Analysis and comparisons of LLB variants for high temperature magnetization dynamics

In 1935, Landau and Lifshitz proposed what is now regarded as the first dynamic theory of magnetization, referred to as the LL equation [1]. R. F. Brown later influenced the naming of what eventually grew into a branch of physics, now known as micromagnetics [2]. Two significant evolutions in this sub-centenarian branch of physics include the modification, in 1955, from T. Gilbert, to improve the relaxation description in materials with high damping [3]. Another significant change was initiated by Brown, introducing a method to treat finite temperature of magnetization dynamics using the Fokker Planck (FP) equation [4]. Both of these steps set the stage for development in two important directions; understanding relaxation and effects of elevated temperature simultaneously. We are now at a juncture where a good understanding of both these aspects is pivotal to the advancement of hard disk drive technologies, reaching of overcome superparamagnetism [5]. Heat assisted magnetic recording (HAMR) is actively being developed in order to solve fundamental issues present in current HDD systems, to enable continued growth in areal density. However, the need to heat the media beyond Curie using optical energy necessitates understanding magnetization dynamics in elevated temperatures, and all the dominant forms of relaxation. The Landau-Lifshitz-Bloch (LLB) theory aims to meet this demand by forming the bridge between the low temperature LL equation and high temperature magnetization dynamics. Over the past 18 years, at least four LLB theories have been proposed, where the differences lie primarily in the description of the relaxation and/or the details of the stochastic formulation. One of the first LLB formulations appeared in 1997 and extended the approach of Brown, based on the FP equation [6]. The stochastics of the formulation was later modified to a second variant [7] using LLB within the FP method. In 2012, another correction was made in the stochastic terms to ensure - etter conforming to a Boltzmann distribution [8]. In the same year, another formulation was introduced based on the quantum kinetic equation[9]. Note that there was a very recent formulation advanced in 2014 [10], based on the FP, however, the equation of motion for the FP assumes the form of [9], and is therefore variant of it. This recent formulation, however, will not be considered in the analysis presented here. Figure 1 shows a table summarizing the various LLB forms considered. In Table 1, m is the magnetization vector normalized by the m at zero temperature. H_T is the total effective field; m_eq is the time-dependent magnetization equilibrium. h is a stochastic thermal fluctuation field, and h* is also, however, it is of a different form. α is the transverse damping while α_|| is the longitudinal damping. τ_s is the spin relaxation time and γ is the gyromagnetic ratio. For some forms, it is important to understand how different parameters relate, so that a consistent comparison is made. If caution is not taken here, one may erroneously obtain larger variations than expected. This is particularly important for LLB-III and IV. III for example, has Curie temperature T_c and α input parameters, while IV has τ_s and m_eq. Switching dynamics are analyzed in all the LLB forms listed in Table 1. A common and interesting feature observed in all variants is that there are different relaxation trends depending on whether the system is undergoing heating or cooling. Moreover, variant IV is shown to have relaxation coefficients that depends both on temperature T and m. Thus, variant IV uniquely demonstrates additional nonlinearity. Variant IV is also found to have distinct features in switching probability. Figure 1 (bottom) illustrates an example, where form III is compared. Variant III displays a repeatable dip in the SP even above T_c of 700