Analysis of switching times distributions for uniaxial magnetic particles.

Abstract

Magnetization switching in nanoparticles and thin-films is the fundamental issue to deal with in order to obtain high speed and energy-efficient recording devices [1]. The optimization of switching mechanisms is constrained in the framework of the so-called magnetic recording trilemma. On one hand, one would like to have the magnetized bit occupying a smaller area on the recording medium and, at the same time, magnetization remaining stable over long enough time for reliable data retention. These two constraints are competing since thermal stability decreases with decreasing active volume of the magnetic bit. On the other hand, circumventing this issue would require higher coercivity of the magnetic material and, consequently, larger current feeding the write head. However, the maximum current amplitude is constrained by technological limits in the realizations of the pole tips and, thus, one cannot meet the aforementioned requirements. For these reasons, several strategies have been investigated in the last decades to realize fast magnetization switching with greater efficiency, such as microwave-assisted switching [2] and precessional switching [3]. In particular, the latter occurs through the application of a field transverse to the initial magnetization and yields much smaller switching times than conventional switching [4], [5]. However, to achieve successful switching, an extremely precise design of the field pulse is needed to switch off the field at the right moment [6]. Then, the equilibrium magnetization is reached after a relaxation from a high-to low-energy state [7]. This mechanism is probabilistic even when thermal fluctuations are neglected, due to multistability and small dissipation in magnetization dynamics [8]. When also thermal fluctuations are considered, the stochasticity of the switching process is even more pronounced [3]. On the other hand, magnetic recording devices must undergo strict reliability requirements in terms of extremely low write-error rates, which can be realized at expense of the speed of the write process. In this paper, we theoretically analyze the magnetization switching for a single magnetic grain of the recording medium subject to the write head field pulses and room temperature thermal fluctuations, as it is the case of perpendicular magnetic recording. This situation, in the absence of thermal fluctuations and for special symmetry of the magnetic particle, has been studied in a pioneering paper [9] and is referred to as damping switching. In this paper, by using analytical techniques, we derive expressions for the switching times distribution functions in terms of material, geometrical and external field properties. These analytical results provide a tool to quantify the write-error rates as function of design parameters, which may help the optimization of switching processes. To this end, we consider the Landau-Lifshitz-Gilbert (LLG) equation augmented with a thermal field of stochastic nature [10], whose intensity is given by the fluctuation-dissipation theorem. We assume that the magnetization is spatially-uniform during the dynamics, so that the magnetic particle can be described within a macrospin approximation. In the absence of excitation, the energy barrier separating the equilibria is much higher than the thermal energy. This hypothesis is suitable when the uniaxial anisotropy is large enough as it is the case for magnetic recording grains. The external field pulse amplitude is assumed to be above the critical switching field of the device. In this situation, the switching time can be evaluated considering the deterministic magnetization motion acting on a random initial magnetization distribution due to thermal fluctuations. In the absence of external field, the magnetization is distributed according to the stationary solution of the Fokker-Planck equation, which can be simply expressed in terms of the small tilting angle $\theta ( \sin \theta \approx \theta $ with respect to the particle's easy axis. Then, considering a rotationally-symmetric particle (z is the symmetry axis) and neglecting the thermal fluctuations during magnetization evolution, the LLG equation can be integrated by separation of variables [9] in order to determine the switching time $\mathrm {t}_{s}$ defined as the time interval between the application of the field pulse (the initial z-component of the magnetization is $\mathrm {m}_{z0})$ and the time instant where the z-component of the magnetization is equal to a given value $\mathrm {m}_{zf}$. By using appropriate derivation (details will be given in the full paper), it can be shown that the relationship $\mathrm {t}_{s}( \mathrm {m}_{z0} , \mathrm {m}_{zf})$ allows one to derive the probability and cumulative distribution functions for magnetization switching times as function of geometrical, material and excitation parameters. Such functions can be used to compute the write error-rate of the switching process for given switching duration $\mathrm {t}_{s}$. The analytical predictions are compared with macrospin and micromagnetic simulations of magnetization switching (an example computation for a circular nanodot with 30nm diameter, 1nm thickness and perpendicular anisotropy is reported in Fig. 1) in order to show the effectiveness of the approach.