Monte Carlo simulation

In section 6.4 we described a method for extracting the coercive field of the two-dimensional antidot layers. The assumption with this approach is that the layers are independent, however different layers interact via the exchange coupling. A Monte Carlo simulation is used to simulate the reversal behaviour of a stack of exchange-coupled two-dimensional layers. It is assumed that all magnetic moments in one two-dimensional layer point in the same direction, i.e. each layer is treated as a single Stoner-Wohlfarth particle (see figure 6.11).

By taking the computed coercive field $B_c$ from the two-dimensional model as a function of $r/R$, we can determine an effective anisotropy energy $K$ (arising from the adapted Stoner-Wohlfarth model described in appendix B considering anisotropy and Zeeman components) for each Stoner-Wohlfarth layer with holes of the size $r$:

$$K_{(r)} = { m_{(r)} B_{c_{(r)}} \over 18}$$ (6.9)

where $m_{(r)}$ is the magnetic moment of a two-dimensional layer of 5nm thickness with holes of radius $r$. Thus, each Stoner-Wohlfarth layer will -- if it is decoupled from the other layers -- switch at the coercive field as measured in the two-dimensional simulations (see appendix B for the full derivation of equation 6.9).

The layers are coupled by the exchange interaction and the exchange energy can be computed between two neighbouring layers $\mathcal{A}$ and $\mathcal{B}$:

$$\mathcal{E}_{\mathrm{ex}^{\mathcal{A},\mathcal{B}}} = -2 \mathcal{N}\mathcal{J}\ensuremath{\mathbf{S}}_\mathcal{A}\cdot\ensuremath{\mathbf{S}}_\mathcal{B}$$ (6.10)

$$= -2\mathcal{N}\mathcal{J}S^2\cos(\phi_{\mathcal{A}\mathcal{B}})$$ (6.11)

where $\mathcal{N}$ is the number of neighbouring atoms, $\mathcal{J}$ is the exchange integral, $S$ is the atomic magnetic moment and $\phi_{\mathcal{A}\mathcal{B}}$ is the angle between the magnetisation in the neighbouring layers. The exchange energy component yielded by this equation represents the energy between the interface of layers $\mathcal{A}$ and $\mathcal{B}$ (point B in figure 6.11).

Figure 6.11: Overview of Monte Carlo simulation. A `stack-of-spins' model (A) is used as a basis for simulation. The `spins' are coupled by an exchange energy component computed through neighbouring interface atoms (B) between layers. An anisotropy component is computed from the coercivity of the 2D layer (C) and the magnetisation contribution is calculated from the volume of material in each layer. See appendix B for the derivation of the anisotropy component.

By performing a Monte Carlo simulation of the system of coupled
Stoner-Wohlfarth layers using the equations above and input parameters for $B_{c_{(r/R)}}$ from the two-dimensional simulations shown in figure 6.12, we are able to more accurately compute the coercive field with respect to the height of the film, and therefore use the results of the two-dimensional simulations to understand a simplified model of the three-dimensional system.