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 (0less than or equal toR(i)less than or equal toR(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; Nd2Fe14B, 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.