{r}$ and the constant part connected to $K_0$ has been neglected.