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We performed metadynamics Monte Carlo simulations to obtain the free energyas a function of the topological charge in the skyrmion-hosting magnetic modelsystems (Pt$_{0.95}$Ir$_{0.05}$)/Fe/Pd(111) and Pd/Fe/Ir(111), using a spinmodel containing parameters based on ab initio calculations. Using thetopological charge as collective variable, this method allows for evaluatingthe temperature dependence of the number of skyrmionic quasiparticles. Inaddition, from the free-energy cost of increasing and decreasing thetopological charge of the system we determined chemical potentials as afunction of the temperature. At lower temperature, the chemical potential forcreating skyrmions and antiskyrmions from the topologically trivial state isdifferent. This splitting of the chemical potential is particularly pronouncedfor large external magnetic fields when the system is in a field-polarizedphase. We observed a change in the shape of the free-energy curves whenskyrmion-skyrmion interactions become more pronounced.

Chemical potential of magnetic skyrmion quasiparticles in heavy metal/iron bilayers