Supporting equations for the 3D/1D Monte Carlo method

Since the three-dimensional OOMMF model is not computationally feasible for the antidot system, by assuming two-dimensional layers (see section 6.4) we can extract the coercive field as a function of the radius of the antidots.

By blending these figures for coercivity with a Stoner-Wohlfarth-like (see section 2.6.1) approach and Monte Carlo stochastic mathematical simulation methods we can approximate the three-dimensional system.

The coercive field of the 2D layers $B_c$ is a function of the radius of the holes $r$ with a periodicity of $2R$ defined as $r/R$. An induced anisotropy can be derived from two points -- first, the magnetisation within the sample will prefer to shift between the holes rather than across the empty space inside the holes and second, the holes are arranged hexagonally. By assuming that the magnetisation in the sample is a single-domain, we can describe the induced anisotropy with the following equation:

$$\mathcal{U}_{\mathrm{an}} = K \sin^2(3\phi)$$ (B.1)

where $\phi$ is the angle in the plane between the magnetisation and a symmetry axis.

The left-hand side of figure B.1 shows a polar plot of this antidot-induced aniso-tropy, with the solid black line emerging from the centre representative of $\phi=0$. Adding the Zeeman term:

$$\mathcal{U}_{\mathrm{tot}} = \mathcal{U}_{\mathrm{an}} + \mathcal{U}_{\mathrm{Ze}}$$ (B.2)

$$= K \sin^2(3\phi) + mB \cos(\phi)$$ (B.3)

If we assume some overall magnetisation direction in the plane such that it is a single domain, we can use the anisotropy term from equation B.1 in the Stoner-Wohlfarth model described in section 2.6.1. To determine the reversal conditions, we can compute the first derivative of $\mathcal{U}_{\mathrm{tot}}$:

$${\mathrm{d}\mathcal{U}_{\mathrm{tot}} \over \mathrm{d}\phi} = K 2 \sin(3\phi)3\cos(3\phi)-mB\sin(\phi)$$ (B.4)

Figure B.1: Polar plot of the anisotropy energy $\mathcal{U}_{\mathrm{an}} = K \sin^2(3\phi)$ when $K = 1$ (left) and reversal condition determination with competing anisotropy energy ( $\mathcal{U}_{\mathrm{an}}$) and Zeeman energy ( $\mathcal{U}_{\mathrm{Ze}}$) with $B=m=1$, $K=Bm/18$.

The energy barriers dominating the system are shown to be around $\phi=0$ (figure B.1, right). Expanding $\sin$ and $\cos$ around zero:

$$\sin(\phi) \approx \phi$$ (B.5)

$$\cos(\phi) \approx 1$$ (B.6)

when $\phi \approx 0$.

Inserting this into equation B.4 yields:

$${\mathrm{d}\mathcal{U}_{\mathrm{tot}} \over \mathrm{d}\phi} \approx 6K3\phi-mB\phi$$ (B.7)

If we assume $\phi\neq0$:

$$K = {mB \over 18}$$ (B.8)

The right-hand side of figure B.1 shows graphically the resulting energies with $B=1$, $m=1$ and $K=Bm/18$.

The value $B$ in equation B.8 is the coercivity $B_c$ obtained from the
two-dimensional antidot simulation layer dependent on $r/R$. Since $K$ and $m$ are the anisotropy constant and the saturation magnetisation respectively for the whole simulated layer rather than per unit volume:

$$K_{(r/R)} = { m_{(r/R)} B_{c_{(r/R)}} \over 18}$$ (B.9)

These equations, coupled with an exchange energy approximation outlined in equation 6.10, can be used to perform a Monte Carlo simulation on the computed two-dimensional coercivity values. This simulation results in a coercivity oscillation as a function of the thickness of the film which more accurately reflects the experimental results shown in figure 6.3.