# Thermal Characterization of Dynamic Silicon Cantilever Array Sensors by Digital Holographic Microscopy

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## Abstract

**:**

_{1}(T)) and Young’s elastic modulus (E

_{1}(T)) of silicon micromechanical cantilever sensors (MCSs) are measured. To perform these measurements, the MCSs are uniformly heated from T

_{0}= 298 K to T = 450 K while being externally actuated with a piezo-actuator in a certain frequency range close to their first resonance frequencies. At each temperature, the DHM records the time-sequence of the 3D topographies for the given frequency range. Such holographic data allow for the extracting of the out-of-plane vibrations at any relevant area of the MCSs. Next, the Bode and Nyquist diagrams are used to determine the resonant frequencies with a precision of 0.1 Hz. Our results show that the decrease of resonance frequency is a direct consequence of the reduction of the silicon elastic modulus upon heating. The measured temperature dependence of the Young’s modulus is in very good accordance with the previously-reported values, validating the reliability and applicability of this method for micromechanical sensing applications.

## 1. Introduction

_{0}= 298 K to T = 450 K. Based on our measurements, we calculate the change of the Young’s modulus of Si upon heating. Advantageously, the interferometric nature of this detection method allows calculating the absolute values of the vibrational amplitudes of MCSs with a sub-nanometric resolution [27].

^{−6}cm

^{2}) to its mass (nano-grams) leads to both a high sensitivity to surface phenomena and a short response time of sensors (typically μs). Therefore, the MCSs can be used as unique tools for mass sensing both in dynamic and static modes of operation. In the dynamic mode, the MCS resonance frequency varies as a function of mass loading like molecular adsorption, mass loss like molecular desorption, or temperature change.

## 2. Sensing Setup, Data Acquisition and Data Evaluation

_{MCS}(s), as shown in Figure 2a. The illuminated laser pulse has a width of τ and is preliminarily initiated at time t

_{0}. To measure the cantilever’s out-of-plane amplitude within one period, the much shorter laser pulse of τ (τ << P

_{MCS}) is shifted in time with respect to the excitation signal as:

_{max}with j

_{max}is number of desired sampled moments by the laser pulse within one period (or simply sample/period) and t

_{j}

_{−1}indicates the sampling moment within the period of the vibration. In our setup, the electronics enable us to synchronize the timings of the CCD camera shutter, the laser pulse with a minimum width of τ = 7.5 ns, and the PZA. In order to increase the signal-to-noise ratio of the digitally-recorded holograms on the CCD camera, the recorded hologram at each vibration moment, t

_{j}

_{−1}, is integrated over a specific number of pulses (or, consequently, over a specific number of periods). The latter is indicated by the vertical grey lines (Figure 2a). Therefore, an additional shutter is mounted in front of the CCD camera to adjust the integration time (Figure 1). Doing so, enough signals are obtained for producing high-quality holograms. The number of pulses are determined by integration time/pulse length time. In our measurements, for each t

_{j}

_{−1}, the static shutter is set to 453 μs to add the intensity of 2074 pulses with the pulse length time of τ = 167.5 ns.

_{0MCS}and (ii) the quality factor Q

_{MCS}which is defined as:

_{2MCS}and f

_{1MCS}are the lower and higher frequencies at which the amplitude of the micromechanical cantilever vibration drops by −3 dB relative to the maximum values.

_{0MCS}and the quality factor Q

_{MCS}could be estimated directly from reading the Bode plots (Figure 3a). However, since the measurements are carried out with the discrete steps of 5 ± 0.1 Hz in the presence of the inevitable mechanical noise from the environment, such an approach may lead to an error in the order of the discrete step size, i.e., 5 Hz in our case. Under such conditions, the circle-fitting algorithm is a much more accurate and very often used modal analysis method [34].

_{0MCS}, f

_{1MCS}, and f

_{2MCS}at $-\frac{\pi}{2}$, $-\frac{\pi}{4}$, and $-\frac{3\pi}{4}$, respectively [32,33,34]. By using this method, the resonance frequencies of MCSs can be evaluated with accuracy even better than the precision of the waveform generator of the stroboscopic unit, which was 0.1 Hz. However, we limit the accuracy of our data evaluation with the circle-fit algorithm to match the precision of the waveform generator.

_{0}= 298 K. The ROIs labeled with A, B, C, and D include all of the moving parts of each MCS. The E- and F-labeled ROIs, however, are selected within the D region on MCS number 4 at the tip and very close to the clipping part, respectively. For all selected ROIs, the plot of the amplitude versus frequency characterizes the system’s response to different input frequencies. By fitting the circle on the experimental data on the real-imaginary Nyquist plane (as shown in Figure 3b), the resonance frequencies of the MCSs, as well as the quality factors for all of the regions A–D were determined with the precision of 0.1 Hz. The results are presented in Table 1.

^{−7}cm

^{3}. Next, by taking into account the density of silicon to be ρ = 2.331 g/cm

^{3}, the mass of the typical MCS is calculated to be (0.24m) = 5.1 × 10

^{−10}kg. By inserting the values of k(T) and m into Equation (4), the calculated resonance frequency is f

_{0MCS_Cal}= 29.1243 kHz. For simplicity, we assume that l and w are constant, but the nominal value of the thickness h = 5 µm might vary by up to the value of ±Δh. In this case, the measured resonance frequencies suggest the variation of Δh

_{1}= 74.4 nm, Δh

_{2}= 63.6 nm, Δh

_{3}= 69.1 nm, and Δh

_{4}= 68.3 nm in the thicknesses of the MCSs, respectively. These values are in the expected range of the Δh = ±300 nm for the thickness variation, reported by the manufacturer (Micromotive GmbH, Mainz, Rhineland-Palatinate, Germany). Further, the evaluated value of f

_{0MCS4}for both E and D regions are equal with each other (f

_{0MCS4}= 29.4959 kHz). This, in return, suggests that selecting only a small ROI, like region E, at the tip of each MCS is enough for the evaluation of the resonance frequencies. In region F, however, the Bode graph shows only a relative movement with low amplitude and without a resonance peak. We expect such a behavior since this ROI is next to the clamped part of the MCS number 4 cantilever.

## 3. Evaluation of the Temperature Coefficient of Resonance Frequencies of Heated MCSs

_{0}= 298 ± 0.1 K to T = 450 ± 0.1 K, with temperature steps of 25 K. The cantilevers are uniformly heated and the temperature is controlled with a home-built oven (Figure 4a). The heater is a low- and constant-resistance solenoid around a hallow brass cantilever holder. The temperature is read out from the resistance change of a platinum resistive (Pt) sensor, positioned in the hallow brass cylinder inside the solenoid. The miniature Z-stage provides the fine distance adjustments for the hologram formation.

_{0MCS}is the resonance frequency of a MCS, $f\left(T\right)=\frac{1}{2\pi}\sqrt{\frac{k\left(T\right)}{0.24m}}$, $k\left(T\right)=\frac{E\left(T\right)w{h}^{3}}{4{l}^{3}}$, and T is the temperature. In fact, both of the above mentioned factors are included in the change of the spring constant, k(T), of the MCSs with the temperature. By substituting Equation (5) into Equation (6) one obtains [35]:

_{0}= 298 K. Additionally, we know that the isotropic thermal expansion coefficient is given by [35]:

_{0}= 298 K to T = 450 K, only the first-order terms are used in Equation (9). The goal is to determine which term on the right-hand side of Equation (9) is the dominating term for the decrease of the resonance frequencies of MCSs upon heating. To do so, we first need to obtain the ${S}_{1}\left(T\right)$ values. This is simply done by plotting our experimental data in the form of $\partial f/{f}_{0MCS}={f}_{iMCS}-{f}_{0MCS}/{f}_{0MCS}$ versus ΔT = T

_{i}− T

_{0}(Figure 4b). Here, f

_{0MCS}is the value of the resonance frequency at T

_{0}= 298 K and f

_{iMCS}is the resonance frequency measured at the given temperatures of T

_{i}. The tangent of a line fitted to each graph yields: ${S}_{1}\left(T\right)=\partial f/(\mathsf{\Delta}T{f}_{0MCS})$. The ${S}_{1}\left(T\right)$ values for the MCSs labelled with the numbers 1–4 are: −23.0 ± 5.3 × 10

^{−6}K

^{−1}, −22.6 ± 3.7 × 10

^{−6}K

^{−1}, −25.5 ± 4.5 × 10

^{−6}K

^{−1}, and −24.4 ± 2.7 × 10

^{−6}K

^{−1}, respectively. The mean value of S

_{1}(T) for all four cantilevers is 25.75 ± 1.94 × 10

^{−6}K

^{−1}. These values are in perfect agreement with the former reported values for Si(100), lying between −23.6 × 10

^{−6}K

^{−1}and −26.6 × 10

^{−6}K

^{−1}[36]. The reported range of values for the isotropic thermal expansion coefficient of Si in the temperature range of T

_{0}= 298 K to T = 450 K is α = 2.555 × 10

^{−6}K

^{−1}to 3.453 × 10

^{−6}K

^{−1}[37]. By taking the value of α(T = 450 K) = 3.453 × 10

^{−6}K

^{−1}for pure Si at the maximum measured temperature of T

_{i}= 450 K, with a precision of 10

^{−8}K

^{−1}[37], the first temperature coefficient of the elastic modulus, ${E}_{1}\left(T\right)$, can be easily calculated. To perform the calculation, we re-write Equation (9).

^{−6}K

^{−1}for the temperature range of 200–290 K [36]. Additionally, the value of α(T = 450 K) = 3.453 × 10

^{−6}K

^{−1}is only 6% of the value of ${E}_{1}\left(T\right)$ = $-54.\text{}95\pm 3.88\text{}\times {10}^{-6}\text{}{\mathrm{K}}^{-1}$ for Si. Therefore, the decrease of the Young’s modulus of Si upon heating has the dominant contribution in the decrease of S

_{1}(T) value versus temperature and, consequently, in the reduction of the resonance frequency with the increase of temperature. In this case if one neglects the effect of Si thermal expansion, then ${E}_{1}\left(T\right)\approx 2{S}_{1}\left(\mathrm{T}\right)\approx -51.50\pm 3.88\text{}\times {10}^{-6}\text{}{\mathrm{K}}^{-1}.$

_{0}= 298 K to T = 450 ± 0.1 K is equal to Δf

_{MCS}

_{1}= (f

_{450 K}− f

_{298 K})

_{MCS}

_{1}= −119.1 ± 0.1 Hz, Δf

_{MCS}

_{2}= (f

_{450 K}− f

_{298 K})

_{MCS}

_{2}= −113.3 ± 0.1, Δf

_{MCS}

_{3}= (f

_{450 K}− f

_{298 K})

_{MCS}

_{3}= −118.5 ± 0.1 Hz, and Δf

_{MCS}

_{4}= (f

_{450 K}− f

_{298 K})

_{MCS}

_{4}= −117.2 ± 0.1 Hz where the subscript number shows the MCS‘s number. Moreover, in these measurements the quality factor shows less than 2% of change when the temperature changes from T

_{0}= 298 K to T = 450 K ± 0.1 K. Therefore, they are not discussed further.

## 4. Summary and Conclusions

_{0}= 298 K to T = 450 K. Our experiments have demonstrated successful remote sensing measurements of: (a) the shift in the resonance frequency; and (b) the reduction of the Young’s elastic modulus of the MCSs upon heating.

_{i}− T

_{0}and fitting a line to the experimental data, we have evaluated the values of S

_{1}(T) and consequently determined the values of E

_{1}(T) as described in Section 3. For the calculation of E

_{1}(T) value, the isotropic expansion of silicon microcantilever sensors was taken into account (Equation (11)). In particular, to avoid the deformation of the micromechanical cantilever sensors by the thermal stress, the cantilever array sensor was heated with a rate of 1 K/s until the desired temperature was reached. Under these conditions, we can safely assume that the silicon MCSs expand isotropically upon the heat load. Our results demonstrate that the reduction of Young’s modulus upon heating is the dominant effect for the decrease of the natural resonance frequency of MCSs upon heating. With regard to the applications of our technique, although the expansion of cantilevers is isotropic, our sensing setup based on stroboscopic reflective-DHM can also be readily applied to study systems composed of different materials which expand non-isotropically upon thermal load. Examples include the effect of thermo-mechanical loads in: (a) the IC-fabrication and optimization process; and (b) the manufacturing and packaging of complex MEMS and micro-components [14,15]. Furthermore, the results also indicate that our technique can be reliably used for future mass sensing applications at elevated temperatures, which is required, for example, for the thermogravimetric analysis of minute amounts of materials.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

^{®}used in the frame of this study.

## References

- Khmaladze, A.; Kim, M.; Lo, C.-M. Phase imaging of cells by simultaneous dual-wavelength reflection digital holography. Opt. Express
**2008**, 16, 10900–10911. [Google Scholar] [CrossRef] [PubMed] - Khmaladze, A.; Restrepo-Martínez, A.; Kim, M.; Castañeda, R.; Blandón, A. Simultaneous dual-wavelength reflection digital holography applied to the study of the porous coal samples. Appl. Opt.
**2008**, 47, 3203–3210. [Google Scholar] [CrossRef] [PubMed] - Rappaz, B.; Barbul, A.; Hoffmann, A.; Boss, D.; Korenstein, R.; Depeursinge, C.; Magistretti, P.J.; Marquet, P. Spatial analysis of erythrocyte membrane fluctuations by digital holographic microscopy. Blood Cells Mol. Dis.
**2009**, 42, 228–232. [Google Scholar] [CrossRef] [PubMed] - Mendoza, F.; Aguayo, D.D.; Manuel, H.; Salas-Araiza, M.D. Butterflies’ wings deformations using high speed digital holographic interferometry. Proc. SPIE
**2011**, 8011, 80116Y. [Google Scholar] - Khmaladze, A.; Matz, R.L.; Epstein, T.; Jasensky, J.; Banaszak Holl, M.M.; Chen, Z. Cell volume changes during apoptosis monitored in real time using digital holographic microscopy. J. Struct. Biol.
**2012**, 178, 270–278. [Google Scholar] [CrossRef] [PubMed] - Toy, M.F.; Richard, S.; Kühn, J.; Franco-Obregón, A.; Egli, M.; Depeursinge, C. Enhanced robustness digital holographic microscopy for demanding environment of space biology. Biomed. Opt. Express
**2012**, 3, 313–326. [Google Scholar] [CrossRef] [PubMed] - Roy, M.; Seo, D.; Oh, S.; Yang, J.-W.; Seo, S. A review of recent progress in lens-free imaging and sensing. Biosens. Bioelectron.
**2017**, 88, 130–143. [Google Scholar] [CrossRef] [PubMed] - Archbold, E.; Ennos, A.E. Observation of surface vibration modes by stroboscopic hologram interferometry. Nature
**1968**, 217, 942–943. [Google Scholar] [CrossRef] - Coppola, G.; Ferraro, P.; Iodice, M.; Nicola, S.D.; Finizio, A.; Grilli, S. A digital holographic microscope for complete characterization of microelectromechanical systems. Meas. Sci. Technol.
**2004**, 15, 529. [Google Scholar] [CrossRef] - Ferraro, P.; Coppola, G.; Nicola, S.D.; Finizio, A.; Grilli, S.; Iodice, M.; Magro, C.; Pierattini, G. Digital holography for characterization and testing of mems structures. In Proceedings of the IEEE/LEOS International Conference on Optical MEMs, Lugano, Switzerland, 20–23 August 2002. [Google Scholar]
- Seebacher, S.; Osten, W.; Baumbach, T.; Jüptner, W. The determination of material parameters of microcomponents using digital holography. Opt. Lasers Eng.
**2001**, 36, 103–126. [Google Scholar] [CrossRef] - Jueptner, W.P.O.; Kujawinska, M.; Osten, W.; Salbut, L.A.; Seebacher, S. Combined measurement of silicon microbeams by grating interferometry and digital holography. Int. Soc. Opt. Photonics
**1998**, 3047, 348–357. [Google Scholar] - Watrasiewicz, B.M.; Spicer, P. Vibration analysis by stroboscopic holography. Nature
**1968**, 217, 1142–1143. [Google Scholar] [CrossRef] - Grosser, V.; Bombach, C.; Faust, W.; Vogel, D.; Michel, B. Optical measurement methods for mems applications. In Proceedings of the International Conference on Applied Optical Metrology 340, Balatonfured, Hungary, 29 September 1998. [Google Scholar]
- Pagliarulo, V.; Miccio, L.; Ferraro, P. Digital holographic microscopy for the characterization of microelectromechanical systems. Proc. SPIE
**2016**, 9890, 989002. [Google Scholar] - Kim, M.K. Principles and techniques of digital holographic microscopy. J. Photonics Energy
**2010**, 1, 018005. [Google Scholar] [CrossRef] - Thundat, T.; Wachter, E.A.; Sharp, S.L.; Warmack, R.J. Detection of mercury vapor using resonating microcantilevers. Appl. Phys. Lett.
**1995**, 66, 1695–1697. [Google Scholar] [CrossRef] - Berger, R.; Lang, H.P.; Gerber, C.; Gimzewski, J.K.; Fabian, J.H.; Scandella, L.; Meyer, E.; Güntherodt, H.J. Micromechanical thermogravimetry. Chem. Phys. Lett.
**1998**, 294, 363–369. [Google Scholar] [CrossRef] - Takahito, O.; Masayoshi, E. Mass sensing with resonating ultra-thin silicon beams detected by a double-beam laser doppler vibrometer. Meas. Sci. Technol.
**2004**, 15, 1977. [Google Scholar] - Toffoli, V.; Carrato, S.; Lee, D.; Jeon, S.; Lazzarino, M. Heater-integrated cantilevers for nano-samples thermogravimetric analysis. Sensors
**2013**, 13, 16657–16671. [Google Scholar] [CrossRef] - Torres, F.; Uranga, A.; Riverola, M.; Sobreviela, G.; Barniol, N. Enhancement of frequency stability using synchronization of a cantilever array for mems-based sensors. Sensors
**2016**, 16, 1690. [Google Scholar] [CrossRef] [PubMed] - Weigold, J.W.; Najafi, K.; Pang, S.W. Design and fabrication of submicrometer, single crystal si accelerometer. J. Microelectromech. Syst.
**2001**, 10, 518–524. [Google Scholar] [CrossRef] - Ang, W.T.; Khoo, S.Y.; Khosla, P.K.; Riviere, C.N. Physical Model of a MEMS Accelerometer for Low-g Motion Tracking Applications. In Proceedings of the 2004 IEEE International Conference on Robotics and Automation, ICRA ’04, New Orleans, LA, USA, 26 April–1 May 2004. [Google Scholar]
- Donald, B.R.; Levey, C.G.; McGray, C.D.; Paprotny, I.; Rus, D. An untethered, electrostatic, globally controllable MEMS micro-robot. J. Microelectromech. Syst.
**2006**, 15, 1–15. [Google Scholar] [CrossRef] - Juan, W.H.; Pang, S.W. Controlling sidewall smoothness for micromachined Si mirrors and lenses. J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom.
**1996**, 14, 4080–4084. [Google Scholar] [CrossRef] - Qiu, Y.; Gigliotti, J.; Wallace, M.; Griggio, F.; Demore, C.; Cochran, S.; Trolier-McKinstry, S. Piezoelectric Micromachined Ultrasound Transducer (PMUT) Arrays for Integrated Sensing, Actuation and Imaging. Sensors
**2015**, 15, 8020–8041. [Google Scholar] [CrossRef] [PubMed] - Cuche, E.; Marquet, P.; Depeursinge, C. Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms. Appl. Opt.
**1999**, 38, 6994–7001. [Google Scholar] [CrossRef] [PubMed] - Etayash, H.; Khan, M.F.; Kaur, K.; Thundat, T. Microfluidic cantilever detects bacteria and measures their susceptibility to antibiotics in small confined volumes. Nat. Commun.
**2016**, 7, 12947. [Google Scholar] [CrossRef] [PubMed] - Fabian, J.-H.; Berger, R.; Lang, H.P.; Gerber, C.; Gimzewski, J.K.; Gobrecht, J.; Meyer, E.; Scandella, L. Micromechanical thermogravimetry on single zeolite crystals. In Proceedings of the uTAS ’98 Workshop Micro Total Analysis Systems ’98, Banff, AB, Canada, 13–16 October 1998. [Google Scholar]
- Iervolino, E.; van Herwaarden, A.W.; van der Vlist, W.; Sarro, P.M. Thermogravimetric device with integrated thermal actuators. In Proceedings of the 2010 IEEE 23rd International Conference on Micro Electro Mechanical Systems (MEMS), Hong Kong, China, 24–28 January 2010. [Google Scholar]
- Berger, R.; Delamarche, E.; Lang, H.P.; Gerber, C.; Gimzewski, J.K.; Meyer, E.; Güntherodt, H.-J. Surface stress in the self-assembly of alkanethiols on gold. Science
**1997**, 276, 2021–2024. [Google Scholar] [CrossRef] - De Silva, C.W. Vibration: Fundamentals and Practice, 2nd ed.; CRC Press LLC: New York, NY, USA, 2006. [Google Scholar]
- Kennedy, C.C.; Pancu, C.P.D. Use of vectors in vibration measurement and analysis. J. Aeronaut. Sci.
**1947**, 14, 603–625. [Google Scholar] [CrossRef] - Ewins, D.J. Modal Testing: Theory and Practice, 2nd ed.; Research Studies Press: Baldock, UK, 2000. [Google Scholar]
- Ziegler, C. Cantilever-based biosensors. Anal. Bioanal. Chem.
**2004**, 379, 946–959. [Google Scholar] [CrossRef] [PubMed] - Boyd, E.J.; Li, L.; Blue, R.; Uttamchandani, D. Measurement of the temperature coefficient of Young’s modulus of single crystal silicon and 3c silicon carbide below 273 K using micro-cantilevers. Sens. Actuators A Phys.
**2013**, 198, 75–80. [Google Scholar] [CrossRef] - Swenson, C.A. Recommended values for the thermal expansivity of silicon from 0 to 1000 K. J. Phys. Chem. Ref. Data
**1983**, 12, 179–182. [Google Scholar] [CrossRef] - Lang, H.P.; Berger, R.; Andreoli, C.; Brugger, J.; Despont, M.; Vettiger, P.; Gerber, C.; Gimzewski, J.K.; Ramseyer, J.P.; Meyer, E.; et al. Sequential position readout from arrays of micromechanical cantilever sensors. Appl. Phys. Lett.
**1998**, 72, 383–385. [Google Scholar] [CrossRef]

**Figure 1.**In a R-DHM, a coherent laser beam is split into the OB and the RB. The OB, which is the light reflected off the MCSs interferes with the RB on the CCD camera to form holograms. The phase and intensity images are extracted from the holograms. With the addition of the stroboscopic unit control, the time synchronization between three components of (1) the positions of the piezoelectric actuator (PZA) vibration, (2) the ON-time of the stroboscopic source (laser) pulse, and (3) the CCD camera shutter are controlled. The phase images are used to calculate the amplitudes of the vibrations. The measurements are performed in a closed chamber and through an optical window. The cantilever chip image shown at the bottom as mounted on top of the piezoelectric actuator is taken with a scanning electron microscope.

**Figure 2.**(

**a**) As an illustrative example, three periods of a cantilever vibration are plotted in a diagram of amplitude (A) versus time. For a typical driving frequency applied to the cantilever, f

_{MCS}, the period of movement is calculated as P

_{MCS}= 1/f

_{MCS}. (

**b**) For each P

_{MCS}, the piezo-actuator excites the cantilevers periodically while a laser pulse with the pulse length time of τ = 167.5 ns is shifted in time with respect to the excitation signal to sample j

_{max}(here 10) number of different moments (t

_{j}

_{−1}). (

**c**) For each moment, the signals from 2074 pulses are integrated by keeping the CCD camera shutter open (Figure 1). Finally, by fitting a sine function (dash line) to the j

_{max}= 10 desired measured moments or the so-called samples/period (dots), the vibrational amplitude for any arbitrary period is reconstructed.

**Figure 3.**(

**a**) Bode graph for six different ROIs on the MCSs at T = 298 K. At the resonance frequency of f

_{0MCSs}, the cantilever response is in phase quadrature (i.e., θ = $-\pi /2$) with the excitation signal. On the vertical axes on left- and right-hand sides, the amplitude and phase values for A, B, C, D, and E ROIs are shifted by the values of 30 dB and 160 degrees with respect to each other, respectively. For ROI F, the amplitude values are shifted by the value of 60 dB. The region shown as Ref. is the reference area; (

**b**) The real-imaginary Nyquist plane of the experimental data for MCS2 from the ROI B (gray dots) with the circle fit (dashed black line). The evaluated resonance frequency f

_{0MCS2}at θ = $-\pi /2$ (square) was determined with 0.1 Hz precision as f

_{0MCS2}= 29.4946 kHz. The quality factor is also calculated by evaluating the bandwidth, Δf

_{MCS}

_{2}= f

_{2MCS2}− f

_{1MCS2}(hexagrams). In this case, the quality factor is ${Q}_{MCS2}=\frac{{f}_{0MCS2}}{{f}_{2MCS2}-{f}_{1MCS2}}=\frac{29.4946}{29.5530-29.4321}\text{}=$ 243.9946 ± 0.0001.

**Figure 4.**(

**a**) The schematic shows the experimental chamber used for the uniform heating of the MCSs in quasi-sealed condition. The schematic is not to scale; (

**b**) For each MCS (in black) and their corresponding mean values (in gray) the measured values of $\frac{\partial f}{{f}_{0MCS}}=\frac{{f}_{iMCS}-{f}_{0MCS}}{{f}_{0MCS}}$ is plotted versus ΔT = T

_{i}− T

_{0}. Here, f

_{0MCS}is the initial resonance frequency measured at T

_{0}= 298 K and f

_{iMCS}is the resonance frequency measured at desired temperatures of T

_{i}during the uniform heating from T

_{0}= 298 K to T = 450 K. A fit on the mean values graph of all four cantilevers yields the mean slope of ${S}_{1}\left(T\right)=\frac{\partial f}{\mathsf{\Delta}T{f}_{0MCS}}=-25.75\text{}\pm \text{}1.94\text{}\times {10}^{-6}\text{}{\mathrm{K}}^{-1}$ with the intersection value of −0.0039 × ${10}^{-6}$ with the vertical axis. All resonance frequency values are evaluated from the circle fit method with the precision of 0.1 Hz.

**Table 1.**The measured resonance frequency f

_{0}of each micromechanical cantilever sensor (MCS) at the reference temperature of T

_{0}= 298 K together with their corresponding quality factors, Q

_{MCS}.

MCS | 1 | 2 | 3 | 4 |
---|---|---|---|---|

f_{0MCS} (kHz) ± 0.0001 | 29.5777 | 29.4946 | 29.5267 | 29.4959 |

Q_{MCS} | 243.9946 | 242.2243 | 214.8007 | 237.4964 |

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**MDPI and ACS Style**

Zakerin, M.; Novak, A.; Toda, M.; Emery, Y.; Natalio, F.; Butt, H.-J.; Berger, R.
Thermal Characterization of Dynamic Silicon Cantilever Array Sensors by Digital Holographic Microscopy. *Sensors* **2017**, *17*, 1191.
https://doi.org/10.3390/s17061191

**AMA Style**

Zakerin M, Novak A, Toda M, Emery Y, Natalio F, Butt H-J, Berger R.
Thermal Characterization of Dynamic Silicon Cantilever Array Sensors by Digital Holographic Microscopy. *Sensors*. 2017; 17(6):1191.
https://doi.org/10.3390/s17061191

**Chicago/Turabian Style**

Zakerin, Marjan, Antonin Novak, Masaya Toda, Yves Emery, Filipe Natalio, Hans-Jürgen Butt, and Rüdiger Berger.
2017. "Thermal Characterization of Dynamic Silicon Cantilever Array Sensors by Digital Holographic Microscopy" *Sensors* 17, no. 6: 1191.
https://doi.org/10.3390/s17061191