1.1.4.2 Cubic anisotropy

This is the case when the anisotropy energy density has cubic symmetry, mostly due to spin-lattice coupling in cubic crystals. Basically it happens that three privileged directions exist. A typical expansion of the anisotropy energy density in this case is, in cartesian coordinates:

$$f_ (\textbf{{m}})=K_0+K_1(m_x^2 m_y^2+m_y^2 m_z^2+m_z^2 m_x^2)+ K_2 m_x^2m_y^2m_z^2+\ldots$$ (1.44)

As before, let us neglect terms of order grater than fourth (i.e. $K_2=0$, etc.). When $K_1>0$, there are six equivalent energy minima corresponding to the directions $x,y,z$, both positive and negative (see Fig. 1.4). Conversely, when $K_1<0$ a more complex situation arises. In fact, there are eight equivalent minima along the directions pointing the vertices of the cube (e.g. the direction [1,1,1]) and the coordinate axes directions become now hard axes.

Figure: Cubic Anisotropy energy density. (left) coordinate axes are easy axes ($K_2>0$). (right) coordinate axes are easy axes ($K_2<0$).

This case has been inserted for sake of completeness, but in the sequel cubic anisotropy will be not considered anymore. It is important to underline that the character of anisotropy interaction is local, that is, the anisotropy energy related to an elementary volume $dV_{\textbf{r}'}$ depends only on the magnetization $\textbf{M}(\textbf{r}')$.