Entropic effects and solitons in thermally activated magnetic transitions INVITED

Abstract

The rate of thermally activated transitions can generally be described by the Arrhenius law, as k=f0 exp(-ΔE/kBT). In the previous expression, ΔE is the internal energy barrier, and f0 is a prefactor that encompasses both dynamical and entropic contributions. In this work, we show how the activation entropy can play a crucial role in thermally activated magnetic transitions. To that end, we study two types of transitions involving solitons either as the stable state or the transition state, namely, skyrmion annihilation, and domain-wall mediated magnetization reversal in nanodisks.The aforementioned transitions pertain to the class of rare events, i.e., they are unlikely to arise in the course of a Langevin simulation, because the mean waiting time between events is much larger than the integration timestep. In this case, one possible approach is to use a form of reaction rate theory to calculate the energy barrier and the Arrhenius prefactor. To this end, we apply Langer’s theory [1], which is an extension of the Kramers method to many dimensions. Alternatively, one may use a path sampling method such as forward flux sampling [2] in order to sample the transition path ensemble from the system’s dynamics and extract a transition rate. In this work, we employ both methods in conjunction to study the thermal stability of magnetic skyrmions and nanodisks, for which the two methods show a good agreement.Magnetic skyrmions are particle-like, two-dimensional, non-collinear solitonic spin textures with a nontrivial topology. In ultrathin films and multilayers, small skyrmions are stabilized by an interfacial form of the Dzyaloshinskii-Moriya interaction (DMI). Skyrmion-based designs currently appear very promising for novel spintronics devices, e.g., for racetrack memories and logic gates, reservoir computing, or probabilistic computing [3]. Isolated skyrmions exist as metastable excitations of the collinear magnetic ground state and can be longed-lived. For applications, a stability of up to 10 years at room temperature is a technological requirement, and the ability to accurately predict that stability is therefore crucial. We find that skyrmions can be stabilized not only by an internal energy barrier, but also by a large activation entropy. This stems from the fact that the metastable skyrmion state possesses a large entropy compared to the transition state, because skyrmions exhibit stable internal modes of deformation [4]. When applying a destabilizing magnetic field perpendicular to the skyrmion core, the skyrmion lifetime decreases quickly with the field amplitude. This results from a shrinking of the skyrmion, which simultaneously reduces the the energy barrier, as well as the activation entropy through the suppression of the stable internal modes (Fig. 1) [5].Next, we compute the mean waiting times between thermally activated domain-wall mediated magnetization reversals in a perpendicularly magnetized nanodisk. We use parameters that resemble that of a free CoFeB layer, as typically used in magnetoresistive random access memories [6]. By varying the perpendicular anisotropy and the interfacial DMI, we find that the Arrhenius prefactor can take extreme, seemingly non-physical values up to 1021 Hz, which is orders of magnitude beyond the typically assumed value of 109 Hz, and vary drastically as a function of material parameters. We show that the prefactor behaves like an exponential of the energy barrier, which stems from a linear variation of the activation entropy with the energy barrier (Fig. 2) [7]. This phenomenon, known as the Meyer-Neldel rule, or entropy-enthalpy compensation, has been reported across diverse fields of the natural sciences for over a century—in semiconductors, chemical reactions, biological death rates, etc [8]. In magnetism, it was only recently observed in the decay of the skyrmion lattice [9]. This result implies that the transition state, which, in the case of the magnetization reversal, is a domain wall found in the center of the disk, tend to possess a very large entropy compared to the collinear ground state. This is verified by the existence of gapless, low energy modes propagating along domain walls [10].These findings suggest that the Arrhenius prefactor, which is typically taken as an attempt frequency within the characteristic timescale of a magnetic system's dynamics, may in some cases contain a very large entropic contribution which should not be neglected. This seems to be particularly true in magnetic transitions involving solitons. **