2.3 Uniform mode magnetization dynamics

The Stoner-Wohlfarth model has been extensively used to explain the occurrence of hysteresis loops in the $M-H$ relationship for magnetic recording media (see Fig. 2.3). Nevertheless, as far as magnetic storage devices are required to be faster and faster, and on the other hand the dimensions of magnetic media decrease more and more, the inclusion of dynamical effects in the switching analysis becomes unavoidable. For this reason, we start our analysis from uniform mode dynamics, which is governed by the Landau-Lifshitz-Gilbert equation (1.95) rewritten in case of spatial magnetization uniformity:

$$\frac{d\textbf{{m}}}{dt} = - \textbf{{m}}\times \textbf{h}_{\text{eff}}+ \alpha\textbf{{m}}\times \frac{d\textbf{{m}}}{dt} \quad,$$ (2.18)

where the effective field and the free energy have now the simple expressions:

$$\textbf{h}_{\text{eff}}(\textbf{{m}},t) =-\frac{\partial g}{\partial \textbf{{m}}}=-D_x m_x\mathbf{e}_x - D_y m_y\mathbf{e}_y - D_z m_z\mathbf{e}_z + \textbf{h}_ (t) \quad,$$ (2.19)

$$g(\textbf{{m}},\textbf{h}_ ) =\frac{1}{2}D_x m_x^2+\frac{1}{2}D_y m_y^2+ \frac{1}{2}D_z m_z^2-\textbf{h}_ \cdot\textbf{{m}}\quad.$$ (2.20)

The coefficients $D_x,D_y,D_z$ take into account shape and crystalline anisotropy. Assuming that the uniaxial anisotropy is along the $x$ axis, the relationship of the $D$ coefficients with material parameters and demagnetizing factors is the following:

$$D_x=N_x-\frac{2K_1}{\mu_0 M_s^2} \quad,\quad D_y=N_y \quad,\quad D_z=N_z \quad.$$ (2.21)

In this framework the LLG equation defines a dynamical system evolving on the unit-sphere $\vert\textbf{{m}}\vert=1$, according to property (1.97). If we assume that the external field $\textbf{h}_$a is constant, LLG equation (2.18) describes an autonomous dynamical system whose phase space is 2D, and therefore, it cannot exhibit chaotic behavior [22,23]. Moreover, by recalling the Lyapunov structure (1.102) of LLG equation for constant field, which states that energy is a decreasing function of time ($\alpha >0$), one can immediately conclude that the only steady solutions are fixed points. The number of these fixed points is at least two and in any case is even, due to Poincaré index theorem [43]. Thus, any bifurcation of fixed points involves two equilibria at the same time. The fixed points of the dynamics can be computed from the following equations:

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

in the four scalar unknowns $\textbf{{m}}=(m_x,m_y,m_z)$ and $\lambda$. Conversely, if the applied field is time-varying the onset of chaos and self-oscillating behavior cannot be excluded in principle [24], but there exist particular cases in which the non-autonomous system can be reduced to an autonomous one by means of appropriate change of the reference frame. An example of this occurrence will be examined in section 2.5 when the dynamics of rotationally-symmetric particles subject to circularly polarized field will be analyzed. In the following sections, extensively use of the phase portraits [43] of dynamical system (2.18) will be made. In many cases it is convenient to project the unit-sphere on the plane to visualize the structure of the LLG vector field. This can be done by means of the stereographic projection which maps the coordinates $m_x,m_y,m_z$ onto $w_1,w_2$ according to the following transformation:

$$w_1=\frac{m_x}{1+m_z} \quad , \quad w_2=\frac{m_y}{1+m_z} \quad .$$ (2.23)

This stereographic projection has a geometric interpretation, sketched in Fig. 2.4 for the case of $\alpha=0$. The points along $x-$axis $\textbf{{m}}=(\pm 1,0,0)$ are mapped to $(w_1,w_2)=(\pm 1,0)$, while the points along $y-$axis $\textbf{{m}}=(0,\pm 1,0)$ are mapped to $(w_1,w_2)=(0,\pm 1)$. The north pole $\textbf{{m}}=(0,0,1)$ is mapped to $(w_1,w_2)=(0,0)$, whereas the south pole $\textbf{{m}}=(0,0,-1)$ is mapped towards infinity onto the plane. Moreover, image through Eq. (2.23) of closed curves on the unit-sphere remain closed, and also angles are preserved.

Figure: Stereographic projection of the unit-sphere on the plane for the case of $\alpha=0$.

In the derivation of phase portraits of the dynamical system (2.18) we will need to perform numerical integration of LLG equation. We will adopt the numerical semi-implicit scheme proposed in Ref. [25], which in spite of low computational effort, preserves the magnetization magnitude conservation property (1.97). Geometric integration of Landau-Lifshitz-Gilbert equation will be discussed deeply in chapter 4.