Dynamics of impurity attraction and repulsion of an intrinsic localized mode in a driven 1-D cantilever array

Because of the many applications of laser beam deflection measurement of a microcantilever, a number of studies have explored how the laser spot may change the mechanical properties of the cantilever^1–4 and affect the accuracy of its vibrational frequency measurement.^5,6 It is known that Joule heating, via both a temperature dependent Young's modulus and thermal expansion, shifts the vibrational frequency to the red for silicon and silicon nitride.^7,8 At low laser power, the expected blue shift from radiation-induced pressure change in the vibrational frequency is usually too small to be important.⁵ Recently, we have observed a visible laser-induced blue frequency shift for the vibrational frequency of SiN cantilevers, which may be the result of a surface stress generated by photo-excited carriers. This last experimental observation has opened up the opportunity to quantify a nonlinear dynamical effect that was reported some years back.^9,10

The earlier discovery that localized dynamical energy can be stabilized in a perfect nonlinear lattice by its discreteness had demonstrated that both extended and localized waves were necessary features of such systems.^11–13 One end result was that dynamically driven intrinsic localized modes (ILMs) provided a new window into the nonlinear dynamics structure of discrete systems.¹⁴ Our qualitative observation for a micromechanical array was that when heat from a focused IR laser was applied to a cantilever near a driven ILM on a 1-D lattice, the ILM would move to the next lattice site, away from the laser spot, if the lattice had hard nonlinearity (with increasing amplitude the frequency increases); while, if the lattice had soft nonlinearity (the converse), the ILM would hop toward the IR laser spot.^9,10,15 With sufficient laser power, the ILM would track the moving laser spot.¹⁵ The recent discovery of the visible laser-induced blue shift of the cantilever resonance in SiN now makes possible a detailed investigation of the dynamics behind the IR laser-induced pushing or visible laser pulling of the ILM in the same lattice.

Experimentally, a stationary ILM can be maintained in steady state using a driver to compensate for damping.^9,14,16,17 In this case, the ILM is frequency locked, and an auto-resonance (AR) state^18–23 is achieved, where the driver frequency controls the ILM amplitude. Previously, it has been demonstrated that a linear response measurement in the driven 1-D array is a powerful tool for studying bifurcation dynamics of an ILM.^24–26 

Here, this same technique is used to follow both the release and capture of an ILM by laser generated impurities. With application of the IR laser, the soft impurity produces a nonlinear coupling between the natural frequency (NF) of the ILM and a nearby even linear local mode (LLM)²⁷ associated with the ILM, giving rise to parametric amplification of both of these modes. Sufficient amplitude can be generated to induce translational ILM motion. When the IR laser spot is replaced by a visible one so that a hard impurity is generated the same mixing process again provides the necessary translational amplitude. The sign of the translation is controlled by the frequency of the impurity mode relative to the plane wave band. Our numerical simulations indicate that when it is a soft impurity mode, the center of gravity of the ILM moves away from the impurity, while the opposite occurs for a hard impurity mode.

Section II illustrates the experimental setup and describes the measurement procedure for the micromechanical array. Section III first presents the experimental results for the visible laser hardening of a single vibrating cantilever and then shows the linear response measurements for the driven nonlinear di-element lattice using a pump-probe method, while either an infrared or visible laser generates a soft or hard impurity, respectively. In Sec. IV, simulation results are compared with experiments. It is verified that the transition from stationary to moving ILM occurs when the NF and the even-LLM frequencies are symmetrically located about the driven ILM frequency. It is also demonstrated with a harmonic impurity-impurity model that the sign of the effect is consistent with the observed impurity-induced repulsion and attraction. The mechanism of cantilever hardening is discussed and the conclusions follow.

Figure 1(a) shows the experimental setup. The driven micromechanical array contains 152 di-element SiN cantilevers coupled together by the common overhang on a silicon wafer. Because of positive nonlinearity of the cantilever array, the ILM is generated above the top of the linear dispersion curve. A di-element array was used so that the highest frequency normal mode could be excited by the application of a uniform driver, which is a piezoelectric transducer (PZT). Because of the mass difference between short and long cantilevers, different forces are felt from the uniform acceleration. A cw pump oscillator feeds energy to the array maintaining the ILM in the large amplitude AR state. Generation and movements of ILMs were monitored by a camera [not shown in Fig. 1(a)]. By increasing the driver frequency above the top of the dispersion curve, ILMs can be generated. For the IR laser repulsion case, the IR laser beam is placed to one side of the ILM and then the IR laser power is increased until the ILM moves away from the laser spot. A visible laser at a low power level (0.06 mW) is used to monitor the motion of one cantilever at the other side of the IR laser spot as shown in Fig. 1(b). The driver frequency is fixed during this transition measurement process. For the visible laser attraction case, a higher power (1 mW) visible laser is used. Then, the laser beam is placed to one side of the ILM and the driver frequency decreased until the ILM is attracted to the laser beam. The reflection from the visible laser beam is used to monitor the motion of the cantilever.

For linear response measurements, an additional weak probe driver of variable frequency is combined with the strong pump so the perturbation is applied uniformly across the lattice. With an ILM present, the motion of a nearby single cantilever is monitored using the visible laser and a position sensitive detector (PSD, S3932 Hamamatsu Photonics). A lock-in amplifier selectively detects the vibration that is caused by the probe oscillating at a given frequency. A response spectrum is measured by scanning the probe frequency, while other conditions, such as the pump frequency and laser power are held fixed. Vibrational modes are recorded as either positive or negative peaks in the imaginary part of the response spectra.

The spectrum of linear local modes associated with a large amplitude ILM is now fairly well understood.^25,27 One mode is the NF of the ILM. This NF is for an ILM with the same vibrational amplitude pattern but without driver or damping. The driven ILM also gives rise to nearby LLMs.

With a fixed boundary condition, or by symmetry breaking by the AR state, all linear modes can be classified into odd and even symmetries in terms of the vibrating spatial pattern and appear alternatively when ordered by their mode frequency. For example, the ILM shown in Fig. 1(b) is odd and the LLM is even. Usually, our driving method couples to the odd mode vibrational pattern (… −0.4, 1, −0.4, …) that have a finite coupling to the uniform acceleration of the PZT with the di-element mass pattern (… m_a, m_b, m_a, …). Normally, even modes are not recorded in these spectra but if an impurity exists at one side of the ILM, the symmetry is broken and the even mode becomes observable in our experiment.

Since laser radiation is used to produce the impurity, we have checked the influence of the radiation on the lowest frequency mode response curve of a single cantilever made from the same SiN material as the array as a function of different laser powers. For the IR laser, the effect is simply to soften the linear spring constant so that the cantilever frequency decreases. The visible laser result for different laser powers is more interesting. Figure 2 shows both the linear and nonlinear response curves versus visible laser power. In both cases, the response curves are shifted upwards in frequency as the laser power increases. Since both curves shift the same way, we can conclude that the application of visible laser radiation increases the linear resonance frequency. This shift can be converted into an increase of a harmonic spring constant (assuming a ball and spring model.) From the equation Δk/k = 2Δf/f, the increment is estimated as 0.04%.

The display in Fig. 3(a) shows the initial condition for the repulsion measurement. First, the soft-focused IR laser beam at very low power level is placed to one side of the ILM. The probe laser is placed on the opposite side of the ILM with respect to the IR laser, as shown in Fig. 3(b). By changing the IR laser current in a stepwise fashion, the linear mode properties can be monitored as the hopping transition is approached. Measurement of the spectrum and step increment of the IR laser current are repeated alternatively till the ILM is repelled by the IR laser as shown in Fig. 3(c).

Figure 4 shows the typical cosine and sine linear response spectra versus probe frequency. The central peak is the large amplitude ILM vibration. The upper positive peak is the NF of the ILM. The NF is accompanied by its four-wave mixing partner NF^*, showing a negative peak in the sine spectrum on the low frequency side, at the same distance from the central peak. In addition to the NF and band modes, the even LLM is identified. The soft impurity is to one side of the ILM, where the even LLM signal is larger.

Linear response curves versus IR laser currents are presented in Fig. 5. With increasing laser current, the even LLM frequency approaches the NF^* peak. Just before the transition (1.00 A), the NF is pulled toward the even-LLM. After the transition takes place (1.02 A), the NF returns to the original position and the even-LLM returns to slightly above the band modes with very small peak height.

Figure 6 demonstrates the attractive interaction produced by the visible laser impurity. In this case, the IR laser is turned off and the power of the visible laser is increased to 1 mW. (Note the same laser was operated at 0.06 mW in the previous IR laser experiment.) Instead of increasing the laser power further, we change the pump driver frequency (ILM frequency) because the effect of the visible laser is so weak, the transition only takes place when the even-LLM frequency is placed at the symmetric position with respect to the NF peak. These modes change their frequencies as the driven ILM frequency is changed.

We obtain very similar spectra to those found for the IR repulsion case. Figure 7 presents the sine spectra for several driver frequencies. The even-LLM is made observable by the symmetry change induced by the visible laser near the ILM. Its frequency approaches the ILM frequency as the driver frequency is decreased. In this figure, the driver frequency decreases from bottom to top. The transition takes place when the even-LLM frequency is coincident with the NF^*. After the transition, the even-mode is no longer detected because the laser focus is now at the center of the ILM and lattice symmetry is recovered.

A lumped element model is used for the follow-on simulations with the equations of motion of the form

where i is the site number of the cantilever, m_i is the mass, τ is the relaxation time, k_2O,i and k_4O are harmonic and quartic onsite spring constant, $k2I(j)$ is the harmonic spring constant for the intersite connection up to j-th neighbor, and k_4I is the quartic spring constant for the intersite connection. The right hand side is the driving term. Here to match experiment α_pump = 1000 m/s² is the pump acceleration and Ω is the pump frequency. The second term is for the probe driver at frequency ω and acceleration amplitude α_probe = 0.01 m/s². The specific lattice parameters are listed in Table I. Those are determined by experimental observation and comparison with simulations. Numbers are the same as those in Ref. 9. Fixed boundary conditions are used and the total number of cantilevers is 100.

Figure 8 summarizes the repulsion transition as determined from simulations. A harmonic impurity located to one side of the ILM repels it when the impurity ratio $k′2O,i/k2O,i$ exceeds a threshold, where k_2O,i and $k′2O,i$ are the onsite linear spring constant for the regular and impurity sites, respectively. When the cantilever nonlinearity is positive, $k′2O,i/k2O,i<1$ is required to repel the ILM. For this particular case, the threshold is 0.74. The harmonic spring constant is changed as a function of time as shown in Fig. 8(a). Figure 8(b) presents the energy density plot versus time and shows the displaced ILM after the spring constant ratio reaches the threshold. The energy difference between sites 48 and 50 is displayed in Fig. 8(c) with the magnified density plot around the transition. Clearly, the transition is due to the amplification of lateral oscillation with time, not by a smooth rectified translational motion of the ILM.

For the attractive transition by the hard impurity, the threshold of the hard impurity is $k′2O,i/k2O,i=1.0165$ for the same driving condition as the soft impurity case. The impurity spring constant is increased with time as shown in Fig. 9(a). The hard impurity is located at site 47, the same site as for the soft impurity case, but now the ILM is attracted to it as shown in Fig. 9(b). The energy difference between sites 48 and 50 again oscillates before the transition as shown in Fig. 9(c). This signature indicates that the transition mechanism is the same as for the soft impurity. Note that the energy difference is mostly negative in this case, while it is mostly positive in Fig. 8(c). This observation implies that the center of the ILM vibration is slightly shifted towards the impurity in Fig. 9(c), while it is shifted away from the impurity in Fig. 8(c).

To find the movement of the linear modes in the spectrum, the probe response spectra are calculated the same way as with experiments. The uniform probe driver is used. Displacement at next short cantilever site of the ILM is analyzed by multiplying the response by $sin ωt$ or $cos ωt$⁠, like a lock-in amplifier, and then time-averaging for sine and cosine responses. Two sets of simulations with opposite probe phase are made, and then subtracted to eliminate the large amplitude ILM vibration from consideration. The resultant sine and cosine spectra are very similar to the experiments shown in Fig. 4. Only the sine components are shown in Fig. 10 for various impurities, from 1.00 (pure) to 0.75 (before the bifurcation). As the bifurcation point is approached, NF^* and the even LLM move close to each other, and both peaks are enhanced. The simulation and experimental results are in good agreement, demonstrating that the hopping transition is due to the softening of the cantilever resonance frequency by the IR laser. (A decrease in the Young's modules of silicon nitride with heating is reported by Bruls et al.²⁸)

The repulsion is triggered by the interaction between the NF and the even LLM. The enhancement of the response peaks in Fig. 10 can be explained as follows. The signal at the upper sideband peak of the NF generates four-wave mixing signal at the NF^* frequency. If the NF^* coalesces with the even-LLM mode frequency, the four-wave mixing signal of the NF resonates with the even-LLM. This signal is four-wave mixed again and returns to the NF frequency. The total loop gain can be far larger than without the coalescence. If the loop gain overcomes the loss, the coupled NF and even-LLM grow without any driving source. Thus, the translational vibration of the ILM diverges. This scenario is consistent with Figs. 8(c) and 9(c), where the lateral motion is growing with time.

Clearly, the even-LLM is a key element. The difference frequency between the driven ILM and the LLM is the ILM lateral vibration frequency. There could be two ways to approach this lateral transition of the ILM: (1) continuously soften the lateral vibration frequency so that the final translation is described by a frozen vibration of the lateral motion, or (2) a diverging amplitude of the lateral motion at a fixed frequency that overcomes the pinning barrier. The second case is observed in our experiments. When decreasing the driver frequency, the even-LLM frequency continuously decreases but the transition takes place when the even LLM coalesces with NF*, before the even-LLM completely softens. In contrast for the first possibility, the even-LLM peak would pass through the NF* and continuously approach the central ILM peak at the transition.

The hopping mechanism for the visible attraction of the ILM is the same as for the IR laser repulsion case, e.g., generation of the translational vibration at the symmetric frequency condition of the two linear modes. An experimental detail is that because of the very weak hardening of the spring constant, the translation can be seen only by scanning the driver frequency. At the symmetric even-LLM and NF conditions, the transition takes place even if the hardening effect is very small. In our case, the effect is so small that the translational motion happens only by controlling the ILM frequency to tune the even-LLM and NF to satisfy the nonlinear mixing condition f_NF + f_even–LLM = 2f_ILM. In simulations, we have succeeded in reproducing the translational oscillation plus the parametric effect with the impurity level $k′2O,i=1.0165k2O,i$⁠, while the experimental value is much smaller: $k′2O,i=1.0004k2O,i$⁠. This difference is most likely due to our approximate simulation model.

It should be emphasized these hopping transitions appear similar to the lower bifurcation transition of the AR region without the impurity when the pump frequency is used as a control parameter.²⁵ In that case, NF^* coalesces and interacts with the top band mode at the transition.²⁵ In our present picture, this would correspond to the NF and the band mode frequencies being symmetrically placed with respect to the ILM frequency. So both for intrinsic and impurity cases, the transition is controlled by the NF and the other linear mode of interest being symmetrically placed about the ILM frequency. They both satisfy $fNF+f′=2fILM$⁠, where $f′$ is the frequency of the even-LLM or the band mode. It is a parametric process that drives the auto oscillation mechanism in both cases.

To examine the difference between repulsion and attraction of the ILM, we tested a linear harmonic lattice eigenvector calculation by constructing an ILM-like impurity mode using inter-site spring constant changes. The impurity spring constants were estimated from the ILM eigenvector for a pure case. In this way, the increment of the spring constant from the regular intersite spring $k2I(1)$ is proportional to the square of the difference displacement (x_i – x_i–1)², where x_i is the eigenvector. The particular set of values used for simulations are Δk_2I,i = k_ILM(…, 0.003, 0.017, 0.104, 0.368, 1.00, 1.00, 0.368, 0.104, 0.017, 0.003,…) where the ILM center is between two 1.00 s, and a parameter adjusting the ILM effect $kILM=0.2k2I(1)$ is used. The intersite springs are changed from those in Eq. (1) by using this parameter set so $k2I(1)→k2I(1)+Δk2I,i$⁠. The onsite impurity, which is either soft or hard, is placed at the next short cantilever site just as before, by changing the onsite spring constant at that particular site. The final linearized equation for calculating the eigenvector is

The eigenvectors are then calculation in the usual way with the ILM-like mode the highest frequency one.

Figure 11 shows the result of calculation for (a) the hard impurity, (b) no on site impurity, and (c) soft impurity. A symmetric eigenvector is obtained for the pure case as expected. For the hard impurity case shown in (a), the ILM-impurity mode has a larger amplitude at the impurity site and a smaller amplitude on the opposite side, indicating that the ILM center gravity is shifted towards the impurity. On the other hand, for the soft impurity case in (c), the mode has smaller amplitude at the impurity site and larger amplitude at the opposite side, indicating that the ILM center gravity has moved away from the impurity. These results are consistent with the experiment and also with the nonlinear simulations.

Here, we have used relatively stronger impurities: $k′2O,i/k2O,i=1.2$ for the hard case and $k′2O,i/k2O,i=0.6$ for the soft case to see the effects more clearly. In the real system, the nonlinear effect on intersite springs should change as the eigenvector changes, so it is expected that the effect of the impurities on the eigenvector shown in Fig. 11 will be smaller than for the nonlinear simulations, because the intersite impurity spring constants are fixed at the pure ILM value. Thus, the direction of motion is the same in this linear calculation and the nonlinearity will only increase the movement.

Here, we review the possible mechanisms that can increase the cantilever vibration frequency under visible laser radiation. One is due to a light force.⁵ But we find that for our situation, the estimated spring constant increase by this mechanism is very small ∼1 × 10⁻⁹% at P = 1.4 mW, too small to be measured with our setup.

Photostriction has been reported in silicon,²⁹ while expansion was observed in germanium.³⁰ Cantilevers made from silicon have been studied experimentally¹ and theoretically² for gas or chemical sensors, which bend cantilevers under light irradiation. Shrinkage or expansion can be determined by the sign of the pressure dependence of the band gap. A positive sign of the pressure dependence ∂E/∂p means positive strain under photo irradiation. Unfortunately, there is no report on the pressure dependence of the band gap of silicon nitride.

On the other hand, it has been shown theoretically and experimentally that a total surface stress of upper and lower surfaces of a cantilever can modify the resonance frequency, while the difference surface stress between these sides gives rise the bending.^31–34 To explain our positive frequency shift by this effect, a negative total surface stress would be expected. If the pressure dependence of the band gap is negative, the blue shift by visible laser irradiation can be explained.

Another possibility, and the one we favor, is stress generation by photo-excited excess carriers. The visible laser (658 nm, 1.9 eV) has larger photon energy than that of the IR laser (808 nm, 1.54 eV). The band gap of the silicon nitride is from 1.8 to 5.1 eV, depending on stoichiometry. Since the visible laser causes hardening, while IR laser causes softening, this difference is most likely due to photo-excitation of electrons and holes in the nitride cantilever.

No doubt if we used a much higher power visible laser, there would be sufficient heat generated that the temperature effect would overcome the photo-carrier effect. In this case, the visible laser would repel the ILM. Since the photon energy of the IR laser is smaller than the band gap of the SiN, the radiation may not be absorbed strongly by the cantilever; however, the power of the IR laser was much larger than for the visible case. In addition, the silicon substrate also absorbs the IR radiation because its band gap 1.1 eV is smaller than the photon energy. Since cantilevers and overhangs rest on the silicon substrate, and the IR laser is softly focused, the heat absorbed by the silicon may warm up the local cantilevers.

We have found that visible laser radiation hardens the silicon nitride cantilever, most likely due to photo-excited excess carriers, so that its impurity mode attracts the ILM. The attractive hopping transition of the AR-ILM by the visible laser and the repulsive transition induced by the IR-laser observed in the 1-D micromechanical cantilever array have been examined experimentally and numerically. At the transition points for both kinds of impurities, the NF and the even-LLM are symmetrically positioned with respect to the ILM frequency so that the transition occurs when f_NF + f_even–LLM = 2f_ILM. This signature condition produces parametric excitation of the NF and even-LLM and drives the lateral motion of the ILM, and this amplified lateral motion causes the translational hopping transition. Interestingly numerical simulations with a harmonic model containing impurities correctly give the sign of the observed interaction but our nonlinear simulations demonstrate that the magnitude of the effect is enhanced in a nonlinear lattice. Finally, casting these results in a larger framework, the laser-induced impurity and its impurity mode simply provide a way to connect the ILM to the external world so that its intrinsic translational dynamics can be explored. Should a different technique be applied to tickle a driven ILM in a general 1-D nonlinear lattice with onsite and intersite coupling, to explore its translational transition, then the intrinsic NF and even-LLM excitations described here would still play the same defining dynamical role.

M.S. was supported by JSPS-Grant-in-Aid for Scientific Research No. 25400394. A.J.S. was supported by Grant No. NSF-DMR-0906491. M.S. thanks Dr. M. Kimura at University of Shiga Prefecture for fruitful discussions. A.J.S. acknowledges the hospitality of San Diego State University and Kanazawa University, where some of this work was completed.