<p> CHAPTER 4</p><p>THERMOPHYSICAL PROPERTIES</p><p>4.1 T-MEMS Curvature Measurements</p><p>The thermally induced curvature of T-MEMS beams were measured from room temperature to ~ 850 °C. A representative data set from a beam measuring 100 m with top-to-bottom width ratio of 0.41 is shown in Figure 4.1. The left y-axis of the graph</p><p>0.0015 7.5</p><p>0.001 5</p><p>0.0005 2.5 h )</p><p>1 (</p><p>-  m m 0 0  ) (</p><p>-0.0005K -2.5</p><p>-0.001 -5</p><p>-0.0015 -7.5 0 200 400 600 800 1000 temperature (°C)</p><p>Figure 4.1 Beam curvature and deflection at high temperatures. </p><p>Lbeam = 100 m, width ratio = 0.41.</p><p>37 indicates curvature, while the right y-axis shows corresponding beam tip deflection in microns. The graph plots curvatures taken during one heating and cooling cycle. The absence of data near zero-deflection indicates the region where the curvature of beam became smaller than the minimum curvature measurable by the system. The beam had an initial positive deflection of 7 m at room temperature, and reached a negative deflection of nearly 6 m at a maximum temperature of 850 °C. </p><p>The data shows good agreement between beam curvature during heating and cooling, and short-term exposure to high temperature does not induce relaxation or other permanent effects in the beam. Effects of prolonged exposure to high temperature is discussed in later sections. </p><p>4.2 Determination of Thermophysical Properties</p><p>Table 4.2 summarizes the properties used to find Si and SiO2 using T-MEMS. The choice of these parameters are discussed in Chapter 3. </p><p>Table 4.2 Input parameters to numerical model</p><p> property value ref. 11 6 3 2 ESi 1.6806×10 – 8.2225×10 T – 5.9816×10 T Pa [24] ESiO2 64 GPa [27] -7 -1 SiO2, up to 300 °C 5×10 °C [28]</p><p>Thermal Expansion Coefficient of Polycrystalline Silicon</p><p>At temperatures below 300 °C, the thermal expansion coefficient of SiO2 was assumed to</p><p> be constant. Linearly varying Si was found by finding average thermal expansion</p><p>38 coefficients over five temperature intervals. The analysis were repeated over 7 trials, and the results are summarized in Table 4.3. Trials were done on one T-MEMS die, on two</p><p> different columns having varying width ratios: 0.54 and 0.41. Because Si was assumed to be linear for temperatures below 300 °C, the average values found for any of the five</p><p> temperature range corresponds to the Si value at the center point of the interval.</p><p>Therefore, the average Si found at temperature interval 50 – 100 °C can be translated</p><p> into Si at 75 °C. The results of this translation are plotted in Figure 4.2. The error bars in the figure corresponds to the standard deviation in data.</p><p>Table 4.3 Average Si for low temperature ranges</p><p> temperature range average Si standard (°C) (10-6 °C–1) deviation 50 – 200 3.34 0.79 50 – 300 3.97 0.58 150 – 250 4.29 0.80 150 – 300 4.65 0.65 200 – 300 4.68 0.79</p><p>6</p><p>) 5 1 - C °</p><p>6</p><p>- 4 0 1 (</p><p> i</p><p>S 3 </p><p>2 50 100 150 200 250 300 temperature (°C)</p><p>Figure 4.2 Thermal expansion coefficient of poly-Si at low temperatures</p><p>39 The figure also shows the result of a first-order fit through the four points. The four</p><p> points all consistently lie near the linear fit. Extrapolation of the fit results in Si at room temperature and 300 °C of 2.20 × 10-6 °C-1 and 5.38 × 10-6 °C-1, respectively. For</p><p> temperatures above 300 °C, Si was approximated to be proportional to the specific heat of silicon, as described in Chapter 3. The result of this approximation is plotted in Figure</p><p>4.3, along with the thermal expansion coefficient for bulk crystalline silicon. The room temperature value agrees well with reported value of 2 × 10-6 °C-1 for thermal expansion</p><p> coefficient for polycrystalline silicon [28]. At higher temperatures, Si becomes substantially larger than that of crystalline silicon. </p><p>40 7 6 5</p><p>4 3 2 0 200 400 600 800 1000</p><p>Figure 4.3 Thermal expansion coefficient of silicon for high temperatures. (—) polycrystalline Si, () bulk crystalline Si</p><p>Thermal Expansion Coefficient of Silicon Dioxide</p><p>41 The thermal expansion coefficient of SiO2 above 300 °C was determined based on Si</p><p> found above. From 300 °C to 1000 °C, the variation in SiO2 was assumed to be linear.</p><p>The values for SiO2 from room temperature to 1000 °C is plotted in Figure 4.4. The</p><p> graph shows an over three-fold increase in SiO2 from 300 °C to 1000 °C, with values changing from 5×10-7 °C-1 to 1.83×10-6 °C-1, respectively. These values were calculated over three trials, performed on same sample with width ratio of 0.54. The standard deviation of the four trials was 0.12×10-6 °C-1 at 1000 °C.</p><p>The curvature of bending T-MEMS calculated numerically using these properties are plotted along with representative experimental results in Figure 4.5. The two show</p><p>2</p><p>1.5 ) 1 - C °</p><p>6</p><p>- 1 0 1 (</p><p>0.5</p><p>0</p><p>0 200 400temperature600 (°C)800 1000</p><p>\</p><p>Figure 4.4 Thermal expansion coefficient of SiO2 thin films</p><p>42 0 ) 1 - -1000 m</p><p> (</p><p>K</p><p> -2000</p><p>-3000 0 200 400 600 800 1000 temperature (°C)</p><p>Figure 4.5 Thermally induced curvature in T-MEMS beam.</p><p>(—) numerical fit, () experimental, Lbeam = 100 m, width ratio = 0.54.</p><p> excellent agreement. Again, the lack of data in the mid-section corresponds to the region where the beam deflection was nearly zero (flat). </p><p>43</p>