ABSTRACT A model is developed that describes the temperature with
respect to time of a silicon wafer as it undergoes latent image activation. The wafer is lowered to a distance just above a hot-plate and allowed to come to steady state temperature, after which point it is removed. For simplistic solutions, the energy equation is a good estimate of both rise time and steady-state temperature, but in order to take into account inaccuracies in the angle of the silicon wafer, a discreet Euler forward differential equation was derived in order to be sure that the surface of the wafer would retain less than a 2K surface temperature gradient. The results show that angle is a very important factor in silicon wafer heating, as even a very small angle can cause a large temperature gradient despite the high thermal conductivity of silicon. The height of the wafer above the hot-plate is also an important factor in the rise time of the wafer to steady-state temperature.
INTRODUCTION In order to produce advanced microcircuits quickly and effectively the silicon wafers they are made from must go through a heat treat process in order to activate the latent image – which when etched results in a layer of the microcircuit. This heating process happens between the lithography step and the etching step in the circuit creation process. The silicon wafers must be heated uniformally (less than 2.0 C variation over the entire surface) to about 180 C for about two seconds in order to activate the latent image. The current processes cause a bottleneck in the production of advanced microcircuits. By developing a better understanding of the factors affecting the wafer properties through thermodynamic modeling, the wafer can be brought up to temperature more quickly, which will allow for increased production rates.
The current method used to heat the wafers is through the use of a hot-plate which is heated to a temperature above the desired temperature of the wafer. A wafer at ambient temperature is quickly lowered to a distance d from the hot-plate for a specific amount of time – after which the wafer is removed from the hot-plate. Associated Physical Phenomena While it is true that the rate of temperature increase of the wafer is indirectly proportional to its distance from the hot plate, a non-zero angle between the hot-plate and the wafer (see
Figure 1: An image of a silicon wafer, which has been etched with micro-circuitry.
Figure 2) will cause the temperature variation requirement (less than 2.0 C variation over the entire surface) to be unsatisfied if the wafer is moved too close to the hot-plate. In other words, if the wafer is at an angle to the hot-plate one part of it will heat up much faster than another. The intensity of this temperature variation is based upon the distance between the wafer and the hot-plate. For this reason the following model will seek to find a compromise between the maximum possible angular accuracy available and the minimum possible heating time of the wafer based on maximizing the proximity of the wafer to the hot-plate without invalidating the temperature variation requirement.
Figure 2: The model involves a wafer set a distance h initially at TS = TAMB being heated by a hot-plate at THOT. Take note of the inclusion of the angle α, which changes the heating conditions.
NOMENCLATURE THOT Temperature of the Hot-Plate TAMB Ambient Air Temperature TS Wafer Temperature α Angle between Wafer and Hot-Plate cv Heat Capacity of Air Const. Volume cP Heat Capacity of Air Const. Pressure mwafer Mass of Wafer Gr Grasshoff Number Pr Prandtl Number NuL Nusselt Number µAIR Dynamic Viscosity for Air kAIR Convective Coefficient for Air kSi Convective Coefficient for Silicon kAl Convective Coefficient for Aluminum g Gravitational Constant ν Kinematic Viscosity for Air ATOP Area based on Characteristic Length
L Characteristic Length of Plates (Diameter) LGAP Distance between Wafer and Hot-Plate tSi Thickness of Silicon Wafer RaL Raleigh Number hCONV Convective Heat Transfer Coefficient hRAD Radiative Heat Transfer Coefficient εSi Radiative Emissivity of Silicon εAl Radiative Emissivity of Aluminum σ Stephan-Boltzmann Constant ρAIR Density of Air Rcondbottom Thermal Resistance to Conduction Rconvbottom Thermal Resistance to Convection Rradbottom Thermal Resistance to Radiation Rradtop Thermal Resistance to Radiation Rconvtop Thermal Resistance to Convection
THE MODEL The heat transfer to the silicon wafer is restricted by the rate at which it travels from the hot-plate to the wafer, and from the wafer to the surrounding air. The wafer can be heated via conductive, convective, and radiative heat transfer, and each have their own associated thermal resistances to heat flow based on temperature difference. The convective, conductive and radiative heat transfer between the wafer and the plate all occur in parallel, and the convective and radiative (conductive is negligible) heat transfer between the wafer and the air also occur in parallel, but these two parallel occurrences are in series with each other (see Figure 3)
Figure 3: This thermal circuit diagram shows the heat transfer rate layout associated with heat transfer from the hot-plate to
the silicon wafer.
Conservation of Energy The temperature change of the silicon wafer is a function of the rate of heat transfer in and out of the wafer, or
The radiation heat transfered into the bottom of the silicon wafer can be expressed as
!
˙ Q RADBOT = hrad AS (THOT "TS ) . (2) where
!
hRAD
="
1
#Si
+1
#AL
$1
(TS
+ THOT)(T
S
2+ T
HOT
2) (3)
is the heat transfer coefficient. For convection, the heat transferred is
!
˙ Q CONVBOT = hC AS (THOT "TS ) (4) where
L
kNuh
F
C
*= (5)
is the convective heat transfer coefficient. The Nusselt number can be found by the correlation
!
NuL
=1+1.44 1"1708
RaLcos#
$
% &
'
( ) 1"
1708 sin(1.8#( )1.6
RaLcos#
* + ,
- ,
. / ,
0 ,
+Ra
Lcos#
5830
1
2 3
4
5 6 1/ 3
"1$
% &
'
( )
•
(6)
For conduction between the plates the transferred heat can be expressed as
!
˙ Q CONDBOT =kAIR ATOP
LGAP
(THOT "TS ) . (7)
The heat transfer rate off of the top of the silicon wafer
is a function of only convection and radiation, as the conduction term is negligible. For convection, the heat transfer rate is
!
˙ Q CONVTOP = hC ATOP (TS "TAMB ) (8) and hc is the same as in Equation 5. In this case, the Nusselt number is
4/1
54.0LL
RauN != (9)
and the Raleigh number based on the characteristic length – in this case the diameter – is
rPGrRaL
!= (10)
which is a function of the Grashof number,
2
)(
!
" AMBS TTgGr
#= , (11)
where β is
)(
)1(
AMBS
AMB
S
TT
T
T
!
!
!"# (12)
and the Prandtl number,
k
crP
pµ
= . (13)
For radiative heat transfer, the rate can be expressed by
!
˙ Q RADTOP = hrad ATOP (TS "TAMB ) (14) where head can be expressed as
))((22
AMBSAMBSTOPRADTTTTAh ++= !" . (15)
Wafer Temperature Gradients from Angle Effects Equation 1 is only capable of calculating the temperature change of a perfectly horizontal wafer, but in real-world scenarios, this is not possible. Slight angles will induce heat flow within the wafer. A differential control volume was used to analyze the heat blow within the wafer (see Figure 4).
Figure 4: Along with the differential heat transfer to the top and bottom of the wafer, the differential heat flow inside the wafer is necessary to determine the effects of angle on the temperature gradient.
The area of the wafer exposed to the hot-plate can be represented by the value LCHORD, where
22xrL
CHORD!= . (16)
Using the first two terms of the Taylor polynomial expansion,
!
˙ q (x + dx) = ˙ q (x) +d ˙ q (x)
dxdx , (17)
and combining it with the energy equation yields:
Tdmcqdqddxdx
xqdVTOPBOT
&&&&
=!+"#
$%&
'!
)(. (18)
In Equation 18 the
BOTqd & andTOPqd & values are calculated using
Equations 2 – 15 as before, only using the physical properties of the differential element for evaluation. The differential heat transfer rate through the wafer is
!"
#$%
&=
dx
dTAkq XSECTIONSiCOND
& (19)
which, when combined with Equation 18 yields:
!
kSi LCHORDtSi *d
2T
dx2
"
# $
%
& ' dx + d ˙ q BOT ( d ˙ q TOP = cV dm ˙ T (20)
EXPERIMENTAL METHODS The model was validated using a hot-plate and silicon wafer set up described pictorially below in Figures 7 and 8. The lowering mechanism consisted of a bearing on a structural beam in order to prevent the wafer from encountering the hot-plate at an angle. A thermocouple was attached to the center of the hot plate in order to measure its temperature.
d
LSi
TPLATE
TS
Lowering
Mechanism Silicon
Wafer
Hot Plate Figure 5: Diagramatic view of the experimental setup.
The wafer at steady-state is quickly lowered to a specified distance above the higher temperature, steady-state hot-plate. The distance is calibrated by varied number of 74µm thick paper. MODEL ANAYLSIS RESULTS Using both a standard ODE solver and a Discreet Euler Forward partial differential equation solver, the behavior or the silicon wafer during heating was classified in several ways. First, the “no-tilt” wafer heating was analyzed to show insights into the behavior of different factors associated with the system. Figure 5 shows lines of constant THOT-TCOLD on a graph of wafer height above the hot-plate versus time for the wafer to reach a steady state temperature.
Figure 7: A graph of wafer height (d) versus the time it takes the to a reach steady state temperature.
Figure 6: Photograph of experimental setup. Notice the thermocouple in the center of the silicon wafer.
Next, similar calculations were done in an attempt to understand how the tilt angle and the height of the wafer affects the temperature gradient across its surface. As seen in Figure 6, the wafer’s temperature gradient is highly sensitive to angle, which is difficult to recover by increasing the wafer’s height.
Figure 8: Lines of constant alpha (in radians) on a graph of height above the hot plate versus temperature difference across the surface of the silicon wafer.
EXPERIMENTAL RESULTS AND DISCUSSION Figure 9 below shows the experimental validation data in comparison with the predicted result by the model. The time to steady state is accurately predicted by the model.
Figure 9: Wafer temperature versus time. Notice that the predicted steady state temperature (line) is higher than the
actual steady state temperature. The wafer is at a height of 66 microns.
In the experimental setup the hot-plate undersized with respect to the silicon wafer. This cause radial fin effects in the fin dynamics, which lowered the actual steady-state temperature of the wafer. In an industrial setting it is assumed that the heating surface will be at least as large as the wafer, and in this case the experimental results will more closely match those predicted by the model. Preliminary calculations were done with fin resistances in play to determine if the severity of the effect. In Figure 10 below a more experimentally friendly model was created which takes fin dynamics into account.
Figure 10: The steady state temperature of this graph was decreased by adding a fin energy loss term into the governing energy equation.
CONCLUSIONS As the model suggests and the experimental results confirm, the wafer temperature is very sensitive to both height above the hot-plate as well as angle to the hot plate. Even a slight angle can cause a large temperature gradient across the surface. The model accurately predicts the rise time and steady state temperature of the wafer in an industrial setting, REFERENCES
Cravalho, E., Smith, Jr., J.L., Bisson II, J.G., McKinley, G.H., 2005, “Thermal Fluids Engineering,” MIT/Pappalardo Series in Mechanical Engineering, Oxford University Press.
Incropera, F.P., DeWitt, D.P., 2002, Fundamentals of Heat
and Mass Transfer, John Wiley and Sons, Hoboken, NJ, p811.
