Nonlinear Theory of Microwave Generation by Spin-Polarized Current

An approximate analytic theory of microwave spin wave generation by spin-polarized direct current in magnetic nano-contacts magnetized in an arbitrary direction is developed. For sufficiently large density of spin-polarized current the damping in the magnetic ``free'' layer is compensated and a quasi-uniform precession of magnetization about the direction of the \textit{internal }bias magnetic field \textbf{H} (which differs from the direction of the \textit{external} bias field \textbf{H}$_{e}$ applied to the ``free'' layer) is excited. The precession amplitude is subsequently limited by the positive nonlinear dissipation of the same precession. With the increase of the current magnitude $I $the angle of precession increases, making precession \textit{nonlinear}, and \textit{reducing the projection M}$_{z}$ of the precessing magnetization vector on the axis of precession ($z$-axis). This reduction of $M_{z}$ is responsible for the observed frequency shifts of the generated microwave oscillations and for the limitation of their amplitudes. Due to the influence of demagnetizing fields in the ``free'' layer the nonlinear frequency shifts have different magnitudes and signs for different orientations of the external bias field \textbf{H}$_{e}$. The theory gives good qualitative and even partly quantitative explanation of the majority of recent experimental results on microwave generation by direct current in nano-contacts.