**3. Experimental measurement**

Meaningful experimental measurements of scale-dependent elastic behavior from microdown to nano-scales have been shown to be one of the major challenges in the field of experimental micro/nanomechanics [24, 45]. As a consequence, a majority of research con‐ cerning the scale-dependent elastic response has focused on developing experimental methods to study the phenomena [24, 41, 42, 8, 32]. One of the most common approaches to measure *Ê* is based on the resonance response of the system. *Ê* is extracted via a simple Euler-Bernoulli beam equation from a resonance frequency response acquisition [6, 46]. The main advantages of this method, which make it popular and widely used, are 1) ease of use; besides measuring

**Figure 6.** (a) Optical microscope image of a cantilever with hidden anchor due to under etching of the base of the cantilever. (b) SEM of a hidden anchor. The cantilever was forcibly removed in order to see the hidden anchor. The inset shows the bonding surfaces of the SOI wafers were in the middle of the BOX layer. (c) Close-up view of bonding interface and silicon device layer.

the resonance frequency no additional modifications on the system are required, and 2) the method is very fast, while being applicable in various environments, including vacuum, gaseous and liquids. The method has some limitations, especially when being used at nano‐ scale. The experimental results include errors due to the uncertainty of the boundary condi‐ tions [30]. The non-ideal boundary conditions lead to a lower resonance frequency, which leads to lower estimates of *Ê*. Moreover, the resonance frequency depends on both the stiffness and the mass [39, 33]. It is therefore difficult to decouple solely by a resonance response the stiffness from the mass changes caused by surface contamination, native oxide and other adsorbed layers [33, 39, 34]. The smaller the nanosystem, the bigger this effect will be. In order to show the extent to which the effects of the extra mass due to contamination or native oxide are significant, we investigated the effects of surface contaminations on the resonant behavior of the cantilevers fabricated in Section 2. We have shown that the mass and the stiffening effects of the contaminations can cause significant shifts in the resonance frequency of the cantilevers, and, thus, the subsequent estimate of the effective stiffness.

**3. Experimental measurement**

cantilevers, (g) covered cantilevers and (h) double clamped beams, respectively.

162 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

pending cantilever structure.

Meaningful experimental measurements of scale-dependent elastic behavior from microdown to nano-scales have been shown to be one of the major challenges in the field of experimental micro/nanomechanics [24, 45]. As a consequence, a majority of research con‐ cerning the scale-dependent elastic response has focused on developing experimental methods to study the phenomena [24, 41, 42, 8, 32]. One of the most common approaches to measure *Ê* is based on the resonance response of the system. *Ê* is extracted via a simple Euler-Bernoulli beam equation from a resonance frequency response acquisition [6, 46]. The main advantages of this method, which make it popular and widely used, are 1) ease of use; besides measuring

**Figure 5.** SEM of (a) 1019 nm, (b) 340 nm, (c) 93 nm, (d) 57 nm, (e) 40 nm thick cantilevers. SEM of (f) paddle-shaped

**Figure 4.** Fabrication process of the cantilevers. (a) photoresist is patterned. (b) Using the photoresist as an etching mask, the silicon underneath is etched with SF6 plasma. (c) Residue of the resist is stripped in nitric acid and the device is submerged in HF solution to etch the oxide. The device is then put in a critical point drier (CPD) to release the sus‐

**Figure 7.** Reconstructed 3D image of (a) a cantilever which is approximately flat, (b) a curved double clamped beam and (c) a curled cantilever using white light interferometry. (d) curvature profile measurement of the cantilever in (c).

The resonance frequency of the cantilevers was measured with an optical laser detection setup equipped with a vacuum chamber [39]. The cantilevers are in resonance due to Brownian motion caused by thermal noise. The resonance frequency and its shift were obtained from the Lorentzian fit of the frequency-bandwidth curves [39]. If the surface of the cantilever is clean without any contamination, such as adsorbed water and gas molecules, then the theoretical effective Young's modulus of the cantilever can be calculated as [39]

$$
\hat{E} = \frac{0.0261 f\_0^2 L^2 \rho}{t} \tag{1}
$$

where *f0* is the resonance frequency, *L* is the length of the cantilever, *ρ* is the density of the cantilever material and *t* is the thickness, respectively. In order to determine the variations in *Ê* associated with the surface contamination, the resonance frequency of cantilevers with contaminations and with minimized contaminations were compared. Assuming that the contamination is distributed evenly along the cantilever, the variations in *Ê* can be calculated using (only for small variations)

$$
\Delta \hat{E} = 2 \frac{\hat{E}}{f\_0} \Delta f \tag{2}
$$

where *ΔÊ* is the variation in *Ê*, and *Δf* is the variation in the resonance frequency due to the contaminations. To show experimentally the effect of ambient contamination, we analyzed the resonance frequency shift of cantilevers by the following procedure (similar to the work in [39] for inaccuracies in frequency-based mass sensors). After fabrication of the cantilevers, their resonance frequencies, *fbv*, were measured in ambient air, to allow for the adsorption of differ‐ ent gases and water molecules and their saturations. At this stage, the masses and stiffness from surface contaminations are added to the newly fabricated cantilevers and are referred to as the added mass-stiffness effects. The cantilevers were then placed into a vacuum (10-6 mbar) for a sufficiently long time to allow for degassing and desorption at the surfaces of the cantilever (reduced added mass-stiffness effects). Further, the resonance frequencies were measured in a vacuum (*fv*). Then, the vacuum chamber was vented with ambient air and the resonance frequencies were measured immediately after the pressure reached atmospheric pressure (*fav*). The measurements were done within a few minutes in order to ensure minimal re-adsorption, whileprovidingthe same amount of airmass loadingcomparedto themeasurements inthe first step (*fbv*). This procedure was repeated for 1019, 340 and 93 nm thick cantilevers. A typical resonance frequency measurement with the procedure above is illustrated in Fig. 8 for a 93 nm thick, 8 µm wide and 8 µm long cantilever. A 20.9 KHz resonance frequency shift was ob‐ served between the resonance frequency before and after vacuum. For thicker cantilevers the resonance frequency shift was less, due to their lower surface-to-volume ratio and consequent‐ ly their being less sensitive. Table 1 shows the results of the resonance frequency measure‐ ments, their shifts and, consequently, the variations in *Ê* extracted from equation 2.

The resonance frequency of the cantilevers was measured with an optical laser detection setup equipped with a vacuum chamber [39]. The cantilevers are in resonance due to Brownian motion caused by thermal noise. The resonance frequency and its shift were obtained from the Lorentzian fit of the frequency-bandwidth curves [39]. If the surface of the cantilever is clean without any contamination, such as adsorbed water and gas molecules, then the theoretical

**Figure 7.** Reconstructed 3D image of (a) a cantilever which is approximately flat, (b) a curved double clamped beam and (c) a curled cantilever using white light interferometry. (d) curvature profile measurement of the cantilever in (c).

2 2

where *f0* is the resonance frequency, *L* is the length of the cantilever, *ρ* is the density of the cantilever material and *t* is the thickness, respectively. In order to determine the variations in *Ê* associated with the surface contamination, the resonance frequency of cantilevers with contaminations and with minimized contaminations were compared. Assuming that the contamination is distributed evenly along the cantilever, the variations in *Ê* can be calculated

> 0 <sup>ˆ</sup> <sup>ˆ</sup> <sup>2</sup> *<sup>E</sup> E f <sup>f</sup>*

r

= (1)

D= D (2)

<sup>0</sup> 0.0261 <sup>ˆ</sup> *f L <sup>E</sup> t*

effective Young's modulus of the cantilever can be calculated as [39]

164 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

using (only for small variations)

**Figure 8.** The influence of surface contamination on the resonance frequency of a 93 nm thick silicon cantilever.

The shift in frequencies is caused by the combination of both the additional adsorbed mass and the stiffening effect on the cantilever [39]. Therefore, the true stiffening effects of the contamination cannot be obtained by simple subtraction of the shifts in the resonance fre‐ quencies. The results from Table 1 clearly show that the experimental measurements of the scale-dependent *Ê* via the resonance frequency method lead to a considerable number of uncertainties.


**Table 1.** The resonance frequency of cantilevers in air and vacuum and the variations in extracted *Ê*. The resonance frequency data are in KHz.

Another popular method which is used to characterize the scale-dependence is the bending tests with the use of force spectroscopy in a scanning probe microscope (SPM) [11]. SPM has very high force and displacement resolution and sensitivity, yet it has significant uncertainties in its measurement and interpretations. Measuring the absolute deflection requires a pains‐ taking calibration of the deflection sensor, as well as the risk of tip slippage [24] and indentation of the probe tip [41]. Another important factor which degrades the accuracy of the method is the lack of knowledge in the precise force. In a typical SPM measurements, the force can only be obtained from a calibrated cantilever with known spring constant. The spring constant of an SPM cantilever usually deviates from its designed nominal value by a significant amount. The lack of an accurate spring constant calibration method, together with the above issues result in inaccuracies which can amount to more than 26% error [11].

In order to avoid the aforementioned issues in experimental investigation of the scaledependence, we implemented the recently developed measurement method, which uses the quasi-static electrostatic pull-in instability (EPI) phenomenon [32]. The distinctive advant‐ age of the EPI lies in its well-known sharp instability and the possibility of applying an SItraceable force along the length of the beam [32]. The details of the phenomenon can be found elsewhere [44]. Due to the quasi-static nature of the method, it does not suffer from massloading effects. The beauty of the methods lies also in its simplicity and ease of use. The required actuation voltage can be precisely measured using standard electrical test equip‐ ment and a microscope. Fig. 9 shows a schematic illustration of the EPI setup. It consists of a voltage source and a semiconductor testing probe station. The pull-in voltage was measured by slowly increasing the voltage difference between the cantilever and the substrate. Visual observation of the color changes and the sudden snap-in of the cantilever were used to determine the pull-in voltage.

Mechanics of Nanoelectromechanical Systems: Bridging the Gap Between Experiment and Theory http://dx.doi.org/10.5772/55440 167

The shift in frequencies is caused by the combination of both the additional adsorbed mass and the stiffening effect on the cantilever [39]. Therefore, the true stiffening effects of the contamination cannot be obtained by simple subtraction of the shifts in the resonance fre‐ quencies. The results from Table 1 clearly show that the experimental measurements of the scale-dependent *Ê* via the resonance frequency method lead to a considerable number of

166 Advances in Micro/Nano Electromechanical Systems and Fabrication Technologies

**L×w×t (µm3) fbv fv fav Δf =fav- fbv ΔÊ (GPa) ΔÊ/Ê (%)**

96 × 8 × 1.019 123.65 147.968 128.85 5.2 0.80 8.4

24 × 8 × 0.340 583.73 607.108 585.48 1.75 0.30 0.6

8 × 8 × 0.093 994.53 1090 973.6 -20.93 -3.90 -4.2

**Table 1.** The resonance frequency of cantilevers in air and vacuum and the variations in extracted *Ê*. The resonance

Another popular method which is used to characterize the scale-dependence is the bending tests with the use of force spectroscopy in a scanning probe microscope (SPM) [11]. SPM has very high force and displacement resolution and sensitivity, yet it has significant uncertainties in its measurement and interpretations. Measuring the absolute deflection requires a pains‐ taking calibration of the deflection sensor, as well as the risk of tip slippage [24] and indentation of the probe tip [41]. Another important factor which degrades the accuracy of the method is the lack of knowledge in the precise force. In a typical SPM measurements, the force can only be obtained from a calibrated cantilever with known spring constant. The spring constant of an SPM cantilever usually deviates from its designed nominal value by a significant amount. The lack of an accurate spring constant calibration method, together with the above issues

In order to avoid the aforementioned issues in experimental investigation of the scaledependence, we implemented the recently developed measurement method, which uses the quasi-static electrostatic pull-in instability (EPI) phenomenon [32]. The distinctive advant‐ age of the EPI lies in its well-known sharp instability and the possibility of applying an SItraceable force along the length of the beam [32]. The details of the phenomenon can be found elsewhere [44]. Due to the quasi-static nature of the method, it does not suffer from massloading effects. The beauty of the methods lies also in its simplicity and ease of use. The required actuation voltage can be precisely measured using standard electrical test equip‐ ment and a microscope. Fig. 9 shows a schematic illustration of the EPI setup. It consists of a voltage source and a semiconductor testing probe station. The pull-in voltage was measured by slowly increasing the voltage difference between the cantilever and the substrate. Visual observation of the color changes and the sudden snap-in of the cantilever were used to

result in inaccuracies which can amount to more than 26% error [11].

uncertainties.

frequency data are in KHz.

determine the pull-in voltage.

**Figure 9.** Schematic view of the measurement setup for EPI. The driving voltage on the cantilever is applied through a probe contact and the substrate is grounded. Reprinted with permission from [32]. Copyright 2010 American Institute of Physics.

For each cantilever, the pull-in voltage and geometry of various cantilevers with different lengths were measured. Fig. 10 shows the experimental measured pull-in voltages versus the length of cantilevers with different thicknesses. By solving the nonlinear differential equation of the electromechanical system, knowing the pull-in voltage and the geometry of the canti‐ lever, one can extract *Ê* for each thickness [32]. Fig. 11 shows the extracted *Ê* as a function of the cantilever thickness. Error bars were calculated according to [32]. The maximum calculated error due to uncertainties in the cantilever's geometry measurements and the measured voltage was 12% [32]. The main error contributions comes from the geometry, and specifically, the length. Results in Fig. 11 show a considerable scale-dependent behavior in the effective stiffness for bending, *Ê*.

Above 150 nm the effective modulus is constant and converges to the bulk value of [110] silicon elastic modulus. Somewhere near 150 nm it starts to decrease monotonically. It can be seen that at 40 nm, the *Ê* is about half of the bulk value. So the question to be answered is: what causes this scale-dependent behavior?

**Figure 11.** The scale-dependent *Ê* for bending. Reprinted with permission from [32]. Copyright 2010 American Insti‐ tute of Physics.
