2.4.2 Damping switching

The traditional mode to realize the switching is the one sketched in Fig. 2.6 and it is here referred to as ``damping switching'' following a terminology introduced by Mallinson [26]. The switching is realized by applying the external field in the direction opposite to the initial magnetization state. If the field is strong enough (the threshold field can be computed by the Stoner-Wohlfarth theory) the initial magnetization state becomes unstable and magnetization dynamics tends to relax toward the new minimum of energy in the direction of the applied field. In the following, the analytical treatment of damping switching proposed in Ref. [27] will be presented.

Figure 2.6: Typical mode of operation in damping switching: the field is applied against the initial magnetization.

This is possible indeed only in the special case in which the magnetic body is rotationally symmetric around a certain axis and the external field is applied exactly along the symmetry axis. If the symmetry axis is $\mathbf{e}_z$, the effective field is given by the formula

$$\textbf{h}_{\text{eff}}= -D_\bot (m_x\mathbf{e}_x +m_y\mathbf{e}_y) - D_z m_z\mathbf{e}_z + \text{h}_{az}\mathbf{e}_z$$ (2.24)

Here, coefficients $D_\bot=N_\bot$ and $D_z=N_z-(2K_1)/(\mu_0 M_s^2)$ account for demagnetizing fields and crystalline anisotropy, while h$_{az}$ is the applied field, which is assumed to be constant during the pulse duration. R. Kikuchi [30] considered a similar problem for the case of isotropic ferromagnetic sphere when the effective field is defined by the formula:

$$\textbf{h}_{\text{eff}}= -D \textbf{{m}}+ \text{h}_{az}\mathbf{e}_z$$ (2.25)

The difference in the mathematical forms of the effective fields (2.24) and (2.25) results in the profound difference in the physical phenomena of magnetization switching. In the case of effective field (2.25), there exists an infinite set of equilibrium states for h$_{az}=0$ and no critical field is required to switch from one equilibrium state to another. In the case of effective field (2.24), there are only two equilibrium states for h$_{az}=0$ with $m_z=1$ and $m_z=-1$, respectively, and the switching from one equilibrium state to another is only possible if the applied field h$_{az}$ exceeds some critical field h$_{c}$. J. Mallinson [26] studied the problem with the effective field given by the formula (2.24). His analysis is based on the solution of LLG equation in spherical coordinates. Conversely, the following approach exploits the rotational symmetry of the problem. It is apparent that the mathematical form of LLG equation with the effective field equation given by Eq. (2.24) is invariant with respect to rotations of coordinate axes $x$ and $y$ around $z$ axis. As a results of this rotational symmetry, it is expected that $dm_z/dt$ depends only on the $z$-component of $\textbf{{m}}$. Indeed, by using simple algebra, it is easy to find that:

$$(\textbf{{m}}\times \textbf{h}_{\text{eff}}) \cdot \mathbf{e}_z =... ...f}}) \right]\cdot \mathbf{e}_z= (\text{h}_{az}- \text{h}_{c}m_z)(1-m_z^2) ,$$ (2.26)

where

$$_{c}= D_\bot- D_z=h_ \quad ,$$ (2.27)

is the classical Stoner-Wohlfarth field (see Eq. (2.17)). Thus from LLG equation and from Eqs. (2.26), we derive the following equation

$$\frac{dm_z}{dt}= \frac{\alpha}{1+\alpha^2} ( _{c}m_z- _{az})(1-m_z^2)$$ (2.28)

It is clear from Eq. (2.28) that the magnetization switching from the state $m_z=1$ to the state $m_z=-1$ (or vice versa) is driven exclusively by damping: in the conservative case $\alpha=0$ the $z$-component of magnetization remains constant. In this sense, this switching can be regarded as ``damping'' switching. It seems from Eq. (2.28) that no switching is possible if magnetization is in equilibrium state $m_z=1$. However, due to thermal effects, magnetization $\textbf{{m}}$ slightly fluctuates around the above equilibrium state. As a result, the value of $m_z$ at the instant when the applied field is turned on may be slightly different from 1 and the switching process can take place. This argument justifies the solution of Eq. (2.28) with the initial condition:

$$m_z(t=0)= _{z0}$$ (2.29)

where m$_{z0}$ is close to $1$. It is apparent from Eq. (2.28) that if h$_{az}>$h$_{c}$ then $dm_z/dt<0$ and the switching to the equilibrium state $m_z= –1$ will proceed for any m$_{z0}$. On the other hand, if h$_{az}<$h$_{c}$, then for m$_{z0}$ sufficiently close to 1 it can be found from Eq. (2.28) that $dm_z/dt>0$ and no switching is possible. This clearly reveals that h$_{c}$ has the meaning of critical field. In the sequel, it is assumed that h$_{az}>$h$_{c}$. By separating variables in Eq. (2.28), we obtain:

$$\int_{\text{m}_{z0}}^{m_z} \frac{d m_x}{(1-m_z^2)(\text{h}_{c}m_z-\text{h}_{az})}= \frac{\alpha}{1+\alpha^2} t .$$ (2.30)

Performing integration, one obtains:

$$\frac{1}{2(\text{h}_{az}-\text{h}_{c})} \ln \frac{1-m_z}{1-\text{m}_{z0}} -\frac{1}{2(\text{h}_{az}+\text{h}_{c})} \ln \frac{1+m_z}{1+\text{m}_{z0}}+$$

$$+\frac{\text{h}_{c}}{\text{h}_{c}^2-\text{h}_{az}^2} \ln \frac{\... ...}m_z}{\text{h}_{az}- \text{h}_{c}\text{m}_{z0}}= \frac{\alpha}{1+\alpha^2} t$$ (2.31)

By using the last equation, the minimal pulse time needed for switching can be found. Indeed, if the duration of magnetic field pulse is such that a negative value of $m_z$ is reached, then the magnetization will be in the basin of attraction of the equilibrium state $m_z=-1$, and the switching will be achieved. This can be clearly seen by setting $m_z=0$ into Eq. (2.28), which implies $dm_z/dt<0$ for h$_{az}>h_c$. Thus, the minimal time can be found from Eq. (2.31) and the condition $m_z=0$. By taking into account formula (2.31) and the fact that time and magnetic field in this formula are normalized by $\gamma$   M$_$s and M$_$s , respectively, we derive the following expression for the minimal pulse time $T_p$ mentioned in section 2.4.1:

$$T_p= \frac{1+\alpha^2}{\alpha \gamma} \left[ \frac{\ln(1-\cos\t... ...} \ln\frac{\text{H}_{az}}{\text{H}_{az}-\text{H}_{c}\cos\theta_0} \right] .$$ (2.32)

Here H$_{c}=(D_\bot - D_z)$   M$_$s and m$_{z0}=\cos\theta_0$, where $\theta_0$ is the angle formed by the initial magnetization with z-axis. It is interesting to point out that for the typical case of small angles $\theta_0$, the minimal pulse time $T_p$ is very close to the actual switching time $T_s$ (see section 2.4.1) at which $m_z$ reaches a value almost equal to $-1$. This is because, for sufficiently small $m_z$ (large angles $\theta$), $m_z$ decreases much faster (see Eq. (2.28)) than when $m_z$ is close to its equilibrium values. This assertion is supported by numerical calculations, performed by using the analytical expression for $m_z$ extracted from formula (2.31) and shown in Fig. 2.7.

Figure: Evolution of $m_z$ with time for different initial angles $\theta _0=0.3^o, 1^o, 3^o$. H$_{az}/H_c=1.2$.

It is apparent from this figure that the initial (near equilibrium) dynamics of $m_z$ is very slow and takes most of the time, while the magnetization dynamics away from equilibrium is very fast. Thus, the switching time is close to the minimal pulse field time, calculated above:

$$T_s\simeq T_p \quad.$$ (2.33)

It is also apparent from formula (2.32) that for the typical case of very small initial angles $\theta_0$, the first term in the right hand side of formula (2.32) is dominant. Thus, by neglecting two other terms and using simple trigonometry, one can derive the following expression for the minimal pulse time (switching time):

$$\frac{1}{T_p}=\frac{1}{\ln(\sqrt{2}/\theta_0)} \frac{\alpha \gamma}{(1+\alpha^2)} ( _{az}- _{c}) .$$ (2.34)

It turns out from Eq. (2.34) that, for short pulse duration $T_p$, the value of applied field needed for switching increases inversely proportional to $T_p$, i.e. H$_{az}\sim 1/T_p$. In this sense, one may say that the dynamic (short-time) coercivity appreciably exceeds the static coercivity H$_{c}$. The last formula is also similar to the one that has been observed in numerous experiments [31], [32], [33]. It is usually written in the form:

$$\frac{1}{T_p}= \frac{1}{S} ( _{az}- _{c}) .$$ (2.35)

The switching time (see formulas (2.32), (2.34)) depends on the value of initial angle $\theta_0$. An expected value of this angle can be evaluated by using Maxwellian equilibrium distribution for $\theta_0$:

$$\rho(\theta_0)= c \sin\theta_0 \exp{\left[\frac{-g(\textbf{{m}})}... ...\mu_0 (D_\bot -D_z)\text{M}_\text{s}^2 V_0 \cos\theta_0} {k_B T}\right] } ,$$ (2.36)

where $k_B$ is the Boltzmann constant, $c$ an appropriate constant to normalize the integral of the distribution, $\mu_0$ the vacuum magnetic permeability, and $V_0$ is the volume of the magnetic body. If we assume that the magnetic body is a Permalloy film with dimension $(0.5 \mu$m$, 0.5 \mu$m$, 10$nm$)$, the typical expected value of $\theta_0$ is $0.5^o$. The expected value of $\theta_0$ is increased as the volume dimensions are decreased.