4.1 T-MEMS Curvature Measurements

CHAPTER 4

THERMOPHYSICAL PROPERTIES

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

0.0015 7.5

0.001 5

0.0005 2.5 h )

-  m m 0 0  ) (

-0.0005K -2.5

-0.001 -5

-0.0015 -7.5 0 200 400 600 800 1000 temperature (°C)

Figure 4.1 Beam curvature and deflection at high temperatures.

Lbeam = 100 m, width ratio = 0.41.

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.

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.

4.2 Determination of Thermophysical Properties

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.

Table 4.2 Input parameters to numerical model

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]

Thermal Expansion Coefficient of Polycrystalline Silicon

At temperatures below 300 °C, the thermal expansion coefficient of SiO2 was assumed to

be constant. Linearly varying Si was found by finding average thermal expansion

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

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

temperature range corresponds to the Si value at the center point of the interval.

Therefore, the average Si found at temperature interval 50 – 100 °C can be translated

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.

Table 4.3 Average Si for low temperature ranges

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

) 5 1 - C °

- 4 0 1 (

S 3 

2 50 100 150 200 250 300 temperature (°C)

Figure 4.2 Thermal expansion coefficient of poly-Si at low temperatures

39 The figure also shows the result of a first-order fit through the four points. The four

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

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

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

coefficient for polycrystalline silicon [28]. At higher temperatures, Si becomes substantially larger than that of crystalline silicon.

40 7 6 5

4 3 2 0 200 400 600 800 1000

Figure 4.3 Thermal expansion coefficient of silicon for high temperatures. (—) polycrystalline Si, () bulk crystalline Si

Thermal Expansion Coefficient of Silicon Dioxide

41 The thermal expansion coefficient of SiO2 above 300 °C was determined based on Si

found above. From 300 °C to 1000 °C, the variation in SiO2 was assumed to be linear.

The values for SiO2 from room temperature to 1000 °C is plotted in Figure 4.4. The

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.

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

1.5 ) 1 - C °

- 1 0 1 (

0.5

0 200 400temperature600 (°C)800 1000

Figure 4.4 Thermal expansion coefficient of SiO2 thin films

42 0 ) 1 - -1000 m

 -2000

-3000 0 200 400 600 800 1000 temperature (°C)

Figure 4.5 Thermally induced curvature in T-MEMS beam.

(—) numerical fit, () experimental, Lbeam = 100 m, width ratio = 0.54.

excellent agreement. Again, the lack of data in the mid-section corresponds to the region where the beam deflection was nearly zero (flat).