Critical sizes for ferromagnetic spherical hollow nanoparticles

The total magnetic energies of zero-field magnetization states of ferromagnetic nanospheres with a non-magnetic core are analytically calculated and compared with each other. The particle radius $R_e$ and the thickness of the spheroidal shell $R_e-R_i$ ($0⩽R_i⩽R_e$, $R_i$: hole/core radius) are systematically varied for different hard and soft magnetic materials. Based on the results, the corresponding phase diagrams of the lowest-energy configurations are derived. The phase diagrams identify two phases for materials of uniaxial anisotropy (Co: single domain, curling vortex; ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$, FePt: single domain, two domain) and materials of cubic anisotropy (permalloy, Fe: single domain, curling vortex). In the case of hard magnetic materials the critical diameter for which the single-domain state becomes energetically unfavorable can become more than doubled for ultrathin spheroidal shells $(\epsilon =R_i∕R_e>0.85)$ compared to the corresponding bulk sphere of the same particle size. As the stray field in a single-domain spherical hollow nanoparticle is inhomogeneous, the nucleation process takes place inhomogeneously starting at the magnetic poles of the inner surface at $R_i$.