1 Phenomenological expression for the effective anisotropy constant in a system with surface anisotropy

The influence of the surface manifests itself in the fact that the values $ \Delta E/V$ are often found experimentally to be different from that of the bulk, i.e. there is an effective anisotropy $ \mathcal{K}^{eff}$ that is not exactly proportional to the particle's volume V. One can expect that the effect of the surface reduces when the particle size increases. The volume dependence of the effective anisotropy is often analyzed on the basis of a simple model:

$\displaystyle \mathcal{K}^{eff}=\mathcal{K}_{\infty}+\frac{S}{V}\mathcal{K}_{S}$ (46)

where $ \mathcal{K}_{\infty}$ is the anisotropy constant for an infinite system, presumably it is equal to the bulk anisotropy constant, and $ \mathcal{K}_{S}$ is the effective surface anisotropy constant, $ S$ and V are the surface and the volume of the system respectively. For spherical particles with diameter $ D$ it has been suggested that the effective anisotropy constants also adjust to a similar phenomenological expression [36]:

$\displaystyle \mathcal{K}^{eff}=\mathcal{K}_{\infty}+\frac{6}{D}\mathcal{K}_{S}$ (47)

We can find a modified version of this formula. For example, Luis et.al., in the study of the magnetic behavior of fcc Co nanoparticles embedded in different non-magnetic matrices [10], supposed the following relation between the effective anisotropy and the bulk and surface anisotropies, corrected by the fact that the surface spins do not feel the bulk anisotropy.

$\displaystyle \mathcal{K}^{eff}=(1-f)\mathcal{K}_{V}+f\cdot \mathcal{K^*}_{S}$ (48)

where $ f$ is the surface atom fraction, $ f\approx1-(1-a/D)^3$ and $ a$ is the lattice parameter. Here $ \mathcal{K^*}_{S}$ represents the surface anisotropy contribution to the effective anisotropy. Note that it has different dimensions than the surface anisotropy in Eq. (3.5).

Rocio Yanes