2.2 The static model. Stoner-Wohlfarth theory

We start our discussion from the static model proposed by Stoner and Wohlfarth [77] in 1948. Basically it can be obtained from the study of the Brown's equations in the case of appropriate single domain particle. Below we summarize the basic hypotheses of this model:

1. Single domain particle
2. Spheroidal geometry
3. Uniaxial anisotropy along the rotational-symmetry axis.

First, assuming uniform magnetization within the body, the exchange energy (1.38) gives zero contribution to the free energy. Next, the ellipsoidal geometry permits a significant simplification in the computation of magnetostatic field, since it can be shown that it can be expressed by a straightforward tensorial relationship with magnetization [12]:

$${\mathbf{H}_\text{m}}=-\mathcal{N}\cdot\textbf{M}\quad,$$ (2.1)

where $\mathcal{N}$ is the so-called demagnetizing tensor which is always positive semidefinite. By expressing $\mathcal{N}$ with respect to its principal axes $x,y,z$, which coincide with the principal axes of the ellipsoid, one can rewrite Eq. (2.1) in the following way:

$$\left(\begin{array}{c} H_x H_y H_z \end{array} \right)=... ...) \cdot \left(\begin{array}{c} M_x M_y M_z \end{array} \right) \quad,$$ (2.2)

where $N_x,N_y,N_z$ are the demagnetizing factors such that $N_x+N_y+N_z=1$. Then, the assumption of uniaxial anisotropy implies that the corresponding energy term is quadratic. For instance, if the easy axis is the $z-$axis, then $\mathbf{e}_$an$=\mathbf{e}_z$ and the anisotropy energy (1.43) becomes:

$$F_ (\textbf{{m}})= K_1 (1-m_z^2) V_0 \quad,$$ (2.3)

where $V_0$ is the volume of the spheroidal particle. Finally, the hypothesis of rotational symmetry implies that

$$N_x=N_y=N_\bot \quad.$$ (2.4)

Under these assumptions, the expression of the free energy is the following:

$$G(\textbf{{m}},\mathbf{H}_ )=K_1 (1-m_z^2) V_0+\frac{1}{2}\mu_0 M_s^2\textbf{{m}}\cdot \mathcal{N}\cdot \textbf{{m}} V_0-\mu_0 M_s \textbf{{m}}\cdot \mathbf{H}_ V_0\quad.$$ (2.5)

From now on, we will carry out the derivation with dimensionless quantities. Thus, by dividing both sides by $\mu_0 M_s^2 V_0$ and remembering that $1-m_z^2=m_x^2+m_y^2$, one obtains:

$$g(\textbf{{m}},\textbf{h}_ )=\frac{K_1}{\mu_0 M_s^2} (1-m_z^2)+ \frac{1}{2}N_\bot (1-m_z^2) + \frac{1}{2}N_z m_z^2 - \textbf{{m}}\cdot \textbf{h}_ \quad ,$$ (2.6)

where the expression of magnetostatic energy has been explicitly developed. By collecting terms in $m_z^2$ one ends up with:

$$g(\textbf{{m}},\textbf{h}_ )=\frac{K_1}{\mu_0 M_s^2}+\frac{1}{2}N_\bot +\left(\frac{1}{2}N_... ...}{\mu_0 M_s^2}- \frac{1}{2}N_\bot\right) m_z^2 - \textbf{{m}}\cdot \textbf{h}_ \quad.$$ (2.7)

By neglecting constant terms (which disappear in a minimization procedure) and by factorizing the expression in parenthesis we end up with:

$$g(\textbf{{m}},\textbf{h}_ )=-\frac{1}{2}\left( N_\bot+\frac{2K_1}{\mu_0 M_s^2}-N_z\right) m_z^2 - \textbf{{m}}\cdot \textbf{h}_ \quad.$$ (2.8)

With the position:

$$k_ =N_\bot+\frac{2K_1}{\mu_0 M_s^2}-N_z \quad,$$ (2.9)

the latter equation assumes the simple form:

$$g(\textbf{{m}},\textbf{h}_ )=-\frac{1}{2} k_ m_z^2 - \textbf{{m}}\cdot \textbf{h}_ \quad.$$ (2.10)

It is important to notice that, in the case of rotationally-symmetric ellipsoidal particle, magnetostatic interaction energy is a quadratic form in $m_z$ as uniaxial anisotropy energy. For this reason it is often said in literature that the quantity $k_$eff takes into account shape and crystalline anisotropy, although they have very different physical meanings^2.1. We observe that, for symmetry reasons, at the equilibrium the magnetization lies in the plane defined by the easy axis $\mathbf{e}_z$ and the applied field $\textbf{h}_$a. At this point, it is useful to introduce the spherical angles $\theta$, $\theta_h$ between $\textbf{{m}}$, $\textbf{h}_$a and $\mathbf{e}_z$, respectively. In this respect, one has:

$$m_z=\cos \theta \quad,\quad \textbf{{m}}\cdot\textbf{h}_ =h_a\cos(\theta_h-\theta) \quad.$$ (2.11)

By using these expressions in Eq. (2.10) one obtains:

$$g(\theta,\theta_h) =-\frac{1}{2} k_ \cos^2 \theta-h_a\cos(\theta_h-\theta)=$$

$$=-\frac{1}{2} k_ \cos^2 \theta - h_a\cos\theta_h \cos\theta- h_a\sin\theta_h \sin\theta =$$

$$=-\frac{1}{2} k_ \cos^2 \theta - h_{az} \cos\theta- h_{a\bot} \sin\theta \quad,$$ (2.12)

Figure: Free energy as a function of angle $\theta$, $k_$eff$>0$. (a) for $h_a=0$ two minima $\theta =0,\pi$ and two maxima $\theta =\pm \pi /2$ exist. (b) for small $h_a$ with given $\theta_h\neq 0$ there still exist two minima and two maxima. (c) a critical value $h_$SW$(\theta_h)$ of $h_a$ exist such that a saddle point appears in place of one minimum and one maximum. (d) for $h_a>h_$SW$(\theta_h)$ only one minimum and one maximum remain.

where $h_{az}$ and $h_{a\bot}$ are respectively the parallel and perpendicular component of the applied field with respect to $z-$axis. Let us now suppose that no field is applied and that the particle is initially magnetized along the positive $z-$axis ($\theta=0$). In this situation, the particle minimizes its energy and therefore this is an equilibrium position, as well as the reversed orientation $\theta=\pi$ (see Fig. 2.1a). The energy is instead maximum for $\theta =\pm \pi /2$. Now, if we apply a small external field, opposite to the initial magnetization ( $\theta_h=0$), the free energy (2.10) will still have two minima and two maxima. By further increasing $h_a$ we arrive at a critical configuration for which one minimum and one maximum disappear. We call $h_$SW the applied field value corresponding to this critical situation. If we increase $h_a>h_$SW only one minimum and one maximum of the free energy will exist. This means that for fields $h_a<h_$SW the particle will remain in the initial configuration along $z$, whereas as soon as $h_a>h_$SW the magnetization will switch to the only remaining energy minimum, corresponding to the reversed orientation. It is important to underline that, in the general case (see Fig. 2.1), the critical field $h_$SW will be a function of $\theta_h$. Thus, the idea of Stoner-Wohlfarth model, is to represent, in the control plane ( $h_a,\theta_h$ in polar coordinates, or equivalently $h_{az},h_{a\bot}$ in cartesian coordinates), the separating curve between the region where two minima exist and the region where only one minimum remains. This bifurcation line justifies the switching behavior. It can be found analytically by means of the following relationship:

$$\frac{\partial g}{\partial \theta}=0 \quad,\quad \frac{\partial ^2 g}{\partial \theta^2}=0 \quad,$$ (2.13)

which determines the saddle points of the free energy in the $h_{az},h_{a\bot}$ plane. By imposing the above conditions, one ends up with the following equations:

$$\left\{\begin{aligned} \frac{h_{a\bot}}{\sin\theta}&-\frac{h_{... ...sin^3\theta}&+\frac{h_{az}}{\cos^3\theta}=0 \end{aligned}\right.$$

Figure: The Stoner-Wohlfarth astroid in the $h_{az},h_{a\bot}$ plane.

By solving these equations one ends up with the parametric expression of the bifurcation line, which is referred to as the Stoner-Wohlfarth astroid:

$$\left\{\begin{aligned}h_{az}&=-k_\text{eff}\cos^3\theta h_{a\bot}&=k_\text{eff}\sin^3\theta \end{aligned}\right.$$

The curve defined by the latter equations is represented in Fig. 2.2. The polar representation can be also found:

$$h_a=k_ (\sin^{2/3}\theta_h+\cos^{2/3}\theta_h)^{-3/2} \quad.$$ (2.16)

In the particular case of $\theta_h=0$, one can easily verify that

$$h_ =k_ =N_\bot+\frac{2K_1}{\mu_0 M_s^2}-N_z \quad.$$ (2.17)

Thus, when the external field has components such that the point $(h_{az},h_{a\bot})$ lies outside the astroid, only one minimum of the free energy is present and magnetization will end up there. For example, in the case of initial magnetization $\textbf{{m}}=\mathbf{e}_z$, the application of an external field along $\mathbf{e}_z$ with amplitude $h_a$ greater than the critical value computed from Eq. (2.17), the switching of the particle occurs. In fact, the initial configuration is no longer stable and the only minimum of the free energy (stable equilibrium) remains $\textbf{{m}}=-\mathbf{e}_z$. Conversely, as soon as the field is such that the point $(h_{az},h_{a\bot})$ lies inside the astroid, the situation is more complicated since there exist still two minima of the free energy, namely two stable equilibria. Which one will be reached by magnetization depends on the dynamics of magnetization motion, which is not described by the Stoner-Wohlfarth model. In this situation, one can say that the switching from one minimum to the other is not precluded, but is not guaranteed. Recently, the possibility to obtain magnetization switching with field amplitudes below the Stoner-Wohlfarth limit has been investigated. We will discuss this possibility in the following sections and in chapter 3.

Figure: Stoner-Wohlfarth model. Picture of hysteresis loops from the original paper [77] for different applied field directions. The external field in abscissa is measured in units of $k_$eff. Magnetization in ordinate is measured in units of $M_s$.