4.7 Finite differences spatial discretization of LLG equation

Up to this point the considerations we made about the properties and the implementation of mid-point rule were rather independent from the spatial discretization used. In the following, to test the method we have chosen a specific technique based on finite-differences method. The magnetic body is subdivided into a collection of rectangular prisms with edges parallel to the coordinate axes. The edge lengths are $d_x,d_y,d_z$. In this framework, it is convenient to identify each cell by three indices $i,j,k$ instead of using the index $l$ as we did before. The magnetization $\textbf{{m}}_{i,j,k}$ is assumed to be uniform within the generic $(i,j,k)$ cell. With this notation, the effective field in the generic $(i,j,k)$ cell can be expressed as

$$\textbf{h}_ =\textbf{h}_ +\textbf{h}_ +\textbf{h}_ +\textbf{h}_$$ (4.57)

The exchange field (4.8) is computed by means of a 7-point laplacian discretization, which is second order accurate in space. In the generic ``internal'' cell $(i,j,k)$, the exchange field can be expressed as follows:

$$\textbf{h}_ =\frac{2A}{\mu_0 M_s^2}\bigg[\frac{\textbf{{m}}_{i+1,j,k}+\textb... ...-1,j,k}}{d_y^2}+\frac{\textbf{{m}}_{i,j+1,k} +\textbf{{m}}_{i,j-1,k}}{d_x^2} +$$

$$+\frac{\textbf{{m}}_{i,j,k+1}+\textbf{{m}}_{i,j,k-1}}{d_z^2}-\lef... ...ac{2}{d_y^2}+\frac{2}{d_x^2}+ \frac{2}{d_z^2}\right)\textbf{{m}}_{i,j,k}\bigg]$$ (4.58)

A similar expression holds for the boundary cells where the Neumann boundary condition (4.6) has to be taken into account. Since the exchange interaction is a first-neighbors interaction, one can easily observe that the matrix ${\underline{\textrm{C}}}_$ex is a block-diagonal matrix. As seen in section 3.1, the magnetostatic field (4.9) can be expressed as discrete convolution:

$$\textbf{h}_ =\sum_{i'\neq i}\sum_{j'\neq j}\sum_{k'\neq k} N_{i-i',j-j',k-k'}\cdot\textbf{{m}}_{i',j',k'} d_x d_y d_z$$ (4.59)

where $N_{i-i',j-j',k-k'}$ is the $3\times3$ block of the magnetostatic interaction matrix ${\underline{\textrm{C}}}_$m which describes the magnetostatic interaction between the cells $i,j,k$ and $i',j',k'$. The discrete convolution (4.59) is computed by means of 3D Fast Fourier Transform (FFT) using the zero-padding algorithm [64]. The kernel of the convolution is obtained in exact analytical form by means of a slight generalization of the one proposed in Ref. [93] for cubic cells. As far as anisotropy is concerned, we assume that the body has uniaxial anisotropy defined by the anisotropy constant $K_1$. Therefore, the anisotropy field is

$$\textbf{h}_ =\frac{2K_1}{\mu_0M_s^2} (\textbf{{m}}_{i,j,k}\cdot\mathbf{e}_x)\mathbf{e}_x \quad.$$ (4.60)

and the matrix ${\underline{\textrm{C}}}_$an is a diagonal matrix.

Figure: (Left) Thin-film geometry for $\mu$-mag standard problem n. 4. (Right) Initial equilibrium S-state.