Lagrangian formulation of the linear autonomous magnetization dynamics in spin-torque auto-oscillators

• Published in: Applied mathematics and computation Vol. 217; no. 21; pp. 8204 - 8215
• Main Authors: Consolo, G., Gubbiotti, G., Giovannini, L., Zivieri, R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.07.2011; Elsevier
• Subjects: Auto-oscillators; Autonomous; Autonomous dynamics; Complex generalized non-Hermitian Eigenproblem; Dissipation; Dynamical systems; Dynamics; Exact sciences and technology; Excitation; Global analysis, analysis on manifolds; Lagrange equations; Landau–Lifshitz–Gilbert equation; Magnetization; Mathematical analysis; Mathematical models; Mathematics; Micromagnetics; Numerical analysis; Numerical analysis. Scientific computation; Rayleigh dissipation function; Sciences and techniques of general use; Spin-transfer torque; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Lagrange equations; Complex generalized non-Hermitian Eigenproblem; Spin-transfer torque; Landau–Lifshitz–Gilbert equation; Autonomous dynamics; Auto-oscillators; Rayleigh dissipation function; Micromagnetics; Lagrangian; Complex generalized non-Hermitian; Lagrange equation; Numerical analysis; Landau―Lifshitz―Gilbert equation; Applied mathematics; Dissipative system; Eigenvalue problem; Eigenproblem; Steady state solution; Landau Lifshitz equation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.02.043

A Lagrangian formalism is used to find steady-state solution of the Landau–Lifshitz–Gilbert–Slonczewski equation corresponding to the linear autonomous dynamics of a magnetic auto-oscillatory system subject to the action of a spin-polarized electric current. In such a system, two concurrent dissipative mechanisms, arising from the positive intrinsic dissipation and the negative current-induced one, take place simultaneously and make the excitation of a steady precessional motion of the magnetization vector conceivable. The proposed formulation leads to the definition of a complex generalized non-Hermitian Eigenvalue problem, both in the case of a macrospin model and in the more general case of an ensemble of magnetic particles interacting each other through magnetostatic and exchange interactions. This method allows to identify the spin-wave normal modes which become unstable in the presence of the two competing dissipative contributions and provides an accurate estimation of the value of the excitation threshold current.