Nonlinear NEMS

During the exceptional growth of interest in NEMS among research and engineering communities, it was quickly discovered that these tiny mechanical structures, among other remarkable characteristics, possess strong easily reachable nonlinear properties. As a consequence, a significant part of our work involves deeper and detailed understanding of the origins of that nonlinearity and the resulting limitations for sensing applications. However, it was shortly demonstrated that due to the ease of fabrication, high frequencies and great level of control of major parameters, nonlinear properties not only could be used to substantially improve the performance of key NEMS-enabled applications but also emerge as exceptional tools for studying complex nonlinear dynamics. Their small size, high frequencies and easily accessible controllable nonlinearities provide a physical regime where inherently rich dynamics can be studied within reasonably short timescales. fast as to miss critical, individual events in the time domain.

One important example of a useful nonlinear-enabled physical phenomenon is parametric amplification¹. Parametric resonance, unlike an ordinary driven resonance, is excited by time-varying modification of a system parameter. A classic example is a swinging gymnast who stands and squats twice during each period of oscillation—thereby modulating the resonator’s effective length—to sustain the oscillatory motion. An increase in amplitude and effective quality factors can be achieved if appropriate phase relation is coordinated between the driving and parametric pumping signals. This mechanism proves to be especially useful for NEMS since at the nanometer scale signal transduction is often limited by the noise floor of the preamplifiers employed as electronic readouts. A fundamental solution to this challenge is the direct amplification of a device’s signal in the mechanical domain by parametric phenomena, before any electronic amplification is employed. In our work we have demonstrated various implementations of mechanical parametric amplifier with surprisingly high linear gain along with substantial quality factor enhancement.

Figure 1. Nanomechanical parametric amplifier overcomes the difficult problem of communication to the large scale world by providing significant mechanical amplitude enhancement as well as combats dissipations by enhancing effective quality factor (inset).

In addition to enabling practical applications, parametric effects introduce an entirely new type of nonlinear system with its own nontrivial dynamical properties. Such intricate phenomena as wide hysteresis and different type of bifurcations can be observed and investigated in detail. The complexity of the system can be further increased when coupled nonlinear and parametric effects are combined.

Coupled mode NEMS

The interest in complex nonlinear behavior in a variety of systems has been prominent ever since Ed Lorenz’s discovery of enormous changes in the outcome caused by infinitesimal variations of the input parameters in a seemingly simple weather prediction model. It has long been believed that investigating systems of many nanomechanical resonators will significantly deepen our knowledge about coupled and nonlinear phenomena in general.

Even a system of two interacting nonlinear nanomechanical vibrating structures² despite its deceptive simplicity demonstrates inherently rich dynamics, including the tuning of the nonlinearity of one mode by the other, bistability and hysteresis as control parameters are tuned, the onset of spontaneous amplitude modulation oscillations, and the first strong evidence for deterministic chaos in nanoscale mechanical systems. In addition, the development of a quantitative theoretical model that describes the complex physical phenomena to high precision proves the ability to predict intricate nonlinear dynamics and engineer systems with the desired properties to demonstrate and exploit this behavior. Together, these show that nanomechanical structures provide an excellent tool to study nonlinear dynamics, and open up the potential to study the complex dynamics of large arrays of coupled nonlinear devices with excellent experimental control and theoretical understanding.

Figure 2. The system of two coupled nonlinear resonators with which the first deterministic chaos in nanomechanical systems was observed. The inset shows period doubling and chaotic behavior phase trajectories.

Personnel
Rassul Karabalin, Staff Scientist
Matt Matheny, Graduate Student
Guillermo Villanueva, PostDoc

References

1. R. B. Karabalin, X. L. Feng, M. L. Roukes, "Parametric Nanomechanical Amplification at Very High Frequencies", Nano Letters, to be published.
2. R. B. Karabalin, M. C. Cross, M. L. Roukes,"Nonlinear Dynamics and Chaos in Two Coupled Nanomechanical Resonators", Phys. Rev. B 79, 165309 (2009).