Injection locking at 2f of spin torque oscillators under influence of thermal noise

Experiments, numerical simulations and an analytic model were developed to elucidate the effects of noise in the synchronized state of a tunnel junction based spin torque nano oscillator (STNO). It is demonstrated that in the in- plane magnetized structure, while the frequency is locked, much higher reference currents are needed to reduce the noise by phase locking. Our analysis shows that it is possible to control the phase noise by the reference microwave current (IRF) and that it can be further reduced by increasing the bias current (IDC) of the oscillator, keeping the reference current in feasible limits for applications.

I. INTRODUCTION

A spin polarized current passing through a magnetic multi-layered nanosystem can drive its magnetization into large amplitude periodic oscillations 1,2,3 when the spin polarized current is large enough to compensate the natural damping. These spin transfer driven magnetization oscillations, together with their particular nonlinear properties4 spurred the interest in STNO’s for several applications in current controlled microwave devices5. Nevertheless, one of the main issues that remains to be addressed for these spin STNO’s is their relative large linewidth. One possibility to reduce the linewidth is to couple either different layers within an oscillator6, or to couple several oscillators. For this second case, several options were proposed, experimentally and theoretically: current mediated coupling7,8, dipolar coupling9,10 or spin wave coupled nanocontacts11,12. In order to understand the conditions for electric synchronization of several oscillators by their own emitted RF current, we studied the synchronization of an STNO to a reference current source, with known spectral specifications. Here we focus on standard uniform in plane magnetized oscillators (in-plane polarizer and in-plane free layer, IP), for which an in-plane precession (IPP) mode is stabilized. The injection locking of such an STNO to a reference current at two times the generated frequency (2f) was demonstrated both numerically and by experiments13. However, the linewidth in the locked regime was reduced only by a factor of seven, while a reduction to the linewidth of the microwave source (several Hz) was expected. These large linewidths are associated to the thermal noise that induces fluctuations which can drive the phase from an equilibrium state to a neighbouring one, with an associated phase slip of  2 which can be envisaged as non- syncronization and re-synchronization events. Zhou et al14 demonstrated that the particular way the phase approaches its

2 synchronized value has consequences in the transients that may limit the modulation of an STO. Recent works investigated the mechanisms of the so called pure phase locking state in double vortex based STNO: Robust synchronization was experimentally shown, with a 105 linewidth reduction15 and the role of the phase slips in the synchronized state was investigated16. In this work we study the injection locking at 2f to an external reference current of an uniform IP magnetized STNO under the influence of thermal noise. We performed both experiments and numerical simulations, together with an analytic model to describe the transients to the locked regime in the IPP geometry. Our results show the key features of electric synchronization of a uniform in plane magnetized STNO. II. ANALYTIC MODEL The effect of thermal fluctuations on the transient behaviour of the synchronized state of an STNO is analyzed in the frame of a generic model of a non-linear auto oscillator4 that we extended for the IPP mode synchronized by an RF current at 2f (details in Appendix). Since STNO´s are non-linear (non-isochronous) oscillators, the power and the phase of the oscillator are not independent, leading to a system of coupled equations.

] Im[ 1 2     op p N dt d      (1)  ] Re[ 2 cos 2 2 0     o p p p p dt p d      F (2)

Here ψ(t) = 2 - ωextt is the phase difference between the STNO phase  and the phase of external source ωextt, N is the coefficient of non-linear frequency shift, F is a real parameter proportional to the reference current, Γp is the damping rate of the power fluctuations and  has the statistical properties of the Gaussian thermal noise17. Linearizing the equations (1) and (2) around a stable solution po (without considering thermal noise) allows us to s

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