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

In this paper, we apply a digital holographic microscope (DHM) in conjunction with stroboscopic acquisition synchronization. Here, the temperature-dependent decrease of the first resonance frequency (S1(T)) and Young’s elastic modulus (E1(T)) of silicon micromechanical cantilever sensors (MCSs) are measured. To perform these measurements, the MCSs are uniformly heated from T0 = 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.

The light beam back-reflected from the microcantilever interferes with the reference beam at the hologram plane on the CCD camera to create a hologram. When this digitally-recorded hologram is numerically multiplied by the complex amplitude of the same reference beam used to record it, the 3D topography of the cantilever array sensor is reconstructed with the interferometric vertical resolution (typically 0.3 nm with the current setup). By using Fourier filtering analysis, resolution down to a few pico-meters can be achieved. Here, n is the refractive index of the immersion medium (here n = 1 for air), and λ is the wavelength of the laser source (here λ = 689.5 nm).   Figure 1). Finally, by fitting a sine function (dash line) to the jmax = 10 desired measured moments or the so-called samples/period (dots), the vibrational amplitude for any arbitrary period is reconstructed. 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.
The optical microscope used for this experiment, inserted into the object beam path, has an objective with 5× magnification. Its numerical aperture and working distance are 0.12 and 14 mm, respectively, resulting in a measurement with the lateral resolution of 2.85 µm. With such a setup, 3D images of MCSs can be routinely recorded in real-time. In the case of MCSs, the R-DHM measures a length of l = 485.94 ± 2.85 µm and a width of w = 90 ± 2.85 µm, respectively. The nominal thickness of the cantilever sensors, provided by the manufacturer (Micromotive GmbH, Mainz, Rhineland-Palatinate, Germany) is h = 5.0 ± 0.3 µm. Since we use a 5× objective, four out of eight cantilever sensors lie in the field of view ( Figure 1). We refer the interested reader to the paper of Cuche et al. for the detailed explanation of the digital recording and reconstruction of the holograms [27].
MCSs are often operated in the dynamic mode, i.e., the vibrational response of the device changes upon a recognition event, such as the change of the mechanical property or the mass. For MCSs a recognition event can induce changes in its vibrational amplitude upon the shift in the resonance frequency. In order to measure the amplitude of the out-of-plane vibration of the MCSs, the R-DHM is operated in combination with a stroboscopic control unit. The stroboscopic unit is connected to a piezoelectric actuator (PZA) to externally excite the out-of-plane vibrations of the cantilever. In addition, the stroboscopic unit controls the time synchronization between the positions of the PZA vibration, the ON-time of the stroboscopic source (laser) pulse which is made by the introduction of a shutter in front of the laser and the CCD camera shutter (Figure 1). In the following paragraph we Exemplarily, the amplitude, A, of the cantilever movement is plotted versus time for one period, P 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: where, j = 1, 2, 3, . . . , j 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.

4
Here, n is the refractive index of the immersion medium (here n = 1 for air), and λ is the wavelength of the laser source (here λ = 689.5 nm).  consisting of only one material, ∆A value is directly calculated from the value of ∆φ with the help of the following equation [9]: Here, n is the refractive index of the immersion medium (here n = 1 for air), and λ is the wavelength of the laser source (here λ = 689.5 nm).
In our experiments, the cantilevers are excited externally with a piezo-actuator (AE0203D04F, Thorlabs, Munich, Bavaria, Germany) using a sine wave. In order to measure the first natural resonance frequencies of the MCSs, the piezo-actuator frequency is scanned with the discrete steps of 5 ± 0.1 Hz from 29.4000 to 30.0000 kHz. The frequency resolution of 0.1 Hz is the precision of the waveform generator in the stroboscopic unit control for the full frequency range.
The so-called MEMSTool analysis tool package (Lyncee Tec, Lausanne, Switzerland) was used to calculate the absolute values of the vibrational amplitudes, ∆A, of MCSs from the ∆φ values. ∆φ values are evaluated from the recorded phase images (Equation (2)).
By choosing the desired region of interests (ROIs) on the phase images, the results can be visualized in two different ways. First, in the form of Bode graphs (Figure 3a), depicting the amplitude (in dB relative to 1 nm) and the phase of the externally excited MCSs response (displacement) as a function of frequency [32]. Second, in the form of Nyquist graphs (Figure 3b) in which the imaginary part of the transfer function is plotted as a function of its real part [33]. The following two parameters are then estimated for each set of measured data: (i) the resonance frequency f 0MCS and (ii) the quality factor Q MCS which is defined as: Sensors 2017, 17, 1191 5 of 11 In our experiments, the cantilevers are excited externally with a piezo-actuator (AE0203D04F, Thorlabs, Munich, Bavaria, Germany) using a sine wave. In order to measure the first natural resonance frequencies of the MCSs, the piezo-actuator frequency is scanned with the discrete steps of 5 ± 0.1 Hz from 29.4000 to 30.0000 kHz. The frequency resolution of 0.1 Hz is the precision of the waveform generator in the stroboscopic unit control for the full frequency range.
The so-called MEMSTool analysis tool package (Lyncee Tec, Lausanne, Switzerland) was used to calculate the absolute values of the vibrational amplitudes, ΔA, of MCSs from the Δφ values. Δφ values are evaluated from the recorded phase images (Equation (2)).
By choosing the desired region of interests (ROIs) on the phase images, the results can be visualized in two different ways. First, in the form of Bode graphs (Figure 3a), depicting the amplitude (in dB relative to 1 nm) and the phase of the externally excited MCSs response (displacement) as a function of frequency [32]. Second, in the form of Nyquist graphs (Figure 3b) in which the imaginary part of the transfer function is plotted as a function of its real part [33]. The following two parameters are then estimated for each set of measured data: (i) the resonance frequency f0MCS and (ii) the quality factor QMCS which is defined as: Here, f2MCS and f1MCS are the lower and higher frequencies at which the amplitude of the micromechanical cantilever vibration drops by −3 dB relative to the maximum values.
In this study, the resonance frequency f0MCS and the quality factor QMCS 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]. Here, f 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.
In this study, the resonance frequency f 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].
The circle-fit method takes the advantage of the fact that the frequency response function of a single-degree-of-freedom (SDOF) system, when plotted on the real-imaginary Nyquist plane, is a circle [33]. This method, although developed for single-degree-of-freedom systems, can be used for separate modes of the multiple-degree-of-freedom systems, such as the MCSs vibrating in their first mode [33,34]. The circle-fitting algorithm consists of drawing the best circle possible through the measured points around the resonance frequency using a least squares circle fit (Figure 3b). Furthermore, it allows one to estimate the possible time delay introduced by the latency of the measurement chain by centering the circle to its canonical position as well as setting f 0MCS , f 1MCS , and f 2MCS at − π 2 , − π 4 , and − 3π 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. Figure 3a shows the Bode graphs for six selected ROIs, specified with A-F on the cantilever array sensor at T 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. We calculated the resonance frequency of a typical MCS with: For a comparison against the measured values. Here, m is the mass of the cantilever. The spring constant of a rectangular MCS, k(T) at a given temperature can be calculated from: where E(T) is the temperature-dependent Young's modulus of the cantilever material. w, h, and l are the width, the thickness, and the length of the cantilever, respectively. For typical values of E = 169 GPa for silicon, l = 485.9400 µm, w = 90.0000 µm, h = 5.0000 µm for a MCS, one obtains k(T) = 4.1420 N/m. The volume of the MCS is additionally calculated to be V = 2.2 × 10 −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 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.

Evaluation of the Temperature Coefficient of Resonance Frequencies of Heated MCSs
In the next step, the MCSs out-of-plane amplitudes are recorded during the frequency sweeps at different temperatures, ranging from T 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. pure Si at the maximum measured temperature of Ti = 450 K, with a precision of 10 −8 K −1 [37], the first temperature coefficient of the elastic modulus, ( ), can be easily calculated. To perform the calculation, we re-write Equation (9).
which, in return, results in a value of −54.95 3.88 10 K for all four Si cantilever sensors in the range of 298-450 K. This value is in very good agreement with the previously reported values of −52.6 ± 3.45 × 10 −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 ( )= −54. 95 3.88 10 K for Si. Therefore, the decrease of the Young's modulus of Si upon heating has the dominant contribution in the decrease of S1 (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 ( ) 2 (T) −51.50 3.88 10 K .
is plotted versus ∆T = 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 geometrical dimension increase of the MCSs as the result of the thermal expansion of Si. The relative change of the resonance frequency with the temperature can be written as [35]: where, again, f 0MCS is the resonance frequency of a MCS, , 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]: The subscript zero indicates the variable at the initial temperature of T 0 = 298 K. Additionally, we know that the isotropic thermal expansion coefficient is given by [35]: Inserting Equation (8) in Equation (7) gives: Here, the term In addition, 1 is the first temperature coefficient of the Young's elastic modulus showing the decrease of elastic modulus upon heating. Now we can rewrite Equation (9) as: It is noteworthy to mention that since Si properties exhibit an almost linear dependence upon the increase of temperature in the range of T 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 (T) values. This is simply done by plotting our experimental data in (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 (T) = ∂ f /(∆T f 0MCS ). The S 1 (T) 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 (T), can be easily calculated. To perform the calculation, we re-write Equation (9).  [36]. Additionally, the value of α(T = 450 K) = 3.453 × 10 −6 K −1 is only 6% of the value of E 1 (T) = −54. 95 ± 3.88 × 10 −6 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 (T) ≈ 2S 1 (T) ≈ −51.50 ± 3.88 × 10 −6 K −1 .
For the MCSs labelled 1-4, the decrease of the resonance frequency upon heating from 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.

Summary and Conclusions
In this paper, a stroboscopic reflective-DHM was used to measure the simultaneous out-of-plane vibrational amplitudes of four externally-actuated MCSs operating in a temperature range of T 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.
One of the key advantageous of the DHM is the 3D recording of the information over the full field of view with only one single hologram acquisition, and without any lateral or vertical scanning of the laser light beam. Therefore our technique realizes a readout which is not affected by lateral drift with regard to changes in temperature. Moreover, advantageously, our laterally-resolved holographic images do not experience crosstalk effects between the adjacent cantilever sensors. For an example of this type of crosstalk, we refer the reader to [38] in which parts of the laser beam hit adjacent structures in a way that results in artifacts when reading out the movement of the cantilever sensors.
Finally, by plotting ∂ f f 0 versus ∆T = T 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.