2 Localized spin (atomistic or Heisenberg) model

We consider a magnetic nanoparticle of ${\cal N}$ spins in the many-spin approach, i.e., taking account of its intrinsic properties such as the lattice structure, shape, and size. This also includes the (nearest-neighbor) exchange interactions, single-site core and surface anisotropy. The magnetic properties of such a multi-spin particle (MSP) can be described by the anisotropic Heisenberg model of classical spins $\mathbf{S}_{i}$ (with $\vert\mathbf{S}_{i}\vert=1$).

$$\mathcal{H}=-\frac{1}{2}J\sum_{i,j}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\mathcal{H}_{\mathrm{anis}}.$$ (18)

where J is the exchange parameter.

The anisotropy energy $\mathcal{H}_{\mathrm{anis}}$ will be different if we are working with core spins or surface spins. For core spins, i.e., those spins with full coordination, the anisotropy energy $\mathcal{H}_{\mathrm{anis}}$ is taken either as uniaxial with easy axis along $z$ and a constant $K_{c}$ (per atom), that is

$$\mathcal{H}_{\mathrm{anis}}^{\mathrm{uni}}=-K_{c}\sum_{i=1}^{N_c} S_{i,z}^{2}$$ (19)

or cubic,

$$\mathcal{H}_{\mathrm{anis}}^{\mathrm{cub}}= \frac{1}{2} K_{c}\sum_{i=1}^{N_c} \left(S_{i,x}^{4}+S_{i,y}^{4}+S_{i,z}^{4}\right)$$ (20)

where $N_c$ is the number of core spins in the particle. For surface spins the anisotropy is taken according to the Néel's surface anisotropy model (referred to in the sequel as NSA), expressed as:

$$\mathcal{H}_{\mathrm{anis}}^{\mathrm{NSA}}=\frac{K_{s}} {2}\sum_{... ...s}\sum\limits_{j=1}^{z_{i}}\left(\mathbf{S}_{i}\cdot\mathbf{u}_{ij}\right)^{2},$$ (21)

where $N_s$ is the number of surface spins, $z_{i}$ the number of nearest neighbors of site $i$, and $\mathbf{u}_{ij}$ - a unit vector connecting this site to its nearest neighbors labeled by $j$.

Dipolar interactions are known to produce an additional "shape" anisotropy. However, in the atomistic description, their role in describing the spin non-collinearities is negligible as compared to that of all other contributions. In order to compare particles with the same strength of anisotropy in the core, we assume that the shape anisotropy is included in the core uniaxial anisotropy and neglect the dipolar energy contribution to the spin non-collinearities. We also assume that in the ellipsoidal nanoparticles the anisotropy easy axis is parallel to the elongation direction.

All physical constants will be measured with respect to the exchange coupling $J$ (unless explicitly stated otherwise), so we define the reduced constants,

$$k_{c}\equiv K_{c}/J,\;\; k_{s}\equiv K_{s}/J.$$ (22)

The core anisotropy constant will be taken as $k_{c}\simeq0.01$, and $k_{c}\simeq0.0025$. The latter constant in real units corresponds to $K_{c}\simeq3.2\times10^{-17}$ erg/atom and is similar to cobalt fcc value. On the other hand, the surface anisotropy constant $k_{s}$ is unknown "a priori" and will be varied.