1.3.5.2 Energy balance equation

It is convenient to recast the normalized Landau-Lifshitz-Gilbert equation (1.95) in the following form:

$$\frac{\partial \textbf{{m}}}{\partial t} = - \textbf{{m}}\times ... ...f{h}_{\text{eff}}- \alpha \frac{\partial \textbf{{m}}}{\partial t} \right) .$$ (1.98)

Now by scalar multiplying both sides of Eq. (1.98) by $\textbf{h}_{\text{eff}}-\alpha \frac{\partial \textbf{{m}}}{\partial t}$ one ends up with:

$$\frac{\partial \textbf{{m}}}{\partial t} \cdot \left(\textbf{h}_{\text{eff}}-\alpha \frac{\partial \textbf{{m}}}{\partial t}\right)=0 \quad.$$ (1.99)

The effective field and the time derivative of the free energy are related by the following relationship:

$$\frac{dg}{dt} =\int_\Omega \left[\frac{\delta g}{\delta \textbf{{m}}}\cdot\fra... ...tbf{h}_\text{a}}\cdot\frac{\partial \textbf{h}_\text{a}}{\partial t}\right]dv =$$

$$=\int_\Omega \left[-\textbf{h}_{\text{eff}}\cdot\frac{\partial \t... ...textbf{{m}}\cdot\frac{\partial \textbf{h}_\text{a}}{\partial t}\right]dv \quad.$$ (1.100)

By integrating Eq. (1.99) over the body volume $\Omega$ and by using the latter equation, one obtains:

$$\frac{dg}{dt}=-\int_\Omega \alpha\left\vert\frac{\partial \textbf... ... \textbf{{m}}\cdot \frac{\partial \textbf{h}_\text{a}}{\partial t} dv \quad.$$ (1.101)

Equation (1.101) is the energy balance relationship for magnetization dynamics. An interesting case occurs when the applied field is constant in time and, therefore, $\frac{\partial \textbf{h}_\text{a}}{\partial t}=\mathbf{0}$. The energy balance equation becomes:

$$\frac{dg}{dt}=-\int_\Omega \alpha\left\vert\frac{\partial \textbf{{m}}}{\partial t}\right\vert^2 dv \quad,$$ (1.102)

meaning that the free energy is a non-increasing function of time, since $\alpha\geq 0$. This property is often referred to as Lyapunov structure [82] of LLG equation. In particular, for $\alpha=0$, one can observe that the free energy conservation holds:

$$g(t)=g(t_0) \quad \forall t,t_0 \quad.$$ (1.103)

The properties expressed by (1.97), (1.101) and (1.103) are very important constraints for magnetization dynamics. Since the solution of LLG equation cannot be obtained in exact analytical form, except some very particular cases, it is fundamental to derive numerical models that can reproduce this properties also in discrete dynamics. This issue will be addressed in detail in chapter 4.