STATIONARY MODE OF COUPLED OSCILLATIONS OF MAGNETIC VORTICES IN A MULTILAYER SPIN-TRANSFER NANOSCILLATOR

The coupled dynamics of magnetic vortices in spin-transfer nanooscillators is investigated. A spin-transfer nanooscillator is a structure in which the precession of magnetization is excited due to the effect of spin moment transfer between the spin-polarized current and magnetization. The case of a spin-transfer nanooscillator, which is a three-layer nanocylinder, where two magnetic layers are separated by a non-magnetic conducting interlayer, has been well studied previously. In permalloy nanocylinders of certain sizes, a magnetic vortex can be realized as the ground state. For two-vortex spin-transfer nanooscillators, two frequencies for the gyrotropic mode of stationary coupled oscillations of vortices are found. One of these frequencies is higher than the frequency obtained for the case of a single vortex, and the second is lower. In this paper, we consider a case of a cylindrical spin-transfer nanooscillator consisting of four permalloy soft magnetic layers of arbitrary thickness separated by three non magnetic conducting layers. Each of the magnetic layers contains a magnetic vortex as a ground state. For an ap proximate analytical study of the stationary dynamics of such vortices of the same chirality, one can use the effec tive Thiel equations for the vectors of the centers of magnetic vortices. In the potential energy of such a system, in the approximation of pair interaction, the quasi-elastic energy of the deviation of the vortex from the equilibrium position and the energy of interaction of vortices associated with magnetostatic interaction are taken into account. Solutions to the effective equations of motion were sought under the assumption that they have the form of har monic oscillations in the steady-state mode. This allows us to reduce the system of differential equations to a sys tem of algebraic equations. An analytical solution to such a system of equations can be obtained for the case of magnetic and non-magnetic layers of the same size. A comparison of the obtained four frequencies of such oscil lations with the case of a vortex spin-transfer nanooscillator with three, two and one free magnetic layers is carried out. It is shown that for four magnetic layers with identical geometric and physical parameters, there is a maxi mum natural frequency, which is greater than the corresponding maximum frequencies for systems with one, two or three magnetic layers.