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 better 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 700K. Thus, switching probabilities may reduce to an extent, even in super-Curie regime. Notice, however, that variant IV does not exhibit the dip at all, only increasing as a function of temperature. We will show that this difference is arising, in-part, from the differences in the stochastic formulations. Part of the goal in this work is also to compare to measurements. To achieve this, pump-probe experiments were carried out on 40nm thick Co films. Using the pump-probe experiments, each formulation was used to simulate the experiment. Figure 2 shows the one of the best comparisons found, with IV. Very good agreement is observed between experiment and LLB-IV. In conclusion, this work underscores the differences in the existing variants of the LLB. While important features are found in common in the variants considered, differences are indeed present, and it is shown that these differences can result in hierarchy in these variants, where it is found that variant IV agrees, most completely, to our pump-probe experimental data. The underlying reasons for the observations are also discussed.