Measurements

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Figure 4.5: Difference ∆(i, j) ≡ (Ω_i,j−Ω^meas_i,j )/2π between the measured frequency spectrum in Figure 4.4 and the ideal model Ωij = Ω11

p(i²+j²)/2, where Ω11is the measured value. The trend is roughly consistent with a toy model in which the membrane is allowed to be rectangular with one dimension≈0.077% larger than the other (model shown in light gray).

angularity, in which case the model dispersion relation becomes Ωi,j = 2π s

T 2ρ

i d_x

² +_j

d_y

² . From this model we infer an effective rectangularity of|1−d_y/d_x| ≈8×10⁻⁴.

In the upper frame of Figure 4.4, we hone in on the fundamental drum mode of the membrane.

An effective bandwidth of B = 200 mHz is used and three measurements are averaged. We have found it difficult to obtain a good linear least-squares fit of this peak to a Lorentzian because the sub- Hz-wide feature tends to drift by more than a linewidth during the coarse of a single measurement (taking ∆t = 2/B = 10 seconds). In the figure we show a fit to a Lorentzian whose linewidth is obtained using a ringdown measurement, as described in the next section.

4.6.2 Ringdown Measurement of Q

_m

An example of a ringdown measurement is shown in Figure 4.6. This measurement was made with the same membrane, optical probe, and etalon configuration described in the previous section. For ringdown measurements, however, the photodetector signal is directed to the input of the HP 4395A network analyzer (Section 4.5.4.2).

The ringdown procedure begins by exciting the membrane. To do this, the network analyzer source is connected to the piezo beneath the etalon mirror and the source frequency is centered at the mechanical frequency, Ωs ≈ Ωm. The resulting vibration of the piezo resonantly excites the membrane to an energyW0>> kBTroom. After a steady-state amplitude is reached (in practice this

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residuals (x1000)

time (seconds)

�_m/2�= 0.82351 MHz

�_m = 532 ± 1 ms

Q_m =1.377e+06 ± 2e+03

Figure 4.6: Example of a ringdown measurement for the (1,1) mode of the {dm, wm} = {50 nm,500µm} membrane characterized in Figure 4.4. Here we use a piezo shaker to excite the fundamental mode of the membrane to an energyW0>> kBTroom. After switching off the resonant drive, we monitor the amplitude of the oscillator decay. The amplitude inferred from the power re- flected from the etalon incorporating the membrane, producing a photodetector signalV(t). VΩ_m(t), the slowly varying amplitude ofV at carrier frequency Ωm/2π= 823 kHz, is monitored using a net- work analyzer in zero span mode. The measurement is fit to a decaying exponential (black curve).

Amplitude residuals are shown in pink. A value of Qm≈1.38×10⁶ is obtained from the formula Qm= Ωm/2τm, whereτmis the amplitude e-folding time inferred from the fit.

involves patiently tuning Ω_sto track slow drift in Ω_m), we abruptly disconnect the network analyzer source from the piezo. We monitorVΩ_m(t), the slowly decaying envelope of the oscillation at carrier frequency Ωm, using the network analyzer in “zero-span” mode, as described in Section 4.5.4.2. To obtain a sufficiently sampled decay curve, we choose an IF bandwidth 2πBIF ∼100×Ωm/Qm, with typical values between∼100− −1000Hz for the measurements discussed here and in later sections.

A ringdown measurement of the 823 kHz fundamental mode corresponding to the thermal noise peak in Figure 4.4 is shown in Figure 4.6. We use an IF bandwidth ofBIF = 300 Hz. The decaying magnitude of V_Ωm(t) is fit by linear least-squares to an exponential. Residuals of this fit (defined as the difference between the measured and fitted values) are shown in the lower, pink trace, and are less than 1% of the starting value of the decay curve. From the fit, we infer a quality factor ofQ_m= 1.380±0.005×10⁶, where the error bar is computed from the chi-square value of the fit.

The repeatability of this measurement, however, is closer to 10%, for reasons we don’t understand.

The noise in the measurement at long times is due to Brownian motion of the membrane. For this measurement, the initial amplitude of the membrane is ∼100 times larger than thermally driven amplitude, corresponding to an initial amplitude of ∼1 nm. Because the starting energy for the

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Quality fa ctor

Figure 4.7: Set of ringdown measurements for all membrane modes (i, j) such that Ωij/2π <4.6 MHz, using the same membrane with dimensions {dm, wm} = {50 nm,500µm} as in Figure 4.4.

The reduced mechanical quality for the fundamental mode relative to Figure 4.4 is due to a slight modification of the chip mounting. For higher order membrane modes, the value of Qm is roughly constant at∼4×10⁶, corresponding to a monotonically increasingQ×f product (f = Ω/2π) which surpasses the critical value of 6×10¹² Hz (shaded gray).

ringdown is substantially larger than the thermal energy, it is worth comparing the result to a more careful measurement of the linewidth of the thermal response. This is done in the next section.

4.6.2.1 Q-factors of Higher-Harmonics — A Single Trial

In Figure 4.7, we have carried out the ringdown measurement for all internal modes of a{d_m, w_m}= {50 nm,0.5 mm}Norcada membrane from 800 kHz to 4.6 MHz, using the etalon configuration shown in the upper left inset of Figure 4.3. We separate vibrational resonances of the piezo and membrane chip (some of which have quality factors in excess of 10⁴) from vibrations of the membrane by examining the thermal noise spectrum, as shown in the lower plot. Remarkably, for higher harmonics the Q×f product exceeds 1.5×10¹³ > kBTroom/h (several years ago this was believed to be unique among NEM/MEMs oscillators at room temperature [14]). Roughly a factor of two variation in quality factor is observed between higher order drum modes. We have yet to systematically investigate the role of mode order in determining of mechanical Q_m, but note that a recent study

has been performed at Cornell in which it was found that different mode shapes may experience different support-related clamping losses [58]. What we have found is that the result shown in Figure 4.7 varies widely depending on the details of the membrane window and membrane chip geometry, as well as the way that the chip is attached to the etalon. A large collection of measurements of the sort shown in Figure 4.7 was taken in order to sort out this behavior phenomenologically. We discuss this study in the next section.