1.3.3 Landau-Lifshitz-Gilbert equation

An in principle different approach was proposed by Gilbert [18] in 1955, who observed that since the conservative equation (1.80) can be derived from a Lagrangian formulation where the role of the generalized coordinates is played by the components of magnetization vector $M_x,M_y,M_z$. In this framework, the most natural way to introduce phenomenological dissipation occurs by introducing a kind of `viscous' force, whose components are proportional to the time derivatives of the generalized coordinates. More specifically, he introduces the following additional torque term:

$$\frac{\alpha}{M_s} \textbf{M}\times\frac{\partial \textbf{M}}{\partial t} \quad,$$ (1.82)

which correspond to the torque produced by a field $-\frac{\alpha}{\gamma M_s} \frac{\partial \textbf{M}}{\partial t}$, where $\alpha >0$ is the Gilbert damping constant, depending on the material (typical values are in the range $\alpha=0.001\div0.1$). We observe that, similarly to the case of Landau-Lifshitz equation, the additional term introduced by Gilbert preserves the magnetization magnitude^1.4. In the following section, when we will analyze the fundamental properties of magnetization dynamics, we will show that the Gilbert damping is connected to the assumption of a suitable Rayleigh dissipation function. Therefore, the precessional equation (1.80), modified according to Gilbert's work, is generally referred to as Landau-Lifshitz-Gilbert equation:

$$\frac{\partial \textbf{M}}{\partial t}=-\gamma\textbf{M}\times\te... ...{\alpha}{M_s} \textbf{M}\times\frac{\partial \textbf{M}}{\partial t} \quad.$$ (1.83)

There is substantial difference between Landau-Lifshitz and Landau-Lifshitz-Gilbert equations although they are very similar from mathematical point of view. For instance, Landau-Lifshitz equation (1.81) can be obtained easily from Gilbert equation. In fact, by vector multiplying both sides of Eq. (1.83) by $\textbf{M}$, one obtains:

$$\textbf{M}\times\frac{\partial \textbf{M}}{\partial t}=-\gamma\te... ...alpha}{M_s}\textbf{M}\times\frac{\partial \textbf{M}}{\partial t}\right) \quad;$$ (1.84)

remembering the vector identity $\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b})$ and observing that $\textbf{M}\cdot\frac{\partial \textbf{M}}{\partial t}=0$ (see section 1.3.5), one ends up with:

$$\textbf{M}\times\frac{\partial \textbf{M}}{\partial t}=-\gamma\te... ...textbf{H}_{\text{eff}})-\alpha M_s\frac{\partial \textbf{M}}{\partial t} \quad.$$ (1.85)

By substituting the latter equation in the right hand side of Landau-Lifshitz-Gilbert equation (1.83), one has:

$$\frac{\partial \textbf{M}}{\partial t}=- \gamma\textbf{M}\times\t... ...textbf{H}_{\text{eff}})-\alpha^2 \frac{\partial \textbf{M}}{\partial t} \quad.$$ (1.86)

The latter equation can be appropriately recast to obtain the following expression:

$$\frac{\partial \textbf{M}}{\partial t}=- \frac{\gamma}{1+\alpha^2... ...\alpha^2)M_s}\textbf{M}\times(\textbf{M}\times \textbf{H}_{\text{eff}}) \quad,$$ (1.87)

which is commonly referred to as Landau-Lifshitz equation in the Gilbert form. One can immediately notice that Eq.(1.87) and Eq. (1.81) are mathematically the same, provided that one assumes:

$$\gamma_L=\frac{\gamma}{1+\alpha^2} \quad,\quad \lambda=\frac{\gamma\alpha}{1+\alpha^2} \quad.$$ (1.88)

Moreover, the work of Podio-Guidugli [82] has pointed out that both Landau-Lifshitz and Landau-Lifshitz-Gilbert equations belong to the same family of damped gyromagnetic precession equations. Nevertheless some considerations about the meaning of the quantity $\gamma$, which indeed is the ratio between physical characteristics of the electrons like mass and charge, are sufficient to say that Eqs. (1.81) and (1.83) express different physics and are identical only in the limit of vanishing damping. Moreover, first Kikuchi [30] and then Mallinson [29] have pointed out that in the limit of infinite damping ( $\lambda\rightarrow\infty$ in Eq. (1.81), $\alpha\rightarrow\infty$ in Eq. (1.83)), the Landau-Lifshitz equation and the Landau-Lifshitz-Gilbert equation give respectively:

$$\frac{\partial \textbf{M}}{\partial t}\rightarrow\infty \quad,\quad \frac{\partial \textbf{M}}{\partial t}\rightarrow 0 \quad.$$ (1.89)

Since the second result is in agreement with the fact that a very large damping should produce a very slow motion while the first is not, one may conclude that the Landau-Lifshitz-Gilbert (1.83) equation is more appropriate to describe magnetization dynamics. In this thesis, from now on, we will use the Landau-Lifshitz-Gilbert equation (1.83).