Chapter 5: Coupling Phenomena in D-NEMS
5.3 Strong Elastic Coupling Through Supports
It turns out that mass matrix M′ is also diagonal due to its proportionality to a unity matrix. The damping matrix Γ′ has small off-diagonal terms although they are a factor of 60 smaller than the diagonal ones and can thus be ignored.
beams 0.9μm apart is defined. The resonators are chosen to be identical because this is a strong coupling case when D>>δ. The ledge length is chosen to be 1μm. A static analysis is used to estimate the strength of the elastic coupling. A large uniform force is applied to the first beam causing it to move by x1. As a result the stress propagates through the ledge so that the second beam is affected by the interaction force ξ(x2-x1) and displaces by x2. The displacement color map is shown in figure 5.4.
Figure 5.4 Finite element simulations of coupling through ledge. The force applied to one of the beams exerts the interaction force on the other one. Calculated displacements are used to estimate the coupling constant.
The estimated interaction spring constant using this method is ξ=1.40 N/m, (regular beam stiffness constant is k=28 N/m). In addition, a modal analysis is performed using the same finite element analysis package. Symmetric and antisymmetric modes were found with resonance frequencies of 13.8 and 14.5 MHz respectively. Using equation (5.6) we estimate the coupling constant using this method ξ=1.38 N/m. The results indicate good agreement, which means finite
element analysis is a proper method to investigate elastic coupling phenomena in mechanical devices.
Modal analysis for three strongly coupled resonators reveals three modes; their shapes are shown in figure 5.5. The first mode is symmetric, and therefore it is expected to be easily detectable in the experiment. On the other hand the second asymmetric mode, with two beams vibrating out of phase while the middle beam rests, is heavily suppressed by both actuation and detection, similar to the simpler two-resonator case. Therefore it is not likely to be detectable in the measurements. The third mode is not fully asymmetric, and therefore we may be able to detect a small resonance peak associated with this mode.
Figure 5.5 Mode shapes of a system of three strongly coupled resonators, calculated by finite element analysis.
Three strongly coupled 6-micron-long beams were fabricated and measured. As was predicted, only two modes are visible, and the first mode is much more evident than the other. Figure 5.6 shows the SEM image of the 3-beam system and its spectral response. The first symmetric mode has to be driven well above the onset of nonlinearity in order to make the third asymmetric mode barely visible.
Figure 5.6(a) SEM image of three strongly coupled D- NEMS resonators (b) the spectral response with two modes visible
Piezoelectric actuation and optical detection turn out to be a remarkably powerful method for characterizing the coupled modes of arrays of nanoresonators. A great demonstration of the coupled mode phenomena is the observation of a variety of modes in an even larger number of resonators. To show this we fabricated and analyzed an array of ten nanomechanical resonators.
Finite element analysis was employed to calculate the shapes of the ten coupled modes. SEM image and calculated mode shapes are shown in figure 5.7. The result of modal analysis shows a set of ten modes that represent an analog the standing wave patterns with progressively decreasing wavelength.
Figure 5.7 An SEM image of ten 5μm long resonators and its coupled mode shapes calculated via finite element analysis.
In the measurement setup the laser beam spot size was deliberately increased so that all the resonators are in its area of illumination. The first mode with resonance frequency of around 22 MHz is very easily detectable, but the second mode is suppressed when the first one is optimized.
Therefore the laser spot was shifted so that the second mode is at its maximum optical detection efficiency.
After all the optical adjustments, the resonance scan over a wide frequency range was performed. As a result nine out of ten modes in figure 5.7 were observed, the missing mode possibly due to its extreme actuation and detection suppression. The amplitude of the drive was set to a high level so that all the modes are visible as small peaks. Hence the first three peaks are excited to very high amplitude, above the onset of nonlinearity. A wide spectral response is shown in figure 5.8. Each mode’s resonance peak was carefully examined and its frequency and quality
factor was measured reducing the drive for over-driven modes in the figure. The result is plotted in the inset in figure 5.8. The resonance frequency predictably steadily increases with the mode number while the quality factor starts at 1530 for the first two modes, then rises as high as 3040 for mode number 7, and then slightly drops for modes eight and nine. This observation is unusual in comparison with previous observations [44], in which the quality factor typically decreases as the mode number is increased
Figure 5.8 The spectral response of the system of ten strongly coupled D-NEMS resonators. Nine out of ten peaks were observed. The inset shows the dependence of resonance peak’s frequency and quality factor on mode number.
The observed phenomenon is an excellent demonstration of a well-known tuning-fork effect. In an asymmetric mode, two adjacent beams vibrate out of phase, and therefore the net motion of the
support between them is heavily suppressed, causing only a limited amount of energy to be dissipated as elastic waves into the bulk substrate. As a result the quality factors of asymmetric modes are larger than those for symmetric ones even though their frequencies are also higher. This phenomenon is promising for designing the architecture for arrays of large number of resonators.