3.1 Discretization of the exchange energy

For our problem of calculating the effective field and integrating the Landau-Lifshitz equation, we have to discretize time and space into regular lattices. For the space discretization a regular cubic lattice has been chosen, because it allows the simplest implementation and irregular lattices are more efficiently handled with the finite element method (cf. chapter ). The time integration is also done on a regular lattice and described in detail in chapter .

In order to calculate the contribution of the exchange interaction to the effective field, we have to discretize (). The first partial derivative

$$(\nabla m_x)^2 = \left( \frac{\partial m_x}{\partial x} \r... ... \right)^2 + \left( \frac{\partial m_x}{\partial z} \right)^2$$

is approximated by the finite difference

$$\left( \frac{\Delta_x m_x}{\Delta x} \right)^2 + \left( \f... ...t)^2 + \left( \frac{\Delta_z m_x}{\Delta z} \right)^2 \quad,$$ (3.1)

where $\Delta x, \Delta y, \Delta z$ denote the lattice spacing in the three dimensions of space, which is identical on a regular cubic lattice and we will simply write $\Delta x$. $\Delta_x, \Delta_y, \Delta_z$ indicate the finite difference operators in $x, y,$ and $z$ direction, respectively.

Let us now consider the magnetization vector $\mathbf{m}_i$ at lattice point $i$ and $\mathbf{m}_{i+1}$ on the neighbouring lattice point in $+x$ direction. Then, we can rewrite the first term in () as

$$\left( \frac{m_{i+1,x}-m_{i,x}}{\Delta x} \right)^2 = \fra... ...{i+1,x}^2 - 2 m_{i+1,x}m_{i,x} + m_{i,x}^2}{\Delta x^2} \quad.$$

From the second and third term in () we get the contribution of the $y$ and $z$ component of the magnetization vectors. Since $\vert\mathbf{m}\vert=1$ we can simplify and get for the finite difference

$$\frac{1}{\Delta x^2}(2-2 \mathbf{m}_i \cdot \mathbf{m}_j) \quad.$$ (3.2)

From the second and third term in (), which represent the contributions of the nearest neighbours in $y$ and $z$ direction, we get analogous expressions. The same procedure can be carried out for the opposite neighbours in the negative directions of space. Then we take the arithmetic mean of the two opposite nearest neighbours.

Our final result for the exchange energy density is

$$\omega_\mathrm{exch}= \frac{A}{\Delta x^2} \sum_i \sum_{j \in NN}(1-\mathbf{m}_i \cdot \mathbf{m}_j) \quad,$$ (3.3)

where $NN$ stands for the indices of the nearest neighbours.

The approximation of the partial derivatives by finite differences is only valid for small arguments, and in our case for small angles between neighbouring magnetization vectors. Other exchange energy representations have been suggested and compared [21], but none of them has significant advantages over the one derived above.

In a discretized form [22] the effective field is defined by

$$\mathbf{H}_\mathrm{eff}=\frac{\partial \omega}{\partial \mathbf{M}} \quad.$$

For the ``exchange field'' at lattice site $i$ we find

$$\mathbf{H}_{\mathrm{exch},i}= \frac{2 A}{M_\mathrm{s}\Delta x^2} \sum_{j \in NN}\mathbf{m}_j \quad.$$