Thermally activated switching at long time scales in exchange-coupled magnetic grains

Abstract

Rate coefficients of the Arrhenius-N\'eel form are calculated for thermally activated magnetic moment reversal for dual layer exchange-coupled composite (ECC) media based on the Langer formalism and are applied to study the sweep rate dependence of $MH$ hysteresis loops as a function of the exchange coupling $I$ between the layers. The individual grains are modeled as two exchange-coupled Stoner-Wohlfarth particles from which the minimum energy paths connecting the minimum energy states are calculated using a variant of the string method and the energy barriers and attempt frequencies calculated as a function of the applied field. The resultant rate equations describing the evolution of an ensemble of noninteracting ECC grains are then integrated numerically in an applied field with constant sweep rate $R=\ensuremath{-}dH/dt$ and the magnetization calculated as a function of the applied field $H$. $MH$ hysteresis loops are presented for a range of values $I$ for sweep rates ${10}^{5}\phantom{\rule{0.28em}{0ex}}\mathrm{Oe}/\mathrm{s}\ensuremath{\le}R\ensuremath{\le}{10}^{10}\phantom{\rule{0.28em}{0ex}}\mathrm{Oe}/\mathrm{s}$ and a figure of merit that quantifies the advantages of ECC media is proposed. $MH$ hysteresis loops are also calculated based on the stochastic Landau-Lifshitz-Gilbert equations for ${10}^{8}\phantom{\rule{0.28em}{0ex}}\mathrm{Oe}/\mathrm{s}\ensuremath{\le}R\ensuremath{\le}{10}^{10}\phantom{\rule{0.28em}{0ex}}\mathrm{Oe}/\mathrm{s}$ and are shown to be in good agreement with those obtained from the direct integration of rate equations. The results are also used to examine the accuracy of certain approximate models that reduce the complexity associated with the Langer-based formalism and which provide some useful insight into the reversal process and its dependence on the coupling strength and sweep rate. Of particular interest is the clustering of minimum energy states that are separated by relatively low-energy barriers into ``metastates.'' It is shown that while approximating the reversal process in terms of ``metastates'' results in little loss of accuracy, it can reduce the run time of a kinetic Monte Carlo (KMC) simulation of the magnetic decay of an ensemble of dual layer ECC media by $2--3$ orders of magnitude. The essentially exact results presented in this work for two coupled grains are analogous to the Stoner-Wohlfarth model of a single grain and serve as an important precursor to KMC-based simulation studies on systems of interacting dual layer ECC media.