Nanomagnetic structures and devices are playing an increasingly important role in science and technology. This is due to the fact that, among other features, they exhibit ultrafast dynamic behaviour on time scales of the order of picoseconds and less. This interest has received a tremendous boost in recent years after the discovery that ultra-fast dynamic phenomena can be induced by applying not only time-varying magnetic fields, but also DC or AC electric currents. In fact, flowing electrons acquire a certain degree of spin polarization when traversing magnetic multilayers, and the mutual exchange of angular momentum (so-called "spin transfer" effect) between the magnetic layers and the electron flow may lead to large and fast magnetization changes in the device.
Spin-transfer-driven magnetization dynamics is a topic of fascinating richness and complexity, resulting from the highly nonlinear, far-from-equilibrium nature of the problem. A wealth of features (stationary states, steady-state oscillations, spatio-temporal chaos) may appear, depending on the particular choice and nature of the excitation conditions (DC or AC external magnetic fields, DC or AC injected electric currents).
Two classes of phenomena, in particular, have attracted great fundamental and technological interest. Current injection may produce magnetization switching, a discovery which is leading to the development of spin-transfer non-volatile magnetic memory cells in which data control is all-electrical in nature, i.e., it is based on spin transfer (for writing) and magnetoresistance (for reading). On the other hand, spin transfer may lead to the appearance of steady-state magnetization oscillations at microwave frequencies. This has potential applications for current-controlled nano-oscillators amenable to natural integration with semiconductor electronics, for example as new types of clocks for synchronization of electronic devices.
The synchronization of spin-transfer oscillator arrays is instrumental for obtaining enhanced emitted power and reduced oscillation linewidth. The value of the natural oscillation frequency of individual nano-oscillators is not exactly controllable, due to unavoidable small differences in geometry and physical properties. One possibility is to attempt synchronization and locking of nano-oscillator arrays to some precisely defined frequency by means of an external ac-current or radio-frequency-field source.
At the INRIM Electromagnetism Division various aspects of these problems are currently under study, in close collaboration with the University of Maryland, USA, and the "Federico II" and "Parthenope" Universities in Napoli.
Two problems, in particular, have been addressed recently. The first is the development of novel analytical and numerical tools for the determination of magnetization normal resonant modes and frequencies in magnetic nanostructures with different geometries (fig. 1).

Fig. 1 Numerically computed resonant modes and corresponding frequencies for a square magnetic thin-film (100 nm by 100 nm by 3 nm), with negligible in-plane magneto-crystalline anisotropy. The colour plot represents the rms value of the pointwise magnetization oscillation amplitude.(after Ref. [1]).

The determination of these resonance modes is important for the interpretation of the magnetization dynamics driven by external radio-frequency magnetic fields. At the same time, the analysis of normal oscillation modes around equilibrium yields insightful information for the description of thermal fluctuations. This is a crucial point in applications. Thermal fluctuations can by no means be neglected, because the small dimensions of nanomagnetic devices make the energy scale of the problem often comparable with that of thermal energy.
The second area of research has been that of synchronization of spin-transfer nano-oscillators subject to time-harmonic external microwave magnetic fields. If the frequency of the microwave source is sufficiently close to the natural frequency of the spin-transfer oscillator, synchronization can be obtained by very low microwave power. We were able to prove the existence of phase-locking between current-induced magnetization precession and the microwave field, and to reproduce the distortions in the signal spectral density (so-called frequency pulling effect) experimentally observed in these devices. A particularly interesting conclusion of our analysis is that phase-locking must exhibit hysteresis as a function of the spin-polarized current as soon as the microwave field amplitude exceeds a characteristic threshold (fig. 2).

Fig. 2 Analytically predicted phase-locking diagram for a spin-transfer nano-oscillator subjected to an external microwave magnetic field. The diagram is represented in the plane (Δω, h_a⊥) (normalized dimensionless quantities are used), where h_a⊥ is the amplitude of the external microwave field, while Δω is the so-called detuning, that is, the difference between the frequency of the microwave field and the natural frequency of the spin-transfer oscillator, the latter being proportional to the intensity of the injected spin-polarized DC current. Phase locking occurs in regions labeled by P, while Q identifies regions of unlocked oscillations. In regions P/Q, phase-locking depends on past history of external microwave field and DC current. The horizontal dotted line marks the microwave field threshold above which this hysteresis effect becomes detectable (after Ref. [2]).

We are currently investigating under which experimental conditions this effect might become observable.

[1] M. d'Aquino, C. Serpico, G. Miano, G. Bertotti: "Computation of Resonant Modes and Frequencies for Saturated Ferromagnetic Nanoparticles", IEEE Trans. Magn. Vol. 44, pp. 3141-3144, 2008.
[2] R. Bonin, G. Bertotti, C. Serpico, I. D. Mayergoyz, M. D'Aquino: "Analytical treatment of synchronization of spin-torque oscillators by microwave magnetic fields", Eur. Phys. J. B, in press.