Linear and Autonomous Magnetization Dynamics in Spin-Torque Auto-Oscillators: a Lagrangian Approach -- Presentazione orale by R. Zivieri - Conferenza internazionale

We present the generalization of the Dynamical Matrix Method developed for magnetic systems in the presence of conservative forces [1] to the dissipative case. To this aim, in order to include the effects arising from non-conservative contributions, we adopt a Lagrangian formalism based upon generalized Lagrange equations. We first derive the explicit expressions for the kinetic and the potential energies for a magnetic system in which each spin describes, during its precessional motion, an elliptical orbit about the equilibrium configuration. Then we deduce, from simple physical arguments, the dissipation functions corresponding to the intrinsic Gilbert damping torque and to the one connected with the spin-transfer torque. The analysis is performed in the “linear” regime, valid under the assumption of small oscillations of the spins around their equilibrium positions. Therefore, the energy terms appearing in the generalized Lagrange equations are expressed as Taylor series truncated at the second perturbative order. Within this framework, we can characterize the linear and autonomous magnetization dynamics of a magnetic auto-oscillatory system subject to the action of an external bias dc field and a dc spin-polarized electric current. The resulting linearized Lagrange equations are recast as a complex generalized non-Hermitian Eigenvalue problem which has to be solved numerically. According to this analysis it is possible to identify the eigenmodes which become unstable under the action of a spin-polarized current by taking into account the change of sign of their imaginary part. Within this formalism it is also possible to provide an accurate estimation (error less than 1%) of the value of the excitation threshold current, which fully agrees with results of both standard finite-differences micromagnetic codes and analytical theory [2]. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007.2013) under Grant Agreement n228673 (MAGNONICS). [1] M. Grimsditch, L. Giovannini, F. Montoncello, F. Nizzoli, G.K. Leaf, and H.G. Kaper, Phys. Rev. B 70, 054409 (2004). [2] G. Consolo, G. Gubbiotti, L. Giovannini, and R. Zivieri, Appl. Math. Comp. 217, 8204 (2011). -- Presentazione orale by R. Zivieri-Conferenza internazionale