1 Atomistic model

We consider a magnetic system of $\mathcal{N}$ spins in the classical many-spin approach, i.e., taking account of its intrinsic properties such as the lattice structure and system size. The magnetic properties of such system are described by the anisotropic Heisenberg model for classical spins $\mathbf{S}_{i}$ (with $\vert\mathbf{S}_{i}\vert=1$). The Hamiltonian includes the (nearest-neighbor) exchange interactions, single-site bulk ($K_{c}$) and Néel surface anisotropy ($K_{s}$):

$$\mathcal{H}=-\frac{1}{2}J\sum_{i,j}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\mathcal{H}_{\mathrm{anis}}.$$ (53)

where J is the exchange parameter.

The anisotropy energy $\mathcal{H}_{\mathrm{anis}}$ will be different if we are working with bulk spins or surface spins. For bulk 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 anisotropy constant $K_{c}$ (per atomic volume), that is

$$\mathcal{H}_{\mathrm{anis}}^{\mathrm{uni}}=-K_{c}\sum_{i=1}^{N_c} V_{i}S_{i,z}^{2}$$ (54)

or cubic,

$$\mathcal{H}_{\mathrm{anis}}^{\mathrm{cub}}= -\frac{1}{2} K_{c}\sum_{i=1}^{N_c} V_{i}\left(S_{i,x}^{4}+S_{i,y}^{4}+S_{i,z}^{4}\right)$$ (55)

where $V_{i}$ is the volume of the atomic atom i, $N_c$ is the number of bulk spins in the thin film. For surface spins the anisotropy is taken according to the Néel's model, expressed as:

$$\mathcal{H}_{\mathrm{anis}}^{\mathrm{NSA}}=\frac{K_{s}} {2}\sum_{... ...m\limits_{j=1}^{z_{i}}V_{i}\left(\mathbf{S}_{i}\cdot\mathbf{u}_{ij}\right)^{2},$$ (56)

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 $j$.

The magnetic systems modeled in this chapter are thin films. The magnetic parameters of the system under study are presented in table 4.1.

Table 4.1: Lattice parameter (a), zero-temperature "on-site" magnetic anisotropy constant $K_{c}$, exchange constant J and saturation magnetization $M_{S}$ at $T=0K$, used in the simulations.

Lattice	a(nm)	$K_{c}$ ( $10^{6} erg/cc$)	J( $10^{-14} erg$)	$M_{S}$($emu/cc$)
sc	$0.3548$	$4.16$	$10$	$1260.38$
fcc	$0.3548$	$4.16$	$5.6$	$1260.38$