CHAPTER 4
THERMOPHYSICAL PROPERTIES
4.1
T-MEMS Curvature Measurements
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
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.
0.0015
7.5
0.001
5
2.5
0
0
-0.0005
h (m)
K (m-1)
0.0005
-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
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
ESi
ESiO2
SiO2, up to 300 °C
value
1.6806×1011 – 8.2225×106T – 5.9816×103T2 Pa
64 GPa
5×10-7 °C-1
ref.
[24]
[27]
[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
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
38
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
(°C)
50 – 200
50 – 300
150 – 250
150 – 300
200 – 300
average Si
(10-6 °C–1)
3.34
3.97
4.29
4.65
4.68
standard
deviation
0.79
0.58
0.80
0.65
0.79
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
Si (10-6 °C-1 )
6
5
4
3
2
50
100
150
200
temperature (°C)
250
300
Figure 4.2 Thermal expansion coefficient of poly-Si at low temperatures
39
7
 (10-6 °C-1)
6
5
4
3
2
0
200
400
600
800
1000
temperature (°C)
Figure 4.3 Thermal expansion coefficient of silicon for high temperatures.
(—) polycrystalline Si, () bulk crystalline Si
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.
Thermal Expansion Coefficient of Silicon Dioxide
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
2
 (10-6 °C-1)
1.5
1
\
0.5
0
0
200
400
600
800
1000
temperature (°C)
Figure 4.4 Thermal expansion coefficient of SiO2 thin films
40
DK (m -1)
0
-1000
-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.
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
excellent agreement. Again, the lack of data in the mid-section corresponds to the region
where the beam deflection was nearly zero (flat).
41