Anisotropy in ferromagnetic nanoparticles: Level-to-level fluctuations of a collective effect

We calculate the mesoscopic fluctuations of the magnetic anisotropy of ferromagnetic nanoparticles; that is, the effect of single-particle interference on the direction of the collective magnetic moment. A microscopic spin-orbit Hamiltonian considered as a perturbation of the much stronger exchange interaction first yields an explicit expression for the anisotropy tensor. Then, assuming a simple random matrix model for the spin-orbit coupling allows us to describe the fluctuation of such a tensor analytically. In the case of uniaxial anisotropy, we calculate the distribution of the anisotropy constant for a given number of electrons, and its variation upon increasing this number by one. The magnitude of the latter scales inversely with the number of atoms in the particle and is sufficient to account for the experimental data.