1 The Néel model of the surface magnetic anisotropy

The modeling of the surface anisotropy contribution is a complex field of research. Néel proposed a phenomenological model of the surface anisotropy called after that the "Néel surface anisotropy (NSA) model". The model assumes an origin of the anisotropy basing on the lack of the atomic bonds on the surface of a crystal. The surface anisotropy contribution is described as a pair-interaction of spins in the following way:
$\displaystyle \mathcal{H}_{i}^{NSA}$ $\displaystyle =$ $\displaystyle \frac{L(r_{ij})}{2}\sum_{j=1}^{z_{i}}(\overrightarrow{s}_{i}\cdot \overrightarrow{e}_{ij})^{2},$ (8)
$\displaystyle \overrightarrow{e}_{ij}$ $\displaystyle =$ $\displaystyle \frac{\overrightarrow{r}_{ij}}{r_{ij}},$  
$\displaystyle \overrightarrow{r}_{ij}$ $\displaystyle =$ $\displaystyle \overrightarrow{r}_{i}-\overrightarrow{r}_{j} .$  

Here $ z_{i}$ is the number of nearest neighbors of the surface spin i, (known as the coordination number), $ \overrightarrow{s}_{i}$ is a unit vector pointing along the magnetization direction,$ r_{ij}$ is the distance between spin i and j, $ \overrightarrow{e}_{ij}$ is the unit vector connecting the spin i to its nearest neighbor j, the factor $ 1/2$ is added in order not to count twice the pair interaction and $ L(r_{ij})$ is the pair-anisotropy coupling constant, also called the Néel surface anisotropy constant, which depends on the distance between spins [47], see Fig. 1.3.

If $ L<0$ the surface anisotropy is locally in-plane and is out-of-plane if $ L>0$. L depends on the interatomic distance $ r_{ij}$ (in the following we will omit the subindex i and j) according to the expression:

$\displaystyle L(r)=L(r_{0})+ \Bigl( \frac{dL}{dr}\Bigr)_{r_{0}} r_{0}\eta$ (9)

where $ r_{0}$ is the unstrained bond length and $ \eta$ is the bond strain.

$ L(r_{0})$ and $ r_{0}\Bigl( \frac{dL}{dr}\Bigr)_{r_{0}}$ depend on the magnetostriction and elastic constants, therefore $ L(r)$ also depends on these magnitudes. All these relations make obvious that the NSA depends on the orientation of the local magnetization with respect to the surface, the orientation of the surface with respect to the crystalline axes, and the loss of neighbors. This way, atoms located at different positions at the surface can possess a different surface anisotropy value. For example, in Fig. 1.4 (from Ref. [14]) we show the neighborhood of atoms located at different positions in a truncated octahedral nanoparticle, which have different directions of the local easy axes and strengths of the surface anisotropy [14].

Figure 1.4: The different environment of the atoms in a fcc octahedral nano-particle ( from Ref. [14]).
Within this model the magnetic surface anisotropy of a given atom can be calculated summing pair-interactions with its nearest neighbors. Despite the fact that this model provides an adequate description of the symmetry of the surface magnetic anisotropy it doesn't provide a physical understanding of its origin since it has a phenomenological character. With the aim to go deeper in the physical origin of the magnetic surface anisotropy we have to consider the effects of the spin-orbit coupling and the ligand field.

Rocio Yanes