Introduction
Introduction and Motivation
A possible solution discussed in this thesis is to amplify the signal in the mechanical field using the idea of parametric oscillations before applying an electrical transducer. A subsequent part of the thesis presents a first step towards large-scale integration by carefully investigating a large variety of physical phenomena in arrays of interacting nanomechanical resonators.
Overview
The result is shown in figure 3.3(b); the blue straight line is a theoretical prediction using the geometric parameters of the fabricated device. The maximum heights of the peaks (at Vin(DC)=0) depend on the signal amplitude Vin(AC).
Parametric Effects Theory
Elastic Beam
For the particular device shown in Figure 2.1, the fundamental resonant mode corresponds to out-of-plane bending vibration at a frequency of 35 MHz. The first important correction to the simple model just described is the effect caused by the longitudinal tensile stress in the beam.
Parametric Amplification (Mathieu Equation)
Another important effect is caused by the initial stress experienced by beams that arise either from fabrication or are externally applied. We now rewrite the equation of motion (2.5) in a simplified form, which, in the first pass, ignores the Duffing nonlinearity.
Gain and Q Enhancement
The greatest improvement is obtained when there is π/4 phase difference between the drive and the pump, while at –π/4 the amplitude is weakened. After replacing the expression for A* from the first of the above equations with the second, we obtain a differential equation for the amplitude A.
Gain-Dynamic Range Tradeoff
The operation of a parametric amplifier is different from that of a linear amplifier, as described above. In the above derivation of the parametric amplifier gain, equation (2.18) implies that the gain diverges as the pump amplitude approaches the threshold value.
Bistability and Parametric Tongue
The inset shows the dependence of the frequency shift on the amplitude of the second peak. The graph shows the dependence of the modulation frequency on the driving frequency of the first mode.
Lorentz Force VHF Parametric Amplifier
Lorentz Force Parametric Amplifier Design
By applying a periodic force to the outer edges of the pump beams, we effectively change the signal beam's tension, thus changing its spring constant. Rpump is the ohmic resistance of the pump jet electrode (typically 50-100 Ω) and VDC is a voltage drop across it.
Measurements Results
As shown in figure 3.3(a), the beam resonance frequency is tuned down when DC bias current is passed through the pump beams, regardless of the bias current polarization. Peak narrowing as well as an increase in peak height was observed (see figure 3.5) due to the linear part of the frequency tuning.
Heating Model
Then the frequency decreases with the amplitude of the second peak, shown by the red arrow.
D-NEMS Parametric Amplifier
D-NEMS Piezoelectric Frequency Tuning
Parametric Gain and Q Enhancement
Parametric Amplification and Frequency Stability
Parametric Actuation
Simple linear interaction leads to complex nonlinear terms in the equations of motion of the system.
Coupling Phenomena in D-NEMS
Variety of Coupling Mechanisms in D-NEMS
The nature of the coupling and its strength strongly depends on the architecture of the nanomechanical resonator. To begin with the fabrication of the equipment discussed in the previous chapter involves etching the wet suspension. When one resonator is in motion, the strain causes a voltage to develop in the depletion region of the resonator, and this voltage in turn affects the other resonator.
Preliminary calculations show that in most cases the piezo repulsion has a negligible effect due to either the large capacitance of the wire bond electrode or the intentional isolation of the resonator's top electrodes.
Strong Coupling Model
If the two resonators are equal to δ=0, the first mode is perfectly symmetrical with both beams vibrating in phase with the same amplitude at the angular frequency ωI=ωm, and the second mode is antisymmetric when the movement of the resonators is not in phase and its frequency. is higher ωII2=ωm2+2D. In this case, in the first mode, the resonators vibrate approximately in phase, although the first beam (lower frequency) has a large amplitude, while the second mode is mainly an out-of-phase motion, while the second beam in this case has a larger amplitude. The system of equations for the modes as two independent resonators in the form similar to (5.3) is obtained by the dediagonalization procedure (explained in detail in [43]).
It turns out that the mass matrix M′ is also diagonal due to its proportionality to the unity matrix.
Strong Elastic Coupling Through Supports
The first mode is symmetric and therefore expected to be easily detectable in experiment. As predicted, only two ways are apparent, and the first way is much more obvious than the other. As a result, nine of the ten modes in Figure 5.7 were observed, the missing mode probably due to extreme activation and detection suppression.
In asymmetric mode, two adjacent beams vibrate out of phase, so there is a common beam motion.
Frequency Bridge
This query gives the equations of motion for the amplitudes of modes AI and AII:. The inset in graph (b) shows the dependence of the frequency shift on the amplitude of the second mode shift. When the second resonator starts to oscillate, the summed response of the two resonators follows the upper branch.
Using the geometry of the D-NEMS resonators used in the experiment, the electrostatic force.
Coupling Of Two Duffing Resonators
Theoretical Analysis
The amplitudes AI and AII are the scaled equivalent displacements of the modes of vibration. The time derivatives of the displacements above have a form similar to equation (2.11), while the cubic terms are more complicated. After substituting the last three expressions in equation (6.1), O(ε1/2) terms cancel, because they are linear resonator terms.
The numerical solutions of the system of equations (6.9) were obtained with a computer program written in Mathematica.
Coupling Parameters of the System
It also means that the non-linear coefficient for a specific region in the input power and frequency of the second mode vanishes. A network analyzer is used to monitor the spectral response of the first mode while a separate function generator drives the second mode. The spectral response of the first mode is shown in black while the corresponding second mode amplitude is shown as the red curve.
The results of the modulation frequency measurements performed on the first device in Table 6.1 are shown in Figure 6.11(b).
Nondemolition Measurements and Linearizaiton
Spectral Responses
If the drive level of the first mode is increased to an extent comparable to that of the second mode, a variety of different effects are predicted and measured. This phenomenon is caused by the fact that when the amplitude of the first mode is small, its effective nonlinearity is negative, so it jumps to the higher state as the frequency goes up. For the last three graphs, the second mode vibration state is unstable and undergoes upward and downward transitions during the sweeps.
In the absence of the second coupled mode, the response is a regular Duffing curve, but when the second mode is excited at a high level, the spectral responses take on more complicated shapes, shown in figure 6.8.
Mechanical State Instabilities and Chaos
The peaks are not symmetrical, which is consistent with an anharmonic form of modulation. Under specific experimental conditions, this effect induces chaotic behavior in the two modes of the nanoelectromechanical system. This balance is changed in favor of the lower branch for Vin(DC)>0 and in favor of the upper branch for Vin(DC)<0, as shown in Fig. 7.2(d) for Vin(DC)=-3mV.
The inset shows the spectrum analyzer measurement of the non-degenerate parametric peak width.
Bifurcation Topology Amplifier
Parametric Actuation of Two Weakly Coupled Resonators
The phase of the response, which is not plotted, is determined only within a π phase. The situation changes when a non-zero coupling is introduced between the resonators, after which the roots of the coupled equations (7.5) are calculated numerically. Finite coupling D≠0 changes the topology of the bifurcation diagram, causing a distortion of the perfect pitchfork (the bifurcation would be imperfect).
This dependence of the bifurcation topology on the effective coupling D between the resonators can be used as an amplification scheme by setting D to be proportional to the input signal Vin that we wish to amplify.
Voltage-Dependent Bifurcation Diagram
After demodulation with the reference of the form VLcos(ω(t)t+ϕL), the expression for the down-converted signal takes the form. It is equal to the average of the V2(t) over the frequency sweep period with an appropriate numerical coefficient. The behavior of the amplitudes resembles that of a single parametric resonator in degenerate mode.
This phenomenon must be considered in combination with an investigation of the relative frequency shift between the two modes.
Bifurcation Topology Amplifier Measurements
Bifurcation Topology Amplifier Characterization
Nondegenerate Parametric Effect
Parametric Actuation of Two Strongly Coupled Resonators
The voltage between the top and bottom electrodes adjusts the resonant frequency of the resonator. The ratio of the two slopes in each plot must be the square ratio of the components of the eigenvectors, which is. The effects of the conventional parametric terms hp11 and hp22 for two independent resonators have been discussed in detail in previous chapters.
These equations are invariant under the transformation aI → aI eiφ, aII → aII e-iφ, so only the sum of the phases φI+ φII is defined.
Nondegenerate Parametric Resonance Measurements
Several long frequency sweeps were performed while monitoring the vibration amplitude of the main and harmonic peaks. In the non-degenerate parametric case, the magnitudes of the harmonic peaks can grow as high as the height of one of the modes (Figure 8.4(b)), up to 100 μV. Furthermore, careful analysis of the plot in Fig. 8.6(b) reveals some other spike-like structures occurring during the sweep, although not observed in the amplitude dependence plot.
The generation of satellite peaks opens another dispersive channel in the motion dynamics of an individual mode.
Harmonics in Nondegenerate Parametric Resonance
Frequency Shifts and Satellites
Therefore, we conclude that around 15.8 MHz in the long parametric frequency sweep the quality factor of the first mode increases relative to the one of the second mode. This observation corresponds to a sign of change in the growth of the amplitudes of the modes, with the height of the first mode peak increasing while the second one decreases. It suggests an explanation for the outcome, but not for the origin of the underlying physical phenomenon.
Another important topic that is thoroughly investigated in this thesis is the behavior of the interacting systems of nanomechanical resonators.
Conclusions And Future Directions
Final Thoughts
