Magnetization orientation dependence of the quasiparticle spectrum and hysteresis in ferromagnetic metal nanoparticles

Abstract

We use a microscopic Slater-Koster tight-binding model with short-range exchange and atomic spin-orbit interactions that realistically captures generic features of ferromagnetic metal nanoparticles to address the mesoscopic physics of magnetocrystalline anisotropy and hysteresis in nanoparticle quasiparticle excitation spectra. Our analysis is based on qualitative arguments supported by self-consistent Hartree-Fock calculations for nanoparticles containing up to 260 atoms. Calculations of the total energy as a function of magnetization direction demonstrate that the magnetic anisotropy per atom fluctuates by several percents when the number of electrons in the particle changes by one, even for the largest particles we consider. Contributions of individual orbitals to the magnetic anisotropy are characterized by a broad distribution with a mean more than two orders of magnitude smaller than its variance and with no detectable correlations between anisotropy contribution and quasiparticle energy. We find that the discrete quasiparticle excitation spectrum of a nanoparticle displays a complex non-monotonic dependence on an external magnetic field, with abrupt jumps when the magnetization direction is reversed by the field, explaining recent spectroscopic studies of magnetic nanoparticles. Our results suggests the existence of a broad cross-over from a weak spin-orbit coupling to a strong spin-orbit coupling regime, occurring over the range from approximately 200-to 1000-atom nanoparticles.2

Transition-metal dimers and physical limits on magnetic anisotropy

“…1a and b. Up to double-counting corrections, the total anisotropy energy is simply the sum of the spin orientation dependent shifts in occupied orbital energies 16 . The first column of levels in the righthand panels of Fig.…”

Orbital and spin contributions to thegtensors in metal nanoparticles

“…has been introduced in a study of the quasiparticle properties in ferromagnetic metal nanoparticles [14]. Here we give only a brief description of the terms in Eq.…”

“…In the limit of weak spin-orbit interaction τ so can be calculated perturbatively by Fermi golden rule. Here, however, we use a more pragmatic approach: we define τ so in terms of the average spin-orbit quasiparticle energy shift [14]h…”

Voltage control of magnetic hysteresis in a nickel nanoparticle

“…To motivate this form of the single electron anisotropy, we note that the discrete electron-box levels in a transition metal ferromagnetic particle are anisotropic with respect to the direction of the total magnetization, and they fluctuate on the order of ǫ so =h/τ so ≈ 1meV due to the so-interaction. 14,19 Here τ so is the so-flip time and is estimated to be 0.58 ps for Ni particles of this size. 14 Therefore, upon the addition of a tunneling electron onto a discrete level of the particle, an anisotropy energy shift ǫ so (which is played by the role of ǫ and ǫ z ) will be added to the particle Hamiltonian.…”

“…The effects of so-shifts (ǫ so ) of discrete energy levels were not considered. Since ǫ so in transition metal particles (∼ 1meV) is much larger than the magnetic anisotropy energy (per spin, ∼ 0.01meV), 10,11,14 the model does not apply to realistic transition metal ferromagnetic particles. In this work, we extend the model from Ref.…”