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The nonautonomous dynamics of spin-torque oscillators in presence of both microwave current and field has been numerically studied in nanostructured devices. When both microwave current and field are applied at the same frequency, integer phase locking at different locking ratio is found. In the locking region, a study of the intrinsic phase shift between the locking force (current or field) and the giant magnetoresistive signal as a function of the bias current is also exploited.

In the last years, the effects due to a direct transfer of the spin angular momentum [

The synchronization phenomenon is based on the well-known property that when the external frequency

We recently studied the injection locking phenomenon based on the application of a microwave field on perpendicular materials [

In this work, by means of a micromagnetic study, the nonautonomous dynamical behavior of STO in presence of microwave signal composed by the simultaneous application of microwave current density

We studied the dynamical behavior of exchange bias spin-valves composed by IrMn(8 nm)/Py(10 nm) (polarizer)/Cu(10 nm)/Py(4 nm) (free layer) with elliptical cross-sectional area

(a) Frequency and (b) power amplitude versus dc current density when an out-of-plane field of 250 mT tilted 10° versus

For a complete model description of the numerical techniques see also [^{2}) and a microwave field linearly polarized at

In order to characterize the device behavior, first of all we analyzed the STO in the free running regime. We observe dynamical regime in a wide range of current density for bias field larger than 180 mT. Here we discuss in detail data for a bias field of 250 mT, but qualitative similar results have been also observed for 200 and 300 mT.

In order to characterize the oscillator regime of the device, we swapped the dc current exceeding the critical current value (to obtain dynamics regime, threshold current density was ^{2}) up to current values, where oscillation regime is degraded by noise. Frequency and power behavior of the nano-oscillator with respect to the current density is shown in Figures ^{2}, where the dynamics is characterized by an in-plane oscillation axis. For

Figure ^{2} is about constant.

We systematically studied the locking to the first harmonic (the same of the self-oscillation) in the blue shift region as a function of the ^{2}). We found different locking regions at the locking ratio 1 : 1, 2 : 1, and 3 : 1 (in the last case only when the microwave field component is applied, not shown here). Typically, the locking region is much larger when the microwave force is a field (or a combination of current and field). In fact, in the case of current we found a locking region of about 150 MHz (1 : 1) and 50 MHz (2 : 1), whereas no locking on the third harmonic is found. Microwave field provides a locking region larger than 1 GHz and, since driving force breaks the oscillation symmetry, the 1 : 1 synchronization region has a specific asymmetric shape. Then, whereas for small forcing signal the synchronization region can be described by an analytical theory (symmetric tongue where the locking region increases linearly with force amplitude) [

Response of the STO self-oscillation for ^{2} as a function of the microwave source frequency when (a) ^{2}, (b) ^{2} and RF field

Figure ^{2}) computed up to ^{2} (^{2} held unchanged. The border lines have been computed considering the lower (in the left part) and higher (in the right part) microwave frequency where the phase locking is achieved. In the low regime of microwave source, the Arnold tongue is related to the only application of the microwave current, which can be considered as a “weak” microwave signal. In fact, such a signal gives rise to symmetric synchronization region (no hysteresis is observed) with a locking band linearly dependent on the force locking.

(a) Arnold tongue computed for ^{2} varying the magnitude of RF current and field. (b) Phase difference between external force (current or field) and output magnetization as a function of dc current when an RF source at the same frequency of the free precessional one is applied.Filled-red circle: RF current, white circle: RF field.

When both microwave current and field are applied simultaneously at the same frequency, the nonautonomous response becomes more complicated. The presence of an additional weak microwave field gives rise to increasing of the locking region from 150 MHz at

Figure ^{2}. After that current value, phase shift jumps about 180°, typically this is due to the different oscillation mode from in-plane to out-of-plane mode. Lastly, the phase difference gradually increases following the frequency slow rising with dc current magnitude [

In the locking region an intrinsic phase shift

Figure

Intrinsic phase shift ^{2}.

In summary, we have studied micromagnetically the nonlinear behavior of spin-torque nano-oscillators in locking regime driven by microwave current and field. We found a large locking region at different harmonics when RF field is applied. The effect of the static applied field is also studied. Finally, we showed and explained the intrinsic phase shift due to the difference between microwave source and magnetization precession inside the locking region, also comparing our numerical data with a recent analytical theory.

This paper was supported by Spanish Project under Contract no. MAT2011-28532-C03-01. The authors would like to thank Sergio Greco for his support with this paper.

_{90}Fe

_{10}/Ni

_{80}Fe

_{20}point contacts

^{3}. The time step used was 32 fs,