2 Implementation of the exchange tensor interaction

Unfortunately, to take into account all exchange contributions in atomistic simulations is very time consuming for a reasonable system size. Due to the properties of the exchange tensor interactions (oscillatory character) it is necessary to take into account not only first neighbors in the calculations. At the same time the exchange is a short-range interaction which falls off sufficiently fast with distance, as clearly seen in the left side of Fig. 6.7, so that only a finite number of neighbors $j$ has to be considered. In other words, we consider that the exchange double sum is over the neighbors whose distance is below a certain cutoff, in our case the cutoff is $5\cdot (a_{2D})$, being $a_{2D}$ the lattice parameter in the 2D fcc lattice. We call this approximation the Truncated Exchange Tensor (TET).

The widely used approximation in the literature [183,172] is the use of the Isotropic exchange constant (IEC). In this approximation it is considered that the exchange tensor interaction $\mathcal{J}_{ij}$ is replaced by an exchange constant $J_{ij}$, in a similar way as it was used by L. Szunyogh et al. in Ref. [173]. This way $\mathcal{H}^{ex}$ could be written in the next form:

$$\mathcal{H}^{ex}=-\frac{1}{2}\sum_{i,j}J_{ij}\mathbf{S_{i}}\mathbf{S_{j}}$$ (124)

where $J_{ij}$ is the isotropic exchange parameter (see section 6.3.4). The $J_{ij}$ constant depends on the pair of spins ${i,j}$. Also in this case we take as neighbors of spin i all other spins j, which are located at a distance smaller than $5\cdot (a_{2D})$.

This approximation involves a modification in the uniaxial anisotropy constant. With the aim to reproduce the correct anisotropy of the system, we replace the value of the on-site anisotropy constant by the effective anisotropy constant evaluated in the same form that in the case in the section 6.3.5 and having contributions from the anisotropic exchange tensor interaction.