Temperature dependence of the effective anisotropies in magnetic nanoparticles with Neel surface anisotropy

We discuss the physical concept of the effective anisotropy in magnetic nanoparticles with surface anisotropy. A recently developed constrained Monte Carlo method allows evaluation of the temperature dependence of the energy surface in the whole temperature range, from which the effective anisotropy is determined. We consider nanoparticles of different shapes with cubic or uniaxial core anisotropy and Neel surface anisotropy. We demonstrate that at low temperatures surface effects can be dominant, leading to an overall cubic effective anisotropy even in spherical nanoparticles with uniaxial core anisotropy. This cubic anisotropy contribution decreases more rapidly with increasing temperature than the uniaxial core anisotropy, leading to a temperature-induced reorientation transition. We discuss the scaling behaviour of the effective anisotropy with magnetization in nanoparticles with surface anisotropy contribution. The scaling exponent deviates from that expected from Callen-Callen theory due to increased fluctuations of the surface spins.