4.2 The mathematical model

It is very useful for the following discussion, to recall the dimensionless form (1.95) of LLG equation:

$$\frac{\partial \textbf{{m}}}{\partial t} = - \textbf{{m}}\times ... ...\textbf{{m}},t) - \alpha \frac{\partial \textbf{{m}}}{\partial t} \right) ,$$ (4.1)

with the usual normalized quantities introduced in section 1.3.4. The LLG equation (4.1) is implicit with respect to $\partial \textbf{{m}}/\partial t$, and it can be transformed in the equivalent normalized Landau-Lifshitz form of Eq. (1.87):

$$\frac{\partial \textbf{{m}}}{\partial t} = - \frac{1}{1+\alpha^2... ...{m}} \times ( \textbf{{m}}\times \textbf{h}_{\text{eff}}(\textbf{{m}},t)) ,$$ (4.2)

where $\partial \textbf{{m}}/\partial t$ is explicitly expressed. This form of LLG equation is the most commonly used for numerical integration. As seen in chapter 1, the normalized effective field $\textbf{h}_{\text{eff}}$ can be defined by the variational derivative $\textbf{h}_{\text{eff}}= -\delta g / \delta \textbf{{m}}$ of the normalized micromagnetic free energy functional, formed by the sum of normalized exchange, magnetostatic, anisotropy and Zeeman energy, respectively:

$$g(\textbf{{m}};\textbf{h}_ )=\int_\Omega \left[ \frac{A}{\mu_0 M_s^2} (\nabla \textbf{{m}}... ...f{e}_\text{an})^2 \right]-\textbf{h}_\text{a}\cdot \textbf{{m}}\right] dv ,$$ (4.3)

where $A$ is the exchange constant, $K_1$ is the uniaxial anisotropy constant, $\mathbf{e}_$an is the easy axis unit-vector and $\textbf{h}_$m is the magnetostatic (demagnetizing) field, which is the solution of the boundary value problem:

$$\nabla\cdot \textbf{h}_ =-\nabla\cdot\textbf{{m}} \Omega ,$$ (4.4)

$$\textbf{n}\times\left[\textbf{h}_\text{m}\right]_{\partial \Omega... ...ft[\textbf{h}_\text{m}\right]_{\partial \Omega}=\textbf{n}\cdot\textbf{{m}} .$$ (4.5)

In Eqs. (4.4)-(4.5), we have denoted with $\textbf{n}$ the outward normal to the boundary $\partial \Omega$ of the magnetic body, and with $[\textbf{h}_$m$]_{\partial \Omega}$ the jump of the vector field $\textbf{h}_$m across $\partial \Omega$. The magnetization $\textbf{{m}}(\textbf{r},t)$ is also assumed to satisfy the following condition at the body surface:

$$\frac{\partial \textbf{{m}}}{\partial \textbf{n}}=\mathbf{0} ,$$ (4.6)

which is related to the presence of first (laplacian) term in Eq. (4.3).