Analytical solutions for ultra-fast precessional switching in inertial magnetization dynamics

Abstract

Here we consider the magnetization dynamics in ferromagnetic nanoparticles or films driven by external magnetic field pulses directed transverse to the initial equilibrium state. The excitation pulse drives large-angle ultra-fast magnetization dynamics that may eventually end up in the reversed equilibrium realizing successful precessional switching. We consider ultra-short pulse duration (fractions of picosecond) and large external field intensities (several Tesla) which may be relevant for the realization of faster energy-efficient memory cells. For such short time scales, we include inertial effects in the theoretical description by considering the inertial Landau-Lifshitz-Gilbert equation which requires to be treated as a system of singularly perturbed ODEs for small values of the inertia. By using suitable perturbation approach based on multiple time scales analysis, we develop an approximate closed-form solution for the switching dynamics as well as formulas for switching time and its safety tolerances to obtain the successful switching as function of physical parameters of the particle and strength of magnetic inertia. The developed analytical solution is validated by numerical simulations.

Figures (4)

• Figure 1: Sketch of ultra-fast precessional magnetization switching via ultra-short transverse external field pulses. When the duration $T_{sw}$ is below picoseconds, inertial effects set in appearing as THz nutation (thin red line) superimposed to magnetization precession.
• Figure 2: Comparison between the numerically computed solutions (continuous line) of (\ref{['eq:illgsixd']}) and their approximation provided by (\ref{['eq:apporiginalone']}) and (\ref{['eq:mdot']}) (dashed line) for $\mu_A$ in panels (a), (b) and $\mu_B$ in panels (c), (d), respectively. Let us recall that (\ref{['eq:apporiginalone']}) and (\ref{['eq:mdot']}) are meant to be plotted via the time transformation (\ref{['eq:tau']}). Panels (a) and (c) show the components of $\boldsymbol{m}(t)$ whilst the corresponding derivatives $\boldsymbol{v}(t)$ are shown in panels (b) and (d).
• Figure 3: In panel (a), the system evolution for $t \in [0, 2 T_{sw}]$, where $T_{sw}=0.0635$ is the switch-off instant of the applied field according to (\ref{['eq:switchoff']}) with $t^*=T_{sw}$. Panel (b) shows the decay of the function $\mathcal{W}$ during the system relaxation for $t \in [T_{sw},100 T_{sw}]$ ($\log$ scale).
• Figure 4: Comparison between the numerically computed solutions (continuous line) of (\ref{['eq:illgsixd']}) and their approximation provided by (\ref{['eq:apporiginalone']}) and (\ref{['eq:mdot']}) (dashed line) for $\boldsymbol{m}(t)$ in panel (a) and $\boldsymbol{v}(t)$ in panel (b). Similarly to Fig. \ref{['fig:validation2']}, panel (c) shows the behaviour of the computed solution $\boldsymbol{m}(t)$ over the time interval $[0, 2 T_{sw}]$ where $T_{sw}=0.6283$. Finally, the decay of the function $\mathcal{W}$ over the interval $[T_{sw},100 T_{sw}]$ is reported in panel (d) in $\log$ scale. Although the approximation is not good as in the examples of sec. \ref{['subsec:one']}, panel (a) still shows a "good approximation" of $m_3(t)$ and, most importantly, $m_1(t)$, i.e. the "switching" variable.

Theorems & Definitions (14)

• Definition 2.1
• Remark 3.1
• Remark 3.2
• Proposition 3.1
• Remark 3.3
• Remark 3.4
• Proposition 4.1
• Remark 4.1
• Lemma 5.1
• Remark 5.1