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. In a remarkable step toward a better understanding of ferromagnetism at the nanometer scale, individual electronic states within ferromagnetic nanoparticles were recently observed using electron tunneling spectroscopy [1,2]. Several features of the tunneling excitation spectra could not be explained within the simple electron-in-a-box picture, unlike the normal metal case [3]. Although this “failure” was anticipated due to the many-body character of ferromagnetism, several particular aspects were largely unexpected: (1) the non-linear magnetic field dependence of the energy levels close to the switching field (Bsw, the field at which the magnetization suddenly changes its orientation), (2) very small g-factors, and (3) excitation energies much smaller than the mean level spacing associated with the finite particle size. While we briefly return to the latter two in closing, we focus on the first effect as representative of the peculiarities of nanoscale ferromagnetism. A non-zero Bsw implies the presence of a magnetic anisotropy energy barrier: the (large) magnetic moment of the particle points in a preferred direction. This preference originates from spin-orbit coupling and/or the shape of the grain. At Bsw, there is an abrupt change in the grain’s total energy, leading to “jumps” in the tunneling spectrum. Refs. [1,2,4] show that such jumps can be described by a simple uniaxial anisotropy model: ĤA = −k Ŝ z/S, where k is the anisotropy constant—which is an intensive quantity in the bulk—S is the total spin and ẑ is the anisotropy axis. However, to account for the non-linear behavior and the magnitude of the jumps, it was crucial to assume that k depends on both the number of electrons