Magnetic skyrmion entropy calculation based on a statistical thermodynamic approach

Magnetic skyrmions have a leading role in low-dimensional magnetic systems for their suitable physical properties and potential applications. New techniques in ferrimagnets and micromagnetic simulations show that skyrmions exhibit changes of size and deformations with time [1]. The purpose of this study is thus the determination of configuration entropy due to skyrmion changes of size and deformations as observed in micromagnetic simulations using a statistical thermodynamic approach. This approach is different from the ones of previous studies based on classical thermodynamics [2,3]. The method is based on two main ansatz: 1) the skyrmion energy is fitted via a parabola in the vicinity of the minimum and 2) the skyrmion diameters population follows a Maxwell-Boltzmann (MB) distribution. Concerning 1), the skyrmion energy is written as E = a (Dsky - D0sky) 2 +b and has a parabolic dependence on skyrmion diameter Dsky, with D0sky the equilibrium diameter, a the curvature and b = Emin the minimum energy. Regarding 2), from the comparison between micromagnetic and analytical results, we have found that the skyrmion diameters distribution is of the form: dn/dDsky = C Dsky 2 exp(-a(Dsky- D0sky) 2 /kBT) with C a constant, kB the Boltzmann constant and T the temperature. This has allowed us to make a strict analogy between the skyrmion diameters population and the MB distribution of particles in an ideal gas and to calculate the skyrmion entropy S = -kB H0 from the Boltzmann H0 order function at thermodynamic equilibrium. In the special case, the configuration entropy of a magnetic Néel skyrmion in a Co circular nanodot with out-ofplane magnetization of radius Rd = 200 nm and thickness t = 0.8 nm has been computed. Figure 1 shows the analytically calculated skyrmion configuration entropy as a function of T using the following parameters at T = 0 K: saturation magnetization MS=600 kA/m, exchange stiffness A=20 pJ/m, iDMI parameter D=3.0 mJ/m2 , uniaxial anisotropy constant Ku=0.60 MJ/m3 , Gilbert damping coefficient a =0.01. Magnetic parameters are scaled with T [4]. S increases with increasing T and decreases with increasing the external bias field at fixed temperature