Discretisation

When a particular geometry is decided upon for simulation, this must be discretised into lots of smaller cuboidal cells to be able to use the finite difference method. Each cell is considered to be homogeneously magnetised, i.e. within a micromagnetic simulation all of the atomic magnetic moments inside this cellular domain are thought to behave as a single particle. This is an acceptable assumption because at an atomic length scale the exchange interaction, responsible for the homogeneous alignment of magnetic moments, is overwhelmingly the most significant energy term. These smaller cells can then be used to perform the simulation. The separate simulation cells represent a certain amount of magnetic material. Obviously in this instance a finer discretisation mesh -- a smaller simulation cell size -- is more desirable than a coarser mesh, particularly when there are curved surfaces in the geometry.

Figure 2.7: The effect of altering the number of cells in a geometry, in this instance a sphere. $4^3=64$ cells (left) gives poor shape resolution for the sphere. Increasing this to $9^3=729$ cells (centre) improves the resolution but $19^3=6859$ cells (right) gives a much more ``spherical'' representation

Figure 2.7 demonstrates the effect of altering the number of cells in a geometry. In the case of extremely coarse discretisation using the finite difference method, a sphere can resemble more a cuboid than a sphere (figure 2.7, left). A poor representation of the shape in the discrete model can affect the influence of the shape anisotropy (see section 2.3.2) on the magnetisation, and subsequently negatively affect the results.

Figure 2.8 shows the discretisation of a sphere using both fixed size cubic cells (finite difference) and variable sized tetrahedral cells (finite element). In this sphere example, there are four times fewer cells in the finite element example yet it is aesthetically more sphere-like.

The exchange length is a length scale over which the direction of $\ensuremath{\mathbf{M}}$ does not change significantly, as across this length the exchange energy is overwhelmingly the dominant component and other influences have little effect. A coarse mesh will not allow the software to resolve the exchange length properly, so independent domains will not form correctly. The exchange length is calculated by considering (Seberino and Bertram, 2001, Kronmüller and Fähnle, 2003):

$$\lambda_{\mathrm{ex}} = \sqrt{ {A} \over {{1 \over 2}\mu_0 M_s^2} }$$ (2.40)

where $A$ is the exchange energy (measured in $\mathrm{J} / \mathrm{m}$), $\mu_0$ is the magnetic constant ( $4\pi10^{-7}$ $\mathrm{T}\cdot\mathrm{m}\cdot\mathrm{A}^{-1}$) and $M_s$ is the magnetisation in $\mathrm{A} / \mathrm{m}$. $\mu_0$The magnetic constant, $4\pi \cdot 10^{-7}$ $\mathrm{T}\cdot\mathrm{m}\cdot\mathrm{A}^{-1}$ $\lambda_{\mathrm{ex}}$The exchange length of a material in metres (m); computed as a function of $A$ and $M$. See equation 

The exchange length $\lambda_{\mathrm{ex}}$ therefore gives us a quantitative measure for the required mesh resolution.

Table 2.3: Properties of some common ferromagnetic materials

material	exchange energy	magnetisation	anisotropy	exchange length
	A (J/m)	$M_s$ (A/m)	$K_1$ (J/m$^3$)	$\lambda_{\mathrm{ex}}$ (nm)
nickel	$9 \times 10^{-12}$	$4.9 \times 10^5$	$-5.7 \times 10^3$ (cubic)	$7.72$
iron	$2.1 \times 10^{-13}$	$1.70 \times 10^6$	$4.8 \times 10^4$ (cubic)	$3.40$
cobalt	$3.0 \times 10^{-13}$	$1.40 \times 10^6$	$5.2 \times 10^5$ (uniaxial)	$4.94$
supermalloy	$1.05 \times 10^{-13}$	$8.0 \times 10^5$	0	$5.11$
permalloy	$5.85 \times 10^{-12}$	$1.11 \times 10^{6}$	0	$2.76$
Ni$_{50}$Fe$_{50}$
permalloy	$1.30 \times 10^{-13}$	$8.6 \times 10^{5}$	0	$5.29$
Ni$_{80}$Fe$_{20}$
iron-palladium	$1.5 \times 10^{-11}$	$1.36 \times 10^{6}$	$3.5 \times 10^6$ (uniaxial)	$3.59$
iron-platinum	$1.0 \times 10^{-11}$	$1.14 \times 10^{6}$	$7.7 \times 10^6$ (uniaxial)	$3.50$

Figure 2.8: Finite difference (left) and finite element (right) meshes. For adequate shape resolution, the finite difference model requires more cells than the finite element model; in this case 27000 and 5000 respectively

The derivation of the exchange energy in the micromagnetic theory uses the Taylor series expansion of the cosine between two moments (equation 2.19) to the second-order. It is crucial that the maximum angle between these two adjacent moments is not high (Donahue and McMichael, 2002) -- indeed if the angle becomes larger than $\pi/2$ radians, then the results of the simulation are highly inaccurate as the torque between the two spins begins to decrease when the angle is further increased; this could potentially lead to the scenario where the angle between two adjacent spins is $\pi$ radians -- according to the second-order Taylor expansion of the cosine, this would be a perfectly legitimate low-energy state, although this is clearly not the case as the exchange energy and consequently the torque between these two spins in this state would be extremely large.

Incidentally, it is worth noting that since the simulation is not atomistic, (i.e. it doesn't compute the exchange energy using equation 2.2), the use of the discretised version of the micromagnetic expression for the exchange energy 2.26 is always slightly inaccurate from a quantitative perspective, however if the angle between two spins is greater than $\pi/2$ radians then the behaviour becomes qualitatively wrong.

The answer to these problems is of course to create a finer mesh; however if one makes the mesh $n$ times as fine, then the number of the cells in the simulation increases by $n^3$ (since the system is three-dimensional) and this results in a massively increased computational overhead.