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Understanding the dynamics of magnetic vortices has emerged as an importantchallenge regarding the recent development of spin-torque vortex oscillators.Either micromagnetic simulations or the analytical Thiele equation approach aretypically used to study such systems theoretically. This work focuses on theprecise description of the restoring forces exerted on the vortex when it isdisplaced from equilibrium. In particular, the stiffness parameters related toa modification of the magnetic potential energy terms are investigated. Amethod is proposed to extract exchange, magnetostatic and Zeeman stiffnessexpressions from micromagnetic simulations. These expressions are then comparedto state-of-the-art analytical derivations. Furthermore, it is shown that thestiffness parameters depend not only on the vortex core position but also onthe injected current density. This phenomenon is not predicted by commonly usedanalytical ans\"atze. We show that these findings result from a deformation ofthe theoretical magnetic texture caused by the current induced Amp\`ere-Oerstedfield.

Quantitative and realistic description of the magnetic potential energy of spin-torque vortex oscillators