1.3.2 The Landau-Lifshitz equation

The first dynamical model for the precessional motion of the magnetization was proposed by Landau and Lifshitz in 1935. Basically, this model is constituted by a continuum precession equation (1.79), in which the presence of quantum-mechanical effects and anisotropy is phenomenologically taken into account by means of the effective field $\textbf{H}_{\text{eff}}$ given by Eq. (1.68). Then, the Landau-Lifshitz equation is:

$$\frac{\partial \textbf{M}}{\partial t}=-\gamma\textbf{M}\times\textbf{H}_{\text{eff}}\quad.$$ (1.80)

First of all, we observe that if the magnetization rate of change $\partial \textbf{{m}}/\partial t$ vanishes, Eq. (1.80) expresses the equilibrium condition given by the first of the Brown's equations (1.71). In addition, since Eq. (1.80) is an integro-partial differential equation, the Neumann boundary condition given by the second Brown's equation is used [4]. We observe that Landau-Lifshitz equation (1.80) is a conservative (hamiltonian) equation. Nevertheless, dissipative processes take place within dynamic magnetization processes. The microscopic nature of this dissipation is still not clear and is currently the focus of considerable research [16,17].

Figure 1.5: (left) Undamped gyromagnetic precession. (right) Damped gyromagnetic precession.

The approach followed by Landau and Lifshitz consists of introducing dissipation in a phenomenological way. In fact, they introduce an additional torque term that pushes magnetization in the direction of the effective field (see Fig. 1.5). Then, the Landau-Lifshitz equation becomes:

$$\frac{\partial \textbf{M}}{\partial t}=-\gamma\textbf{M}\times\te... ...\lambda}{M_s} \textbf{M}\times(\textbf{M}\times\textbf{H}_{\text{eff}}) \quad,$$ (1.81)

where $\lambda>0$ is a phenomenological constant characteristic of the material. It is important to observe that the additional term is such that the magnetization magnitude is preserved according to the micromagnetic constraint $\vert\textbf{M}\vert=M_s$. This can be seen by scalar multiplying both sides of Eq. (1.81) by $\textbf{M}$.