Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays

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 (ILMs) 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.

Figure 1

Mass-and-spring nonlinear lattice model. (a) A and B represent two different masses per unit cell, each with nearest-neighbor intersite and onsite restoring force springs. Both types of springs possess a harmonic and hard quartic potential energy. (b) Dispersion curve (solid) for a mono-element lattice for the harmonic limit with dimensionless parameters $m=1$, $k_{2O}=1$, $k_{2I}=1$ in Eq. (1). The frequency gap below the $k=0$ mode is produced by the onsite energy term. Dotted curves: Linear dispersion relation for a di-element harmonic lattice with parameters $m_A=1.1$, $k_{2A}=1.1$, $m_B=0.9$, $k_{2B}=0.9$. Superimposed on the graph are two examples showing the real space shapes of nonlinear ILM eigenfunctions. These functions were calculated for the mono-element lattice such that they have the same dynamical energy and the same nonlinearity placed at one of two possible lattice locations. With only onsite nonlinearity a broad eigenmode appears at $\omega ∕{\omega }_0=2.26$, while with only intersite nonlinearity a sharp eigenmode mode appears at $\omega ∕{\omega }_0=2.89$.

Figure 2

Fabrication of a micromechanical array with hard nonlinearity. (a) Silicon nitride deposition of 300 nm thickness on a silicon substrate, (b) definition of cantilever geometry by photolithography and ${\mathrm{CF}}_4$ plasma etching of the nitride layer, and (c) KOH wet etching of the silicon substrate producing the cantilever and the overhang. The length, pitch, and overhang of a di-element array are all roughly the same, on the order of $50\mu \mathrm{m}$. (d) Microscope image of the top view of a finished sample. Bright gray, bare silicon nitride thin film; dark gray, film over silicon substrate.

Figure 3

Fabrication of a micromechanical array with soft nonlinearity. (a) Electron microscope image of a gold-coated mono-element cantilever lattice so that a dc voltage can be applied. The applied bias is used to modify the sign of the lattice nonlinearity. (b) Potential energy of a cantilever for different bias voltages. (c) Frequency-dependent amplitude vs voltage showing both hard and soft nonlinearity for the uniform mode of a mono-element array with the gold electrode. Increasing (solid) and then decreasing (dotted) the driving frequency gives rise to hysteresis in the amplitude response. For 0 V the potential is hard, for 40 V the potential is soft. At 20 V the potential is nearly harmonic. Parameters are listed in Table , column 3, except now $\alpha =200\mathrm{m}∕{\mathrm{s}}^2$.

Figure 4

A simulation of ILM production, dynamics, and decay for a di-element array with hard anharmonicity. Cantilever simulation parameters are listed in Table , column 1. (a) Driver frequency as a function of time for this simulation. The driver is on from $t=0$ to 5000 periods. The driver frequency starts just below the optic mode frequency $f_4$, increases up to $1.027f_4$ linearly until 2500 periods, and then remains at that frequency. The amplitudes of the ILMs increase with time until the end of the chirp. The magnitude of the acceleration is kept constant, at $\alpha =1.4\times {10}^4\mathrm{m}∕{\mathrm{s}}^2$. (b) A density plot of the vibrational energy distribution vs time. Dark regions identify large amplitude localized modes. During the time that the cw driver is on five locked stationary ILMs appear. ILM hopping is identified in the left oval marker. Once the driver is turned off, the ILM amplitude decreases and as its width increases it becomes unpinned from a particular lattice site. ILMs are seen to interact and repel one another. (c) Magnified image of the ILM hopping event identified in the oval in (b). The resting position is always on a short lattice site (horizontal dotted lines). Other observations: When the ILM amplitude is small, $t<2500$ periods, the lattice pinning effect is weak and ILM collisions permit them to hop among stable sites. When the ILM amplitude becomes large the pinning effect is strong, and an ILM oscillates about a particular trapping site.

Figure 5

Evolution of ILM energy vs time in both real and reciprocal space. (a) Spatial distribution of vibrational energy density vs time in real space. The simulation parameters are the same as for Fig. 4. (b)–(g) Time-space Fourier transform of the displacement for a number of time windows. The time windows for these frames are indicated by the letters $b$ through $g$ above frame (a). The ILM time development during the initial chirped driver are shown in frames (b) and (c), frame (c) is particularly interesting since it shows nonhorizontal straight lines associated with running ILMs as well as excitations in the top of the optic branch. The straight line slope gives the ILM velocity. (d),(e) The horizontal line associated with the locked state, and after the cw driver is turned off the decaying ILM state is shown in (f) and (g).

Figure 6

Experimental setup for observing ILMs. (a) A low power laser beam is used to probe the motion of the cantilevers. A cylindrical lens produces a line focus on the tips of the cantilevers. The reflected image is directed by a lens (not shown) onto a 1D CCD camera. The image from a highly excited cantilever will increasingly miss the CCD producing a dark spot. (b) Complete optical arrangement for three different kinds of measurements including the linear frequency response of the system, the ILM observing setup shown in (a), and the production of an optically induced impurity mode. The sample in vacuum is driven by a piezoelectric transducer (PZT). A HeNe beam is focused onto a cantilever, reflected from it, and directed to a position sensitive detector to measure the linear vibrational response. A fiber pigtailed infrared laser diode is used to locally heat a few elements and produce a movable, gap impurity mode in the cantilever array. (c) Schematic representation of the three kinds of laser images on a di-element cantilever surface.

Figure 7

Measured frequency spectrum produced by the linear normal modes of a di-element cantilever array. OFF: The response spectra of one cantilever for (a) opticlike and (b) acousticlike bands are shown. ON: The spectra for a local laser-heated case described in Sec. 5 is also shown (shifted up for clarity). Here a linear impurity mode is created below the optic branch, as indicated by the arrow. After 88.

Figure 8

Cantilever excitation vs time showing the production, interaction, and decay of ILMs. These experimental results for (a)–(c) 248-element and (d)–(f) 152-element arrays are taken with the 1D CCD camera. (a) The PZT driver has a fixed frequency at the top of the optic branch. Running modes are produced. (b),(c) Identical starting conditions. The PZT frequency is now chirped from 0.9986 to 1.016 times the optic branch frequency. (d) The driver frequency is again fixed at the top of the optic branch (136.3 kHz). In (e) and (f) the driver frequency is chirped from 1.000 to 1.011 times the optic branch frequency. (e),(f) Identical starting conditions. The chirp ends at 16.2 ms for (b) and (c), 14.2 ms for (e) and (f). The dark regions identify localized excitations. The lighter regions correspond to times when the driver is on. Some localized excitations become trapped during the cw phase. The driver duration is 48.9 ms for (a)–(c) and 72.7 ms for (d)–(f).

Figure 9

Summary of the impurity mode-ILM interaction results obtained from different simulations. (a) Linear system. A local mode (LM) appears above or below the plane-wave band states depending on the sign of the linear onsite spring constant change. (b) For the nonlinear array with hard nonlinearity a stationary ILM is created above the band states. When the impurity mode is either below the band states or above the ILM frequency, the ILM is repelled, as represented by the shaded regions. When the impurity mode is above the band states but below the ILM frequency, the ILM is attracted. (c) For the nonlinear array with soft anharmonicity the ILM appears below the band states. The conditions for the ILM repulsion or attraction interaction are now inverted with respect to case (b).

Figure 10

Laser manipulation of an ILM in an array with hard anharmonicity. (a) Experimental observation of a laser-induced ILM hopping event. Periodic horizontal white lines are images of stationary cantilevers. The dark ILM location is centered at site 73 and the white laser spot is located at site 76. The laser spot is placed nearby a locked ILM and then the IR laser power is gradually increased. When an ILM hops away from the laser spot, it is accompanied by wave-packet emission, as labeled by $A$ and $B$. Reflected waves from the edges are seen to the right of $C$ and the left of $D$. The canted white lines are guides to eye of the traveling waves. The speed of these waves are $16.3\times {10}^3,12.8\times {10}^3,16.2\times {10}^3$, and $12.3\times {10}^3(p∕\sec )$ for $A,B,C$, and $D$, respectively, where $p$ is the cantilever pitch. An ILM is stable at shorter cantilever sites (odd numbered sites in this figure). When one hops onto such a site its lateral momentum produces an oscillation that decays with time. (b) Successive manipulation of the ILM. The images are constructed from a number of chronologically ordered pictures. The two arrows at sites 105 and 78 indicate the initial locations of the laser spot and the ILM, respectively. The darkest stripe in each frame corresponds to a highly excited ILM. The white rectangles identify the heating laser spot, which produces a linear impurity mode. As the laser spot approaches, the ILM is repelled and hops away. When the laser spot is far removed, the ILM remains fixed as shown in the frames around 10, 40, and 52. When the laser power is low, it can pass through the ILM without interaction (around frame 23). At the last frame, the ILM was pushed against an impurity site near the end of the array, and disappeared.

Figure 11

Observed laser manipulation of an ILM in a soft nonlinear lattice. Demonstration of an attractive interaction for a wide linear bandwidth, soft nonlinear mono-element array. The two arrows identify the initial location of the ILM and the laser spot. As the laser spot approaches an ILM, it is attracted and captured by the impurity mode, see frame 5. As the laser spot moves, the ILM moves with it. When the laser is turned off, the ILM stays in a fixed position (frames 10, 18, and 21). Due to the larger bandwidth of the lattice, the size of an ILM is wider than in the case of the hard nonlinear lattice case. Driving the soft mono-element wide bandwidth array also generates running wave packets, which appear as the noisy line structure throughout the figure. After 89.

Figure 12

Simulated time-dependent manipulation of an ILM by an optically induced impurity mode. One linear impurity is placed at lattice site 49 and its onsite spring constant $k_{2O}$ is changed to a defect spring constant with the impurity force constant ratio changing with time as shown in the lower trace of each panel. The energy density of the resulting ILM in the lattice is the upper feature in each panel. Different kinds of ILM manipulation are then demonstrated: (a) Repulsion with a low-frequency impurity mode, (b) repulsion with a high-frequency impurity mode, (c) seeding an ILM using a high-frequency impurity mode, (d) destruction of an ILM using a high-frequency impurity mode, and (e) capturing or tweezing an ILM with a low-frequency impurity mode. Parameters are listed in Table , column 2.

Figure 13

Plots of the linear impurity mode frequency as a function of the impurity force constant ratio for a di-element array. (a),(b) The lattice under consideration has a hard nonlinearity. Gray area; the top and bottom of the upper opticlike band. The dotted horizontal line at 139 kHz corresponds to the cw driver frequency used in Fig. 12. Solid circles identify the linear impurity mode frequencies where an ILM attractive interaction (tweezing) is observed. Open circles identify the linear impurity mode frequencies where an ILM repulsive interaction is observed. An ILM seeding effect can be produced when the impurity force constant ratio is varied in the range from 1.14 to 1.20 with the impurity strength changed adiabatically. (c),(d) The lattice under consideration has a soft nonlinearity. Open circles and closed circles are linear impurity frequencies for the repulsive and tweezing effects, respectively. Unlike the hard nonlinear case, the repulsive interaction frequency goes between the band edge and the ILM frequency as shown in (d). Open squares, described in the text, identify the more precise nonlinear impurity mode frequencies associated with ILM repulsion.

Figure 14

Simulated time-dependent manipulation of an ILM in a soft nonlinear lattice with different impurity modes. These impurity modes are made by changing the onsite linear spring constants over 11 lattice sites, as described under Eq. (9). The central position of the 11 sites vs time is identified by the straight dotted line. (a) The seeding effect. Bottom trace: impurity force constant ratio vs time. The linear impurity frequency is resonant with the driver when the force constant ratio is 0.9. The impurity mode is fixed at $\text{site}=100$. (b) The repulsive interaction with the high-frequency impurity mode. The impurity force constant ratio is 1.1, and it is moved from the lower left corner to the upper right corner. The ILM is repelled. It reappears at the bottom, due to the periodic boundary condition. (c) The tweezing effect by a low-frequency impurity mode. The impurity force constant ratio is 0.99. (d) The repulsive interaction produced by a lower frequency impurity mode. The impurity force constant ratio is 0.90. Parameters are listed in Table , column 3.