Statistical Thermodynamics of Chiral Skyrmions in a Ferromagnetic Material

Abstract

Solitons are a challenging topic in condensed matter physics and materials science because of the interplay between their topological and physical properties and for the crucial role they play in topological phase transitions. Among them, chiral skyrmions hosted in ferromagnetic systems are axisymmetric solitonic states attracting a lot of attention for their dazzling physical properties and technological applications. In this paper, the equilibrium statistical thermodynamics of chiral magnetic skyrmions developing in a ferromagnetic material having the shape of an ultrathin cylindrical dot is investigated. This is accomplished by determining via analytical calculations for both Néel and Bloch skyrmions: (1) the internal energy of a single chiral skyrmion; (2) the partition function; (3) the free energy; (4) the pressure; and (5) the equation of state of a skyrmion diameters population. To calculate the thermodynamic functions for points (2)–(5), the derivation of the average internal energy and of the configurational entropy is crucial. Numerical calculations of the thermodynamic functions for points (1)–(5) are applied to Néel skyrmions. These results could advance the field of materials science with special regard to low-dimensional magnetic systems.

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

“…In this section, the main results of this work applied to an outwardly Néel skyrmion hosted in a Co dot of radius R d = 200 nm and t = 0.8 nm having the core magnetization m z (r = 0) = −1 (S = −1) and H ext applied along + z are discussed. The magnetic parameters are the ones used in the analytical and numerical calculations in [13,14]. In particular, the following magnetic parameters at T = 0 K were used: saturation magnetization M s = 600 KA/m, exchange stiffness constant A = 20 pJ/m, IDMI parameter D = 3.0 mJ/m 2 , K u = 0.6 MJ/m 3 (for example, Co).…”

“…The magnetic parameters A, D and K u at non-zero temperature are scaled from their zero temperature values, by using the scaling laws A(m) = A (T = 0 K) m (T) 3/2 , D (T) = D(T = 0 K) m(T) 3/2 , and K u (T) = K u (T = 0 K) m(T) 3.6 [15]. In Table 1 are summarized the values of a and of D 0sky used in the numerical calculations [13,14]. Table 1 does not contain the values of a and D 0sky at T = 0 K since the shown calculations are for T ≥ 50 K. Table 1.…”

“…Table 1 does not contain the values of a and D 0sky at T = 0 K since the shown calculations are for T ≥ 50 K. Table 1. Calculated a and D 0 sky used in the numerical calculations [13,14]. In Figure 2, the 2D skyrmion diameters distribution calculated via Equation (2) (solid black lines) is compared to the 3D distribution (dashed red lines) [13] obtained by means of Equation (1) for three different temperatures, T = 100 K, 200 K and 300 K, respectively and, at fixed T, for three values of the external bias field, µ 0 H ext = 0 mT, 25 mT and 50 mT, respectively.…”

“…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].…”

“…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.…”