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 Rₑ and the thickness of the spheroidal shell Rₑ-Rᵢ (0RᵢRₑ, Rᵢ: 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 (ε=Rᵢ∕Rₑ>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 Rᵢ.