Modeling and Temperature Control of Rapid Thermal Processing

Modeling and Temperature Control of
Rapid Thermal Processing
Eyal Dassau, Benyamin Grosman and Daniel R. Lewin †
PSE Research Group, Dept. of Chemical Engineering, Technion I. I. T.,
Haifa 32000, Israel
Abstract.
In the past few years, Rapid Thermal Processes (RTP) have gained acceptance as
mainstream technology for semi-conductors manufacturing. These processes are
characterized by a single wafer processing with a very fast ramp heating of the silicon
wafer (up to 200 o C/sec). The single wafer approach allows for faster wafer processing
and better control of process parameters on the wafer. As feature sizes become smaller,
and wafer uniformity demands become more stringent, there is an increased demand
from rapid thermal (RT) equipment manufacturers to improve control, uniformity and
repeatability of processes on wafers. In RT processes, the main control problem is that of
temperature, which is complicated due to the high non-linearity of the heating process
(radiation), process parameters that change significantly during a single wafer process
and between processes, and difficulties in measuring temperature and edge effects. In
work carried out in cooperation with Steag CVD Systems, we developed algorithms for
steady state and dynamic temperature uniformity. These algorithms are designed to
ensure uniform temperature in RTP equipment. The steady-state algorithm involves the
reverse engineering of the required power distribution, given a history of past
distributions and the resulting temperature profile. The algorithm for dynamic
temperature uniformity involves the development of a first-principles model of the RTP
chamber and wafer, its calibration using experimental data, and the use of the model to
develop a controller.
Keywords: Rapid thermal processing (RTP); Non-linear Model Predictive Control
(NMPC); Genetic Algorithm (GA); Genetic Programming (GP).
Submitted to Computers and Chemical Engineering, July 2005.
† Author to whom all correspondence should be addressed. Email: dlewin@tx.technion.ac.il. http://pse.technion.ac.il
1

1. INTRODUCTION
Integrated circuits are at the heart of all electrical appliances. These are based mainly on
semiconductor devices, which are fabricated in a sequence of batch chemical processes such
as chemical vapor diffusion (CVD), oxidation, nitration, ion implantation, and annealing.
Incremental improvements in integrated circuit technology, together with increased
performance demands from semiconductor devices, have gradually led to requirements that
the variation in the key quality variables be reduced and to the increased yields afforded by
larger diameter silicon wafers. This in turn has increased the reliance of the microelectronics
industry on advanced process control (APC) strategies, and to seek new fabrication methods.
Thermal processes play an important role in the fabrication of semiconductor chips in
the microelectronics industry. Shrinking device dimensions to the sub-micron range make
stringent demands on the thermal processing of semiconductor wafers. The wafer should
spend the minimal time close to the process temperature to reduce the solid-state diffusion of
dopants introduced in the previous fabrication steps. The drive to reduce this “thermal
budget” and the tight quality demands gave birth to a new technology: single wafer
processing (SWP). SWP systems must heat up and cool down quickly in order to compete
economically with multi-wafer technology, and this has led to the development of rapid
thermal processing (RTP).
RTP involves the processing of single silicon wafers, and is used for various
processes for the manufacture of semiconductor devices, such as rapid thermal annealing
(RTA), rapid thermal oxidation (RTO), rapid thermal chemical vapor deposition (RTCVD)
and rapid thermal nitration (RTN). A typical RTP operating cycle consists of three phases:
(1) rapid heating to the desired operating temperature, (2) the processing phase, in which
temperature is held constant, and (3) rapid cooling to ambient conditions. The drive to reduce
the “thermal budget” makes RTP an attractive alternative to conventional methods of thermal
processing. This goal forces a stiff constraint on the control of the process temperature and
thickness uniformity. As feature sizes become smaller, and wafer uniformity demands
become more stringent, there is an increased demand from rapid thermal (RT) equipment
manufacturers to improve control, uniformity and repeatability of processes on wafers. In RT
processes, the main control problem is that of temperature regulation, which is complicated
due to the high non-linearity of the heating process (radiation), process parameters that
change significantly during a single wafer process and between processes, and difficulties in
2

measuring temperature and edge effects. The controller should be able to track ramped set
point trajectories of between 50 and 200 o C/sec, and subsequently, to maintain a uniform
temperature across the wafer. The rapid heating is made possible using clusters of high
powered lamps, with the lamp configuration defining the structure of the RTP system, and the
number of pyrometers or other temperature measuring techniques defining the character of
the control configuration that can be implemented on the RTP system.
To meet the control objectives, a number of alterative approaches have been
suggested. The proposed strategies involve decoupled decentralized control (Balakrishnan et
al., 1999), learning control (Yangquan et al., 1997 and Jin Young and Hyun Min, 2001),
adaptive control (Morales and Dahhou, 1998), Internal model control (IMC: Schaper et al.,
1994), model predictive control (MPC: De Keyser and Donald, 1999), nonlinear MPC
(NMPC: Breedijk at al., 1993), and quadratic dynamic matrix control (QDMC: Breedijk at
al., 1994). A review on the state-of-the-art in RTP control is provided by Edgar et al. (2000).
In this paper, two alternative control strategies are developed for temperature
uniformity in RTP. The first strategy is applicable in cases where the RTP system has only a
single temperature measurement positioned at the center of the wafer, and involves the
optimal selection of two sets of heating lamp zone ratios, one of which is applied in the rapid
heating stage, and the second in the constant-temperature processing stage. A robustly tuned
PI controller ensures that the measured center-point temperature is maintained on its setpoint
during the entire trajectory. The second strategy, applies nonlinear model predictive control
to regulate the entire RTP temperature trajectory for uniformity. This option can only be
implemented in RTP systems in which a number of temperature measurements are available,
but gives superior performance.
This paper is structured as follows. Section 2 provides a brief description of the
commercial RTP system setup used in this work. Next, the mathematical model developed to
describe the process and its calibration is detailed. Sections 4 and 5 describe the algorithm
developed for temperature uniformity, and its application in a single-loop control strategy.
Finally, in Section 6, we describe the application of a novel nonlinear model predictive
controller for temperature uniformity, relying on empirical discrete models generated using
genetic programming (Grosman and Lewin, 2002).
3

2. PROCESS DESCRIPTION
This work was carried out in cooperation with Steag CVD Systems, a vendor of RTP
processing chambers. Steag’s Integra Pro RTP-CVD system, used in this study and shown
schematically in Figure 1, involves a heating system consisting of 64 1.5 kW halogen lamps,
arranged in five concentric banks, each of which can be adjusted independently to assist the
uniform processing of 8″ wafers.
Table 1 provides details of the arrangement of the lamps in these five banks,
henceforth referred to as zones, and their locations. A single pyrometer is positioned at the
center of the wafer, which sends temperature measurements to a PID controller that
manipulates the total energy supply to the array of lamps. The total energy supplied in each
zone therefore depends on the number of lamps in the zone, the fraction of the total power set
for the zone, and the total power defined by the PID controller. Thus, for example, if zone
two, consisting of six lamps, is set to have a power fraction of 30%, and the controller sets a
total power of 50% of the full range, the actual power generated would be 1,500×6×0.3×0.5 =
1,350 W.
The reaction chamber is closed from above by a quartz window, which allows for
radiative heating of the wafer by the heating lamp array, while at the same time permitting
wafer processing under vacuum. The heating lamps and chamber are cooled to a temperature
of 15 o C by channels conveying chilled water through the chamber matrix, and the quartz
window is cooled by a stream of air. The wafer is placed by a robot arm on a support,
positioned 99.5 mm under the heating array, which spins during processing to enhance
uniformity. The system is computer-controlled, allowing for automation and data-acquisition.
In the commercial system, an optical pyrometer provides the only temperature measurement,
which is positioned underneath the center of the wafer. However, on the experiments carried
out to enable the development of a mathematical model, the radial wafer temperature profile
is measured by running the equipment using TC wafers of various emissivities, on which five
thermocouples are attached radially, at the center, and positioned 2.5, 5, 7.5 and 9.5 cm from
the center.
4

Concentric Lamp Array
Quartz window
Measuring system - TC Wafer
Optical Pyrometer
Figure 1 - Integra Pro RTP-CVD system.
Table 1. Heating System Arrangement for Steag’s Integra Pro RTP-CVD System.
Zone Number of Lamps Rin (mm) Rout (mm)
1 1 0 13.5
2 6 18.5 45.5
3 12 50.5 77.5
4 19 82.5 109.5
5 26 114.5 141.5
3. MODELING AND CALIBRATION OF THE RTP SYSTEM
The first step in achieving a control scheme involves the development of a first-principles
model of the RTP chamber and wafer. This is calibrated to match experimental data using
non-linear regression. The dynamic model, expressed as a partial differential equation, has
been approximated by finite differences. It is solved numerically using the implicit Crank
Nicholson scheme with some modifications to handle the non-linear temperature terms that
were included explicitly for simplicity. The subsequent control work relies on this model as a
surrogate for the real RTP process at Steag CVD Systems.
Modeling the Wafer.
An energy balance on the wafer in the RTP chamber gives:
∂T
ρC
= qk
+ qc
+ q
∂t
where ρ, C and T are the wafer density, specific heat and temperature, t is the time, and qk, qc
and qr are the heat transfer rates by conduction, convection and radiation, respectively. A
5
r
(1)

number of simplifications can be made the model that describes the specific equipment under
investigation. Since the system has rotational symmetry, the full three-dimensional model (in
r, θ and z) can be reduced to a two-dimensional one (in r and z). Furthermore, since during
processing, the wafer is positioned on a rotating plate, the entire wafer is represented by
radial chord, leading to a two-dimensional model in Cartesian coordinates (x, z). Furthermore,
it is assumed that the quartz window and cooling water temperatures are constant.
Using these assumptions, the energy balance is expressed as:
∂T
∂ ⎛ ∂T
⎞ ∂ ⎛ ∂T
ρ C ( T)
= ⎜k
( T)
⎟ + ⎜k
( T)
∂t
∂x
⎝ ∂x
⎠ ∂z
⎝ ∂z
The following boundary conditions apply:
T = T at t = 0
init
∂T
k ( T ) = 0 at x = 0
∂x
∂T
k( T ) = −he
wall
∂x
( T − T ) at x = R
4 4 ( T − T ) + h ( T − T ) at z = 0
∂T
k( T ) = F1
ε 1(T)
σ
cool w cool
∂z
⎛ x ⎞
hw = hi
+ ( ho
− hi
) ⎜ ⎟
⎝ R ⎠
( x,t)
( x)
6
4
⎞
⎟
⎠
4 4 ( T − T ) at z Z
∂T
q
k( T ) = ε ( T ) − F ( T )
(8)
2ε
σ
a =
∂z
A
where T is the wafer temperature, Tinit is the initial wafer temperature, he, hi, ho, and hw, are
the convective heat transfer coefficients at the wall, at the center and edge of the wafer, and
the overall coefficient, respectively, Twall is the wall temperature, Tcool is the cooling water
temperature, and Ta is the temperature of the quartz window, C(T) is the heat capacity, k(T) is
the thermal conductivity, σ is the Stephan-Bolzman constant, ε1(T) and ε1(T) are the
emissivities of the lower and upper wafer surface, respectively, F1 and F2 are reflective
coefficients, x and z are the radial and depth coordinates, Z is the wafer thickness, R is the
radial chord length, A(x) is the effective wafer area at the chord position x, and q(x, t) is the
heat transfer rate to a given point at x.
The initial condition in Eq. (3) defines the initial wafer temperature. The boundary
condition in Eq. (4) expresses the symmetry at the center of the wafer (x = 0). At the edge of
the wafer, at x = R, the boundary condition in Eq. (5) relates the conduction in the wafer with
heat losses to the reactor walls by convection. For the z direction, the boundary condition in
(2)
(3)
(4)
(5)
(6)
(7)

Eq. (6) at z = 0 (i.e., below the wafer), relates the conduction in the wafer to heat losses to the
surroundings by radiation and convection. The overall convectional heat transfer coefficient
in Eq. (7) is that proposed by Lord (1988), which accounts for spatial variations. Finally, the
boundary condition in Eq. (8) at z = Z (i.e., facing the heating lamps), relates the heat transfer
in the wafer to the heat supplies from the heating lamps and the heat losses to the quartz
window.
The wide range of operating temperatures (between 25 and 1200 o C) affects the
thermal properties of the silicon wafer. Thus, the effect of temperature on the thermal
conductivity and heat capacity are accounted using the correlations of Borisenko et al.
(1997):
-1.
12
k(
T ) = 802.
99 T
⎡ W ⎤
⎢cmK
⎥
⎣ ⎦
300 −1683K
C
⎡ J ⎤
⎢ ⎥
⎣ gK ⎦
300
The correlation of Virzi (1991) is employed to describe emissivity:
−4
( T ) = 0.
641+
2.
473×
10 T
> K
ε(
T ) = 0.
2662 + 1.
8591T
7
−0.
1996
e
25
1.
0359×
10
-
8.
8328
T
Since the wafer density does not depend strongly on temperature, it was taken as constant (ρ
= 2,330 kg/m 3 ). Furthermore, the weak temperature dependence of the thermal conductivity
and the homogenous nature of the silicon wafer allow the energy balance in Eq. (2) to be
further approximated:
Modeling Heat Transfer to the Wafer.
2 2
∂T
⎛ ∂ T ∂ T ⎞
ρ C(
T)
= k(
T ) ⎜ + ⎟
2 2
∂t
⎝ ∂x
∂z
⎠
The main mechanism that raises the wafer temperature to the desired processing level
is radiation from the lamp array. The lamp array, which is located directly above the wafer, is
arranged in five concentric rings of heating zones. Heat transfer by radiation depends on the
radiation-transfer medium, its wavelength, and the system geometry. An ideal model for heat
transfer by radiation in RTP needs to account for both diffusive and reflective radiation heat
transfer. However, for control purposes, the heat transfer mechanism can be significantly
simplified. Firstly, the radiating body is assumed to be a diffusive gray system, meaning that
the surface emissivity, ε,
and observativity, α, do not depend on the ray direction. A gray
body is a body whose emissivity and observativity are independent of the wavelength, but
may be functions of temperature. This means that each surface will radiate as a black body at
(9)
(10)
(11)
(12)

all wavelengths depending only on its temperature (Siegel, 1981). Secondly the lamp power
must be related to the heat flux transmitted to the wafer, expressed in terms of view factors.
The view factor defines the radiation fraction which is transferred from one surface to the
other and is deriving from the system geometry using the following equation:
F
=
1
∫∫
cosθ
cosθ
dA dA
1 2
1−
2
2 2 1
(13)
A1
πS
AA 1 2
where F1-2 is the radiation fraction transmitted from surface 1 to surface 2, A1 and A2 are the
surface areas respectively, θ1 and θ2 are the normal angles at the surfaces, and S is the
distance between the surfaces. The literature abounds with equations of view factors for
different bodies and systems. In this model, the view factors connect the lamp array to a
differential piece of wafer, which is expressed in terms of a differential definition of the
radiation view factor:
cosθ1
cosθ
2
dF1−
2 =
dA
2
2
(14)
πS
The lamp array is divided in a natural way to five heating rings where each ring radiates to
the wafer slice as shown in Figure 2:
x 0
z 0
(x,y,z)
S
y 0
(x 0 ,y 0 ,z 0 )
8
rout
rin
(b) (a)
Figure 2 – The system geometric for view factor calculation. (a) Differential annular ring from rin to rout.
(b) The plane (x,y,z) is the lamp array plan where (x 0 , y 0 ,z 0 ) is the wafer plane.
By integrating Eq. (13) on a differential annular heating ring for each ring the following
relation is found:
F
1−d
2
⎛
⎜
⎜
1
=
⎜
2 ⎜
⎜
⎜
⎝
x
2
2 2 2
( x + z + r )
in
+ z
2
1−
− r
2
in
4x
2 2
rin
2 2 2
( x + z + r )
in
2
−
x
2
2 2 2
( x + z + r )
out
+ z
2
1−
x
− r
2
out
4x
y
2 2
rout
⎞
( ) ⎟⎟⎟⎟⎟⎟
2 2 2 2
x + z + r
Thus, the heating ring power is related to the heat flux to a differential wafer slice:
out
⎠
(15)

q( x,
t)
= α ⋅
5
∑
j=
1
F
j−
x
( x,
r , r
9
in
out
) ⋅ q(
j)
where j is the ring number, q(x,t) represent the heating ring powers multiplied by their view
factor and α is a tunable parameter that is calibrated against the experimental system at Steag,
which determines the radiation heat transfer that is not diffused.
Model solution
The model was discretized using finite difference approximations and solved
numerically using the implicit Crank Nicholson method. However, nonlinear terms
introduced by radiation in the boundary conditions and temperature dependence of the
process parameters would significantly increase the computation time. To avoid these
problems the nonlinear terms appear explicitly in the approximations used (Haimovich,
2000).
Model Calibration
To enable the use of the developed model for improved design and control of the
Steag RTP setup, several of the parameters in the model need to be calibrated to match
existing process conditions. The parameters are the heat transfer coefficients, top and bottom
emisivity, the F1 and F2 reflective coefficients and α, the tunable parameter that accounts for
undiffused radiation heat transfer. Since our objective is to use the model to drive the
controller for the lamp power during the heating and the process portions of a processing
cycle, and not the cooling stage that involve gas flow to the process for cooling, the heat
transfer by convection is neglected and these parameters are set to zero in the model. The
parameters were calibrated by non-linear regression using a genetic algorithm (GA) as
described by Lewin (1996). The GA allows a wider initial population in the optimization than
a conventional optimization with single initial guess for each tunable parameter. Figure 3
shows comparisons between the model and experimental data at the five wafer locations for a
typical run, for a desired set point temperature of 750 o C. The GA was driven by a desire to
reduce the summed squared prediction error, computed using:
1
( ) 2
= ∑ , − mod,
= 1
n
SSE T dat i T i
n i
where SSE is the summed squared error, T dat, i is the i'th temperature measurement, mod,i
(16)
(17)
T is
the i'th temperature as predicted by the model, and n is the number of measurements. The

model fit is acceptable, with the largest SSE being less then 1.6, equivalent to an average
magnitude of one degree C.
T [K]
T [K]
T [K]
1200
1000
800
TC. # 1
600
0 20 40 60
t [sec]
80 100 120
1200
TC. # 3
1000
800
600
0 20 40 60 80 100 120
t [sec]
1200
TC. # 5
1000
800
600
0 20 40 60
t [sec]
80 100 120
10
T [K]
T [K]
1200
1000
800
TC. # 2
(1) (2)
(3)
(5)
600
0 20 40 60
t [sec]
80 100 120
1200
TC. # 4
1000
800
600
0 20 40 60
t [sec]
80 100 120
Model
Data
Figure 3 – Model results against experimental data: (1) Temperature measurement at 9.5 cm from wafer
center; (2) Temperature measurement at 7.5 cm from wafer center; (3) Temperature measurement at 5
cm from the wafer center; (4) Temperature measurement at 2.5 cm from the wafer center: (5)
Temperature measurement at the wafer center.
Figure 4 presents the normalized temperature profile against time at the five measurement
points. These results show the same non-uniformity as was seen in the real RTP system both
in the rapid temperature raise and the steady state.
(4)

|∆ T| = | Tc(i)-Tc(1) | [ o C ]
12
10
8
6
4
2
0
0 10 20 30 40 50
Time [sec]
60 70 80 90 100
Figure 4 – Temperature transients at the measurements points.
Model Validation
11
Tc(1)
Tc(2)
Tc(3)
Tc(4)
Tc(5)
The last step in approving a model of a physical system is to test whether this model
shows the same dynamic behavior as expected from a study of the literature. This was
accomplished by running several step test simulations to estimate the linearized process gain,
with results presented in Figure 5. Schaper et al. (1994) and Kailath et al. (1996) observe that
the process gain decreases with temperature rise and indeed the RTP model shows this
relation. Furthermore, linearizing the RTP system dynamics leads to a first order model, in
which the process gain is inversely proportional to the third power of temperature. Indeed, as
seen in Figure 5, the step test results indicate the empirical relationship:
K = -3.664⋅ log ( ) + 32.86
(18)
p
T
The small power difference between the empirical and the expected value of −3 is explained
by system nonlinearities, other simplifications and measurement error.

log ( K p )
8.2
8
7.8
7.6
7.4
7.2
7
6.8
6.6
6.4
6.2
6.75 6.8 6.85 6.9 6.95 7 7.05 7.1 7.15 7.2 7.25
log (T)
Figure 5 – The effect of temperature on process static gain.
.
4. ALGORITHM FOR TEMPERATURE UNIFORMITY
12
R 2 = 0.9747
log(T)
linear
In the following, a solution for the temperature uniformity problem is suggested, and
demonstrated in concert with a feedback control scheme on a simulation of the RTP
equipment at Steag CVD Systems. This solution is designed to ensure uniform CVD of
substrates grown in RTP equipment. Our uniformity algorithm involves the reverse
engineering of the required power distribution, given a history of past distributions and the
resulting temperature profile. The algorithm has been realized in MATLAB ® , and a userfriendly
GUI has been developed to make it easy to use.
The uniformity algorithm is based on a linear approximation formulated in terms of
deviation variables of the temperature, and the power from a base case profiles that are
supplied by the user. Since we are using deviation variables and we must set five power
distributions, the user must supply at least six (independent) sets of data. The target
temperature profile is set according to the following:
=
T − T
(19)
T sp
0
0

where T sp is the target profile, T 0 base case temperature profile and T 0 is the average of
T 0 . The target temperature profile was defined in this way to account for temperature non-
uniformity in the base case. We assume a linear model of the following structure:
Y = C ⋅ P
[ T 1 − T 0 � T 2 − T 0 �…
T T 0 ]
[ P − P � P − P �…
P P ]
Y = � n −
P = � n −
1
0
2
where Y is matrix of temperature profiles in terms of deviation variables, P is the power
distribution matrix in terms of deviation variables and C is the linear model coefficient
matrix that is computed by multiple linear regression:
T
−1
T T
( ( ) )
C = P P P Y
where T is the matrix transpose. Having estimated a linear approximation relating power
distribution to temperature profiles, the next step is the minimization of an objective function,
selected according to how the wafer was measured. Thus, for data measured radially, the
objective function needs to be suitably weighed:
2 2
∑ ( r i − r i−1
)( C ⋅ P opt − T )
min sp
Popt
Alternatively, if elipsometrically-measured data is used, no weighting is required:
2
min ( C ⋅ P opt − T sp )
Popt
∑
In Eqs. (24) and (25), P opt is the optimal power distribution. Since Steag required that the first
measurement point measured by the pyrometer should be equal to the base profile, the
optimization is carried out on four power values, with the fifth being fixed using the linear
model.
Algorithm validity
The basic assumption behind the algorithm is that a linear model based on
temperature and power deviation variables will fit the operating conditions. Since the process
is in fact nonlinear, the optimized power distribution obtained in the first iteration is only an
approximation, and will probably result in a temperature profile that is not sufficiently
uniform. However, running the algorithm iteratively, and adding new information as it
becomes available, will lead to convergence. Figure 6 shows the convergence of the
algorithm around set point temperature of 700 o C in six runs with a STD of 2 as a stopping
condition.
0
13
T
2
0
(20)
(21)
(22)
(23)
(24)
(25)

∆T = T set point -T mean [ o C]
10
5
0
1 2 3 4 5
0
6
Number of Iterations
Figure 6 – Algorithm convergence.
The flow diagram for the proposed algorithm is shown in Figure 7, noting that the same
algorithm can accept any other measurement which represents wafer uniformity as an input,
(e.g., elipsometer readings).
14
8
6
4
2
STD

Display optimal
power
Figure 7 – Flow diagram of algorithm.
5. SINGLE-LOOP CONTROL
Process
data and
power
Number of data
sets
Radial or spiral
Measurement
Insert or load data
files and power
distribution
Is the data
singular ?
No
Calculate mean
and STD
Auto set base
Is the base o.k.
Yes
Generate T sp
vector
Generate Y and P
perturbation
matrixes and
Calculate C matrix
Start optimization
15
Yes
No
Replace the i th
data row
Set base
A simulator of the Steag RTP system was developed using MATLAB ® and SIMULINK ® to
assist in the controller design and testing. The Steag control system relies on a PID controller,
which controls the total power to the lamp array (0-100%), with the power distribution of the

heating zones being prespecified. To improve on the performance of the linear controller, the
system is actually run in open loop until the center point temperature attains a temperature
referred to as "cut-back low," at which point the controller is activated to bring the wafer
center point temperature to the set point. It is noted that with this strategy, the predefined
zone ratio is the only means to attain temperature uniformity.
Controller Design
The purpose of the controller is to assure temperature uniformity in the RTP system
mainly in the fast heating the ramp and at temperature steady-state, with the main
requirements being to minimize the overshoot and rise time of the trajectory. The controller
was designed based on Internal Model Control principles (Rivera et al., 1986). A PI
controller was selected since this system has a relatively small delay time relative to the
characteristic time (a delay of 0.2 sec and a characteristic time of 16 sec are typical). In such
cases, there are no advantages in employing PID control. The solution to the uniformity
problem was addressed in two steps: (a) The PI controller was designed to bring the center
temperature to its set point; (b) two distinct optimal heater power zone distributions were
generated: one for the fast ramp and one for the operating temperature. These power
distributions were optimized using the proposed uniformity algorithm.
Control system tuning
The algorithm for the tuning of the control system is shown in Figure 8. This provides
uniform temperature profile in the fast heating zone and in the process temperature, and is
divided into three stages:
(a) PI Controller tuning is defined, including controller parameters such as open loop
power, cut back low, temperature set point, initial zone ratio, process time, zone ratio
temperature switch and controller parameters. After completing this step, the center
point of the wafer will be at the set point.
(b) The uniformity algorithm is invoked to ensure temperature uniformity in the process
operating temperature (at the temperature steady-state).
(c) The uniformity algorithm is again invoked, to derive the optimal power distribution in
the fast ramp, to further improve the temperature uniformity in the fast ramp and the
transient from the open loop to the close loop control by reactivating the uniformity
16

algorithm to find optimal zone ratio for the open loop part of the process. The following
case studies illustrate how the algorithm works at various desired temperature levels.
Zone Ratio
Switch
Initial Zone Ratio
Tune the
controller
Run simulation
No
Is the STD < defined
Is the over
shoot too large
Yes
Yes
Yes
No
Temperature Set
Point
No
Invoke the
uniformity tool on
the steady state
temperature
profile
New steady state
zone ratio
Set the s.s. and
ramp zone ratio
Run simulation
Is the STD < defined
Raise the cut
back low
Open Loop
Power
Start simulation
Is the
controller
tuned
Is the target
temperature profile
uniform
Define Stopping
condition on the STD
Invoke the
uniformity tool
on the knee
temperature
profile
New ramp zone
ratio
Set the ramp
zone ratio
Run simulation
Figure 8 – Flow diagram calibrating the control system.
No
No
17
Cut Back Low
Yes
Define Stopping
condition on the STD
No
Yes
Process time
Is the over
shoot too large
Is the over
shoot too large
No
Is the knee area
temperature profile
uniform
Yes
(1)
Yes
(2)
Controller
parameters
Yes
(3)
No
Raise the cut
back low
Raise the cut
back low
Done

Case study – Process temperature target of 700 o C
T [ o C]
T [ o C]
T [ o C]
800
600
400
200
0
0 10 20 30 40 50 60 70 80 90 100
800
t [sec]
600
400
200
0
0 10 20 30 40 50 60 70 80 90 100
800
t [sec]
600
400
200
0
0 10 20 30 40 50
t [sec]
60 70 80 90 100
Figure 9 – Base temperature profile for set point of 700ºC: (1) Base profile; (2) Profile after setting the
steady state zone ratio; (3) Profile after setting the ramp zone ratio.
In this case study, the temperature set point for the process is 700 o C , the transient
temperature from open to close loop is 680 o C and arbitrary zone ratio powers of
(zr1=0.0446, zr2=0.827, zr3=0, zr4=0.387, zr5=0.95). The first plot in Figure 9 shows the
initial profile of the system which, as can be seen, exhibits uniformity problems in the target
temperature as well as in the fast heating zone. An improved temperature distribution can be
seen in the second plot of Figure 9, which shows the result of implementing a new zone ratio
of (zr1=1, zr2=0.408, zr3=0, zr4=0.795, zr5=1) as calculated by the uniformity algorithm. This
new trajectory achieves a uniformity with an STD of less than 2, as seen also in Figure 10.
This solves the uniformity problem in the steady-state temperature region, but there is still a
problem in the fast heating stage (the transient from open to closed loop), this problem is
resolved by re-running the uniformity algorithm for the fast heating and setting the zone ratio
powers to (zr1=0.17, zr2=0.483, zr3=0, zr4=0, zr5=1) and the transient temperature (from open
to closed loop) to 650 o C. The proposed control presents an improved temperature profile as
can be seen in the third plot of Figure 9, which indicates that the uniformity of the
18
(1)
(2)
(3)

temperature trajectories are indeed significantly improved, but at a cost of a longer start-up
time.
∆T = T set point -T mean [ o C]
10
5
0
1 2 3 4 5
0
6
Number of Iterations
Figure 10 – Convergence of the algorithm for set point of 700ºC: Temperature STD (right) and mean
temperature deviation from the set point (left).
This control strategy, combining a PI controller that is tuned for a specific temperature set
point and a uniformity algorithm, was tested over a wide range of temperature set points with
good results. The main advantages are that the operator needs to perform only few iterations
to tune the controller to a desire operating temperature, where the trade off between
uniformity and the time that is needed to reach the set point is a degree of freedom that
depends on the process. Furthermore, this control scheme can be easily implemented on a
system that has only one temperature measurement, which still providing acceptable
performance.
6. MULTIVARIABLE CONTROL
To improve the control of RTP systems and to meet tighter uniformity specifications, to
reduce the time needed to acquire the set point, and otherwise improve the flexibility of the
process, there is a need for control systems more advanced than that that developed in the
19
8
6
4
2
STD

previous section. However, such a system will require more than one temperature
measurement. To enable the control system to work at different process temperatures without
any need for retuning, we implemented a non-linear model predictive control (NMPC)
system based on non-linear models derived using Genetic Programming (GP), as described
by Grosman and Lewin (2002). The inputs for the controller are three temperature
measurements on the wafer radius. In a real system, these would have to be optical
pyrometers, as the temperature measurement should not interfere with the wafer rotation.
Model Formulation: The first step was to formulate GP models for MPC. In our study, these
models were based on the mathematical model of the Steag RTP system which represents the
real system. By investigating the mathematical model, a response time in the order of a few
seconds were identified, leading to the sampling time selection of half a second. This
sampling rate will provide the necessary information for constructing the GP models. The
system was excited by 12 steps of five seconds in the inputs, in this case, the power ratio for
the heating rings assuming full total power. However, since it is of interest to bring the
system to a given temperature range, this calls for a specific heat loading. Thus, the
perturbation steps are arranged in sets of threes where the first perturbation in each set is free
to activate all the heating rings while limiting the fourth and fifth heating rings to 15% of full
power. In the second and third perturbations, only one heating ring is activated randomly.
Figure 11 shows a typical perturbation sequence derived for the GP and Figure 12 presents
the resulting temperature profiles obtained.
20

% Power
% Power
% Power
% Power
% Power
1
0.5
0
1
0 10 20 30 40 50 60
0.5
0
1
0 10 20 30 40 50 60
0.5
0
1
0 10 20 30 40 50 60
0.5
0
1
0 10 20 30 40 50 60
0.5
0
0 10 20 30
t [sec]
40 50 60
Figure 11 – Typical perturbation of the RTP for the GP. (1) – (5) different heating rings.
T [ o C]
1000
800
600
400
200
0
0 10 20 30
t [sec]
40 50 60
Figure 12 – Temperature profile resulting from the system perturbation.
21
(1)
(2)
(3)
(4)
(5)
x = 0
1
x = 0.01
2
x = 0.02
3
x = 0.03
4
x = 0.04
5
x = 0.05
6
x = 0.06
7
x = 0.07
8
x = 0.08
9
x = 0.09
10
x = 0.1
11

The GP models and their prediction against data are presented below:
T [ o C]
1. Model for the center of the wafer:
1000
900
800
700
600
500
400
300
200
100
T
1
+ 1
(
() i = zr4
( i −1)
+ 1.
72 ⋅ ( zr2
( i −1)
+ 1.
00045 ⋅ zr3
( i)
) +
. 0766 ⋅ zr ( i −1)
) ⋅15.
2924 + 0.
97707 ⋅T
( i −1)
+ 12.
9392
5
0
0 20 40 60
t [sec]
80 100 120
Figure 13 – The behavior of the model for the center point against the process data.
22
1
Data
Model
(26)

T [ O C]
2. Model for the second measurement point (five centimeters from the center):
900
800
700
600
500
400
300
200
100
T
2
() i = (
zr ( i −1)
+ 1.
2417 ⋅ zr ( i −1)
+ 1.
0959 ⋅ zr ( i − 2)
+
+ 1.
1334 ⋅ zr ( i −1)
Data
Model
4
5
3
) ⋅18.
2151 + 0.
97785 ⋅T
( i −1)
+ 11.
8499
0
0 20 40 60
t [sec]
80 100 120
Figure 14 – The behavior of the model for the second point against the process data.
23
2
2
(27)

T [ o C]
3. Model for the third measurement point (9.5 centimeters from the center):
T
3
() i = (
zr ( i −1)
+
+ 1.
7898 ⋅ zr ( i −1)
800
700
600
500
400
300
200
100
3
5
Data
Model
0.
532122
⋅ zr ( i −
2
) ⋅16.
9627 + 0.
97737 ⋅T
( i −1)
+ 10.
4339
24
2)
+
3
2.
0857
⋅ zr ( i −1)
+
0
0 20 40 60
t [sec]
80 100 120
Figure 15 – The behavior of the model for the third point (wafer edge) against the process data.
As can be seen, the GP models are in excellent agreement to the simulated data. It should be
noted that, following Grosman and Lewin (2002), the model predicts ten data points ahead
from one set of known inputs. Hence at every tenth data point, the model is reset to fit the
simulated data and than used to predict the subsequent ten points. Furthermore, it is
interesting to note that the GP eliminates the center heating ring from the model, probably
because it has a negligible effect on the response of the system, since the center ring consists
of a single lamp.
NMPC for the Steag RTP System: The NMPC objective function for this application is:
4
(28)

J = S
+ S
1
+ S
+ S
4
⋅
⋅
⋅
6
n
⋅
∑
i=
1
n
∑
i=
1
m
∑
m
∑
i=
1
m
2
( T () i −T
() i ) + S ⋅ T () i −T
() i
1sp
m
2
∑( 2 2sp
) + S8
⋅∑(
T 3()
i −T
3sp
() i )
i=
1 i=
1
2
2
( zr () i − zr ( i −1)
) + S ⋅ ( zr () i − zr ( i −1)
) + S ⋅ ( zr () i − zr ( i −1)
)
1
∑
i=
1
2
( zr () i − zr ( i −1)
) + S ⋅ ( zr () i − zr ( i −1)
)
4
1
1
4
7
2
5
∑
i=
1
2
2
2
[ ( T 3()
i −T
1()
i ) + ( T 3()
i −T
2()
i ) + ( T 2()
i −T
1()
i ) ]
9
i=
1
where S1-S9 are weighting functions, 1 , T 2,
T 3
n
n
2
5
25
2
5
2
+
3
n
∑
i=
1
3
3
2
+
2
+
(29)
T are temperature vectors, T 1 sp , T 2sp
, T 3sp
are
temperature set point vectors and the index i is the i’th sample point. It is noted that the
objective function consists of three main parts: (a) the sum of squared errors (SSE) between
the controlled variables and their set points, (b) the SSE of the controller movements, and (c)
the SSE of the difference between the three measured temperatures.
The prediction and control horizon are set to five and two, and the objective function
weights are set empirically to the values listed in Table 2.
Table 2– Weights values for the NMPC
Weight
Value
S1
0
S2
6
7×
10
S3
6
6×
10
S4
6
9×
10
S5
6
5×
10
Case study – Process temperature target of 700 o C
As in SISO control using PI control integrated with the uniformity algorithm, the
NMPC is tested on three temperature operating points where the target was to arrive to this
set point as fast as possible with a uniform temperature profile. Figure 16 shows an example
of the NMPC performance for a set point of 700 o C. The main benefit from NMPC is a faster
convergence to the temperature target, which has a direct affect on the overall thermal budget
of the wafer. Furthermore, the deviation of the radial temperature profile is only of the order
of one o C. The small overshot of approximately 30 o C can be overcome by changing the
objective function weights. The fact that one can minimize overshot while keeping the radial
temperature profile uniform on different working temperatures without any need for retuning,
is one of the advantages of NMPC.
S6
70
S7
70
S8
70
S9
400

T [ o C]
∆u
800
600
400
200
0
0 10 20 30
t [sec]
40 50 60
1
0.8
0.6
0.4
0.2
0
0 10 20 30
t [sec]
40 50 60
26
(1)
(2)
zr(1)=0
zr(2)
zr(3)
zr(4)
zr(5)
Figure 16 – GP-NMPC control for set point tracking of 700 ºC: (1) Temperature profile in the wafer;
(2) Controller moves.
7. CONCLUSIONS
Two distinct solutions are presented in this work:
The first one, which could be implemented directly on the Steag CVD RTP system,
involves the implementation of the uniformity algorithm and an IMC-tuned PI controller. The
operating sequence calls first for a heating stage in open loop mode until a pre-defined
temperature, at which point the feedback controller takes over. It has been observed that
significant temperature uniformity occurs both at the processing temperature and during the
fast ramp, or more precisely, in the knee region between the open loop and the close loop
phases. Our solution utilizes the uniformity algorithm to set different zone ratios for the
process region and for the ramp stage. The switch between the zone ratios is made at a predefined
temperature that shifts to reduce the heating rate if the temperatures profile shows
unacceptable overshoot. The set point tracking is achieved by the PI controller that brings the
wafer center point to the set point temperature. By using different zone ratios, the overall

temperature uniformity is kept at ± 2 ºC of the set point. This solution gave acceptable
performance at three distinct operating temperatures.
The second solution involves non-linear model predictive control (NMPC) based on
genetic programming (GP). We have decided to control three points on the wafer: the first is
at the center point as in the standard control scheme, the second is positioned five centimeters
from the center point on a radial line, and the last one is at the wafer edge. These three points
were picked relying on the observation that the highest non-uniformity is located near the
wafer edge. On the real system, all of these temperature measurements would be executed
using pyrometers, in such a way that the rotation of the wafer, common in many RTP
systems, will not affect the measurement. The strength of this approach is that the same set of
tuning parameters can control the RTP system at a range of operating temperature set points
with a very short rise time to the set point and a uniform temperature profile. Although we
have experienced overshoot, it has been observed only for a few seconds and the set point
was maintained accurately.
The two approaches have great potential for resolving real engineering problems
associated with RTP. The simple SISO technique was developed taking into account the
limitation associated with the existing RTP equipment at Steag CVD Systems, and relies on a
single on-line temperature measurement and PID control. In contrast, has been demonstrated
that a nonlinear multivariable approach can significantly improve performance, but relies on
additional on-line temperature measurements. Together, they provide a RTP control package
that represents the state-of-the-art.
REFERENCES
Balakrishnan, K. S., S. Shooshtarian, N. Acharya, P. J. Timans and R. P. S. Thakur
(1999).“Dynamic uniformity control in a rapid thermal processing system,” Advances
in Rapid Thermal Processing. Proceedings of the Symposium. (Electrochemical Society
Proceeding Vol.99-10). Electrochem. Soc.,99-10, 399-406.
Borisenko, V. E. and P. J. Hesketh (1997). Rapid thermal processing of semiconductors.
Perseus Publishing, Cambridge.
Breedijk, T., T. F. Edgar and I. Trachtenberg (1993).“A model predictive controller for
multivariable temperature control in rapid thermal processing,” Proceedings of the
1993 American Control Conference, AACC, 3, 2980-4, Evanston, IL, USA.
Breedijk, T., T. F. Edgar and I. Trachtenberg (1994).“Model-based control of rapid thermal
processes,” Proceedings of the 1994 American Control Conference . IEEE.,1, 887-91,
New York, NY, USA.
27

De Keyser, R. and J. Donald, III (1999).“Model based predictive control in RTP
semiconductor manufacturing,” Proceedings of the 1999 IEEE International
Conference on Control Applications.,2, 1636-41.
Edgar, T. F., S. W. Butler, W. J. Campbell, C. Pfeiffer, C. Bode, S. B. Hwang, K. S.
Balakrishnan and J. Hahn (2000). “Automatic control in microelectronics
manufacturing: practices, challenges, and possibilities,” Automatica, 36(11), 1567-603.
Grosman, B. and D. R. Lewin, “Automated Nonlinear Model Predictive Control using
Genetic Programming,” Comput. Chem. Eng., 26(4-5), 631-640 (2002).
Haimovich, N. (2000). "Oxidation-Oven Physical Model,” B.Sc. Final Year Project,
Technion I.I.T, Haifa
Jin Young, C. and D. Hyun Min (2001). “A learning approach of wafer temperature control
in a rapid thermal processing system,” IEEE Transactions on Semiconductor
Manufacturing, 14(1), 1-10.
Kailath, T., C. Schaper, Y. Cho, P. Gyugyi, S. Norman, P. Park, S. Boyd, G. Franklin, K.
Saraswat, M. Moslehi and C. Davis (1996). "Control for advanced semiconductor
device manufacturing: A case history," The Control Handbook. W. S. Levine.
Lewin, D. R. (1996). “Multivariable feedforward control design using disturbance cost maps
and a genetic algorithm,” Computers & Chemical Engineering, 20(12), 1477-89.
Lord, H. A. (1988). “Thermal and stress analysis of semiconductor wafers in a rapid thermal
processing oven,” IEEE Transactions on Semiconductor Manufacturing, 1(3), 105-14.
Morales, S. and B. Dahhou (1998). “Temperature uniformity in RTP using MIMO adaptive
control,” International Journal of Adaptive Control & Signal Processing, 12(3), 227-
245.
Rivera, D. E., S. Skogestad and M. Morari (1986). “Internal Model Control for PID
Controller Design,” I&EC Proc. Des. Dev., 25, 252-265.
Schaper, C. D., M. M. Moslehi, K. C. Saraswat and T. Kailath (1994). “Modeling,
identification, and control of rapid thermal processing systems,” Journal of the
Electrochemical Society, 141(11), 3200-9.
Siegel, R. (1981). Thermal radiation heat transfer. Hemisphere Pub. Corp. Edition: 2nd ed.,
Washington.
Virzi, A. (1991). “Computer modelling of heat transfer in Czochralski silicon crystal
growth,” Journal of Crystal Growth, 112(4), 699-722.
Yangquan, C., X. Jian-Xin and W. Changyun (1997). “A high-order terminal iterative
learning control scheme RTP-CVD application,” Proceedings of the 36th IEEE
Conference on Decision and Control, 4, 3771-2 vol.
28