Bound on Anisotropy in Itinerant Ferromagnets from Random Impurities

We calculate the anisotropy energy of a single-domain ferromagnetic particle in which the only source of anisotropy is the presence of nonmagnetic impurities. Such anisotropy has easy-axis and easy-plane contributions, with random orientations of the axes. Typically the anisotropy energy is of order $N^{1/2}\hslash /{\tau }_{\mathrm{so}}$, where $N$ is the number of electrons in the ferromagnetic particle and ${\tau }_{\mathrm{so}}$ is the spin-orbit time.

Figure 1

Diagrammatic representation of $C(\theta )$. The dotted lines represent the impurity average. The solid lines are impurity averaged single-electron Green functions.