Configurational entropy of magnetic skyrmions as an ideal gas

Abstract

The study of thermodynamics of topological defects is an important challenge to understand their underlying physics. Among them, magnetic skyrmions have a leading role for their physical properties and potential applications in storage and neuromorphic computing. In this paper, the thermodynamic statistics of magnetic skyrmions is derived. It is shown that the skyrmion free energy can be modelled via a parabolic function and the diameters statistics obeys the Maxwell-Boltzmann distribution. This allows for making an analogy between the behavior of the distribution of skyrmion diameters statistics and the diluted gas Maxwell-Boltzmann molecules distribution at thermodynamical equilibrium. The calculation of the skyrmion configurational entropy, due to thermally-induced changes of size and shape of the skyrmion, is essential for the determination of thermal fluctuations of the skyrmion energy around its average value. These results can be employed to advance the field of skyrmionics. 3 I. INTRODUCTIONMagnetic skyrmions have been gaining an important role in studies of low-dimensional magnetic systems due to their suitable physical properties and potential applications [1][2][3][4]. Skyrmions can be considered as quasi-particles with topologically-protected magnetization texture, characterized by an integer skyrmion number [1,4]. Although skyrmions can be stabilized by the interplay between exchange and dipolar interactions (so called "bubble skyrmion" [1]), much of the interest is devoted to systems where the Dzyaloshinskii-Moriya interaction (DMI) plays a role in this stabilization [5,6].The DMI is a chiral exchange interaction due to lack or breaking of inversion symmetry in bulk crystalline lattices (bulk DMI) [7][8][9] or at the interfaces in magnetic multilayers (interfacial DMI (IDMI)) [10-13]. While other types of DMI can exist (for instance, in D2d structures) [14,15], in this work we focus on the IDMI. This because it promotes the formation of small Néel skyrmions [1,2,4], which are stable at room temperature as isolated skyrmions, and can be nucleated [16-18], manipulated [12,19-21] as well as detected [22-24] by electrical currents. Therefore, Néel skyrmions have become promising for technological applications [25-33]. Fundamentally, because of thermal fluctuations, Néel skyrmions are subject to (i) internal deformations [10,12,34,35], that are responsible for the loss of the circular symmetry; (ii) thermal drift [18,34,35], which leads to a random skyrmion motion throughout the film plane; (iii) thermal breathing modes [35,36] that can induce non-stationary expansion and shrinking of the skyrmion core, i.e. a time-evolution of the skyrmion size. Hence, the effect of thermal fluctuations should be considered for a proper design of skyrmionbased devices and applications [32,35], especially at room temperature.In this work, we show that the thermal fluctuations promote the existence of a number of skyrmions characterized by the same energy, but having different shapes and diameters. This aspect allows us f...

Statistical Properties and Configurational Entropy of a Two-Dimensional Néel Magnetic Skyrmions Population

“…In particular, in the limit for T→0 K it is < D sky (T → 0 K) > = D 0 sky (T → 0 K) as in the 3D model. Equation 6is different from the one found in [13] according to the 3D approach that reads:…”

“…The computation of the 2D configurational entropy is based on the calculation of the skyrmion energy. Details about the numerical calculation of the skyrmion energy can be found in [13,14].…”

“…This statistical ensemble is characterized by microscopic states having a well-defined energy whose thermal fluctuations are very small if compared to the average energy to be considered negligible. For the present case the 3D MB distribution of skyrmion diameters, with no interactions and correlations one with each other, can be written within a microcanonical ensemble, analogously to the 3D MB of an ideal gas, as [13]:…”

“…However, the 2D skyrmion distribution of Equation 2is proportional to D sky and not to D 2 sky as for the 3D distribution taking into account that one deals with a planar structure depending on the in plane coordinates only. From the minimization of Equation (1), it was found that the displacement law as a function of the temperature of the diameters distribution in the 3D model reads [13]:…”

“…In a recent paper, the configurational entropy of a skyrmion diameters population was calculated within a three-dimensional (3D) approach showing that the entropy can be expressed as a function of physical and geometrical parameters and temperature. According to the 3D model, the distribution of the skyrmion diameters population represents the analogous of the Maxwell-Boltzmann (MB) distribution of particles for a dilute ideal gas and, on this basis, the configurational entropy, thermal fluctuations of energy and the statistical thermodynamics of chiral skyrmions were investigated (see [13,14] and references therein). As observed in micromagnetic simulations [13], configurational entropy can be regarded as the number of different skyrmions having the same average energy resulting from the thermal fluctuations that enable the promotion of a number of skyrmions possessing the same average energy.…”