2.5.1 Equation of motion

We consider an uniformly magnetized thin film or spheroidal particle subject to a time-varying external magnetic field. The magnetization dynamics is governed by the LLG equation which is written in the usual dimensionless form (see section 2.3),

$$\frac{d \textbf{{m}}}{d t} - \alpha \textbf{{m}}\times \frac{d \t... ...m}}}{d t} = - \textbf{{m}}\times \textbf{h}_{\text{eff}}(t,\textbf{{m}})\quad.$$ (2.57)

The effective field is given by

$$\textbf{h}_{\text{eff}}(t,\textbf{{m}})= -D_\bot \textbf{{m}}_\bot - D_z m_z \mathbf{e}_{z} +\text{h}_{az} \mathbf{e}_{z} + \textbf{h}_{a\bot}(t)$$ (2.58)

where $\mathbf{e}_{z}$ is the unit vector along the symmetry axis $z$, the subscript ``$\bot$'' denotes components normal to the symmetry axis, $D_{\bot}$, $D_z$ describe (shape and crystalline) anisotropy of the body. The applied field has the dc component h$_{az}$ along the $z$-axis and the time-harmonic component $\textbf{h}_{a\bot}(t)$ uniformly rotating with angular frequency $\omega$ in the plane normal to the symmetry axis:

$$\textbf{h}_{a\bot}(t) = _{a\bot}\left[\cos(\omega t)\mathbf{e}_{x}+ \sin(\omega t)\mathbf{e}_{y}\right] \quad,$$ (2.59)

where $\mathbf{e}_{x}$, $\mathbf{e}_{y}$ are the unit vectors along the axis $x$ and $y$ respectively. The dynamical system defined by Eq. (2.57) is non autonomous ( $\textbf{h}_{\text{eff}}$ explicitly depends on time) and it is characterized by magnetization dynamics with $\vert\textbf{{m}}\vert=1$. In other words, Eq. (2.57) defines a non autonomous vector field on the unit sphere. The analysis of this system is greatly simplified when Eq. (2.57) is studied in the reference frame rotating at angular velocity $\omega$ around the symmetry axis $\mathbf{e}_{z}$. By choosing an appropriate origin of the time, we can obtain that in the rotating frame $\textbf{h}_{a\bot}=$h$_{a\bot} \mathbf{e}_{x}$ and

$$\textbf{h}_{\text{eff}}(\textbf{{m}}) = -D_\bot \textbf{{m}}_\bot... ...hbf{e}_{z} +\text{h}_{az} \mathbf{e}_{z} + \text{h}_{a\bot}\mathbf{e}_{x} .$$ (2.60)

In addition, in passing to the new frame, the derivative of $\textbf{{m}}(t)$ transforms according to the rule

$$\frac{d\textbf{{m}}}{dt}\bigg\vert _ \mapsto \frac{d\textbf{{m}}}{dt}\bigg\vert _ +\omega \mathbf{e}_{z} \times \textbf{{m}}\quad,$$ (2.61)

and thus Eq. (2.57), written in the rotating reference frame, takes the following autonomous form:

$$\frac{d \textbf{{m}}}{d t} - \alpha \textbf{{m}}\times \frac{d \t... ...omega \mathbf{e}_{z} +\alpha \omega \textbf{{m}}\times \mathbf{e}_{z} \right).$$ (2.62)

Equation (2.62) describes an autonomous dynamical system evolving on the surface of the unit sphere $\vert\textbf{{m}}\vert=1$. The fixed points of the dynamics can be computed from the following equations similar to Eqs. (2.22):

$$\left\{\begin{aligned}&\textbf{h}_{\text{eff}}(\textbf{{m}})-\o... ...uad, & \vert\textbf{{m}}\vert=1 \quad. \end{aligned}\right.$$

It is interesting to notice that equilibria in the rotating frame correspond to periodic solutions in the laboratory frame while limit cycles in the rotating frame correspond to quasiperiodic magnetization solutions in the laboratory frame (see Fig. 2.15). The quasiperiodicity derives from the combination of the rotation of the frame with angular frequency $\omega$ and the periodicity of the limit cycle in the rotating frame with angular frequency self-generated by the dynamical system (and in general not commensurable with $\omega$). Notice also that chaos is not permitted in this dynamical system, despite the presence of a driving sinusoidal field, due to the rotational symmetry and the consequent reduction to a dynamical system on a 2D manifold.