LOAC driveDC
3.1 Electrostatic tuning 1
resonators has been irreversibly tuned in situ by metal deposition [57]. Nanomechanical resonators have also been reversibly tuned by bending the substrate [58] and changing the clamping conditions with a scanning tunneling microscope [59].
In our experiments, we have used two frequency tuning techniques: electrostatic tuning using a side gate; and reversible gas-adsorption tuning at low temperatures. They are described in more detail in the following sections.
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DC Gate Voltage (V)
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(b)
4(a)
Figure 3.1: (a) Scanning electron micrograph of the SiC doubly clamped beam used in the tuning experiments. The red arrow indicates the direction of motion for the out-of-plane resonance. (b) Frequency of the fundamental out-of-plane mode versus applied gate voltage. The inset shows the mechanical losses versus the tuning voltage.
resonance frequency. Second, the dependence of the electrostatic attraction on the distance between the beam and the gate can change the effective restoring force for the beam, which can tune the frequency down. The detailed theory of the capacitive tuning in this system was developed by Inna Kozinsky [46,60], and here I will only discuss qualitative aspects of the theory.
These two mechanisms affect the out-of-plane and in-plane modes differently. The out-of-plane mode is primarily affected by the stretching of the beam. The spatial dependence of the attractive force does not affect its frequency because the distance between the beam and the gate is nearly unchanged during out-of-plane oscillation. As a result, the frequency of the out-of-plane mode increases with increasing tuning gate voltage. Figure 3.1(b) shows that the frequency of the out- of-plane mode depends on the voltage applied to the tuning gate. The inset shows that the tuning does not noticeably affect the mechanical quality factor.
For the in-plane mode, the spatial dependence of the attractive force between the beam and the gate is the dominant effect. As a result, the frequency of the beam decreases with increasing gate voltage. At high applied voltages, the gate voltage also significantly reduces the quality factor of the resonance. This is because the changing distance between the beam and the gate modulates the capacitance between them and therefore the charge accumulated on the beam. The modulation of charge induces additional currents in the beam that produce a Lorentz force that provides additional
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DC Gate Voltage (V)
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% F re q u e n c y T u n in g
(a) (b)
Figure 3.2: Same as Figure 3.1 for the fundamental in-plane mode. The blue curve in (b) is the prediction of the theoretical model for the capacitive frequency tuning.
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Output voltage, µV
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Output voltage, µV
Frequency, MHz
(a) (b)
Figure 3.3: Resonance curves of the in-plane mechanical mode for a tuning voltage of zero (a) and 28 V (b). Both quadratures of the signal are shown. The amplitude of the resonance peak is smaller in (b) because of the reduced quality factor.
damping. At the highest applied tuning voltages, the quality factor was less than half the quality factor without any tuning voltage. The broadband electrical noise also depended on the applied voltage, increasing by approximately a factor of five as the tuning voltage increased from 0 V to 28 V (Figure 3.3). Although the reasons for the increase in the broadband noise are not entirely clear at this point, they are most likely related to the parasitic leakage currents between the the gate and the beam at high tuning voltages.
The maximum change in frequency we have observed with capacitive tuning is approximately +4% for the out-of-plane mode and−6% for the in-plane mode. These changes in frequency occurred
when DC voltage of approximately 28 V was applied to the tuning gate. Gate voltages higher than 35 V typically destroyed similar devices either because of snap-in of the beam into the tuning electrode or because of dielectric breakdown. Thus, electrostatic tuning can produce significant changes in resonance frequencies of a doubly clamped beam but has some important limitations. In particular, the in-plane mode suffers from a decrease in the quality factor, and both modes suffer from an increase in broadband electrical noise at high tuning voltages.
The capacitive tuning also becomes less effective for resonators of higher frequencies, as these resonators are stiffer and need larger forces to change the spring constant by the same percentage.
To illustrate this, we have theoretically modeled the response of a beam that is three times shorter than the one described above, i. e., 5µm long, and has all other dimensions the same. According to these calculations, the resonance frequency of the in-plane mode would change from 47.12 MHz at zero tuning voltage to 47.02 MHz at 30 V tuning voltage—a tuning range of only 0.2%. This tuning range can be increased by proportionally reducing the gap between the beam and the electrode.
For example, if we reduce the gap from 300 nm to 100 nm, the same 5-µm long device will tune from 47.12 MHz to 45.18 MHz—a tuning range of 4%, which is comparable to the tuning range demonstrated above. Scaling capacitive tuning to even shorter beams and higher frequencies is problematic, however, as it would require electrode gaps of less than 100 nm, which would be difficult to fabricate using the present e-beam lithography techniques.