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

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