3 Symmetric anisotropic exchange tensor

The second term in equation (6.25) represents the symmetric anisotropic exchange coupling (similar to that described in Refs. [107,173]) and it is known as an "anisotropic exchange" or a "two-site anisotropy". We defined it previously as:

$\displaystyle \mathcal{J}_{ij}^{S}=\frac{1}{2}\bigl( \mathcal{J}_{ij}+\mathcal{J}_{ij}^{T}\bigr)-J_{ij}\mathcal{I}.$    

As it will be seen later, $ \mathcal{J}_{ij}^{S}$ is one of the important contributions to the macroscopic anisotropy. For example, in the case of $ FePt$, $ \mathcal{J}_{ij}^{S}$ term is the key for the understanding of the deviation in the exponent of the power-law dependence of the anisotropy with the magnetization $ K\propto M^{2.09}$ [174,107], from the theoretical $ K\propto M^{3}$ law given by the Callen-Callen theory [129,106].

The contribution of the symmetric part of the exchange matrix interaction to the Hamiltonian ( $ \mathcal{H}^{SExc}$) can be written as follows:

$\displaystyle \mathcal{H}^{SExc}=-\frac{1}{2}\sum_{i,j}\mathbf{S_{i}}\mathcal{J}_{ij}^{S}\mathbf{S_{j}} \: .$    

The double sum is over all $ i\neq j$. Now we try to analyze the effective symmetric exchange contribution of the exchange matrix interaction at a layer $ z_{i}$. For this purpose we define $ \mathcal{J}_{z_{i}}^{S}$ as:

$\displaystyle \mathcal{J}_{z_{i}}^{S}=\sum_{j}\mathcal{J}_{ij}^{S},$ (105)

We have evaluated the effective symmetric exchange contributions for Co(100), $ Co(100)\setminus Ag_{1}$ and $ Co(111)\setminus Ag_{1}$ systems [*].

In the first place, we would like to comment the characteristic features of these contributions: In the case of Co(100) and $ Co(100)\setminus Ag_{1}$, we observe that the symmetric anisotropic exchange matrix has the form:

$\displaystyle \mathcal{J}_{z_{i}}^{S}=A_{z_{i}} \left( \begin{array}{ccc} -\frac{1}{2} &0 & 0 \ 0 & -\frac{1}{2} & 0 \ 0 & 0 & 1 \ \end{array} \right)$ (106)

and could lead to an effective uniaxial anisotropy in the direction perpendicular to the plane of the surface, which could be one of the principal contributions to the effective anisotropy [*].

In the $ Co(111)\setminus Ag_{1}$ system, we observe a little different behavior. In this system the effective symmetric exchange contribution $ \mathcal{J}_{Zi}^{S}$ is a matrix whose non-diagonal elements are in practice negligible, and it has the following form:

$\displaystyle \mathcal{J}_{z_{i}}^{S}= \left( \begin{array}{ccc} A_{11}^{z_{i}} &0 & 0 \ 0 & A_{22}^{z_{i}} & 0 \ 0 & 0 & A_{33}^{z_{i}} \ \end{array} \right)$ (107)

Here unlike what happens in the cases of a semi-infinite Co(100) and a $ Co(100)\setminus Ag_{1}$ systems, the diagonal elements have the property $ A_{11}^{z_{i}} \neq A_{22}^{z_{i}} \neq A_{33}^{z_{i}}$. Also we have observed that $ \mid A_{33}^{z_{i}}\mid> \mid A_{11}^{z_{i}}\mid$ and $ \mid A_{33}^{z_{i}}\mid> \mid A_{22}^{z_{i}} \mid$ and $ A_{33}^{z_{i}}$ has an opposite sign than the other two elements.

Rocio Yanes