2.4.3 Precessional switching

Precessional switching is a new strategy to realize magnetization reversal which has been recently the focus of considerable research [34,37]. In the usual switching process, the external field is applied in the direction opposite to the initial magnetization and the reversal is achieved after several precessional oscillations due to dissipative effects [26]. In precessional switching, the field is applied at a certain angle (usually orthogonal) with respect to initial magnetization in order to use the associated torque to control magnetization precessional motion. In fact, this torque pushes the magnetization out-of-plane, creating a strong demagnetizing field in the direction perpendicular to the film plane.

Figure 2.8: (left) Magnetic thin film subject to in-plane applied field. (right) Example of precessional motion of magnetization for in-plane applied field.

Then the magnetization start to precess around the demagnetizing field (see Fig. 2.8). The reversal is obtained after half precessional oscillation and it is realized by switching the external field off precisely when the magnetization is close to its reversed orientation. This kind of switching is usually much faster and it requires lower applied fields with respect to the traditional switching. However, the switching is realized only if the field pulse duration is accurately chosen. In the following, the derivation of the analytical formula that provide this information is reported [28]. We assume that the magnetic body has a thin-film like shape with $\mathbf{e}_z$ normal to the film plane (see Fig. 2.8). Since the film is assumed to be very thin, the demagnetizing factors in the film plane $N_x,N_y$ and perpendicular to the film $N_z$ are practically equal to zero and $-1$, respectively. This leads to the following expression for the effective field:

$$\textbf{h}_{\text{eff}}(\textbf{{m}}) = (Dm_x-h_{ax}) \mathbf{e}_{x} + h_{ay}\mathbf{e}_{y} -m_z\mathbf{e}_{z} ,$$ (2.37)

where $D=2K_1/(\mu_0 M_s^2)$ accounts for the in-plane $x$-axis anisotropy, and $h_{ax}$, $h_{ay}$ are normalized components^2.2 of the applied magnetic field that is assumed to be constant during the pulse duration.

The magnetic free energy corresponding to the effective field (2.37) has the form

$$g(\textbf{{m}})=-\frac{1}{2}D m_x^2 + \frac{1}{2}m_z^2 + h_{ax}m_x-h_{ay}m_y .$$ (2.38)

The precessional switching process consists of two stages: in the first stage the magnetization precesses under the influence of the applied external field until its orientation is almost reversed, in the second stage the external field is switched off and the magnetization undergoes relaxation oscillations toward the nearby equilibrium point. In the first part of the process, the magnetization dynamics is typically so fast that dissipative effects can be neglected. On the other hand, dissipation has to be taken into account during the relaxation process. For this reason, we shall first analyze the precessional switching dynamics in the conservative case $\alpha=0$.

Figure: Sketch of the conservative phase portraits in the stereographic plane. (left) zero applied field. (right) the external field is applied along the $y$ axis.

The conservative phase portraits of the LLG equation, which can be obtained by plotting the contour lines of the energy function (2.38), can be conveniently represented in the plane by using the stereographic projection defined by Eqs. (2.23). The result is schematically depicted in Fig. 2.9. In the case of zero applied field (Fig. 2.9 on the left), the phase portrait is characterized by 6 equilibrium points: the 4 centers $C_1^+$, $C_1^-$, $C_3^+$, $C_3^-=+\infty$, and the two saddles $S_2^+$, $S_2^-$ doubly connected by heteroclinic trajectories. All trajectories, except the heteroclinic ones, circle around the centers. The two centers in the shaded regions are low energy states ( $\textbf{{m}}$ along the easy axis) while the centers $C_3^+$, $C_3^-=+\infty$ are high energy states. Notice that when no field is applied there is no way to reverse magnetization from one shaded region to the other. In the case of field applied along the $y$ axis, (see Fig. 2.9 on the right), the heteroclinic trajectories break into two homoclinic structures, one for each saddle. In this situation, along with trajectories that circle around centers, a new type of trajectories appears: the ones that encircle the saddle homoclinic structure (e.g. the dashed curve in Fig. 2.9, right). This type of trajectory allows the magnetization to move from one shaded region to another and, thus, to realize the switching. However, for a given initial condition, it is necessary a certain field amplitude (critical field for switching) for realizing the situation that the trajectory starting from that initial condition will enter the target shaded region. This aspect will be discussed later.

Figure: Sketch of the phase portrait on the stereographic plane in the case of zero applied field and $\alpha >0$.

Once that magnetization has entered the target shaded region, the field can be switched off and the magnetization remains trapped around the target equilibrium. After certain time, the relaxation process will bring magnetization to the equilibrium. This stage of the switching process has to be analyzed in the dissipative case $\alpha >0$. In this case, the phase portrait of LLG equation on the stereographic plane is sketched in Fig. 2.10. It can be observed that, with the introduction of damping, the Landau-Lifshitz-Gilbert equation (2.18) has now two stable equilibrium points $F_1^{\pm}$ with $\textbf{{m}}=\pm \mathbf{e}_x$, two saddle points $S_2^{\pm}$ with $\textbf{{m}}=\pm \mathbf{e}_y$, and two unstable equilibrium points $F_3^{\pm}$ with $\textbf{{m}}=\pm \mathbf{e}_z$. In Fig. 2.10, shaded regions are the regions where the magnetic free energy is below the energy of the saddle points, while in white regions the energy is above the energy of the saddle points. Since the dissipation results in the decrease of the magnetic free energy, it can be concluded that the time evolution of magnetization within any shaded region inevitably leads to the focus inside that region. If the magnetization motion starts in white regions, then depending on the initial conditions it may relax to one of the two stable foci in the shaded regions. This is because in the high energy (white) regions magnetization trajectories leading to different foci are closely entangled resulting in entangled basins of attractions [35] (see Fig. 2.11).

By using the phase portraits shown in Figs. 2.9-2.10, the essence of the precessional switching can be summarized as follows.

Figure: Numerically computed entangled structure of the basins of attraction for LLG equation with no applied field. The black region is the Âhigh energy portion of the basin of attraction of $F_1^-$ while the white region is the Âhigh energy portion of the basin of attraction of $F_1^+$.

The applied magnetic field creates the torque that tilts magnetization up (or down) with respect to the film plane. This results in a strong vertical demagnetizing field that forces magnetization to precess in the film plane. When this precession brings magnetization from one shaded region to another, the applied field is switched off. Then, due to the damping, magnetization relaxes to the new equilibrium that coincides with the focus of the latter shaded region. It is clear that the precessional switching is accomplished if the applied field is above some critical field necessary to bring the magnetization from one shaded region to another and if the applied field is switched off at appropriate times. Thus, the knowledge of critical magnetic fields and appropriate duration of applied magnetization field pulses is crucial for proper controlling of precessional switching. Before proceeding to the discussion of critical fields and timing of switching-off, it is worthwhile to note that if magnetic field is switched off when the magnetization is in the high energy (white) regions, the result of subsequent (damping driven) relaxation of magnetization is practically uncertain. This is due to a very convoluted and close entanglement in the high energy regions of magnetization trajectories leading to different equilibria (see Fig. 2.11). The smaller the damping, the more pronounced this entanglement in the high energy regions. This entanglement may lead to seemingly stochastic nature of precessional switching if the applied magnetic field is switched off when magnetization is in the high energy regions. This seemingly stochastic nature of precessional switching has been experimentally observed (see Figure 1 in Ref. [37]). Next, the issue of finding critical fields for precessional switching is addressed. Since the magnetization precession is typically (i.e. for short-time field pulses and small damping) so fast that dissipative effects can be neglected, magnetization motion in the first stage of the switching can be studied by using the conservative LLG equation

$$\frac{d \textbf{{m}}}{d t} = - \textbf{{m}}\times \textbf{h}_{\text{eff}}(\textbf{{m}}) .$$ (2.39)

We recall that this equation has two integrals of motions (see section 1.3.5):

$$m_x^2 + m_y^2 +m_z^2 = 1 ,$$ (2.40)

$$-\frac{1}{2} D m_x^2 + \frac{1}{2} m_z^2 + h_{ax}m_x-h_{ay}m_y= -\frac{1}{2} D +h_{ax} .$$ (2.41)

The second integral of motion is the conservation of the free energy (1.103) in the case where the initial magnetization is at $\textbf{{m}}=\mathbf{e}_x$. From Eqs. (2.40) and (2.41) one can see that on $(m_x,m_y)$-plane, the precessional magnetization motion occurs along the elliptic curve:

$$(1+D) m_x^2 + m_y^2 - 2 h_{ax}m_x+ 2 h_{ay}m_y= (1+D) - 2 h_{ax} .$$ (2.42)

confined within the unit circle:

$$m_x^2 + m_y^2 \leq 1 .$$ (2.43)

The possible elliptic magnetization trajectories on $(m_x,m_y)$-plane are shown in Fig. 2.12. Here, the shaded regions correspond to the low energy (shaded) regions of the stereographic plane (see Fig. 2.9), while two high energy (white) regions of stereographic plane are projected into the same high energy region on $(m_x,m_y)$-plane confined by the following ellipse:

$$(1+D) m_x^2 + m_y^2=1 .$$ (2.44)

The components of the applied field $h_{ax}$ and $h_{ay}$, determine the type of elliptic trajectories. In fact, some elliptic trajectories consist of a single piece of elliptic curves (for example, trajectory 0-1-2), while other elliptic trajectories can be made of two disjoint pieces of elliptic curves (for instance, trajectories 0-5 and 6-7).

Figure: ``Single-piece'' and ``Disjoint'' trajectories on $(m_x,m_y)$-plane.

As far as magnetization motion is concerned, one can see that every piece of elliptic trajectories on $(m_x,m_y)$-plane corresponds to periodic motion on the unit spherical surface. In fact, magnetization oscillates back-and-forth along the curves located on the surfaces of positive ($m_z>0$) and negative ($m_z<0$) hemispheres. In addition, since these curves are symmetric with respect to $(m_x,m_y)$-plane, the back and forth pieces of actual magnetization trajectories are orthogonally projected into the same pieces of elliptic trajectories on $(m_x,m_y)$-plane. Thus, it turns out that the precessional switching may only occur along the ``single-piece'' elliptic trajectories that intersect the unit circle (2.43) at negative values of $m_x$. The ``disjoint'' elliptic trajectories are separated from ``single-piece'' elliptic trajectories by the elliptic trajectory 0-3-4 that is tangential to the unit circle. It can be shown [28] that the condition of tangency of the elliptic trajectory to the unit circle leads to the following relation:

$$D m_xm_y-h_{ax}m_y-h_{ay} m_x= 0 .$$ (2.45)

At the point 3 of tangency (see Fig. 2.12), equations (2.42), (2.43) and (2.45) are satisfied. These three equations define the curve $h_{ay}$ vs $h_{ax}$ that separates the values of $h_{ax}$ and $h_{ay}$ that correspond to single-piece and disjoint elliptic trajectories, respectively. A parametric representation of this $h_{ay}$ vs $h_{ax}$ curve can be found by introducing the polar angle $\theta$ such that

$$m_x= \cos \theta , \quad m_y= \sin \theta \quad .$$ (2.46)

In fact, by substituting Eq. (2.46) into Eqs. (2.42) and (2.45) and solving with respect to $h_{ax}$ and $h_{ay}$, the following parametric representation can be found:

$$h_{ax}=D \cos \theta \cos^2 \frac{\theta}{2} , \quad h_{ay}=D \sin \theta \sin^2 \frac{\theta}{2} ,$$ (2.47)

The separating curve, defined parametrically by Eqs. (2.47), is valid only for positive values^2.3 of $h_{ax}$, which correspond to values of the parameter $\vert\theta\vert\leq \pi/2$. For negative values of $h_{ax}$, it can be shown that all the elliptic trajectories starting from point 0 (Fig. 2.12) intersect the unit circle only once. Thus, the points $(h_{ax},h_{ay})$ in the shaded region of Fig. 2.13(a) correspond to ``single-piece'' elliptic trajectories, while the points $(h_{ax},h_{ay})$ in the white region correspond to ``disjoint'' elliptic trajectories.

Figure: (a) Separating curve on $(h_{ax},h_{ay})$-plane; (b) region corresponding to single-piece elliptic trajectory intersecting the unit-circle at negative $m_x$.

In Ref. [28] the conditions on $h_{ax}$ and $h_{ay}$ that guarantee that ``single-piece'' elliptic trajectories intersect the unit circle (2.43) at negative values are also derived. The appropriate values of $h_{ax}$ and $h_{ay}$ correspond to the shaded regions formed by the intersecting lines (see Fig. 2.13(b)).

$$h_{ay}= \pm \left( h_{ax}-{D}/{2} \right)$$ (2.48)

The values of $h_{ax}$ and $h_{ay}$, that guarantee ``single-piece'' elliptic trajectories intersecting the unit circle at negative values of $m_x$, correspond to the points of $(h_{ax},h_{ay})$-plane that belong to the intersection of the shaded regions shown in Figures 2.13(a) and 2.13(b). This intersection is the shaded region shown in Fig. 2.14. The boundary of this region corresponds to the critical fields for precessional switching. It is useful to notice that, in the case of field applied orthogonally to the $x$ axis, that is $\textbf{h}_$a$=h_{ay}\mathbf{e}_y$, the critical field for precessional switching is

$$h_c=\frac{D}{2} \quad,$$ (2.49)

which is half the critical value provided by Stoner-Wohlfarth theory in the case $N_\bot=0$ (see^2.4 Eq. (2.17)). In this respect, this is an example of magnetization switching with external field below the Stoner-Wohlfarth limit. As we mentioned in section 2.2, the switching is not guaranteed with the only application of the external field, but one has to take care of magnetization motion to realize successful reversal by switching the field off at the right time. Interestingly enough, precessional switching dynamics is less energy-consuming than traditional one.

Figure: Switching region in $(h_{ax},h_{ay})$-plane.

It has been previously mentioned that in the case of precessional switching the timing of switching-off the applied magnetic field is crucial in the sense that there exists a certain time-window during which this switching-off must occur. One can be convinced from Fig. 2.12 that this time-window is the time interval of the back-and-forth motion between the points 1 and 2 on a single piece trajectory, as for instance the trajectory 0-1-2. To find this time-window [28], one can write the Eq. (2.42) of this elliptic trajectory in parametric form

$$m_x= _x + \frac{p}{k} \sin u , \quad m_y= _y + p \cos u$$ (2.50)

where $u\in[0,2\pi]$ is the parameter, $k^2=1+D$, a$_x={h_{ax}}/(1+D)$, a$_y=-{h_{ay}}$, $p^2 = h_{ay}^2 + (1+D) [1 - h_{ax}/(1+D)]^2$. From the conservative LLG equation (2.39), one obtains:

$$\frac{d m_x}{dt}=(m_y+ h_{ay})m_z .$$ (2.51)

By substituting Eq. (2.50) into (2.51), one has

$$\frac{du}{\sqrt{1-(\text{a}_x+(p/k) \sin u)^2 -(\text{a}_y+p \cos u)^2}}= k dt .$$ (2.52)

By using the last formula, the time-window $t_1<t<t_2$ for switching the applied field off can be derived:

$$t_1= \int_{u_0}^{u_1} \frac{du}{k \sqrt{1-(\text{a}_x+(p/k) \sin u)^2 -(\text{a}_y+p \cos u)^2}} ,$$ (2.53)

$$t_2-t_1= 2\int_{u_1}^{u_2} \frac{du}{k\sqrt{1-(\text{a}_x+(p/k) \sin u)^2 -(\text{a}_y+p \cos u)^2}} ,$$ (2.54)

and $u_0$, $u_1$ and $u_2$ correspond to points 0,1 and 2 in Fig. 2.12, respectively. The values of the parameters $u_0,u_1,u_2$ can be determined by finding the intersections of the unit circle (2.43) with the elliptic trajectory (2.42) and by using the parametric representation (2.50) of the ellipse. Moreover, one can derive the time instant at which magnetization is in the closest position with respect to the reversed orientation. This position is determined by the intersection of the single-piece ellipse with the unit disk occurring at negative $m_x$. We denote this instant as

$$T_s = \int_{u_0}^{u_2} \frac{du}{k \sqrt{1-(\text{a}_x+(p/k) \sin u)^2 -(\text{a}_y+p \cos u)^2}}=\frac{t_2 + t_1}{2} \quad.$$ (2.55)

Thus, to summarize, in the conservative case, $t_1$ is the time instant at which the magnetization enters the potential well around the reversed state and $t_2$ is the time instant at which magnetization goes out from that potential well. With the notations introduced in section 2.4.1 we have:

$$T_p=t_1 \quad.$$ (2.56)

In presence of a small damping, the separation between the high energy regions and the low energy regions is very close to that in the conservative case. Therefore, one reasonably expects that switching the applied field off when $t_1<t<t_2$ lets the magnetization relax towards the reversed state. This analysis works very well in the case of uniformly magnetized particles. In Chapter 3 we will remove this simplifying assumption and we will demonstrate with micromagnetic simulations that precessional switching process, for thin-films having dimensions and material parameters of technological interest, is a quasi-uniform process, whereas damping switching is not. Moreover, we will show that the evaluation of the switching time window $t_1,t_2$ with Eqs. (2.53)-(2.54) gives very accurate results also in non-uniform cases.