In this paper, we propose an approximate nonlinear theory of a phase-locked loop (PLL) of the spin torque nano-oscillator (STNO). We study the nonlinear dynamics of a filterless PLL generating microwave oscillations in a broad range of frequencies under the spin-polarized electrical current and external magnetic field. We consider the bifurcation analysis caused by a change in the frequency detuning of synchronized oscillations. We determine the bands of phase locking and quasi-synchronism, which basically distinguish STNOs from other types of microwave oscillators. Finally, we study the amplitude and phase noises of isochronous and nonisochronous STNO-based PLLs and compare them to the analogous characteristics of an autonomous oscillator.

In this paper the mutual phase locking theory of very nonidentical spin-torque nanooscillators, which is based on the Slavin-Tiberkevich model, considering the theory of nonlinear oscillations, is developed. Using generalized Adler equation we calculate phase-locking region of the system with spin-wave coupling in the parameter plane -distance between nanocontacts and radii difference. We describe trajectories of such a system in the phase space and show the effect of a broadband synchronization. We introduce a generalization of this approach to the ensembles of spin-torque nanooscillators. I INTRODUCTIONThe spin-transfer torque effect in the magnetic multilayers was theoretically predicted by J. Slonczewski [1] and L. Berger [2] in 1996. This effect will give a chance to implement new methods of microwave generation in a nanoscale. The scientists previously showed (for more details see perfect review [3]), that electrical current, which is passing through a magnetic multilayered structure becomes spin-polarized and, if the current density is high (near then 107-108 A/cm2), the spin-polarized current can transfer spin angular momentum among this magnetic layers, which leads to the microwave generation of such a structure.Microwave spin transfer torque -based autooscillators, which are called spin torque nanooscillators (STNO), are very attractive for potential applications. They are highly tunable by bias current and magnetic field, they are the smallest oscillators that have ever been developed (more than 50 times smaller than a standard autooscillators), and can be biased at low currents and voltages (less than 1.0 V). The main drawback of STNOs is its very weak output microwave power (less than 1 W for a simple structure of layers). This negative effect creates some difficulties for the development of novel nanosized devices based on STNOs.A suggested solution of this problem is to synchronize several STNOs and to coherently summarize an output power from each device. Latter effect gives the way to coherent addition of microwave signals by partial nanocontacts. The coupling between the STNOs can be caused by different physical mechanisms. One of the first of them is the nonlocal mechanism (not dependent on a distance between contacts) coupling through the common microwave component of the bias current (for details see [4][5][6][7]). Using this mechanism of the mutual phase locking it is rather difficult to achieve impedance matching of each STNO as a part of an ensemble, and it is really possible to achieve synchronization of only a several tens of oscillators that may not be enough for practical realization. Another [3] physical mechanism of mutual coupling of STNOs is the local mechanism which strongly depends on the distance between nanocontacts. Compared with common bias current phase locking mechanism, this mechanism can give a much more size compactification effect in the creation of a large ensembles. One of the simplest local coupling mechanisms of the STNOs is the dipole-dipole mechanism, which takes...

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