CHAPTER 3
NUMERICAL MODELS
3.1
Thermally Induced Curvature
The bending behavior of multilayers have been studied by Townsend, et.al, and the
numerical model developed is used in the analysis [25]. In the model, the layers are first
divided into smaller sub-layers, as shown in Figure 3.1.
The sublayers must be
sufficiently small so that the stress inside can be approximated as a constant; however, it
is not required that the sublayer thicknesses, ti, are the same through the structure.
SiO2
n
ti
.
.
.
poly-Si
t
SiO2
z
0
i
Figure 3.1 Discretization of thin film layers
The curvature, K, is solved by balancing the forces and moments through the sublayers,
as well as satisfying the dimensional constraints at the layer interfaces:
n


E j t j  thermal

j


n
t i  thermal j 1

E





i i
i
n


2
i 1
E jt j



j 1


K
2
n
 t t 


1
Ei t i     t    i 
3 i2  t i2  t 2 

2 12
i 1
 2 3 


22

(3.1)
where the subscripts denote the different sublayers. Ei, ti, and ithermal are the Young’s
modulus, thickness, and thermal strain in each sub-layer.
The neutral plane in the
composite, measured from the bottom, is , and t is the total thickness of the layers. In
the equation, i is defined as:
n
 i    ij t j
(3.2)
j 1
where ij is –1 for j < i, 0 for j = i, and 1 for j > i. The neutral axis is given by:
n

t

2
 Ei  i
i 1
n
ti
2
(3.3)
 Ei t i
i 1
The thermal strain used in Eqn.1 for temperature-dependent thermal expansion
coefficient is given by:
Tf
 itherm al
   i T dT
(3.4)
T0
for a temperature change from T0 to Tf. For beams having different layer widths, the
difference in widths is treated by replacing the Young’s modulus of the layer with an
effective modulus, E*, equal to:
E* 
wi
Ei
wmax
where wi is the width of i-th layer, and wmax is the maximum width of the beam [26].
23
(3.5)
Determining Thermophysical Properties
The design of the bending T-MEMS is such that the Young’s moduli of the layers do not
affect the beam curvature significantly. The result is that their curvature is based solely
on  of the layers; therefore, the structures can be used effectively for determining the
thermal expansion coefficients of Si and SiO2 thin films. However, the interdependency
of Young’s moduli and thermal expansion coefficients in the beam behavior requires that
three of the four properties (ESi, ESiO2, Si, SiO2) to be known in order to determine the
last property. The solution process for finding Si(T) and SiO2(T) begins with using
previously published results for SiO2 properties and ESi to solve for Si.
Room temperature value for ESiO2 was obtained from previously published data for
thermally grown oxide by Weihs, et. al. [27]. The study used photolithographically
fabricated cantilevers, deflected mechanically by a Nanoindenter. The resulting loaddeflection curve was analyzed to give ESiO2 of 64 GPa at room temperature. Textbook
value of 5.0×10-7 °C-1 was used for SiO2 at low temperatures [28].
Young’s modulus of polycrystalline silicon and the variation with temperature was
studied by Kahn, et. al., using lateral resonant structures [24]. Values reported for
temperatures up to ~450 °C were used to generate a second-order polynomial fit:
ESi(T) = 1.6806×1011 – 8.2225×106 T – 5.9816×103 T2
(3.6)
where the temperature, T, is in Celsius and the modulus is calculated in Pa. This curve
was extrapolated to all temperature range included in this study. The resulting value at
24
300
5
4
Si (°C-1)
3
2
1
fi t
room
0
50
100
150
200
temperature (°C)
250
300
Figure 3.2 Linear approximation of low-temperature
thermal expansion coefficient of poly-Si
room temperature is 167.8 GPa, and the modulus decreases monotonically at higher
temperatures.
These data are first used to find the low-temperature value for Si; the solution process
for this range is illustrated in Figure 3.2.
For temperatures below 300 °C, a linear
variation in Si with temperature was assumed. Average thermal expansion coefficients
were found for four temperature ranges: 50 – 100 °C, 50 – 200 °C, 150 – 300 °C, and 200
– 300 °C. Since a linear variation was assumed for these ranges, the average values
correspond to Si at 75, 125, 225, and 250 °C, respectively (labeled 1 – 4 in Fig.3.2). A
linear fit is performed through these points, and extrapolation yields the actual room
temperature value for Si, as well as the value at 300 °C. Since thermal expansion
coefficient of silica glass materials generally remain constant for temperatures below 300
°C, SiO2 was held constant for this temperature range [29].
25
For temperatures higher than 300 °C, the value for Si was approximated based on the
Grüneisen relation, which states that the thermal expansion coefficient is proportional to
the specific heat of the material [30]. This characteristic has been reported for crystalline,
bulk silicon at temperatures above 600 K, and will be applied here for polycrystalline
silicon [31]. This application is a consequence of the nature of polycrystalline Si films.
Poly-Si consists of small crystalline Si regions arranged randomly through the film and
separated by grain boundaries. Within the grain, the atomic structure is periodic and
resembles that of bulk crystalline Si; therefore, the behavior each grain is nearly identical
to that of crystalline silicon [28]. The specific heat of Si as a function of temperature is
approximated by a linear curve-fit [31]:
Cv = 0.748 + 1.68×10-4 T
(3.7)
where T is given in Kelvin. The resulting Si values were used to determine SiO2, which
is expected to undergo larger variation at high temperature range of up to 1000 °C.
3.2
Radiative Properties
When silicon wafers have surface patterns such as thin films and two dimensional
features, radiative properties of the wafer must be determined by considering properties
of silicon, microscale radiative effects at the surface, and two dimensional effects at the
surface. These effects are categorized as coherent or incoherent effects, illustrated in
Figure 3.3. Coherent effects are the results of features that are on the same length-scale
as the wavelengths of radiation, and include thin film interference at the wafer surface.
26
thin films
substrate
incident
radiation
incoherent effects
coherent effects
Figure 3.3 Radiative effects through a wafer
Furthermore, sub-micron two dimensional surface features lead to diffraction and
scattering. Incoherent effects prevail when the length-scale involved are much larger
than the wavelength of radiation, and include phenomena such as partial transparency
through the wafer.
Radiative Properties of Silicon
The radiative properties of silicon are highly dependent on wavelength and temperature.
The variation is due to the three absorption mechanisms of the semiconductor. Silicon
absorbs incoming radiation by bandgap, free-carrier, or lattice absorption mechanisms
[32]. Bandgap absorption occurs when the photon energy is used to move an electron
from the valance to the conduction band. This mechanism is possible only when the
energy of the photon exceeds the bandgap energy of silicon; therefore, it is dominant at
relatively short wavelengths (< 1100 nm at room temperature). In free-carrier absorption,
absorbed energy is used to move an electron to a higher energy level within the
conduction band. This mechanism dominates in the infrared region of the spectrum, at
27
wavelengths where bandgap absorption diminishes, and increases significantly with
temperature. Lattice absorption is the result of photon-phonon interaction in the Si
crystal; the energy absorbed induces changes in vibration of the crystal. This mechanism
is significant only at high IR wavelengths (> 6 m).
The cumulative effect of the three absorption mechanism is that silicon is transparent
(non-absorbing) in certain regions of the spectrum.
At room temperature, bandgap
absorption is dominant only at wavelengths below 1100 nm, and the wafer becomes
transparent at higher wavelengths due to weak free-carrier absorption. This results in
partial transparency effect within the wafer, shown in Figure 3.4. When radiation strikes
the top surface of a wafer, part of it is reflected, and the remaining portion enters the
wafer. In a non-absorbing wafer, the portion of radiation which enters the wafer passes
through unattenuated to the bottom side. At the bottom interface, part of it is reflected
back into the wafer, while the remaining portion transmits through the interface to the
net reflectivity
incident
radiation r1
r2 (1-r1 )2 •
exp(-2a total d)
r1 r2 2 (1-r 1) 2 •
exp(-4a total d)
d
(1-r1 )(1-r2 )exp(-atotal d)
r1 r2 (1-r1 )(1-r2 ) •
exp(-3a total d)
net transmissivity
Figure 3.4 Partial transparency effects
28
exiting medium. This process repeats until all of the radiation exits the wafer through
front or back interface. In such case, the net reflectivity (rnet) is enhanced by the higherorder reflection from the backside of the wafer, and can be found by ray-tracing to be an
infinite series [33]:

r  ,net  r  ,1  1  r  ,1 2  r n ,1r n, 21 exp  2n  1atotal d 
(3.8)
n 0
where r1 and r2 are front and backside spectral reflectivity of the wafer, respectively,
and d is the thickness of the wafer. This study used a plain, double polished Si wafer, for
which case r1 = r2, and a patterned sample with a rough backside, where r2 was
estimated to be 0.25 through all wavelengths. The absorption coefficient, atotal, is the sum
of absorption from all three mechanisms [32]:
atotal = abandgap + afreecarrier + alattice
(3.9)
Each component of the absorption coefficient is calculated based on results by Abramson
[32]. In a similar ray tracing technique, the absorptivity of the wafer is expressed by:


  ,net  1  r  ,1   1  e  atotal d 
1  r
,2 e
 atotal d
 r  ,1r  , 2 e
 2 atotal d
2
 r  ,1r 2 , 2 e 3atotal d

(3.10)

The spectral emissivity of the wafer is equal to the spectral absorptivity of the wafer by
Kirchhoff’s law [34].
29
The total radiative properties are found by integrating the spectral properties with the
appropriate blackbody emissive power function. For example, the total absorptivity of
the wafer is given by:

 total 
  ,net Eb, d
0
(3.11)

 Eb, d
0
where Eb, is the blackbody emissive power function [34]:
E b , 
C1
 exp C 2 / T   1
5
(3.12)
where C1 is 3.742 × 108 W·m4/m2 and C2 is 1.439 × 104 m·K. The blackbody emissive
power function describes the amount energy emitted by a body at some temperature.
Figure 3.5 shows the blackbody emissive power function for a body at 5800 K (sun),
2500 K (lamp temperature), 1100 K (wafer temperature), and 300 K (room temperature).
As temperature increases, Eb, shifts to shorter wavelengths, and the spectral emission
increases in magnitude. The wavelength at which the maximum energy is emitted (max)
is found by Wien’s displacement law [34]:
maxT = 2897.8 mK
30
(3.13)
The area under the blackbody emissive power curve represents the total energy emitted
by the body.
To calculate the absorptivity, the emissive power function is calculated using the lamp
temperature.
The total reflectivity and emissivity of wafer is found evaluated Eb,
corresponding to the wafer temperature. To determine the total radiative property, the
integral in Eqn.3.8 was performed over a wavelength range such that 99.5% of all
incident or emitted radiation was included in the analysis.
Thin Film Interference
When thin films are present on the wafer, microscale optical effects must be considered
in modeling radiative properties. For plain thin films, thin film interference need to be
considered. Thin film interference arise due to interaction of rays reflecting at different
interface between the thin film layers. When the wavelength of incoming radiation is on
the same order as the film thickness, then the rays undergo coherent reflections within the
maxT = 2897.8 mK
0.1
300
K
1100
K
2500
K
1E+8
1E+6
1E+4
1E+2
1E+0
1E-2
1E-4
580
0K
Eb,(W/m2m)
film layers. The interaction of these reflections produce constructive and destructive
1
10
wavelength (m)
100
Figure 3.5 Blackbody emissive power function for various temperatures.
31
interference, and the phases of the rays must be taken into account when calculating the
net reflectivity from thin film surfaces. The spectral radiative property can be calculated
by solving the Maxwell’s equation at each interface between the layers using complex
indices of refraction [33,35]. Thin film interference occurs only when the reflecting
surfaces are within the coherence length of the light (a few wavelengths for ordinary light
sources), it only needs to be considered at the top surface of the wafer.
Incoherent Effects
In MEMS structures, films or airgaps of tens of microns are not uncommon. The
thickness of these layers result in incoherent effects in the structures that must be treated
separately from thin film interference. In the samples studied, this is seen in regions
beneath the beam, where the beam and Si substrate is separated by a 23 m airgap. The
net reflectivity of this region is found by ray tracing and is expressed as an infinite series:

r  ,beam,net  r  ,beam   2 ,beam  r i,1net r i ,beam
(3.14)
i 0
where r,beam,net is the net spectral reflectivity of the region, r,beam and ,beam are the
spectral reflectivity and transmissivity of the beams with thin-film interference, and r,net
is the spectral reflectivity of Si substrate, corrected for partial transparency.
Two Dimensional Effects
It has been found that micron-scale surface patterns, such as gratings, do not significantly
affect the spectral radiative property of wafers at room temperature [36]. The study was
32
region 1
region 2
region 3
region 4
region 5
(a)
(b)
Figure 3.6 a) Top-view of T-MEMS under optical microscope; b) T-MEMS regions.
done using SiO2 stripe patterns on Si wafer, with feature sizes ranging from ~4 m to ~
50 m.
Although diffractive effects were visually observed on the samples, the
experimental results of spectral reflectivity showed that the average radiative property of
the surface can be computed by compiling the product of the weighted area fraction and
the property of each region:
r average   Fi ri
(3.15)
i
where i represents the regions and Fi is the area fraction, Ai/Atotal.
The T-MEMS have a minimum feature size of ~2 m; therefore, the average area
technique was used to find the average radiative property over the die. The die is broken
down into five regions as shown in Figure 3.6. The regions are: substrate (Si substrate
only), three films (SiO2, poly-Si, SiO2, Si substrate), one film (SiO2, Si substrate), three
films and air (SiO2, poly-Si, SiO2, airgap, Si substrate), and one film and air (SiO2,
airgap, Si substrate). Table 3.1 lists the five regions and the corresponding fill-factors for
a typical T-MEMS die.
33
Table 3.1 Fill factors for T-MEMS regions
region
1
2
3
4
5
description
Si substrate
multilayered film (SiO2, poly-Si, Si, Si substrate)
single film (SiO2 on Si substrate)
multilayered film suspended over Si substrate
single film suspended over Si substrate
F
0.198
0.562
0.142
0.045
0.030
The walls of the T-MEMS cavity are slanted at 60 º due to anisotropic etching process.
This angle may have some scattering effect on the radiative property of T-MEMS;
however, this is neglected in the modeling of T-MEMS average properties.
3.3
Temperature Distribution on a Wafer
A two-dimensional heat transfer model was developed to calculate the steady-state
temperature distribution on a patterned wafer during radiative heating [36,37]. The
model includes conduction within the wafer, radiant heating from the topside of the
wafer, and radiative heat loss from all surfaces of the wafer. Rapid thermal processing is
often done in vacuum; therefore, convection at the surface was not included in the model.
The wafer is discretized into equally-sized elements measuring 0.25 mm square. The
heat transfer in and out of an element is shown in Figure 3.7 for an interior cell. The
conduction into a cell from each neighboring cell is:
qcond,n 
kAC
Tn  T0 

34
(3.16)
where k is the thermal conductivity, AC is the
conducting area (equal to dz for interior
cells),  is the distance between the
neighboring nodes, and T0 is the temperature
of the node in question, and Tn is the
temperature of the neighboring node.
The
emission from the top and bottom surfaces of
the wafer are the heat loss from the element:
Figure 3.7 Control volume
qrad ,out   top AT04   bottomAT04
(3.17)
where  is the emissivities of the surfaces, A is the emitting area (equal to ² for interior
cells), and  is the Stefan-Boltzmann constnat, equal to 5.67×10-8 W/m2K4. The radiative
heat input is calculated by estimating the heat source to be a graybody with constant
emissivity:
4
qrad ,in   top f lampATlamp
(3.18)
where top is the total absorptivity on the top surface of the wafer, f is the shape factor
from lamp to the wafer, lamp is the emissivity of the lamp, A is the area of the top surface
of the cell (equal to 2 for interior cells), and Tlamp is the temperature of the lamp. The
tungsten-halogen heat source is approximated to have a temperature of 2200 ºC with an
emissivity of 0.30. The shape factor from the lamp to the wafer is approximated as 0.1
for the entire wafer.
35
For cells located at the edge of the wafer, the above equations are modified to account for
the changes in cell surface area for radiation and convection, as well as the radiative heat
loss from wafer edges. At steady state, the sum of all heat transfer terms in Figure 3.7
must be zero. The steady-state temperature was found by solving the heat balance
equation iteratively using under-relaxation [38]. Only a quadrant of the wafer was
modeled due to symmetry.
The dimensions of the wafer modeled is as shown in Figure 2.3. The die, measuring 4 ×
4 mm, are arranged on a wafer with 1 mm spacing between the dies. The regions of the
wafer outside of the dies are covered by three films: (from bottom) 1.03 m SiO2, 0.54
m poly-Si, and 0.19 m SiO2. Within the die, the radiative properties are calculated by
applying Eqn.3.11. The total radiative properties are nearly constant for temperatures
above 700 ºC; therefore, radiative properties at 800 ºC were used for the numerical
simulation.
36