Effective anisotropies and energy barriers of magnetic nanoparticles with Neel surface anisotropy

Magnetic nanoparticles with Neel surface anisotropy, different internal structures, surface arrangements, and elongation are modeled as many-spin systems. The results suggest that the energy of many-spin nanoparticles cut from cubic lattices can be represented by an effective one-spin potential containing uniaxial and cubic anisotropies. It is shown that the values and signs of the corresponding constants depend strongly on the particle's surface arrangement, internal structure, and shape. Particles cut from a simple cubic lattice have the opposite sign of the effective cubic term, as compared to particles cut from the face-centered cubic lattice. Furthermore, other remarkable phenomena are observed in nanoparticles with relatively strong surface effects. (i) In elongated particles surface effects can change the sign of the uniaxial anisotropy. (ii) In symmetric particles (spherical and truncated octahedral) with cubic core anisotropy surface effects can change the sing of the latter. We also show that the competition between the core and surface anisotropies leads to a new energy that contributes to both the second- and fourth-order effective anisotropies. We evaluate energy barriers Delta E as functions of the strength of the surface anisotropy and the particle size. The results are analyzed with the help of the effective one-spin potential, which allows us to assess the consistency of the widely used formula Delta E/V=K-infinity+6K(s)/D, where K-infinity is the core anisotropy constant, K-s is a phenomenological constant related to surface anisotropy, and D is the particle's diameter. We show that the energy barriers are consistent with this formula only for elongated particles for which the surface contribution to the effective uniaxial anisotropy scales with the surface and is linear in the constant of the Neel surface anisotropy.