It has been known for some time that nonlinearity and discreteness play important roles in many branches of condensed-matter physics as evidenced by the appearance of domain walls, kinks, and solitons. A recent discovery is that localized dynamical energy in a perfect nonlinear lattice can be stabilized by the lattice discreteness. Intrinsic localized modes ͑ILMs͒ are the resulting signature. Their energy profiles resemble those of localized modes at defects in a harmonic lattice but, like solitons, they can move. ILMs have been observed in macroscopic arrays as diverse as coupled Josephson junctions, optical waveguides, two-dimensional nonlinear photonic crystals, and micromechanical cantilevers. Such dynamically driven localized modes are providing a new window into the underlying simplicity of nonequilibrium dynamics. This Colloquium surveys the studies of ILMs in micromechanical cantilever arrays. Because of the ease of sample fabrication with silicon technology and the intuitive optical visualization techniques, these experiments are able to demonstrate the general nature and properties of dynamical energy localization while at the same time providing new information on ILM generation, locking, pinning, and interaction with impurities.

The nonlinear vibrational properties of a periodic micromechanical oscillator array have been measured. For sufficiently large amplitude of the driver, the optic mode of the di-element cantilever array becomes unstable and breaks up into excitations ranging over only a few cells. A driver-induced locking effect is observed to eternalize some of these intrinsic localized modes so that their amplitudes become fixed and the modes become spatially pinned. DOI: 10.1103/PhysRevLett.90.044102 PACS numbers: 05.45.-a, 05.45.Xt, 63.20.Pw, 85.85.+j Intrinsic localized modes (ILMs) which extend over only a few lattice sites of a periodic nonlinear lattice have been examined theoretically in a variety of systems [1][2][3][4]. Although the smallest excitations would be in a quantum lattice [5], there are still interesting issues in classical models to be explored experimentally such as mobility [6], discrete lattice potential [7], and interaction among ILMs [8], as well as the long-time behavior of a driver-induced modulational instability for a dissipative system. Thus the exploration of amplitude-dependent features is one of the important experimental issues at the present time.Although some experimental studies have been reported for large scale mechanical systems [9], for somewhat smaller Josephson-junction arrays [10,11] and optical waveguides [12,13], and for nanoscale lattices [14 -16] none of these systems are, as yet, appropriate for studying the detailed motion of large numbers of ILMs in the presence of a driver. Recently, microelectro-mechanical system (MEMS) silicon technology has matured sufficiently [17] so that we can make a few hundred coupled cantilever oscillators [18]. In this Letter we describe our experimental investigation of ILM creation and locking for a cantilever array in the presence of damping, disorder, and a driver.Our cantilever array is produced from a photoresist mask over a silicon nitride layer on a silicon substrate. This is exposed and then etched via a CF 4 plasma in a reactive ion chamber. The silicon substrate is undercut using an anisotropic KOH etch, thus releasing the silicon nitride cantilevers. A 3D rendition of one unit cell of the resulting array is shown in Fig. 1(a). Such cantilevers have a hard nonlinearity. To achieve the large amplitude uniform mode instability required to produce ILMs one needs to drive the highest frequency uniform mode of the system [19]. In order to accomplish this when using a piezoelectric transducer (PZT) driver, two different length cantilevers per unit cell have been constructed as shown in Fig. 1(a). With this di-element array the dispersion curve is folded over so that the highest frequency vibrational mode is now at the zone center. The room temperature quality factor of this mode is about 8000.Figure 1(b) shows the experimental setup for measuring ILM dynamics and mobility versus time. A cylindrical lens is used to focus a He-Ne laser line image on the static array. The reflected beam is then incident on a 1-D CCD camera. The PZT is driven...

In a strongly nonlinear discrete system, the spatial size of an excitation can become comparable to, and influenced by, the lattice spacing. Such intrinsic localized modes (ILMs)--also called 'discrete breathers' or 'lattice solitons'--are responsible for energy localization in the dynamics of discrete nonlinear lattices. Their energy profiles resemble those of localized modes of defects in a harmonic lattice but, like solitons, they can move (although, unlike solitons, some energy is exchanged during collisions between them). The manipulation of these localized energy 'hotspots' has been achieved in systems as diverse as annular arrays of coupled Josephson junctions, optical waveguide arrays, two-dimensional nonlinear photonic crystals and micromechanical cantilever arrays. There is also some evidence for the existence of localized excitations in atomic lattices, although individual ILMs have yet to be identified. Here we report the observation of countable localized excitations in an antiferromagnetic spin lattice by means of a nonlinear spectroscopic technique. This detection capability permits the properties of individual ILMs to be probed; the disappearance of each ILM registers as a step in the time-dependent signal, with the surprising result that the energy staircase of ILM excitations is uniquely defined.

Intrinsic localized modes ͑ILMs͒ have been observed in micromechanical cantilever arrays, and their creation, locking, interaction, and relaxation dynamics in the presence of a driver have been studied. The micromechanical array is fabricated in a 300 nm thick silicon-nitride film on a silicon substrate, and consists of up to 248 cantilevers of two alternating lengths. To observe the ILMs in this experimental system a line-shaped laser beam is focused on the 1D cantilever array, and the reflected beam is captured with a fast charge coupled device camera. The array is driven near its highest frequency mode with a piezoelectric transducer. Numerical simulations of the nonlinear Klein-Gordon lattice have been carried out to assist with the detailed interpretation of the experimental results. These include pinning and locking of the ILMs when the driver is on, collisions between ILMs, low frequency excitation modes of the locked ILMs and their relaxation behavior after the driver is turned off. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1540771͔An advance of the theory of nonlinear excitations in discrete lattices was the discovery that some localized vibrations in perfectly periodic but nonintegrable lattices could be stabilized by lattice discreteness. The modulational instability of extended large amplitude vibrational modes has been proposed as a mechanism for the realization of dynamical localization on the scale of the lattice constant. Although theoretically a variety of methods to excite the instability of a homogeneous vibrational mode have been proposed, these ideas have yet to be tested experimentally. Since the observation of nanoscale localized vibrational modes still cannot be achieved there is definite advantage to examining a macroscopic array, which is small enough so that the entire time dependence of the instability dynamics occurs in a practical measurement interval. This has been accomplished by using micromechanical silicon technology to fabricate up to 248 identical cantilevers with a 40 micron lattice constant. Optical techniques have been used to track the motion of individual cantilevers in the presence of an inertial driver. In addition to experimentally characterizing the modulational instability and identifying the best method for producing intrinsic localized modes a new discovery is the locking of the local mode amplitude with the driver frequency. Numerical simulations have been used to better understand the nature of this synchronization effect.

Optically-induced real-time impurity modes are used to shepherd intrinsic localized vibrational modes (discrete breathers) along micromechanical arrays via either attractive or replulsive interactions. Adding an electrode to the cantilever array provides control of the sign of lattice anharmonicity, hence allowing both hard and soft nonlinearities to be studied. A number of dynamical effects are demonstrated and explained, including the optical tweezing of localized vibrational energy in a nonlinear lattice. PACS

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An indirect method of producing intrinsic localized modes ͑ILM's͒ is to drive an unstable uniform mode to large amplitude. In this investigation the intimate connection between the shape-dependent demagnetization factor, the modulation instability, and ILM behavior is established. The stability of the antiferromagnetic resonance ͑AFMR͒ against breakup into such ILM's is shown to depend strongly on the frequency difference between the linear AFMR and the long-wavelength limit of the spin-wave dispersion curve. This difference, which stems from the long-range dipole-dipole interactions, depends on the demagnetization factor and hence the sample shape. Here it is demonstrated initially with linear perturbation analysis and later with moleculardynamics simulations that the instability characteristics and ILM production properties depend strongly on the sign and magnitude of the frequency difference. The simulations show that when the AFMR frequency is coincident with the spin-wave band, a spatially coherent train of ILM's appears, but this coherence is lost at long times. When the AFMR frequency is inside the spin-wave band, the Suhl instability populates the degenerate spin waves, whose subsequent modulational instability leads to traveling and decelerating ILM's. When the AFMR frequency is below the spin-wave band, a large-amplitude threshold for the instability appears. Above this threshold ILM's and localized dynamic spin flops occur, confined to specific lattice regions.