resonances (such as those in Fig. 6.1 and Fig. 6.2) have been obtained. While the piezoresistive signal mixing cannot be implemented in simple network analysis mode, here the synchronized ‘bias’ source is generating RF bias which is constantly Δω away from the ‘drive’ frequency. One uniqueness of this scheme is that it is to date the first combination of the magnetomotive excitation, which is particularly powerful for VHF/UHF/microwave NEMS resonators [25], and the piezoresistive detection, which is very convenient and effective for high-impedance (kΩ) devices. We choose the down- conversion intermediate frequency Δω to be high enough to satisfy two considerations: (i) Δω is much larger than the resonance bandwidth, i.e., Δω>>ω0/Q, so that when the
‘drive’ source is sweeping the frequency in a wide band, only the ‘drive’, but not the
‘bias’, hit the device resonance frequency, thus only obtaining one resonance peak for each device; and (ii) Δω is in the frequency range for a convenient RF lock-in detection.
215.0 215.2 215.4
0.0
0.1
0.2
Fig. 6.5 shows the measured piezoresistive down-conversion signal from the 215MHz Si NW device by using an intermediate frequency Δω=20MHz. As shown in Fig. 6.5 inset, the piezoresistive down-conversion signal amplitude has a linear dependence on bias, confirming the piezoresistive effect as described by eq. (6-2).
6.6 Si NW Mechanical Properties Measured in Resonant Mode
Thus far, we have demonstrated that the Si NWs grown across the microtrenches are robust VHF nanomechanical resonators, and that they are also versatile devices in terms of electrical attributes and transduction schemes, either being matched to RF/microwave standard (50Ω) with metallization, or as high-impedance but heavily-doped semiconductor piezoresistors. Since the Si NWs are from single-crystal epitaxial growth, they have pretty high Q’s as bare piezoresistive devices, and also decently high Q’s as metallized devices. All these attributes suggest interesting applications for Si NWs.
An immediate application of the demonstrated Si NW resonators and techniques is towards the study of the basic properties of VLS epitaxial Si NWs. For instance, the measurement of the Young’s modulus of these Si NWs can be made in the resonance mode, instead of using AFM to perform the tedious static bending experiments. While bending Si NWs with AFM tips appears to be straightforward for measuring the strength and elastic modulus (Young’s modulus), the displacement and strain in the Si NWs are usually inferred from the AFM tip’s movement and AFM cantilever’s deflection, and are thus carried out with possibly large error bars. For the fundamental flexural mode of non-metallized doubly-clamped Si NW resonators, the fundamental resonance frequency is
( )
2 20 0.8913
2 4 . 22
L E d A
I E
f = L Y = Y ⋅
ρ ρ
π , (6-3a)
where EY is the Young’s modulus of Si NW, ρ is mass density (2330kg/m3), I=πd4/64 is the momentum of inertia and A=πd2/4 is the cross section of the Si NW, respectively, with d the NW diameter. For metallized Si NWs, the mass loading effect of the metallization should be considered and the resonance frequency is
m m Y
m m m
m L
d f E
⋅ +
⋅
= 2
,
0 0.8913
ρ , (6-3b)
where m is the structural Si NW mass, and mm is the mass of the metallization layers.
The resonance mode measurements have the advantage that the resonance frequency can be measured very precisely and the Young’s modulus can be determined based on eq. (6- 3) even without knowing the details of the displacement and strain of the device. The accuracy will partially rely on the accurate measurement of the Si NW dimensions, which is also essential for the AFM contact mode bending method. Considering that the AFM tip size is usually much larger than the diameter of a Si NW, and that AFM approaching and pushing Si NW at some sweet spot is trial-and-luck based, and difficult and time- consuming, determination of the mechanical rigidity and elastic properties by reliably measuring resonance mode frequency is a valuable technique.
Table 6-2 presents the Young’s modulus extracted from the accurate measurements of the resonance frequencies of the devices, based on eq. (6-3). The Si NWs’ diameters are read out from high-resolution SEM imaging with an error bar of ~2nm, and the length error is within ~2%. The mass density ρ is well-known for single-crystal Si and does not change from bulk Si to Si NWs, as the single-crystal nature of the Si NWs has been verified with STM [21]. Hence the accurate measurement of resonance frequency leads to reliable determination of Young’s modulus.
As shown in Table 6-2, the Si NW Young’s modulus values obtained in this study are very close to those of bulk single crystal Si (111), ~160−200GPa, as widely accepted and taken in literature; and are in very good agreement with a recent elaborate AFM bending study [22]. We note that the Si NWs in this AFM study [22] and the present resonance measurements have been produced with similar processes in the same system in the same lab, but in different batches. There is a noticeable difference in the measured Young’s modulus between the above data and 93−250GPa from another recent AFM study [23].
The difference may be ascribed to such subtleties in Si NWs as defects, and detailed differences in the two kinds of AFM bending experiments and their measurements accuracy and reliabilities [22,23].
Table 6-2 Young’s modulus measured by dynamic method with the resonances of the Si NW resonators, as compared to static measurements with AFM bending experiments (we note that the widely used Young’s modulus of bulk Si (111) is in the range of 160−200GPa).
Experiments and Samples Device Diameter
d (nm) Si NW Length
L (μm) Young’s Modulus EY (GPa)
SiNW-215 81 1.69 170 ± 15
SiNW-80 74 2.77 205 ± 18
SiNW-M-200 142 2.25 187 ± 16
SiNW-M-188 118 2.1 200 ± 17
AFM Bending-Cantilever [22] 120 8 ≈186
AFM Bending-Beam [22] 190 12 ≈207
AFM Bending-Cantilever [23] 140 10 ≈93
AFM Bending-Beam [23] 200 10 ≈150
AFM Bending-Beam [23] 200 10 ≈250
6.7 Frequency Stability and Mass Sensitivity of Si NW Resonators
The Si NWs’ attributes also make them interesting for nanosensor applications. We perform initial experimental investigations of the frequency stability and sensitivity of these Si NWs. We embed the Si NWs resonators into low-noise phase-locked loop (PLL) circuitry for real-time resonance frequency tracking to measure the instantaneous frequency fluctuations, and thus the frequency stability characteristics as a function of averaging time [27,28]. Shown in Fig. 6.6 is the measured real-time frequency
fluctuation with 1 sec averaging time, and the instantaneous frequency fluctuation is about 0.182ppm, for the metallized 200MHz Si NW resonator. In the real-time nanomechanical mass sensing paradigm [27,32], using this device as a mass sensor with its mass responsivity of 1.74Hz/zg, its frequency stability level translates into a mass resolution of 21g (1zg=10-21g). The Si NWs’ frequency stability and mass sensitivity performance are again comparable to the performances of some of the best top-down mass sensors made of single-crystal high-quality SiC. In particular, the 21zg noise floor is almost exactly the same as that achieved in the 100-zeptogram real-time mass loading steps monitored by a 190MHz SiC resonator [27]. The measured results of the frequency stability data and corresponding mass sensitivity for several other Si NWs devices are also collected in Table 6-2.
0 2000 4000 6000 8000 10000 -0.6
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8
<δf 0/f 0> τ=1s, PPM
Time (sec)
0.182PPM
Fig. 6.6 Si NW resonator frequency stability characteristics. Fractional frequency fluctuation of the 200MHz metalized Si NW resonator, measured in real time with an averaging time of τ≈1sec for each readout of the frequency data tracked by the phase-locked loop. The measured noise floor is 0.182PPM, which combined with the device mass responsivity, 1.74Hz/zg, leads to a resonant mass resolution of 21zg.
1 10 100 1000 10-7
10-6
Measured Frequency Stability 200MHz Si NW Resonator
Allan Deviation
Averaging Time (sec)
Fig. 6.7 Measured frequency stability performance, the Allan deviation as a function of averaging time, for the 200MHz Si NW resonator. The optimized Allan deviation achieved is 7.6×10-8 for around 5~10secs averaging time.
6.8 Quality Factor and Dissipation Issue of Si NW Resonators
Just like for the top-down SiC NEMS resonators, the quality factor and dissipation issue in the Si NW resonators are also of great interest and importance. We have conducted careful measurements of the Q’s and hence the dissipation in these VHF Si NWs. Our measurements and analyses follow the logic and approaches we have employed in studying the dissipation of top-down UHF SiC resonators in Chapter 5. Fig. 6.8 shows the measured dissipation as a function of device temperature for both the 188MHz and 200MHz metalized Si NW devices. The weak power law dependency of dissipation on temperature is again clearly visible. The dashed line in Fig. 6.8 shows the Q-1~T0.3 fit to guide the eyes—the Q-1~T0.3 dependency has also been identified in top-down Si MEMS and NEMS resonators (see Chapter 5). Here we see that this dependency fits the experimental data of 200MHz device very well, and fits that of the 188MHz device fairly well in the low-T range, while showing a visible deviation for T>60K.
20 40 60 80 100 120 10-4
10-3
188MHz Si NW 200MHz Si NW T0.3 Fit
Dissipation Q-1
Temperature (K)
Fig. 6.8 Measured dissipation Q-1 as a function of temperature for two metalized Si NW resonators (200MHz and 188MHz devices). The dash line is the Q-1∝T0.3 weak power-law dependency, as has been found in top-down Si MEMS and NEMS resonators.
The almost constant difference (offset) between the two traces in Fig. 6.8 is ascribed to the difference in clamping loss in these two devices, similar to the case of the SiC devices in Chapter 5. In these VHF Si NW resonators, the measurements also manifest that the clamping losses are the most significant and dominating dissipation mechanism. In brief, this is because of the energy radiation (loss) from the resonant mode of the vibrating Si NW to its supporting pads at the two self-welded ends. There is an interesting subtlety worth mentioning here: Although for these as-grown suspended Si NWs there is no etching undercut of the supporting pads, which is inevitable in top-down NEMS resonators, the self-welded clamping joints at the facing microtrench walls are usually fatter than the Si NW itself; this may have an effect resembling that of the etch undercut in the top-down devices and hence comprise the clamping losses.