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MODELLING AND CONTROL OF A VORTEX ARC LAMP FOR
RTP APPLICATIONS
by
Harpreet Singh Grover
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
© Copyright by Harpreet Singh Grover 2014
ii
ABSTRACT
Modelling and Control of a Vortex Arc lamp for RTP Applications
Master of Applied Science 2014
Harpreet Singh Grover
Graduate Department of Electrical and Computer Engineering
University of Toronto
The objective of this thesis is to develop a controller that can ramp the temperature of a
semiconductor wafer in a controllable fashion. The semiconductor wafers are heated using
vortex arc lamps, placed on either side of the wafer. The wafer is heated from room temperature
to an intermediate level of around 900 degrees C in a ramp wise fashion before it undergoes flash
annealing from the top surface. This thesis focuses on the control of the bottom lamps during the
process of heating to the intermediate phase. The challenge in designing this control system is
that the wafer temperature measurements are not available during the initial phase of the ramp
and also that the resulting lamp current profile should be smooth and free of fluctuations. To
achieve this, a vortex arc lamp model, a semiconductor wafer model and a suitable control
strategy has been developed.
iii
Acknowledgements
Firstly, I would like to thank my supervisor Professor Francis Dawson for his great support and
guidance throughout the course of this work. I will never be able to thank Prof. Dawson enough
for his help. It was a privilege being his student. In addition, I would also like to thank Dave Camm
from Mattson Technologies for his extremely valuable technical guidance for the project. I also
want to thank Yann Cressault from Paul Sabatier University in Toulouse, France, for his
contribution of transport and thermodynamic coefficient data. Finally, I would like to thank
Markus Lieberer, Rolf Bremensdorfer from Mattson Technologies and Adrian Amanci from ECE,
University of Toronto.
Without the help and support of these people, this thesis was not possible.
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Table of Contents
List of Figures vi
List of Tables x
List of Symbols x
1. INTRODUCTION…………………………………………………………………………...1
1.1. The Apparatus Setup………………………………………………………………….…5
1.2. The Control System ……………………………………………………………………..9
1.3. Thesis Motivation…………………………………………………………….….……..10
1.4. Thesis Objectives…………………………………………………………………..…..11
1.5. Thesis Outline……………………………………………………………………...…..11
2. LAMP MODELING…………………………………………………………………….….13
2.1. Full Lamp Model of the Bottom lamp……………………………………………..…..13
2.1.1. Region 1: Anode Interfacial Region Model………………………………….…..17
2.1.2. Region 2: Anode Constriction Region Model……………………………….…..20
2.1.3. Region 5: Cathode Interfacial Region Model ………………………………..…..21
2.1.4. Region 4: Cathode Constriction Region Model……………………………...…..23
2.1.5. Region 3: Model of the Positive Column…………………………………….....27
2.1.6. The Full Lamp Model……………………………………………………..……..32
2.1.7. Full Lamp Model………………………………………………………….……..55
3. THERMAL MODELING OF THE WAFER AND QUARTZ SUBSTRATE
HOLDER……………………………………………………………………..….………….56
3.1 Thermal Model of the Wafer………………………………………………………........56
3.2 Thermal Model of the Quartz Substrate…………………………………………...…....60
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3.3 Validation of the Combined Wafer and Quartz Substrate Models………………….…..61
4. CONTROL OF THE SYSTEM………………………………......................................…..66
4.1. Overall System with Controller…………………………………………………....…..66
4.2. Requirements of the Control System…………………………………….………...…..68
4.3. Proposed “Temperature Controller” …………………………………………....……..68
4.4. Performance of the Control System………………………………………………..…..87
4.5. Alternative/ Less Computationally Intense Version of Control System
Architecture…………………………………………………………………..…….…..90
5. CONCLUSIONS……………………………………………………………………...….…..91
References………………………………………………………………………………………..95
Appendix A: Ultra-fast Radiometer………………………………………………………….…..96
Appendix B: Derivation of the Total absorbed heat flux from incident intensity………….…..101
Appendix C: Derivation of the Net emission Coefficient……………………………………....103
Appendix D: Calculation of attenuation of incident intensity in the outer regions of the
lamp……………………………………………………………………………………………..104
Appendix E: Details of the Blocks used in the Temperature Controller…………………….....109
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List of Figures
Figure 1.1: A comparison of the various annealing techniques; where T is temperature, t is time
and d is the distance with respect to the surface of the wafer………………………………….….1
Figure 1.2: The measured temperature profile of the wafer during the RTP process……………..3
Figure 1.3: The wafer topside and bottom side temperature during the millisecond RTP. Inset
shows the zoomed in version where TI and TP are initial and peak temperatures of the wafer top
side during the flash. ……………………………………………………………………………...3
Figure 1.4: The process chamber used for semiconductor wafer annealing……………………....4
Figure 1.5: The measured current through each of the top and bottom lamps during the process.
Note that the current through the top lamps is raised prior to injecting the current pulse. This is
done to increase the arc radius and stabilize it in preparation for the flash. ……………………...4
Figure 1.6: The Process chamber in detail………………………………………………………...6
Figure 1.7: the arc lamp structure showing the circulating contents inside……………………….7
Figure 1.8: The Process chamber in detail………………………………………………………...8
Figure 1.9: Control system overview…………………………………………………………….10
Figure 2.1: The lamp and its circulating contents………….…………………………………….14
Figure 2.2: Five regions of the lamp …………………………………………………………….14
Figure 2.3: Radiation transport through a medium………………………………………………16
Figure 2.4: General structure of the region surrounding an electrode…………………………...18
Figure 2.5: Boundary conditions applied for an axisymmetric representation of the constricted
cathode region……………………………………………………………………………………26
Figure 2.6: Simulation of the cathode constriction region for a cathode spot of radius 3mm and a
current of 300A ………………………………………………………………………………….27
Figure 2.7: Cross section of the lamp where the arc (yellow), cold argon (green) and water wall
(blue) can be seen. ………………………………………………………………………....…….28
Figure 2.8: Cross section of the positive column showing the region simulated and the boundary
condition………………………………………………………………………………………....30
Figure 2.9: Cross section of the lamp showing its temperature profile as predicted by the arc
model for a current of 300 A. Temperatures are in Kelvin………………………………………31
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Figure 2.10: Power density balance in the lamp estimated by the arc model for a current of 300
A. Base Power Density = 81MW/m3…………………………………………………………….31
Figure 2.11: Experimental and corrected estimated model (positive column) resistance for the
shown current profile during a test. Base Current = 450A, Base Temperature = 12,000K……...37
Figure 2.12a: Experimental and corrected estimated model voltage for the lamp current
shown…………………………………………………………………………………………….38
Figure 2.12b: Experimental and corrected estimated model voltage for the lamp current
shown…………………………………………………………………………………………….38
Figure 2.13a: Radius of the Isothermal portion of the Arc vs Current…………………….…….39
Figure 2.13b: Arc temperature profile for various currents..…...…………………………….….39
Figure 2.14: Isothermal Arc core temperature variation with Current…………………….…….40
Figure 2.15: Experimental Radiative Efficiency of the Lamp………………………………..….41
Figure 2.16: The attenuation of emitted radiation by various layers outside the arc. ………..….44
Figure 2.17: The attenuation of emitted radiation by various layers outside the arc. ………..….45
Figure 2.18: The ratio of power in each spectral band to the total power in all bands, as a
function of isothermal arc temperature. Plotted using the NEC (corresponding to an arc radius
(Rp) = 5mm) for each band ……………………………………………………………………...47
Figure 2.19: Radiative efficiencies determined from experiment and modified full lamp model
………………………………………………………………………………………………...….51
Figure 2.20a: Radiative efficiency of the attempted model and the experimental………...…….51
Figure 2.20b: Total Lamp Voltage of the attempted model and the experimental……………....52
Figure 2.21: Cross sectional view of the flow pattern inside the lamp………………………….54
Figure 3.1: The process chamber ……………………………………………………………….56
Figure 3.2: Heat flux balance at the wafer’s surface…………………………………………….58
Figure 3.3: Block diagram of the full system simulation setup………………………………….61
Figure 3.4a: Experimental and model estimated wafer temperature profiles for a 60C/sec ramp
assuming an initial wafer temperature of 255C………………………………………………….62
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Figure 3.4b: Experimental and model estimated wafer temperature profiles for a 70C/sec ramp
for an initial wafer temperature of 132C………………………………..………………….…….63
Figure 3.4c: Experimental and model estimated wafer temperature profiles for a 130C/sec ramp
assuming an initial wafer temperature of 230C………………………………………………….63
Figure 4.1: Overall system block diagram………………………………………………….…….67
Figure 4.2: The step response of the full current loop …………………………………….…….68
Figure 4.3: The step response of the first order low pass equation 4.1…………………….…….68
Figure 4.4: Architecture of the proposed control system………………….……………………..72
Figure 4.5: Full simulation system setup………………………………………………….….….73
Figure 4.6: Lamp voltage and current, when a step current of 225A is applied to the Lamp
Model…………………………………………………………………………………………….74
Figure 4.7: Lamp voltage and current when the outer loop (ICM) is run at the same frequency as
the Lamp Model and controls the lamp current for wafer temperature tracking …………....….75
Figure 4.8: Proposed Temperature Controller with the Current Conditioning block added…….76
Figure 4.9: Lamp voltage and current when the outer loop (ICM) is engaged but the current
setpoint signal is ramp rate limited to 1000A/s. …………………………………………..…….76
Figure 4.10: The impact of limiting the Lamp Current ramp rate to 1000A/s is observed as a
small lag in tracking the requested temperature. ………………………………………….…….77
Figure 4.11: Case 1: Simulation of the ideal scenario where the models in the ICM are assumed
to perfectly describe the real system. ……………………………………………………...…….79
Figure 4.12: Case 2: Simulation of the non-ideal scenario where a 10% higher wafer absorptivity
was assumed for the wafer model in the ICM as compared to the real wafer. ………………….80
Figure 4.13: Case 2: The simulation of the non-ideal scenario with the reset option implemented.
As seen there is no current spike upon engagement of the PI controller at t=2.45sec. ………….82
Figure 4.14: Case 3: Simulation of the ideal scenario where the initial wafer temperature is
unknown. The transition to closed loop is still smooth. ………………………………….…….83
Figure 4.15: Simulation showing the scenario where the P controller’s (Kp = 35) request exceeds
1000A/s which results in a spike in the current profile………………………………………….84
Figure 4.16a: Simulation of the worst case scenario where a 50% error is incorporated in the
ICM which results in a large deviation of the wafer temperature prior to the P controller
engagement and a huge corrective action after its engagement. ……………………………..….85
Figure 4.16b: Zoomed in version of figure 4.16a. ………………………………...…………….85
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Figure 4.17a: Experimentally Recorded Wafer Temperature and Lamp Current for a Ramp Rate
of 130C/s……………………………………………………………………………………...….88
Figure 4.17b: Performance of the Proposed System showing the Wafer Temperature and Lamp
Current for a Ramp Rate of 130C/s. …………………………………………………………….88
Figure 4.18a: Experimentally Recorded Wafer Temperature and Lamp Current for a Ramp Rate
of 70C/s…………………………………………………………………………………….…….89
Figure 4.18b: Performance of the Proposed System Showing the Wafer Temperature and Lamp
Current for a Ramp Rate of 70C/s. ………………………………………………………..…….89
Figure 4.19: Alternative system architecture. Real time lamp voltage measurement replaces the
lamp model. …………………………………………………………………………………..….90
Figure A1: Figure showing the architecture of the UFR. ……………………………………….96
Figure A2: Diagram showing the UFRs in the system and the diagnostic flash setup for the
estimation of the bottom side emissivity……………………………………………………..….99
Figure B1: Geometry for derivation of the Radiative Heat Flux Absorbed………………...….101
Figure D1: Absorption coefficients of argon vs wavelength for various temperatures ……….105
Figure D2: Absorption coefficient of water at room temperature…………………………..….106
Figure E1: the wafer model block showing the inputs and outputs…………………………….109
Figure E2: The lamp model block showing the inputs and outputs………………..…………...110
Figure E3: Specific Heat Capacity (Cp) of the plasma versus temperature……………...…….112
Figure E4: Net Emission Coefficient (εN) for Rp = 5mm of the plasma versus temperature….112
Figure E5: Thermal Conductivity (κ) of the plasma versus temperature………………..…..…113
Figure E6: Electrical Conductivity (σarc)of the plasma versus temperature……………..……113
Figure E7: Density (ρarc)of the plasma versus temperature………..………………………….113
Figure E8: The feedback controller with embedded reset option…………...………………….114
Figure E9: The inner current control loop…………………………………………………..….115
Figure E10: The current conditioning block ………………………………………………..….115
x
List of Tables
Table 2.1: Summary of the Estimated Full Lamp Model…………………………………….….32
Table 2.2 Attenuation factor as a function of isothermal core temperature……………………..48
Table 2.3 Experimental radiative efficiency of the lamp as a function of the input power……..50
Table E1: Radiative efficiency of the lamp at a given input power………………………….…111
List of Symbols
Tarc : Temperature of the of the arc (K)
𝐶𝑝𝑎𝑟𝑐: specific heat capacity at constant pressure (J/(kg. K))
𝜌𝑎𝑟𝑐 : density of the argon and water mixture (kg/m3) 𝜎𝑎𝑟𝑐 : is the electrical conductivity parameter of the argon and water mixture (S/m) 𝐸, 𝐸𝑟 , 𝐸𝑧: Electric field vectors (V/m) 𝐹𝑟 : this is the net radiation flux density at any given point
𝐽: is the current density vector (A/m2)
Φ: is the electric potential (V)
𝜌 : is the gas density (kg/m3)
v : is the flow velocity vector (m/s)
p : is the gas pressure (N/m2)
f : represents the body forces acting on the gas (N)
S: length of the medium under consideration
𝐼𝜆: Intensity of radiation (𝑊
(𝑚2.𝑠𝑟))
Ω: solid angle (sr)
𝜏𝜆: optical thickness (unitless)
𝐹𝑟: Radiative heat flux (W)
𝜅𝜆: is the absorption coefficient of the material (1/m)
𝐼𝑏𝜆: is the spectral intensity of radiation from a black body (W/(sr.m2))
𝜏𝜆′ : is a dummy integration variable
𝜏: time constant of the current controller
𝑟: the radial coordinate variable
𝜀𝑁 : Net Emission Coefficient (W/ (sr.m3))
Rp: radius of the isothermal arc (mm)
𝐺𝑎𝑟𝑐 : the lamp’s conductance ( ∙ 𝑚)
𝑖𝑙𝑎𝑚𝑝 is the current through the lamp (A)
xi
𝑖𝑙𝑎𝑚𝑝𝐼𝐶𝑀: current estimated by the Internal Control Model (A)
𝑣lamp: lamp voltage (V)
𝜅: thermal conductivity of the arc (W/K.m)
𝑃𝜆𝑖: Radiative power in a spectral band I (W)
𝑃𝐵: total radiative power exiting the isothermal arc (W)
𝑃𝑂𝑈𝑇: total radiative power exiting the lamp (W)
𝑃𝑖𝑛: electrical input power to the lamp (W)
𝜂𝑙𝑎𝑚𝑝: radiative efficiency of the lamp
𝐶𝑝: specific heat capacity of the silicon wafer (J/(kg. K))
𝜌: density of the silicon wafer (kg/m3)
κ𝑆𝑖: thermal conductivity of the silicon wafer (W/K.m)
κ𝑁: thermal conductivity of nitrogen (W/K.m)
𝑇: wafer temperature (C)
𝑇𝐶: temperature of the quartz substrate (C)
𝑙𝑔𝑎𝑝: distance between the wafer and the quartz substrate
𝑡ℎ𝑆𝑖: thickness of the wafer
𝜂𝑜𝑝𝑡 : optical efficiency of the process chamber
𝑡ℎ𝐶 : is the quartz substrate thickness
𝐶𝑝𝐶: the specific heat capacity of quartz substrate (J/(kg. K))
𝜌𝐶: density of quartz substrate (kg/m3)
𝛼𝐶: is the total hemispherical absorptivity of the quartz to wafer radiation (unitless)
𝜀𝐶: is the total hemispherical emissivity of quartz (unitless)
𝑆𝑤: surface area of the wafer (m2)
𝛼: total hemispherical absorptivity of the wafer (unitless)
𝜀𝑆𝑖: total hemispherical emissivity of the wafer (unitless)
𝑃𝑟𝑎𝑑: total radiative power emitted by the 2 bottom lamps (W)
Δt: sampling time
t: time (s)
1
CHAPTER 1: INTRODUCTION
Rapid thermal annealing is a process used in the manufacture of semiconductor wafers. In this
process wafers are heated in order to electrically activate the introduced dopants and to control
their positioning within the wafer (diffusion profile). Control of the wafer temperature profile
dictates the doping profile and doping depth. Various types of rapid thermal processing (or RTP)
techniques are used to reach high enough temperatures that satisfy the requirements of the
semiconductor industry. Examples are spike anneals, impulse anneals, laser anneals etc. The key
difference between the RTP techniques is the temperature versus time profile used for processing
the wafer, as demonstrated in Figure 1.1.
Figure 1.1: A comparison of the various annealing techniques; where T is temperature, t is
time and d is the distance with respect to the surface of the wafer
The future generation of wafers will be more densely packed than the current generation of
wafers and will need to be heated in an even more stringent window of time. RTP times (using
2
conventional techniques like impulse anneals, spike anneals) have been getting shorter but these
techniques heat the entire thickness of the wafer, so there is a practical limit on the reduction of
the energy expended (or thermal budget). Laser processing can reduce the processing time to
nanoseconds but many desirable processes require processing times in the order of a millisecond
to be completed. Lasers are not optimized for a millisecond time scales. Moreover, the radiation
intensity offered by lasers is limited thus requiring many lasers if higher power levels are
required. This is not a cost effective option. Finally, lasers generate coherent light therefore the
interaction between the surface and the radiation field may create interference effects which are
deleterious to the thermal processing of the material.
However, the flash-assisted RTP, can give a large improvement (or reduction of thermal budget)
over the standard annealing techniques, with a 1 msec annealing time that satisfies the
requirements of the process. First, the wafer is heated to an intermediate temperature (around 900
degrees C) from the bottom side with a plasma arc lamp. Then an additional flash of energy
(~1msec) is used to heat the top side of the wafer to a final temperature between 1200 C and
1300 C. The temporal evolution of the wafer temperature and the temperature of the bottom and
top side of the wafer are shown in fig 1.2 and 1.3 respectively. The flash duration is short enough
to only make an impact up to a certain depth in the wafer, as shown in fig 1.1. This technique is
what results in a reduction of annealing times to milliseconds. However this can only be
achieved by producing a large amount of heat within a very short period of time in a highly
controlled manner.
3
Figure 1.2: The measured temperature profile of the wafer during the RTP process
Figure 1.3: The wafer topside and bottom side temperature during the millisecond RTP.
Inset shows the zoomed in version where TI and TP are initial and peak temperatures of the
wafer top side during the flash.
Such a system has been developed by Mattson Technologies Inc., as shown in fig 1.4. The
system consists of six plasma arc lamps that can radiate energy; two lamps at the bottom are used
4
to heat the wafer to an intermediate level (around 900C) and then four lamps at the top are used
to flash energy (millisecond annealing) on to the topside of the wafer to achieve the desired
wafer temperature profile. Figure 1.5 shows an example of the current profile through the lamps
during the whole process. Currently, Mattson lamp technology is the only technology that is
capable of achieving the temperature versus time targets required by the future semiconductor
industry.
Figure 1.4: The process chamber used for semiconductor wafer annealing
Figure 1.5: The measured current through each of the top and bottom lamps during the
process. For t< 2.9s, the controller operates in open loop mode. At t=2.9s, the wafer
temperature measurements become available and the controller switches to closed loop
operation.
5
1.1 The Apparatus Setup
The apparatus consists of the semiconductor wafer and its holding unit, the lamps, and the
temperature measuring devices. These parts of the system are described next.
The semiconductor wafer and its holding unit:
The chamber used for the annealing process, as shown in Fig. 1.4, is filled with nitrogen at a
pressure of 1 atm. The wafer is suspended on pins and the pins are connected to a quartz
substrate (the wafer holder). The quartz substrate plate is 7 mm thick and is assumed to be
optically transparent at all wavelengths except below 200nm. The distance between the quartz
plate and wafer is 1.4 mm and this quartz-wafer assembly sits at the center of the chamber. It is
assumed that heat transfer between the wafer and quartz plate is due primarily to radiative
exchange and heat conduction (convection can be ignored). Most radiation irradiating the quartz
is from the wafer and is absorbed on the top surface while the bottom surface is cooled by
radiation, conduction and convection. The wafer is assumed to be optically thick to arc
irradiation and therefore absorption takes place within a thin layer of the wafer. The absorption
on either side of the wafer can differ depending on the type of patterning on either side. The top
of the wafer is exposed to one radiation source (the four top lamps) which for the purpose of this
project is assumed to be producing little or no radiation during the temperature ramp-up phase,
while the bottom surface of the wafer is exposed to two lamps. Also, the radiation from both the
top and bottom lamps must pass through the cooling water inside the lamp and a water window
before it reaches the wafer, as shown in fig 1.6.
6
Figure 1.6: The Process chamber in detail
The purpose of the water window is to filter out any radiation from the lamps at 1450nm and
prevent it from entering the process chamber. It is assumed that any radiation from the wafer
towards the walls of the chamber is fully reflected and that the temperature of the walls is so low
as not to make an impact on radiative exchange on the top side of the wafer.
The Lamps:
The four lamps that are fixed to the top of the chamber are very different from the bottom two
lamps in their pressures and diameters and are designed for purposes of flashing. The six lamps
used in this system and shown in Fig. 1.7, are high pressure argon lamps. The arcs in these lamps
are stabilized by a vortex flow. The lamps are constructed of a cylindrical quartz tube with an
electrode at either end. High pressure argon gas enters the lamp at the cathode end and is swirled
along the central core until it exits the lamp at the anode end.
7
Figure 1.7: the arc lamp structure showing the circulating contents inside
A wall of water is also forced to swirl along the inside wall of the quartz tube, creating a sheet of
water to cool the edge of the arc to ambient temperature. The water wall extends the lifetime of
the lamp by cooling the quartz and carrying away sputtered electrode material. The radius and
length of the bottom lamps is 12.5mm and 290mm respectively while that of the top lamps is
22.5mm and 270mm respectively.
Temperature Measuring Devices:
The chamber also consists of two radiation measuring devices or UFRs (ultra-fast radiometers)
to estimate the temperature of the wafer. Radiometers are devices that measure the irradiation
falling on them. If the emissivity of the wafer surface is known, then Plank’s radiation law can
be used to calculate the wafer surface temperature from the measured irradiation intensity (refer
to appendix A for details).
One UFR is located at the top of the chamber and measures the wafer top side temperature. The
other UFR is located at the bottom of the chamber and measures the bottom side temperature,
with reference to fig 1.8. Initially, when the wafer is being heated to an intermediate temperature,
8
the bottom UFR data is used to calculate the wafer temperature using the emissivity computed
for the bottom side of the wafer.
Figure 1.8: The Process chamber in detail
This bottom emissivity is estimated by measuring the reflection and transmission of a diagnostic
flash, fired from an emissometer. Details are described in Appendix A. Both the UFRs cease
normal operation during the diagnostic flash and are coordinated to capture the flash to estimate
the emissivity. The bottom radiometer uses this estimated value of emissivity to estimate the
wafer temperature, until the next diagnostic flash (50msec later). By measuring the bottom
emissivity in such a periodic manner, the bottom UFR gives us the true temperature. Once the
wafer reaches the intermediate temperature, the wafer top side temperature is estimated from the
known top temperature (approximately the same as the bottom). Thus with the knowledge of the
top side temperature, the top UFR is used to calculate the top side emissivity at this point (refer
9
to appendix A). We assume that the top side emissivity does not change during the flash anneal
and this emissivity value is used to calculate the top side wafer temperature during annealing.
Since UFRs measure the irradiation falling on them, one crucial disadvantage of using them is
that they can only give an accurate measurement when the wafer starts radiating sufficient
energy. At low wafer temperatures, the radiation emitted by the wafer is too low and hence the
signal to noise ratio generated by the UFR is very poor. The wafer begins radiating energy only
after it has reached a certain temperature (blackbody radiation) and therefore there is no
temperature measurement available until the wafer reaches this temperature (approx. 300C). In
addition to two UFRs there is also an IR (infra-red) temperature sensor which measures the
temperature of the quartz wafer holder.
1.2 The Control System
A control system overview diagram for wafer temperature control is shown in fig 1.9. The whole
wafer RTP process is directed by the controllers which are given a wafer temperature profile as a
input. The controllers are responsible for setting the appropriate current for the lamps (see fig
1.5) so that sufficient radiation irradiates the wafer and allows the measured temperature to track
the reference temperature. Initially, when the wafer is “cold” (below 300C), there is no wafer
temperature feedback available, therefore the controllers operate in an open loop mode. After the
wafer is heated to above 300C, the temperature feedback becomes available and the controllers
switch to closed loop control to regulate the wafer temperature.
10
Figure 1.9: Control system overview
1.3 Thesis Motivation
Currently Mattson uses a semi empirical model for the lamp and the wafer. Their attempts to
ramp the wafer temperature to an intermediate level (using their existing control strategy), results
in a lamp current profile which is not smooth and apparently leads to enhanced electrode (anode)
erosion. It has been observed that there is a relation between the rate at which the wafer
temperature is increased and the electrode erosion caused by the resulting lamp current.
Mattson desires a control system which can be used to precisely control the ramp rate of the
wafer temperature upto the intermediate temperature level while making the lamp current follow
a smooth trajectory (be free of rapid fluctuations). In this way, Mattson can experimentally
11
determine an optimum temperature ramp for the wafer (up to the intermediate level), which
would lead to minimum anode electrode erosion.
However, for us to be able to develop such a control system we first need to develop an accurate
dynamic lamp model that predicts the voltage versus current characteristic and the radiated
power versus input power characteristic for the lamp. This lamp model is required for the
development of the controller since the real apparatus is not available for testing and has to be
done by simulation. Also the lamp model may be used in the controller for real time wafer
temperature control due to the lack of availability of real time lamp voltage measurements.
Secondly, we need to develop a dynamic thermal model for the wafer that provides accurate
temperature evolution information as a function of the irradiation from the lamp.
The scope of this thesis therefore, is the development of a lamp model, a wafer model and a
controller to control the radiation from the bottom lamps which are required to ramp the wafer
temperature to an intermediate level just before the millisecond flash occurs, as shown in fig. 1.9.
1.4 Thesis Objectives
The objectives of this thesis are:
To develop a model for the wafer and arc lamp.
To validate the wafer and arc lamp models.
To develop a control strategy for the bottom lamps which controls the temperature of the
bottom wafer up to an intermediate temperature with a specified ramp rate without
generating fluctuations in the arc current.
To test the proposed control strategy using available data provided by Mattson.
12
1.5 Thesis Outline
The thesis is organized as follows:
Chapter 2 describes the electro-thermal-radiation model for the arc lamp.
Chapter 3 describes the thermal model for the wafer and substrate holder.
Chapter 4 describes the control system consisting of an inner current loop and an outer
temperature loop.
Chapter 5 summarizes the contents of the thesis, the contributions and future work.
13
CHAPTER 2: LAMP MODELING
The main focus of chapter is on the model of the bottom lamps. Hence the specifics relate to the
properties of the bottom lamps even though the general approach to analysis taken also applies to
the top lamps.
2.1 Full Lamp Model of the Bottom lamp
The physics of the lamp is complicated by the fact that the contents of the lamp (i.e. plasma and
cooling water) are subject to a continuous swirling motion (see fig. 2.1). The anode electrode
operates at higher heat flux compared to the cathode so its temperature is higher and thus it
erodes at a faster rate. The eroded electrode debris can react with the quartz vessel and produce a
metallic composite on the inner surface of the quartz vessel. The change in the material
properties of the quartz vessel results in a change to the spectral distribution of the exiting
radiation and premature failure of the quartz vessel due to absorption. To prevent this, the
electrode debris must be removed from the water before the water is recirculated. The water wall
which is used to keep the lamp tube at room temperature also maintains the periphery of the
argon gas near room temperature thus creating a large temperature gradient within the plasma
near the water-wall.
14
Figure 2.1: The lamp and its circulating contents
The full lamp model, as shown in Fig. 2.2 is comprised of five regions: 1) the cathode interfacial
non local thermodynamic equilibrium (LTE) region, 2) the cathode constriction region, 3) the
positive column region, 4) the anode constriction region and 5) the anode interfacial non LTE
region. The total power delivered to the arc corresponds to the total power injected into the five
regions. The positive column is the bulk of the lamp. It can be characterized approximately as an
axially-invariant electric arc.
Figure 2.2: Five regions of the lamp
Regions 2, 3 and 4 are near or in local thermodynamic equilibrium and can be described by the
following coupled equations:
Power Balance (the Ellenbaas Heller equation):
15
𝐶𝑝𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) · 𝜌𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) ·
𝜕𝑇𝑎𝑟𝑐
𝜕 𝑡− 𝛻 · (𝜅(𝑇𝑎𝑟𝑐) · 𝛻𝑇𝑎𝑟𝑐) + 𝛻 ∙ (𝐶𝑝𝑎𝑟𝑐
𝜌𝑎𝑟𝑐𝒗𝑇𝑎𝑟𝑐) = 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) ·
‖𝐸‖2 + 𝛻 ∙ (−𝐹𝑟) (2.1)
𝛻 ∙ 𝐽 = 0 (2.2𝑎)
where 𝐽 = 𝜎𝑎𝑟𝑐𝐸 (2.2𝑏)
and 𝐸 = − 𝛻Φ (2.2𝑐)
‖𝐸‖2 = 𝐸𝑟2 + 𝐸𝑧
2 (2.2𝑑)
where:
Tarc : Temperature of the of the arc
Cparc(Tarc), ρarc( Tarc) : are specific heat capacity at constant pressure and density
characteristics of the argon and water mixture respectively σarc(Tarc) : is the electrical conductivity parameter of the argon and water mixture. 𝜅: is the thermal conductivity parameter of the argon and water mixture. 𝐸 = 𝐸𝑟 + 𝐸𝑧: Electric field vector 𝐹𝑟 : this is the net radiation flux at any given point
𝐶𝑝𝑎𝑟𝑐𝜌𝑎𝑟𝑐𝒗𝑇𝑎𝑟𝑐 : is the convection term, where 𝒗 is the velocity vector
𝐽: is the current density vector
Φ: is the electric potential
The first term on the left hand side of equation 2.1 is the thermal inertia associated with the
plasma. The second term on the left hand side represents heat conduction. The third term on
the left hand side represents the heat loss due to convection. The first term on the right hand
side is the electrical energy input to the lamp. The second term on the right hand side is the
expression for the net radiation, emitted at any given point within the plasma.
Momentum balance (Navier-Stokes) Equation:
𝜌 (𝜕𝒗
𝜕𝑡+ 𝒗 ∙ ∇𝒗) = −∇𝑝 + 𝑓 (2.3)
where:
𝜌 : is the gas density
v : is the flow velocity vector
16
p : is the gas pressure
f : represents the body forces acting on the gas
Radiation Transport Equation (ignoring scattering) [16]:
The radiation transport equation is given by equation 2.4 and its pictorial representation is
shown in Fig. 2.3.
Figure 2.3: Radiation transport through a medium
𝑑𝐼𝜆(𝑠, Ω)
𝑑𝑠= ∇𝐼𝜆(𝑟) = 𝜅𝜆(𝑇) ∙ 𝐼𝑏𝜆(𝑇) − 𝜅𝜆(𝑇) ∙ 𝐼𝜆(𝑠, Ω) (2.4)
where:
s: is the path length
𝐼𝜆 : is the spectral intensity of radiation at the distance s (Unit: W/(sr.m2))
𝜅𝜆(𝑇): is the absorption coefficient of the material as a function of temperature T (Unit:
1/m)
𝐼𝑏𝜆(𝑇): is the spectral intensity of radiation from a black body at temperature T (Unit:
W/(sr.m2))
Ω: is the solid angle
For an absorbing and emitting medium of length S, we can solve for 𝐼𝜆 by integrating
equation 2.4 and obtain the following expression:
17
𝐼𝜆(𝜏𝜆, Ω) = 𝐼𝜆(0, Ω)𝑒−𝜏𝜆 + 1
4𝜋 ∫ 𝐼𝑏𝜆(𝜏𝜆′)
𝜏𝜆
0
𝑒−(𝜏𝜆−𝜏𝜆′ )𝑑𝜏𝜆
′ [𝑊
(𝑚2. 𝑠𝑟)] (2.5)
where 𝜏𝜆 = ∫ 𝜅𝜆 𝑑𝑠𝑆
0 is referred to as the optical thickness of the region under
consideration (a unitless quantity) and 𝜏𝜆′ is a dummy integration variable
The first term on the right hand side of equation 2.5 is the attenuation term and the
second term is the source term. By definition, the radiative heat flux absorbed by a
differential volume element dV, located at a distance S, from all directions is given by:
𝐹𝑟𝜆_𝐴𝐵𝑆𝑂𝑅𝐵𝐸𝐷 = 𝜅𝜆(𝑇) 𝑑𝑉 ∫ 𝐼𝜆(𝜏𝜆, Ω) 𝑑Ω4𝜋
0
[𝑊] (2.6𝑎)
The radiative heat flux emitted from dV in all directions is given by:
𝐹𝑟𝜆_𝐸𝑀𝐼𝑇𝑇𝐸𝐷 = 𝜅𝜆(𝑇) 𝑑𝑉 ∫ 𝐼𝑏𝜆(T) 𝑑Ω4𝜋
0
= 4𝜋 𝜅𝜆(𝑇) 𝑑𝑉 𝐼𝑏𝜆(𝑇) [𝑊] (2.6𝑏)
Note that the emission and absorption coefficient are equivalent according to Kirchoff’s
radiation law. A more detailed derivation of these equations is presented in Appendix B.
Using equations 2.6a and 2.6b, we can obtain the divergence of the radiation flux for the
differential volume element dV, which is taken to be positive if emitted radiation is taken
to be positive in the outward direction from the differential volume:
∇ ∙ 𝐹𝑟𝜆 = 𝐹𝑟𝜆𝐸𝑀𝐼𝑇𝑇𝐸𝐷− 𝐹𝑟𝜆𝐴𝐵𝑆𝑂𝑅𝐵𝐸𝐷
= 4𝜋 𝜅𝜆(𝑇) (𝐼𝑏𝜆(𝑇) − 1
4𝜋∫ 𝐼𝜆(𝜏𝜆, Ω) 𝑑Ω
4𝜋
0
) [𝑊
𝑚3] (2.7)
Regions 1 and 5 are not in LTE. Hence they are not described by the above equations and must
be handled in a different way. The physics for these regions along with regions 2-4 will be
discussed in more detail in the five subsequent subsections. Sections 2.1.1-2.1.5 discuss the
specifics of regions 1-5. Section 2.1.6 provides a summary of the full lamp model consisting of
the concatenation of the five regions.
2.1.1 Region 1: Anode Interfacial Region Model
Figure 2.4 shows the general structure of an electrode zone and therefore applies to both the
cathode and anode regions. Ratt is the radius of the arc attachment spot to the electrode surface
and Rarc is the radius of the arc in the positive column region. Ratt will be different for the
cathode and anode regions.
18
Figure 2.4 General structure of the region surrounding an electrode
The modeled anode interfacial region is in the immediate vicinity of the anode electrode
spanning about 1mm axially (see figure 2.4). The anode is 22.4mm in diameter. The difficulty in
modeling this region comes from the fact that the plasma gas cools from the known core arc
temperature of around 10,000K [3] to the anode surface temperature (typically 3000-5000K), but
the electron temperature does not do likewise. Hence a two temperature model is needed to
account for this region. This would require equations describing the neutral species, the ion
species and the electrons. However, these details are not covered since the transport coefficients
involved (assuming a two temperature model) are not well known. Fortunately, this region turns
out to not play a major role in influencing the overall characteristics of the model, hence a
simplified representation of the region is provided.
19
Ultimately, we wish to determine the electrical characteristics of this region and hence we can
use the following approximation referred to as the generalized Ohm’s law to determine the
current density as a function of scalar functions which are meaningful:
𝑗𝑒 = 𝜎𝑒 (𝐸 + 1
𝑒 𝑛𝑒
𝑑𝑝𝑒
𝑑𝑥) + 𝜑
𝑑𝑇𝑒
𝑑𝑥 (2.8)
je is the electron current density, σe the electron electrical conductivity, E the electric field, ne, pe
and Te the electron density, partial pressure and temperature, respectively, and φ the
thermodiffusion coefficient.
The first term in the right hand side of equation 2.8 is the current due to the electric field, the
second term is the current due to a pressure gradient and the third term is the current due to
electron temperature gradient. We can ignore the last term in eq. 2.8 because the electron
temperature does not change significantly in the anode region even though the plasma or heavy
species temperature changes significantly [1]. Now, considering that the total current through
this anode region remains constant, we can write:
𝐼 = 2𝜋 ∫ 𝜎𝑒 (𝐸 + 1
𝑒 𝑛𝑒
𝑑𝑝𝑒
𝑑𝑥)
𝑅
0
𝑟 𝑑𝑟 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (2.9)
where R is the arc radius. It can be seen that the electric field and its contribution to the current
has to reduce for an increase in the current due to the partial pressure gradient for the electrons.
This increased electron partial pressure is provided by ambipolar diffusion and the convective
flow directed towards the anode surface [2]. This leads to deceleration of electrons and an ion
flux to the anode. The outcome of the convective flow is to cause a diffuse attachment of the arc
to the anode surface. The result is an anode voltage drop on the order of 1V which is relatively
20
insensitive to the operating current level [1]. This is the value assumed for modeling the anode
interfacial voltage drop.
There are significant heat fluxes to the anode due to various sources and the excess heat is
removed by water flow internally. The heat flux balance on the anode is given by:
𝑞𝑎 = 𝑗𝑒𝑙𝜑𝑎 + 𝑞𝑒𝑙 − 𝑘𝑒
𝑑𝑇𝑒
𝑑𝑥− 𝑘ℎ
𝑑𝑇ℎ
𝑑𝑥+ 𝑗𝑖(𝐸𝑖 − 𝜑𝑎) + 𝑞𝑅 (2.10)
where qa is the specific anode heat flux, jel the electron current density, φa the anode work
function, qel the heat flux associated with the electron flux into the anode, ke, kh and Te and Th
are the electron and heavy-particle thermal conductivities and temperatures, respectively, ji is the
ion current density, Ei the ionization energy and qr the radiative flux from the arc. A calculation
of this heat flux balance has not been done, however an estimate provided by Mattson indicated
that the combined losses in the anode and cathode regions are approximately 10% of the input
power when the input power is 300KW. The losses on the anode are larger than the cathode, thus
the anode has larger cooling requirements than the cathode.
2.1.2 Region 2: Anode Constriction Region Model
This is the region between the anode region and positive column (see fig. 2.4). Since it is
observed that the arc attachment to the anode surface is diffuse and that the anode diameter and
lamp diameter are 22.4mm and 25mm respectively, the amount of arc constriction is small and
hence the impact of constriction can be neglected. Therefore this region can be considered to be
a part of the positive column region.
21
2.1.3 Region 5: Cathode Interfacial Region Model
The mechanism of electron emission from the cathode in the lamp is thermionic emission. The
cathode is made of tungsten (12.7mm diameter) and has a melting point high enough so that it
can emit electrons at high current densities on the order of 10,000 A/cm2 without melting. Each
emitted electron takes with it an energy equal to the work function when leaving the cathode.
Thus electron emission produces a cooling effect on the cathode. However, to maintain a high
enough temperature, there must be a mechanism to heat the cathode to a temperature required for
thermionic emission. This heat is provided by the ions coming from the plasma as well as
radiation from the plasma (ions being the dominant contributor). Hence the amount of ion
current contribution at the cathode is significant [2].
Similar to the anode, the cathode interfacial model consists of the cathode surface and the region
near the surface where the plasma cools from the arc temperature to the cathode surface
temperature. The attachment of the arc to the cathode is observed to be in the form of a spot.
The region in front of the cathode is split into four layers in order to describe the physics in the
interfacial region [2]. The layers are (starting from the one closest to the cathode): the space-
charge sheath, the ionization layer, the layer of thermal non equilibrium and the layer of thermal
perturbation [2]. Deviations from LTE as seen moving from the positive column towards the
cathode are described next. First encountered is the layer of thermal perturbation where the
power balance between the joule heating and radiated power is no longer the same because of the
heat conduction flux to the electrode. Next, in the layer of thermal non-equilibrium, the
temperature of the heavy particle and electrons are no longer the same. This is because the heavy
particles are cooling due to the lower cathode surface temperature. Moving further towards the
22
cathode takes us into the ionization layer, where ionization equilibrium breaks down i.e. the rate
of ion, electron generation and recombination is not balanced. And lastly in the region adjacent
to the cathode surface, the electron and ion densities no longer remain the same, hence a space
charge sheath is formed.
Ultimately, we desire an electrical model of the cathode and for this it is important to understand
that the ionization level or the charged particle density in the ionization layer due to the heat
conduction flux from the arc is insufficient to maintain current flow from the cathode. Hence
additional energy must be deposited into the space-charge sheath to accelerate the electrons
emitted by the cathode. This additional energy transfer to this region is seen in the form of a
voltage drop and hence the voltage drop of the cathode unlike the anode, is much higher [2]. This
cathode region voltage for a similar geometry has been found from reference [15] to be in the
range of 40-25V for a current ranging from 100-500A. However, these numbers may be slightly
different for our case due to details which are beyond the scope of this thesis and therefore for
simplicity a constant drop of 40V will be assumed for this project.
In addition to the electrical characterization of this region, the power balance at the cathode
surface can be described by the following equation:
𝑞𝑐𝑎𝑡ℎ𝑜𝑑𝑒 = −𝑗𝑒𝑚 (𝜑𝑒𝑓𝑓 + 2.5𝑘𝑇𝑐
𝑒) + 𝑗𝑖 (𝑉𝑐 + 𝐸𝑖 − 𝜑𝑒𝑓𝑓 +
2𝑘𝑇𝑐
𝑒) (2.11)
where ji is the ion current, jem the electron emission current, Vc the sheath potential drop, Ei the
ionization energy of the plasma gas, φeff the effective work function (i.e. the cathode material
work function reduced by the electric field in front of it), Tc the cathode surface temperature and
q the heat flux entering the cathode. Similar to the anode, the cathode also has its own cooling
water system although excess heat produced in the cathode is smaller than the anode.
23
2.1.4 Region 4: Cathode Constriction Region Model
This is the region in between the cathode interfacial region and the positive column. The length
of this region is usually in the order of the arc radius [1]. This is a transition region for the arc
wherein the arc is diverging from its radius corresponding to the cathode surface spot (< 6mm) to
the vortex stabilized radius in the positive column. This results in the arc assuming a constricted
shape throughout this region. The arc in this region is still in local thermodynamic equilibrium
and the radiative losses dominate the region. The key difference between the anode and cathode
constriction regions is that gas entering the anode constriction region is already at the full arc
temperature (hence favoring the existence of a diffuse attachment mode) but the gas entering the
cathode constriction region is cold and hence must also be heated up to the arc temperature.
Therefore, within the transition region, the incoming flow of cold gas is heated up and brought to
the adjacent plasma temperature before entering the positive column.
As fresh swirling gas at 300K enters the lamp, it swirls around the cathode and directly enters the
cathode constriction zone and starts heating up by joule heating due to the flow of current
through it. As this cold gas begins heating up, it starts expanding (initially at constant pressure)
in the core of the lamp. The amount of expansion is determined by the amount of heat added to
the gas. We wish to determine the (radial) mass distribution of the flow just about to enter the
positive column due to this heating. This piece of information will enable us to calculate the
convective loss occurring in this region. Since the arc temperature is known to be around
10,000K, we can estimate the expansion of the gas due to this rise in temperature from 300K to
10,000K using the ideal gas equation. We assume here that the energy addition to the gas does
24
not affect the angular or axial velocity of the gas and hence one can use the following simple
ideal gas law relation for pressure:
p= ρR 𝑇 (2.12)
Initially, the gas expands at a constant pressure, therefore keeping p constant and increasing T to
10,000K, we can estimate that the gas would expand roughly to a density which is roughly 30
times lower than its original density just before it entered the lamp. Moreover, the thermal
expansion of the gas creates a mass distribution pattern such that most of the mass is in the high
density outer region of the lamp. This region is very thin because it has been pushed outward by
the high temperature plasma which occupies most of the lamp volume. By knowing the
approximate density and temperature profiles of the arc, we can evaluate the energy transfer to
the gas in the region per unit mass. This will be equal to [4]:
∫ 𝜌(𝑇) 𝐶𝑃(𝑇)(𝑇−𝑇0)2𝜋𝑟 𝑑𝑟𝑅
0
∫ 𝜌(𝑇) 2𝜋𝑟 𝑑𝑟𝑅
0
𝐽/𝐾𝑔 (2.13)
where R is the radius of the plasma. The gas exiting the lamp takes this energy with it, and is
thus a source of power loss for the lamp. This loss is referred to as convective power loss.
The constricted region has on one side the cathode interfacial region and on the other side the
positive column. As previously discussed, the specifics of the adjacent cathode interfacial region
and the cathode spot size depend on flow fields, cathode shape and net heat flux. Hence
assigning a boundary condition to this region is quite complex and is beyond the scope of this
thesis. For our purpose, an understanding of these details is not necessary. Therefore, even
though the exact cathode spot size and interfacial boundary condition are not known precisely,
the scenario in the cathode constriction region was simulated for a range of spot diameters
25
assuming a stationary scenario (no gas flow) just to obtain a rough estimate of the voltage drop
across the region.
The exact solution of the cathode constricted region involves modeling more complicated
phenomena but for our purpose, equation 2.1 and 2.2 were solved using a commercial finite
element program (COMSOL) and imposing the boundary conditions shown in figure 2.5. The
last term in equation 2.1 representing the contribution of radiation was replaced with an
approximation referred to as the Net Emission Coefficient, which is more useful for describing
the properties of the positive column and will be discussed in the next section. Here we will use
it for describing in a very approximate sense the radiation properties within the constriction zone
even though the quantitative results generated by the simulations cannot be taken at face value.
The radius of the arc is changing axially in this constriction region, therefore the Net Emission
Coefficient should be implemented as a two dimensional lookup table being a function of radius
and temperature. However, for simplicity of implementation, the NEC was chosen for a fixed
radius (of 5mm) and only varied with temperature. The convective term in eq. 2.1 (i.e. the effect
of gas flow on the temperature profile of the region) was ignored in this case, because simulating
the effect of gas flow would require solving the Navier-Stokes equation in parallel. This would
increase the complexity of the model which is not desired since the purpose of this exercise is to
roughly estimate the voltage drop across the region for a given constricted arc shape assuming
stationary contents. The inner radius of the water-wall was chosen to be 10.5mm because
measurements made by Mattson indicate that the thickness of the water-wall is around 2mm.
Simulation results showing the temperature profile in the constriction region are shown in fig.
2.6. The axial length of the region shown in the figure was determined by trial and error to
accommodate only the constricted arc region and not the region where arc is axially invariant.
26
The voltage drop in the cathode constriction region was estimated to be between 5-15V for a
range of currents (100-600A) which is in agreement with an estimate seen in the literature [6].
Simulations were also performed where the second boundary condition (across the arc
attachment spot) ∇𝑇 = 0 was removed and a heat flux (up to 5 MW/m3) was forced towards the
cathode interfacial region through the arc attachment spot. This did change the attachment spot
area temperature from 14,350K (for no flux condition) to 15,050K, however a negligible change
in the voltage drop across the region was observed and therefore the voltage drop appears to be
insensitive to the choice of heat flux imposed on the boundary.
The power dissipated in the cathode constriction region due to the voltage drop across it, will be
equal to the convective power transfer to the gas (thermal energy carried by the gas exiting the
lamp), and radiation power losses and conduction losses to the cathode and plasma periphery.
Figure 2.5: Boundary conditions applied for an axisymmetric representation of the
constricted cathode region
27
Figure 2.6: Simulation of the cathode constriction region for a cathode spot of radius 3mm
and a current of 300A
2.1.5 Region 3: Model of the Positive Column
Figure 2.7 shows a cross section of the positive column. In the positive column, the arc is
assumed to occupy the space up to the inner periphery of the water wall. The water wall is
created by swirling water along the inside of the quartz tube. The water temperature is
approximately 300K. A layer of the plasma is cooled by the water wall, which results in a layer
of “cold” argon that swirls adjacent to the arc and stabilizes it. The water wall and cold argon
appear to rotate together therefore no slip can be assumed.
28
Figure 2.7 Cross section of the lamp where the arc (yellow), cold argon (green) and water
wall (blue) can be seen.
The Arc Model Equations for the Positive Column:
We can assume to first order that field quantities have no axial dependency and that the plasma is
in LTE. This presumes that the axial flow of gas can be neglected for the purpose of modeling
the positive column. Assuming stationary contents makes the modeling problem a much easier
one to solve.
The plasma composition is 98% Argon and 2% water at 5atm. To begin with, we can assume that
the effect of the water-wall which is 2mm thick (indicated by Mattson measurements), is to keep
the outer most edge of the plasma at 300K. It is also assumed that the regions surrounding the arc
inside the lamp are optically thin (i.e. do not participate in radiation emission or absorption),
which we know is not true however this is the first step towards a further refinement to the model
which will be described later.
To avoid solving equation 2.4, we can replace the radiation flux density term 𝛻 ∙ (−𝐹𝑟) with a
simplified representation referred to as a net radiation emission term 4𝜋𝜀𝑁(𝑇𝑎𝑟𝑐). For this
29
simplification we assume that the arc can be represented by an isothermal core with no radiation
impinging from the outside on this core and no absorption or emission occurring outside this
core. A brief derivation of the net emission coefficient (NEC) method is given in Appendix C.
We can model the plasma arc using the net emission radiation term in the energy balance
equation, (eq. 2.1, the Ellenbaas Heller equation):
𝐶𝑝𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) · 𝜌𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) ·𝜕𝑇𝑎𝑟𝑐
𝜕 𝑡− 𝛻 · (𝜅(𝑇𝑎𝑟𝑐) · 𝛻𝑇𝑎𝑟𝑐) = 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) · 𝐸2 − 4𝜋𝜀𝑁(𝑇𝑎𝑟𝑐) (2.14)
where:
𝑇𝑎𝑟𝑐 : Temperature of the arc
𝐶𝑝𝑎𝑟𝑐( 𝑇𝑎𝑟𝑐), 𝜌𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) : are specific heat capacity at constant pressure and density
characteristics of the argon and water mixture 𝜅: thermal conductivity of the arc (W/K.m) 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) : is the electrical conductivity parameter of the argon and water mixture. 𝐸 : Electric field across the lamp 𝜀𝑁(𝑇𝑎𝑟𝑐) : this is the net emission coefficient (NEC) calculated by integrating the net
radiation flux of the isothermal core region (region which is approximately within 90% of
the maximum plasma core temperature) of the arc. The following equation can be found
in reference [7] and is described in more detail in Appendix C:
𝜀𝑁 = ∫ 𝐼𝑏𝜆(𝑇)𝜅𝜆(𝑇)𝑒−(𝜅𝜆(𝑇)𝑅𝑝)𝑑𝜆∞
0 (2.15)
In equation (2.15), 𝐼𝑏𝜆 is the Planck’s radiation function, 𝜅𝜆 is the spectral absorption
coefficient, 𝜆 is the wavelength and Rp is the radius of the isothermal region in the arc.
The NEC is a function of the isothermal arc temperature and the isothermal arc radius
usually denoted as Rp. The units of NEC are W/ (sr.m3).
The first term on the left hand side of equation 2.14 is the thermal inertia of the plasma. The
second term on the left hand side of equation 2.14 represents heat conduction. The first term on
the right hand side of equation 2.14 is the electrical energy input to the lamp. The second term on
the right hand side of equation 2.14 is the net emission representation for the radiated power as a
function of the isothermal temperature. Since the electric field is constant in the positive column
and has no radial component, eqs. 2.2 can be simplified further to give the following expression:
30
𝐸𝑧 =𝑖𝑙𝑎𝑚𝑝
(𝐺𝑎𝑟𝑐)=
𝑖𝑙𝑎𝑚𝑝
2𝜋 ∫ (𝑟 ∙ 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐))𝑑𝑟𝑅
0
(2.16)
where:
𝐸𝑧 is the Axial electric field in units of V/m 𝐺𝑎𝑟𝑐 is the positive column conductance in units of ∙ 𝑚
𝑖𝑙𝑎𝑚𝑝 is the current through the lamp in units of A
We can solve for the radial temperature profile of the positive column while also obtaining the
radiated power, positive column resistance, positive column voltage, as by products using
equations 2.14 and 2.16 and imposing the boundary condition that T= 298K at r= 10.5mm due to
the cooling by the water wall [3]. Since the isothermal arc radius remained more or less constant
with current, the NECs used are chosen for an Rp of 7mm and vary only with temperature.
A pictorial representation of the boundary condition applied in the radial direction is shown in
Fig. 2.8.
Figure 2.8: Cross section of the positive column showing the region simulated and the
boundary condition
The positive column region was simulated using a commercial finite element solver (COMSOL)
and the resulting temperature profile of the lamp is shown in figure 2.9. Fig 2.10 shows that the
31
radiative losses dominate in the positive column and heat conduction only takes place near the
edges where there is a large temperature gradient.
Figure 2.9: Cross section of the lamp showing its temperature profile as predicted by the
arc model for a current of 300 A. Temperatures are in Kelvin
Figure 2.10: Power density balance in the lamp estimated by the arc model for a current of
300 A. Base Power Density = 81MW/m3
32
2.1.6 The Full Lamp Model
The Estimated Full Lamp Model:
The models for regions 1-5 were concatenated to create a single model for the full lamp. Current
is conserved in going from one region to the next and is the only input given to the model. The
total arc voltage is the sum of the voltage drops across the five regions. The outputs from this
lamp model are the total lamp voltage and the radiated power.
The full lamp model was simulated in MATLAB where the positive column region was
simulated by assuming a cylinder made of five concentric shells whose total radius is 10.5mm
and length is 280mm. A minimum of five shells were chosen because this reduced the simulation
computational time considerably and the results were close to those seen using finite element
simulations performed with COMSOL. Eq. 2.14 was applied to each individual shell and eq.
2.16 was used to compute the axial electric field Ez. This resulted in a total of six coupled
equations describing the power balance for the positive column. The resulting estimate of the
positive column voltage was added to the voltage drops across the four other regions to give the
total lamp voltage. Let us refer to this model as the “estimated full lamp model”. Table 2.1 shows
the salient expressions for the estimated full lamp model.
Table 2.1: Summary of the Estimated Full Lamp Model
Region Model
Anode 1V voltage drop
Cathode 40V voltage drop
Anode Constriction
Region
Negligible
Cathode Constriction
Region
10V voltage drop
Positive Column 𝐶𝑝𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) · 𝜌𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) ·𝜕𝑇𝑎𝑟𝑐
𝜕 𝑡− 𝛻 · (𝜅(𝑇𝑎𝑟𝑐) · 𝛻𝑇𝑎𝑟𝑐) = 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐) ·
𝐸𝑧2 − 4𝜋𝜀0(𝑇𝑎𝑟𝑐)
33
𝐸𝑧 =𝑖𝑙𝑎𝑚𝑝
(𝐺𝑎𝑟𝑐)=
𝑖𝑙𝑎𝑚𝑝
2𝜋 ∫ (𝑟 ∙ 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐))𝑑𝑟𝑅
0
Radiation transport Uses Net Emission Coefficient to determine output of the inner isothermal
section of the plasma and assumes the outer cooler regions to be optically thin
The products of the lamp model are the total lamp voltage, the positive column resistance, total
input power and total radiated power from the isothermal section of the core, according to the
following relations:
Total lamp voltage, 𝑣lamp = [Length of Positive column (~280mm) x 𝐸𝑧 (eq. 2.16)] +
Voltage drop across the two electrode interfacial & constriction regions (~50V)
Positive Column Resistance = [Length of Positive column (~280mm) x 𝐸𝑧 (eq. 2.16)] /
ilamp
Total Power Radiated from the isothermal core, 𝑃𝐵 = 4𝜋𝜀𝑁 x Length of Positive Column
x Cross sectional area of isothermal arc
Total Input Power = 𝑣lamp × ilamp
The results of the simulations of the estimated lamp model indicated two major discrepancies
between this estimated full lamp model and the experimental lamp data provided by Mattson.
The first discrepancy (error in the estimated value of Total Lamp Voltage):
A discrepancy was observed between the total lamp voltage predicted by the estimated full lamp
model and the lamp experimental data wherein the voltage predicted by the estimated lamp
model was too low. Pushing the electrode and constriction region voltage drops to their
reasonable maximum limit did not solve the problem. Consequently, it turned out that the
estimated full lamp model developed so far was underestimating the positive column resistance
34
i.e. was estimating lower than the experimental value. A sensitivity analysis was performed on
the arc’s transport coefficients. The resistance of the positive column was found to be insensitive
to all of the arc transport and thermodynamic coefficients that were tested. This meant that the
only other factor that could impact the value of resistance was the radius of the high temperature
isothermal section of the arc. Since an estimate of the experimentally measured isothermal arc
radius is not available, this is an area of uncertainty and flexibility. Lowering the radius of this
isothermal arc would reduce the effective current conducting area and lead to a higher resistance.
As indicated earlier, the meaning of isothermal in this document does not exactly mean
isothermal but specifies a region which is at a temperature which is greater than about 90% of
the maximum arc temperature, based on experience and model validation. To estimate the most
probable radius that gave the required (experimentally determined) total lamp voltage, a test was
performed in which the radius of the isothermal part of the arc was reduced until there was
agreement between the experimental and test results for the total lamp voltage. This was done by
building a new simplified test model of the lamp.
Determining the Radius of the Isothermal Section of the Positive Column:
The test model replicates only the isothermal arc inside the positive column of the lamp where
radiation dominates and the electrical conductivity is axially uniform. The model consists of an
isothermal cylinder of length 280mm and radius Rp. The goal of the test is to determine the
radius Rp of the cylinder which gives the closest fit with the experimental data. For the purpose
of this test, the regions outside the isothermal arc, namely the cold gas, the water-wall and the
quartz tube are ignored and radiation is the only assumed loss heat mechanism from the
isothermal core. Neglecting the impact of the outer regions and other losses is not strictly true,
35
however for the purpose of the test model, the results based on this assumption are still very
informative and should indicate an approximate arc radius which can then be fine-tuned in the
full model.
A test scenario was created where the profiles of the positive column resistance and the output
radiated power were taken from experimental data (assuming the models of the electrode regions
hold). Using the experimental profile of the radiated output power, the temperature of the
isothermal cylinder (or arc) for various isothermal arc radii was determined using the NEC
method. Then using this estimated isothermal temperature, the resistance of the isothermal arc
was determined (using the temperature dependent electrical conductivity coefficient) and
compared with the experimental positive column resistance profile. The radius of the isothermal
portion of the arc that gave the best fit was around 5mm. This radius brought the resistance of the
positive column closer to the experimental value for higher currents but it was observed that for
lower currents an even smaller radius was required to obtain a good fit. For low currents
(<300A), an increase in current would lead to an expansion of the arc (radius) along with an
increase in temperature but at higher currents the radius remained more or less fixed while the
temperature of the arc increased. Using this test it was determined that the radius of the arc starts
from ~2mm at low currents and with increasing current expands to about 5mm and remains fixed
there for higher currents.
In summary, the test model revealed a critical piece of information about the arc namely that that
the effective radius of the isothermal region in the positive column which gives a good
agreement between experiment and theory does not extend to the full radius of the enclosure up
to the water wall. In contrast, the estimated full lamp model predicts that the arc occupies most
of the cylindrical enclosure’s volume and thus has a larger cross sectional area to conduct current
36
and therefore a lower resistance. The correction of radius of the arc was made to the estimated
full lamp model by reducing the cold boundary or water-wall radius from 10.5mm to 7mm and a
very good agreement was seen in the resistance profile of the modified positive column model
and the experimental results (see figure 2.11). The outcome of this correction was that the Total
Lamp Voltage was now in very good agreement with the experimentally recorded profile as
shown in figures 2.12a and 2.12b for two different lamp current profiles. Figure 2.13a shows the
isothermal arc radius versus current and figure 2.13b shows the radial temperature profile of the
lamp for various currents. Figure 2.14 shows the isothermal arc temperature variation with
current. The lowering of radius meant that for the same input power the isothermal arc
temperature was higher. This change of radius was incorporated for the cathode constriction
region simulation but the voltage did not change by a considerable amount and can still be
assumed to be ~10V. The results achieved thus far assume that the voltage drops of the electrode
interfacial regions hold, however, the approach taken here to determine the most likely
isothermal arc radius to give the desired voltage is always valid and can be re-applied to
accommodate any change in electrode voltages.
Having achieved the correct positive column resistance in the model, it became important to
understand why the isothermal arc has a smaller radius than what the estimated full lamp model
predicted. One possibility is the uncertainty of the radius of the water-wall. A measurement by
Mattson, indicated that the thickness of the water-wall was around 2mm. However, this
measurement was done in a test lamp which was not running, and it is suspected that a running
lamp could affect the flows resulting in a different water-wall size. It has also been confirmed by
Mattson that 2-5mm is a reasonable thickness for the water-wall in a fully running lamp. This
37
would imply that the radius of the computational space would be the inner radius of the water
wall which is approximately 7 mm.
It is important to note that the NECs used in the estimated lamp model (with corrected radius)
were chosen corresponding to an Rp of 5mm and varied only as a function of temperature.
However, according to fig. 2.13a, the radius of the isothermal arc changes as a function of
current. This changing radius as a function of current, means that NECs corresponding to
different Rp(s) must be used in the model for different levels of current. The resulting NEC data
then becomes a two dimensional lookup table in the model and can be used to improve the
accuracy of the model even further. However, for our purpose, only currents greater than 200A
are expected. Consequently, a one dimensional lookup table for NEC corresponding to an Rp of
5mm is used. This lookup table is only a function of temperature.
Figure 2.11: Experimental and corrected estimated model (positive column) resistance for
the shown current profile during a test. Base Current = 450A, Base Temperature =
12,000K
38
Figure 2.12a: Experimental and corrected estimated model voltage for the lamp current
shown
Figure 2.12b: Experimental and corrected estimated model voltage for the lamp current
shown
39
Figure 2.13a: Radius of the Isothermal portion of the Arc vs Current
Figure 2.13b: Arc temperature profile for various currents.
40
Figure 2.14: Isothermal Arc core temperature variation with Current
The second discrepancy (error in the arc’s radiative efficiency):
The second discrepancy was observed in the estimated full lamp model’s radiative efficiency.
Radiative efficiency of the lamp denoted by 𝜂𝑙𝑎𝑚𝑝 is defined as the ratio of radiative power
measured just outside the lamp (watts) to the total lamp input power (watts). The experimentally
determined radiative efficiency of the actual lamp, as shown in Fig. 2.15, is just under 50%,
whereas the estimated full lamp model predicted a much higher efficiency (>70%). Moreover,
the estimated full model (with corrected radius) predicted that the radiative efficiency was a
function of the lamp input power, while the experimentally determined radiative efficiency,
remained constant independent of the input current or power level. Therefore some phenomenon
in the positive column is resulting in lower efficiency which is unaccounted for in the estimated
full lamp model.
41
Figure 2.15: Experimental Radiative Efficiency of the Lamp
Reference [4] discusses an experiment conducted on a lamp with a very similar structure to the
lamps adopted by Mattson. In their experiment, the measured thermal wall loading (heat
conducted to the cooling water-wall) was found to be much higher than predicted (by the arc
model discussed in section 2.1.5). Therefore it is expected that the assumption of the outer
regions (namely, the cold gas, water-wall and the quartz tube) being optically thin does not hold,
so a certain amount of power does not escape the lamp (it gets self-absorbed). This absorbed
power is then lost in the form of heat conduction and heat convection hence explaining the
increased thermal wall loading.
Turbulence inside the lamp has been another suspect for the lower efficiency of the lamp.
Turbulence is the chaotic flow of gas. It can occur after the flow has travelled some distance
42
from an entry point and transitions from a solid body rotation pattern to a complex flow pattern
resulting in intensive mixing of the fluid. The impact of radiation transport and turbulence is
discussed next.
Radiation absorption at the periphery of the Arc:
The estimated full lamp model developed so far uses a NEC or Net Emission Coefficient
(equation 2.15) to compute the radiation output of the isothermal plasma. However, this
approach has limitations, since the NEC takes into account the emission and self-absorption of
the isothermal section of the plasma only and does not account for the outer regions (cold
plasma, water and quartz tube) of the lamp which are in general much cooler.
We now know that the assumption that the outer regions are optically thin to the radiation
coming from the inner isothermal arc is not valid. It is therefore necessary to account for the
radiation transport phenomena across the entire radial extent of the lamp up to and including the
exterior wall of the water cooled quartz tube.
This can in principle be done by replacing the NEC method with the full scale estimation of the
radiation transport equation and using it to solve for the divergence of radiative flux. The
radiation transportation equation (eq. 2.4) and its solution was discussed in section 2.1, however
that was for a one dimensional path between two points and not for a particular geometry. To
precisely predict the radiation transport in the lamp, a cylindrical system, must be adopted where
the radiation transport equation eq. 2.4 becomes [8]:
sin 𝜃 [cos 𝜓𝜕𝐼
𝜕𝜏(𝜏, 𝜃, 𝜓) −
sin 𝜓
𝜏
𝜕𝐼
𝜕𝜓(𝜏, 𝜃, 𝜓)] = 𝐼𝑏(𝜏) − 𝐼(𝜏, 𝜃, 𝜓) (2.17)
43
where 𝜓, 𝜃 are the spherical coordinates specifying the direction (azimuthal and polar angle) of
incident intensity at a particular point in a cylinder. The position of the point under consideration
is given in cylindrical coordinates.
There are solutions available to the above equation that require rather involved numerical
analysis techniques. However, this is too complex to be included in the lamp model for real time
implementation in a digital controller. Solving this problem also assumes we have knowledge of
all parameters and that the cylinder is infinite in length, which is not true.
However, there is a simpler alternative in which we can make use of the already known NEC
approximation method in the model. In this method, we use the NEC to predict the radiation
output from the isothermal arc. We then treat the output as the input to the outer regions and
predict the impact of attenuation through the different regions as shown in figure 2.16. Since, the
NEC method assumes that all radiation exiting the isothermal arc is diffuse and hence
independent of direction, it simplifies our analysis greatly. We then simplify things considerably
considering only one ray that is normal to the isothermal core and determine how radiation along
this path is impacted in traversing the outer regions. This can give some quantitative information
as to how the radiation in specific spectral bands is attenuated and thus explain why the
measured radiative efficiency is much lower than the value calculated using the estimated model.
Then using this information we can create empirical multiplicative factors that correct for the
radiation absorption so that the corrected results are in agreement with the experimental results.
44
Figure 2.16: The attenuation of emitted radiation by various layers outside the arc.
Starting with the assumption that the intensity of radiation from the isothermal arc is diffuse and
hence independent of direction we use a simplified solution of the radiation transport equation
(eq. 2.4) in a direction which is normal to the surface of the isothermal core and penetrates the
outer cold regions. This expression, used to trace the radiation along a specific path while
neglecting the emission along the path, can be found in a number of reference texts such as
reference [14] and is given as follows:
𝐼𝜆(𝜏𝜆, Ω) = 𝐼𝜆(0, Ω) exp[− ∫ 𝜅𝜆
𝑆
0
(𝑇) 𝑑𝑠] (2.18)
where S is the thickness of the region under consideration, λ is the wavelength, 𝐼𝜆(0, Ω) is the
intensity of incident radiation pointing outwards from the isothermal core in a particular
direction (taken normal in our case) and 𝜅𝜆 is the spectral absorption coefficient, which will be
assumed constant within a specified frequency band. Applying equation 2.18 for the shortest
45
distance through the medium (i.e. the path having the smallest optical thickness) will give us an
estimate of the minimum amount of attenuation to expect from the region for the spectral band
under consideration. Note that this expression evaluates the intensity at a given point, incident
from a particular direction only and does not give the total intensity at the point from all
directions. Since, the radiation coming from other angles to the point of consideration will be
attenuated even further, the results calculated using eq. 2.18 (assumed taking the shortest path)
are expected to underestimate the attenuation of the signal appearing at the exterior surface of the
quartz wall. Figure 2.17 shows where radiation exiting the isothermal core is selectively
absorbed and converted into internal heat energy and convective energy.
Figure 2.17: The attenuation of emitted radiation by various layers outside the arc.
Since the radiation is assumed to be diffuse, intensity and power are correlated to each other by a
constant factor. Therefore, converting eq. 2.18 into a power equation, we get:
𝑃𝜆𝑖_𝑂𝑈𝑇(𝜏𝜆) = 𝑃𝜆𝑖_𝐼𝑁(0) exp[− ∫ 𝜅𝜆𝑖
𝑆
0
(𝑇) 𝑑𝑠] (2.19)
where i represents a particular discretized spectral band from a total of N discretized bands,
𝑃𝜆𝑖_𝑂𝑈𝑇 represents the power in band i exiting the outer regions of the lamp, 𝑃𝜆𝑖_𝐼𝑁 represents the
46
power in band i entering the outer regions (i.e the power radiated by the isothermal arc). Arriving
at this equation from eq. 2.18 assumes that the intensity remains the same across all the angles.
If we now sum equation 2.19 over all bands we arrive at the following expression for 𝑃𝑂𝑈𝑇, the
total power exiting the outer regions of the lamp:
𝑃𝑂𝑈𝑇 = ∑ 𝑃𝜆𝑖_𝑂𝑈𝑇(𝜏𝜆)𝑁
𝑖=1= ∑ 𝑃𝜆𝑖_𝐼𝑁(0) exp[− ∫ 𝜅𝜆𝑖
𝑆
0
(𝑇(𝑠))] 𝑑𝑠𝑁
𝑖=1 (2.20)
Then dividing both sides by the total radiative power exiting the isothermal arc (which is the
estimated in the same way as in the estimated lamp model with corrected radius):
𝑃𝐵 = ∑ 𝑃𝜆𝑖_𝐼𝑁(0)𝑁
𝑖=1 (2.21)
we obtain:
𝑃𝑂𝑈𝑇
𝑃𝐵= ∑
𝑃𝜆𝑖_𝑂𝑈𝑇(𝜏𝜆)
𝑃𝐵
𝑁
𝑖=1= ∑
𝑃𝜆𝑖_𝐼𝑁(0)
𝑃𝐵 exp[− ∫ 𝜅𝜆𝑖
𝑆
0
(𝑇(𝑠)) 𝑑𝑠𝑁
𝑖=1 ] (2.22)
The ratio 𝑃𝑂𝑈𝑇
𝑃𝐵 in eq. 2.22 is defined as the ‘total transmissivity factor’ of the outer regions and is
the desired quantity.
The ratio 𝑃𝜆𝑖_𝐼𝑁 (0)
𝑃𝐵 is plotted for each band i (8 bands in total) as a function of isothermal core
temperature in figure 2.18. This data was made available to us by our French collaborators. The
integration on the right hand side of equation 2.22 is performed over each band starting at the
isothermal core radius and extending to the outside wall of the quartz tube (see Appendix D).
In absolute units, the radiated output power as a function of the total power emitted by the
isothermal arc is given by:
𝑃𝑂𝑈𝑇 =𝑃𝑂𝑈𝑇
𝑃𝐵∙ 𝑃𝐵 (2.23)
47
Figure 2.18: The ratio of power in each spectral band to the total power in all bands, as a
function of isothermal arc temperature. Plotted using the NEC (corresponding to an arc
radius (Rp) = 5mm) for each band
Fig. 2.14 shows the range of isothermal arc temperatures that need to be considered is
approximately 10,000K-13,000K. The total transmissivity factor 𝑃𝑂𝑈𝑇
𝑃𝐵 was estimated to vary
from 80% at an isothermal arc temperature of 10,000K to 70% at 13,000K. The details of this
calculation are shown in Appendix D.
A lookup table was generated to create points between the lowest and highest temperature as
shown in Table 2.2. Transmissivity factors for temperatures between tabulated values are
48
determined through a linear interpolation process. Also, it was assumed that over the current
range of interest, the isothermal radius is maintained constant at 5mm.
Table 2.2 Total Transmissivity factor as a function of isothermal core temperature
Tisothermal_Arc [K] Transmissivity_Factor
0 1
10000 0.80
10500 0.78
11000 0.76
11500 0.74
12000 0.72
12500 0.71
13000 0.70
This (isothermal arc temperature dependent) transmissivity factor was implemented in the
estimated lamp model (with corrected arc radius) by multiplying it to the power radiated by the
isothermal core, 𝑃𝐵 and this modified model is referred to as the modified full lamp model. The
modified full lamp model data is in better agreement with the experimental data, compared to the
estimated lamp model, as shown in Fig. 2.19.
The remaining discrepancy between the modified model and experimental model above the
1*105 W Input Power level is likely caused by the simplifying assumptions made during the
calculations of equation 2.18. An attempt was made to also understand the discrepancy that is
49
observed in the shape of the curve below the 0.5*105 W Input Power level. A two dimensional
NEC lookup table was implemented in the modified lamp model where the NEC is a function not
only of the temperature but also of the isothermal radius Rp. It was expected that at low input
power levels when the arc has an isothermal radius much smaller than 5mm, the two dimensional
NEC implementation would lead to a better agreement however the improvement was found to
be insignificant. Moreover, the spectral radiation of the arc was also studied for lower arc radii to
see if it differs from the profile at 5mm, but even this could not explain the shape of the curve.
However, it was seen that the shape of the curve below the 5*105 W level is sensitive to the
choice of power loss in the other four regions of the lamp (namely the electrode interfacial and
constriction regions). For example, by assigning these four regions a combined voltage which is
20% of the positive column voltage (which varies rather than staying constant at 50V), the
agreement of the radiative efficiency curve (as shown in fig. 2.20a) with the experimental curve
is much better (note that the discrepancy above 1*105 W still remains and is due to the
simplifying assumptions made during the calculations of equation 2.18). However, the
modification to the combined voltage of the four regions has a negative effect on the Total Lamp
Voltage predicted by the model as shown in fig. 2.20b and does not match with the experimental
profile as close as the estimated lamp model with corrected radius shown before in fig.2.12.
In summary, it appears that at lower currents the specifics of the radiative efficiency curve may
be more influenced by the accuracy of the models for the electrode interfacial and constricted
regions. From an engineering point of view however, it is simpler and more practical to use the
experimentally known radiative efficiency profile in the modified full lamp model instead of the
radiation model developed so far as shown in figure 2.19. Therefore, the known experimental
50
profile for radiative efficiency of the lamp is used to determine the output radiative power in the
lamp model as follows:
𝑇𝑜𝑡𝑎𝑙 𝑃𝑜𝑤𝑒𝑟 𝑅𝑎𝑑𝑖𝑎𝑡𝑒𝑑 (𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑎𝑚𝑝) = 𝑣𝑙𝑎𝑚𝑝 × 𝑖𝑙𝑎𝑚𝑝 × 𝜂𝑙𝑎𝑚𝑝 (2.24)
where 𝜂𝑙𝑎𝑚𝑝 is the experimentally determined radiative efficiency and is given as a function of
input power as shown in table 2.3. 𝜂𝑙𝑎𝑚𝑝 values for input power values that lie in between
tabulated values are determined through a linear interpolation process.
Table 2.3 Experimental radiative efficiency of the lamp as a function of the input power
Lamp input power in Watts Radiative efficiency ( ηlamp)
0 0
6,000 0.1
9,108 0.26
21,169 0.37
34,886 0.47
60,000 0.475
140,000 0.487
705,721 0.487
51
Figure 2.19: Radiative efficiencies determined from experiment and modified full lamp
model.
Figure 2.20a: Radiative efficiency of the attempted model and the experimental
52
Figure 2.20b: Total Lamp Voltage of the attempted model and the experimental
Effect of the Vortex on Transport Coefficients:
Fresh argon gas at room temperature is injected into the lamp in a tangential direction through
openings inside the lamp. As this gas enters tangentially, it has a large rotational (or
angular/azimuthal) velocity and a relatively small axial velocity. As the flow begins its journey
through the positive column, it can be considered to be rotating like a solid body.
Since the gas is swirling, its pressure distribution has to satisfy the centripetal force field in order
to attain an equilibrium rotation. We can apply the Navier-Stokes equation in the radial direction,
in cylindrical coordinates to find this pressure distribution. The pressure distribution within the
positive column which must be satisfied by the vortex is given by:
𝑝 =𝑚𝜔2𝑟
𝐴 (2.25)
53
𝑑𝑝
𝑑𝑟=
𝑚𝜔2
𝐴 𝑟
𝑟 (2.26)
𝑑𝑝
𝑑𝑟= 𝜔2𝜌𝑟 (2.27)
where m is the mass of the gas, 𝜔 is the angular velocity and A is the cross sectional area.
Therefore, the creation of a centripetal force inside the lamp, results in a radial pressure
distribution which is lower towards the center of the lamp and higher near the outer edge of the
lamp. Reference [4] shows the calculation of the radial pressure variation and shows that a
pressure gradient exists but is small. Hence the variation of the transport parameters versus
pressure (like electrical, thermal conductivity, absorption coefficient) is neglected and the vortex
is therefore assumed to play no role in affecting the efficiency of the arc.
Revisiting the "No Axial Flow” assumption:
Until now the radial profiles of the lamp parameters were discussed with the assumption that the
flow remains the same axially throughout the positive column and the contents were hence
assumed stationary. However, in reality the lamp parameters also vary along the axis. The major
reason for this variation is due to the dynamic effects of the fluid flow.
In general the rotational flow of the plasma at the beginning of the positive column can be
assumed to be that of a solid body rotation i.e. the flow is laminar. However due to viscous drag,
the outermost layer of the rotating gas decelerates as it travels through the lamp. Therefore, the
assumption of laminar flow may fail to hold at a certain distance from the cathode end.
Moreover, as the argon gas travels even further it is known to get turbulent. In the turbulent
regime, there is severe mixing and hence the radial pressure distribution explained above for the
54
laminar region becomes flat. This axial variation of the radial pressure profile results in a
pressure difference axially and causes a reverse flow of the plasma. This reverse flow (shown in
figure 2.21) is driven by the pressure difference created by the existence of a slightly lower
pressure core within the laminar region of the positive column.
Figure 2.21: Cross sectional view of the flow pattern inside the lamp
In the past, turbulence has been one of the suspected reasons for the higher than expected
thermal wall loading, however we can use experimental data available from a reference [4] and
see the magnitude of the effect of turbulence. From a literature study [5], it has been observed
that the effect of increasing the azimuthal velocity in a vortex is a more turbulent flow (in other
words, the flow becomes turbulent much earlier than before). Experimental studies showed that
an increased azimuthal velocity, while keeping the mass flow rate constant, lead to a lower lamp
radiative efficiency. Quantitatively, an 80% increase in the azimuthal velocity leads to a decrease
in radiative efficiency by roughly 6-8%. Therefore, we can conclude that although turbulence
does have an effect on the radiative efficiency, it is not very significant in comparison to the total
discrepancy between the estimated full lamp model with corrected radius and the Mattson
provided numbers for lamp efficiency.
55
2.1.7 Summary of the Full Lamp Model
The full lamp model consists of the estimated full lamp model with the inclusion of the
modifications that account for:
for the reduced arc radius
for the radiation absorption in the regions exterior to the isothermal core.
Experimental agreement with theory was achieved by reducing the radius of the inside of the
water-wall to 7mm and using the experimentally determined radiative efficiency profile of the
lamp (eq. 2.24) to get the correct power balance amongst the different loss mechanisms;
conduction, radiation, convection. The proposed full lamp model is capable of accurately
estimating the lamp voltage, and the radiative efficiency (vs input power) for a given input
current.
56
CHAPTER 3: THERMAL MODELING OF THE WAFER AND QUARTZ
SUBSTRATE HOLDER
The purpose of this chapter is to develop the thermal models of the wafer and quartz holder
which can accurately predict the temporal evolution of their temperatures subject to the heat and
radiation fluxes in the process chamber.
3.1 Thermal Model of the Wafer
Fig. 3.1 shows a diagram illustrating the layout of the wafer, the quartz holder and the bottom
lamps.
Figure 3.1: The process chamber
The radiation generated by the bottom lamps is used to bring the wafer to the desired
intermediate temperature before the top lamps are pulsed. We wish to develop a model for the
wafer such that knowing the power radiated by the lamp, we can estimate the temporal
temperature rise of the wafer. The dynamics of the wafer temperature (T) as a function of space
and time can be described with the help of the following power balance equation:
57
𝐶𝑝(𝑇)𝜌(𝑇)𝜕𝑇
𝜕𝑡= ∇ ∙ (𝜅𝑆𝑖(𝑇) ∙ ∇𝑇) + ∇ ∙ (−𝐹𝑟) (3.1)
Where 𝐶𝑝(𝑇), 𝜌(𝑇) are the specific heat capacity and density of the silicon wafer respectively,
𝜅𝑆𝑖(𝑇) is the thermal conductivity, and 𝐹𝑟 represents the net radiation flux at any given point.
Due to the higher loss of heat from the edges of the wafer (relative to the center), the areas near
the edge of the wafer would remain cooler than the central regions of the wafer if not
compensated for. Therefore, to maintain uniform temperature across the wafer, the reflectors in
the chamber are designed to compensate for the loss of heat from the edges by irradiating the
edges slightly more than the central region of the wafer surface. As a result of this non-uniform
irradiation profile, the wafer temperature T can be assumed to be the same throughout the entire
silicon wafer. With this assumption, T loses its spatial dependence, and transforms into a
function of only one variable - time. Integrating equation 3.1 over the wafer volume and
applying the divergence theorem results in the following equation:
𝐶𝑝(𝑇)𝜌(𝑇)𝑡ℎ𝑆𝑖𝑑𝑇
𝑑𝑡= −κ𝑁 ∙
𝑇−𝑇𝐶
𝑙𝑔𝑎𝑝− 2𝜀𝑆𝑖(𝑇)𝜎𝑇4 +
𝑃𝑟𝑎𝑑 𝜂𝑜𝑝𝑡 𝛼
𝑆𝑤 (3.2)
where 2𝜀𝑆𝑖(𝑇) represents the total hemispherical emissivity of the upper and lower surface of the
silicon wafer, 𝜎 is the Stefan–Boltzmann constant, 𝑡ℎ𝑆𝑖 represents the wafer thickness
(=0.775mm), 𝛼 is the total hemispherical absorptivity of the wafer side exposed to the lamp
radiation (~0.65) and 𝑇𝐶 represents the temperature of the quartz substrate. A model of the quartz
substrate will be discussed in more detail later. 𝑙𝑔𝑎𝑝 is the space between the wafer and the
quartz holder (=1.4mm), 𝑆𝑤 is the planar surface area of the wafer (radius=150mm) and 𝑃𝑟𝑎𝑑
represents the total radiated power (in watts) from the two bottom lamps. A portion of the power
radiated by the lamps is also absorbed by the chamber reflectors before reaching the wafer, so
58
this loss is introduced into the model by multiplying 𝑃𝑟𝑎𝑑 by a term 𝜂𝑜𝑝𝑡 which accounts for
these optical losses. This term will be referred to as the chamber efficiency in the rest of the
document.
Fig 3.2 summarizes the heat flux balance at the surface of the wafer. The first term on the right
hand side of equation 3.2 accounts for the heat conducted between the wafer and the quartz
substrate. The space between the wafer and the quartz substrate is filled with nitrogen gas at
1atm pressure which has a thermal conductivity of κN= 0.03W/(m.K). The last two terms on the
right hand side of equation 3.2 account for the total radiation emitted and absorbed by the wafer.
The radiative emission of the wafer surface is modeled as black-body radiation (σT4) with a
correction term, the total hemispherical emissivity (εSi(T)). The radiation from the two bottom
arc lamps, represented by 𝑃𝑟𝑎𝑑 is given by:
𝑃𝑟𝑎𝑑 = 2 𝜂𝑙𝑎𝑚𝑝 · 𝑃𝑖𝑛 = 2 𝜂𝑙𝑎𝑚𝑝 · 𝑣𝑙𝑎𝑚𝑝 · 𝑖𝑙𝑎𝑚𝑝 (3.3)
where 𝑣𝑙𝑎𝑚𝑝 , 𝑖𝑙𝑎𝑚𝑝, 𝜂𝑙𝑎𝑚𝑝 is the lamp voltage, lamp current and lamp radiative efficiency
respectively. The factor 2 is multiplied to the right hand side of eq. 3.3 because there are two
bottom lamps.
Figure 3.2: Heat flux balance at the wafer’s surface
59
Equation 3.3 is substituted into equation 3.2 to obtain an expression for the change in the wafer
temperature:
𝑑𝑇
𝑑𝑡= −κ𝑁 ∙
𝑇−𝑇𝐶
𝑙𝑔𝑎𝑝𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)−
2𝜀𝑆𝑖(𝑇)𝜎𝑇4
𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)+
2 η𝑙𝑎𝑚𝑝 𝑣𝑙𝑎𝑚𝑝 𝑖𝑙𝑎𝑚𝑝 𝛼 η𝑜𝑝𝑡
𝑆𝑤𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇) (3.4)
This is an ordinary differential equation to predict the time evolution of the wafer temperature.
The density and specific heat capacity of silicon 𝐶𝑝(𝑇), 𝜌(𝑇) in equation 3.4 can be
approximated as polynomial functions of temperature [10]. The following approximate functions
can be used to estimate the thermodynamic properties of crystalline silicon, for the temperature
range of interest (25C – 1000C):
ρ(Si) (T) = 2330 − 2.19 * 10−2T kg/m3 (3.5)
Cp(Si)(T) = 836 + 1.08 * 10−1T − 1.04 * 10−5T 2J/K/kg (3.6)
These two terms can be combined together with the wafer thickness into a single polynomial
term denoted as β(T):
β (T ) = thSi ρ(Si)(T) Cp(Si)(T) = thSi(1.95*106
+ 2.33* 102
T − 2.66 * 10−2
T 2 + 2.28 * 10
−7
T 3
) (3.7)
The total hemispherical emissivity of silicon is also a function of temperature and dependent on
its doping profile [11] - [12]. Heavily doped silicon has an emissivity which stays in the 0.7-0.72
range for the temperature interval of 25 – 800C, while lightly doped silicon has an emissivity
which varies from 0.12 to 0.7 for the same temperature interval. The emissivity of lightly doped
silicon (78Ωcm) can be approximated with the following equation:
60
𝜀𝑆𝑖 = 1.53 ∗ 10−3𝑇 − 0.3 𝑓𝑜𝑟 280 < 𝑇 < 660𝐶
0.7 𝑓𝑜𝑟 660𝐶 < 𝑇 (3.8)
3.2 Thermal Model of the Quartz Substrate:
This model is used to estimate the temporal evolution of the quartz substrate temperature 𝑇𝐶 , used
in equation 3.4. The quartz substrate is subject primarily to the conduction and radiation fluxes
from the wafer. The temperature of the quartz is assumed to be uniform throughout its volume.
Similar to equation 3.4, an ordinary differential equation can be written to describe the thermal
model of the quartz substrate:
𝑑𝑇𝐶
𝑑𝑡= κ𝑁 ∙
𝑇−𝑇𝐶
𝑙𝑔𝑎𝑝𝑡ℎ𝐶𝐶𝑝𝐶(𝑇)𝜌𝐶(𝑇)+
𝛼𝐶𝜀𝑆𝑖(𝑇)𝜎𝑇4
𝑡ℎ𝐶𝐶𝑝𝐶(𝑇)𝜌𝐶(𝑇)−
2 𝜀𝐶(𝑇)𝜎𝑇𝐶4
𝑡ℎ𝐶𝐶𝑝𝐶(𝑇)𝜌𝐶(𝑇) (3.9)
Where:
𝑡ℎ𝐶 : is the quartz substrate thickness
𝐶𝑝𝐶(𝑇), 𝜌𝐶(𝑇): are the specific heat capacity and density of quartz substrate respectively
𝛼𝐶: is the total hemispherical absorptivity of the quartz to wafer radiation
𝜀𝐶(𝑇): is the total hemispherical emissivity of quartz
The first term on the right hand side of equation 3.9 represents the heat conduction flux from the
wafer. The second term represents the absorption of radiation from the bottom surface of the
wafer and the third term represents the emission of radiation from the quartz.
61
3.3 Validation of the Combined Wafer and Quartz Substrate Models
Fig 3.3 shows a block diagram of how the models of the wafer, quartz substrate and arc lamp are
interconnected. The wafer and quartz substrate models defined by equations 3.4 and 3.9 were
combined with the full lamp model derived in chapter 2. This full system was simulated using
three different current profiles (as shown in fig. 3.4a,b,c) that were recorded experimentally at
the Mattson facility.
Figure 3.3: Block diagram of the full system simulation setup
Since the lamp model has been independently tested already and the results have already been
discussed in chapter 2, the focus of this test was to validate the performance of the wafer model
only, therefore only the wafer temperature quantities will be shown. The wafer temperature
profile predicted by this system can be compared to the corresponding experimentally recorded
wafer temperature profile and can be used to indicate the accuracy of the wafer and quartz
thermal models.
62
To begin with, the optical efficiency of the process chamber 𝜂𝑜𝑝𝑡 used in the wafer model of
equation 3.4 is unknown and must be determined by trial and error until an agreement is seen
with the experimentally recorded response of the wafer temperature. This had to be done only
once and the value determined was used in all future tests. This optical efficiency of the chamber
𝜂𝑜𝑝𝑡 was found to be ~0.23. Using the value of 𝜂𝑜𝑝𝑡 as 0.23, the system was simulated for three
test cases and the wafer temperature profiles from experiment and model were compared. The
results of this simulation are shown in the following figures for various ramp rates and current
profiles.
Figure 3.4a: Experimental and model estimated wafer temperature profiles for a 60C/sec
ramp assuming an initial wafer temperature of 255C
63
Figure 3.4b: Experimental and model estimated wafer temperature profiles for a 70C/sec
ramp for an initial wafer temperature of 132C
Figure 3.4c: Experimental and model estimated wafer temperature profiles for a 130C/sec
ramp assuming an initial wafer temperature of 230C
64
It should be mentioned that the experimentally recorded temperature in figs. 3.4 (black curves)
are accurate only for temperatures greater than 300 degrees C, for the following reason: This is
because of the limitation of the UFR which estimates the wafer temperature based on the wafer’s
radiative emission. Since the wafer does not emit sufficiently until it has reached around 300
degrees C, the UFR generates poor signal to noise ratios and hence returns invalid measured
temperatures. As a consequence of this, the initial starting temperature of the wafer in the
experiment was unknown. However, for the purpose of simulating the wafer model, an initial
value of the wafer temperature was required as an input. This initial wafer temperature for the
purpose of this test was therefore estimated from the final experimental temperature, the known
ramp rate of the wafer and the total time taken by the wafer to get to the final temperature.
From figures 3.4a,b,c, it can be seen that the estimated wafer temperature from the model shows
good agreement with the experimentally recorded profile for every case. The quantity of interest
is the ramp rate of the wafer temperature and in most cases there is negligible deviation between
model’s estimated temperature ramp rate and the experimental ramp rate. Any deviation seen is a
result of the small deficiencies in models used in the system. However, the difference in
temperature between the experimental and model never exceeds 30 degrees in any case. The
source of this deviation is suspected to be the parameters used in the models such as wafer
absorptivity which can take a different (temperature dependent) profile in reality from the one
assumed in the model and small changes in absorptivity can make a considerable impact.
Moreover, a combination of other factors such as free convection in the chamber, the
contamination of the gas/water in the lamp (leading to deviation of the lamp’s radiative
65
efficiency from the one used), the changing efficiency of the process chamber (η𝑜𝑝𝑡) over a
period of time can also play a role.
In summary, the observed deviation between the model’s estimated wafer temperature and the
experimentally measured temperature is small in most cases. Moreover, because the wafer
temperature measurements are accurate for temperatures greater than 300C, any deviations seen
can be corrected for by a feedback control system. Therefore the accuracy of the wafer model
developed was deemed sufficient for the purpose of the project.
66
CHAPTER 4: CONTROL OF THE SYSTEM
The previous chapters discussed the dynamics of the silicon wafer and arc-lamp. This chapter
introduces the control system which will be used to make the wafer temperature track a reference
temperature profile. Details of individual blocks within the control system diagram are described
in more detail in Appendix E.
4.1 Overall System with Controller
A block diagram of the overall system with a tracking controller denoted as ’Temperature
Controller’ is presented in figure 4.1. The system has an outer temperature control loop for the
wafer and an inner current control loop for the arc lamp. The physical input set by the user is the
required wafer temperature (Treq_wafer). Both the temperature and current controller are discrete
time systems, whose analog measurement inputs, temperature and current, are converted into
discretized signals. The required temperature input (Treq_wafer) is also discretized. The model of
the arc lamp and the wafer prior to discretization is a non-linear deterministic continuous-time
dynamical system.
Figure 4.1: Overall system block diagram
The outer temperature control loop operates at a sampling frequency of 1 kHz, or an equivalent
sampling time Δt of 1 ms. The inner current loop constituting the controller and the 4-phase
chopper was already designed a few years ago [13]. The inner current loop has a first order time
67
constant of 𝜏 =0.3 msec and the effective sampling frequency is 8 kHz, implying that Δt=0.125
ms: note each chopper operates at 2 kHz but because of interleaving the effective sampling rate
is 8 kHz. The inner current loop is replaced in our system by the following equation:
𝑖𝑙𝑎𝑚𝑝[𝑘] = 𝑖𝑟𝑒𝑞[𝑘] (∆𝑡
𝜏 + ∆𝑡) + 𝑖𝑙𝑎𝑚𝑝 [𝑘 − 1] (
𝜏
𝜏 + ∆𝑡) (4.1)
A sample waveform for the full inner loop [13] and a graph of equation 4.1 is shown in Fig 4.2
and Fig. 4.3 respectively. The two results are in good agreement with each other.
Figure 4.2: The step response of the full current loop
Figure 4.3: The step response of the first order low pass equation 4.1
68
4.2 Requirements of the Control System
The Temperature Controller shown in figure 4.1 should be designed to track a temperature ramp.
The temperature of the wafer is ramped up to around 900C at a rate chosen by the user which
would typically be between 50-150 degrees/sec. The initial wafer temperature can be between
25-250C and is not known precisely a priori because of measurement limitations: the measuring
device (the Ultra-Fast Radiometer) generates a poor signal to noise ratio at low wafer
temperatures due to the low amount of radiation emitted by the wafer. The wafer temperature
measurements are not reliable until the wafer reaches around 300C. Consequently, the design of
the controller should take into account the absence of a temperature measurement or feedback
below 300 C.
4.3 Proposed “Temperature Controller”
One solution to provide tracking of the wafer temperature ramp during the absence of feedback
is to use the wafer and lamp models to estimate the lamp current set-point required for wafer
temperature tracking, thereby operating the controller in an open loop manner.
Assume that the required wafer temperature is represented by a function 𝑔(𝑡) which takes on the
form:
𝑇𝑟𝑒𝑞_𝑤𝑎𝑓𝑒𝑟 = 𝑔(𝑡) = 𝑎0 + 𝑎1 · 𝑡 (4.2)
where 𝑎0 is the initial wafer temperature and 𝑎1 is the requested wafer temperature ramp rate.
The equation which models the temperature evolution of the wafer (equation 3.4) was
derived in chapter 3 and is repeated here:
𝑑𝑇
𝑑𝑡= −𝜅𝑁 ∙
𝑇 − 𝑇𝐶
𝑙𝑔𝑎𝑝𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)−
2𝜀𝑆𝑖(𝑇)𝜎𝑇4
𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)+
2 𝜂𝑙𝑎𝑚𝑝 𝑣𝑙𝑎𝑚𝑝 𝑖𝑙𝑎𝑚𝑝 𝛼 𝜂𝑜𝑝𝑡
𝑆𝑤𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇) (4.3)
69
where 𝑇 is implicitly assuming 𝑇(𝑡). We can rearrange equation 4.3 to solve for a future value of
the required lamp current, 𝑖𝑙𝑎𝑚𝑝(𝑡 + ∆𝑡), as follows:
𝑖𝑙𝑎𝑚𝑝(𝑡 + ∆𝑡) =𝑆𝑤𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)
2𝜂𝑙𝑎𝑚𝑝 𝑣𝑙𝑎𝑚𝑝 𝛼 𝜂𝑜𝑝𝑡 [
𝑑𝑇
𝑑𝑡+ 𝜅𝑁 ∙
𝑇 − 𝑇𝐶
𝑙𝑔𝑎𝑝𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)+
2𝜀𝑆𝑖(𝑇)𝜎𝑇4
𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇)]
(4.4)
Substituting 𝑑𝑇
𝑑𝑡= 𝑎1, for the required ramp rate and collecting terms, we obtain:
𝑖𝑙𝑎𝑚𝑝(𝑡 + ∆𝑡) =
𝑎1 ∙ 𝛽(𝑇) + 2𝜀(𝑇)𝜎𝑇4 + 𝜅𝑁 ·( 𝑇 − 𝑇𝑐)
𝑙𝑔𝑎𝑝
2 𝑣𝑙𝑎𝑚𝑝 𝜂𝑙𝑎𝑚𝑝 𝜂𝑜𝑝𝑡 𝛼 (1
𝑆𝑤)
(4.5)
where :
𝛽(𝑇) = 𝑡ℎ𝑆𝑖𝐶𝑝(𝑇)𝜌(𝑇) = 𝑡ℎ𝑆𝑖(1.95*106
+ 2.33* 102
T − 2.66 * 10−2
T 2 + 2.28 * 10
−7
T 3
) (4.6)
Since the temperature controller will be implemented using a discrete time controller, equation
4.5 is represented in discrete time as follows:
𝑖𝑙𝑎𝑚𝑝𝐼𝐶𝑀[𝑘 + 1] =
𝑎1 ∙ 𝛽(𝑇[𝑘]) + 2𝜀(𝑇[𝑘])𝜎𝑇[𝑘]4 + 𝜅𝑁 ·( 𝑇[𝑘] − 𝑇𝑐[𝑘])
𝑙𝑔𝑎𝑝
2 𝑣𝑙𝑎𝑚𝑝[𝑘] 𝜂𝑙𝑎𝑚𝑝[𝑘]𝜂𝑜𝑝𝑡 𝛼 (1
𝑆𝑤)
(4.7)
Equation 4.7 can be used to calculate the required current for obtaining a particular time
dependent temperature change of the wafer. However, the evaluation of eq. 4.7 requires the
knowledge of all the parameters on the right hand side of the equation. While
𝛽(𝑇[𝑘]), 𝑎1, 𝜀(𝑇[𝑘]), 𝜎, 𝜅𝑁 , 𝛼, 𝑆𝑤, 𝑙𝑔𝑎𝑝, 𝜂𝑜𝑝𝑡 are known, 𝑇[𝑘], 𝑇𝑐[𝑘], 𝑣𝑙𝑎𝑚𝑝 [𝑘], 𝜂𝑙𝑎𝑚𝑝[𝑘] are
unknown and must be computed. To determine 𝑣𝑙𝑎𝑚𝑝 [𝑘] and 𝜂𝑙𝑎𝑚𝑝[𝑘] in equation 4.7, the lamp
model can be invoked for the last known value of current, i.e. 𝑖𝑙𝑎𝑚𝑝[𝑘], to predict the lamp
70
voltage and radiative efficiency. 𝑇[𝑘], the wafer temperature, can be estimated using the
discretized version of equation 4.2, equation 4.8a and 4.8b:
𝑑𝑇[𝑘] = [−𝛫𝑁 ∙𝑇[𝑘] − 𝑇𝐶[𝑘]
𝑙𝑔𝑎𝑝𝑡ℎ𝑆𝑖𝐶𝑝(𝑇[𝑘])𝜌(𝑇[𝑘])−
2𝜀𝑆𝑖(𝑇[𝑘])𝜎𝑇4
𝑡ℎ𝑆𝑖𝐶𝑝(𝑇[𝑘])𝜌(𝑇[𝑘])+
2 𝜂𝑙𝑎𝑚𝑝[𝑘]𝑣𝑙𝑎𝑚𝑝[𝑘]𝑖𝑙𝑎𝑚𝑝 [𝑘]𝛼 𝜂𝑜𝑝𝑡
𝑆𝑤𝑡ℎ𝑆𝑖𝐶𝑝(𝑇[𝑘])𝜌(𝑇[𝑘])] ∙ ∆𝑡 (4.8𝑎)
𝑇[𝑘 + 1] = 𝑇[𝑘] + 𝑑𝑇[𝑘] (4.8𝑏)
and the known or guessed initial wafer temperature value. 𝑇𝑐[𝑘], the quartz substrate
temperature, can be calculated using the discretized version of equation 3.9, equation 4.9a and
4.9b, and the known initial substrate temperature.
𝑑𝑇𝐶[𝑘] = [κ𝑁 ∙𝑇[𝑘] − 𝑇𝐶[𝑘]
𝑙𝑔𝑎𝑝𝑡ℎ𝐶𝐶𝑝𝐶(𝑇𝐶[𝑘])𝜌𝐶(𝑇𝐶[𝑘])+
𝛼𝐶𝜀𝑆𝑖(𝑇[𝑘])𝜎𝑇4
𝑡ℎ𝐶𝐶𝑝𝐶(𝑇𝐶[𝑘])𝜌𝐶(𝑇𝐶 [𝑘])−
2 𝜀𝐶(𝑇[𝑘])𝜎𝑇𝐶[𝑘]4
𝑡ℎ𝐶𝐶𝑝𝐶(𝑇𝐶[𝑘])𝜌𝐶(𝑇𝐶[𝑘])] ∙ ∆𝑡 (4.9𝑎)
𝑇𝐶[𝑘 + 1] = 𝑇𝐶[𝑘] + 𝑑𝑇𝐶[𝑘] (4.9𝑏)
Therefore using the full lamp model, and equations 4.8 and 4.9, all the required parameters can
be calculated and there are no more unknowns on the right hand side of equation 4.7. Equation
4.7 can then be used to predict the value of 𝑖𝑙𝑎𝑚𝑝𝐼𝐶𝑀[𝑘 + 1].
Equation 4.7 along with the full lamp model, the wafer model equation 4.8 and the quartz model
equation 4.9 are collectively called the Internal Control Model (or ICM). The ICM is used to
predict the lamp current required to track the desired wafer temperature profile.
If the thermodynamic and transport coefficients (of the wafer and lamp) used in the Internal
Control Model match the real world values and the system of equations accurately describe the
real world devices, then perfect tracking of the requested temperature is obtained. In practice,
this is difficult to achieve since most coefficient values are approximations of the real ones and
the dynamic equations are based on several assumptions and simplifications. Errors in the
parameter estimations will cause the wafer temperature to deviate from the required profile. If
this is the case, then we need to be able to detect this deviation and correct for it. Temperature
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detection and correction using feedback is only possible after the wafer temperature
measurements become available, namely when the wafer reaches approx. 300C. The feedback
controller will compensate for mismatches in parameter estimations and simplifying assumptions
made in deriving the system models.
The corrective term, ∆𝑖𝑙𝑎𝑚𝑝[𝑘 + 1] can be implemented using a P (proportional) controller that is
engaged after the wafer temperature measurements become available and works in conjunction with the
Internal Control Model. The ICM computes an approximate value of the required lamp current and the P
controller adjusts this approximate value based on the difference between the requested and the measured
wafer temperature to produce a lamp current equal to:
𝑖𝑙𝑎𝑚𝑝 [𝑘 + 1] = 𝑖𝑙𝑎𝑚𝑝𝐼𝐶𝑀[𝑘 + 1] + ∆𝑖𝑙𝑎𝑚𝑝[𝑘 + 1] (4.10)
A diagram showing the proposed temperature controller architecture is shown in figure 4.4. In
this diagram, the ICM or Internal Control Model is shown as being comprised of the Wafer
Model and the Lamp Model. The Wafer Model here is a collection of equations: control equation
4.7, the wafer model equation 4.8 and the quartz model equation 4.9. The details of every block
in the temperature controller are presented in Appendix E.
Note that all the blocks inside the temperature controller in fig. 4.4, except the Lamp Model are
sampled at a frequency of 1 kHz. The Lamp Model has to independently be run at a higher
sampling frequency (>10 kHz) in order to obtain a solution which does not contain an aliased
signal. Therefore 16 kHz is taken to be the default sampling frequency of the Lamp Model block
unless otherwise specified. Simulation of the system shown in fig. 4.4 was therefore done at a
global sampling frequency of 16kHz where the temperature controller (excluding the lamp
model) was only updated once every 16 times.
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Figure 4.4: Architecture of the proposed control system
Since the system to be controlled (shown on the right half side of fig. 4.4) consisting of the N-
phase Chopper, the Arc Lamp and the Wafer Assembly is not currently available for real time
testing, the analysis of the system had to be done by simulation. Therefore the above mentioned
systems were replaced by their simulated versions i.e. their models, in the full system. This full
simulation setup is shown in figure 4.5.
Figure 4.5: Full simulation system setup
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Limiting the rate of change of current:
When the system shown in figure 4.5 is run, the lamp (or lamp model) is initially assumed to be
running at a current of 25A. In the first sampling cycle, the ICM initially estimates the current
required for wafer temperature tracking and passes this current value to the current controller,
which results in a step change in the lamp current. Unfortunately, upon applying a large enough
step change in the current to the lamp, a large transient spike is observed in the lamp voltage
waveform. Such a transient behavior, when a step current from 25A to 225A is applied to the
lamp model (with the outer loop disengaged) is shown in fig. 4.6. The response shown in figure
4.6 was obtained using discrete time simulation of the lamp model when simulated at a sampling
frequency of 96 kHz. Note that the higher frequency of 96kHz was used in this case just to
obtain higher output data resolution. The injection of such a step current into the lamp is known
to significantly accelerate electrode erosion and the resulting voltage response from the lamp
model also poses a potential problem for the outer loop as described next.
Figure 4.6: Lamp voltage and current, when a step current of 225A is applied to the Lamp
Model
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Notice how there is an initial large increase in the lamp voltage in response to the applied step
current. The physical mechanism behind this large spike is described as follows: when a large
step change in current is applied to the lamp, the lamp voltage also responds (increases) in a step
manner to a very large value because the resistance of the lamp cannot change instantaneously
due to thermal inertia. This large voltage causes a large amount of power to be injected into the
lamp. After a short time, the arc responds to the large power input by rapidly expanding and
increasing in temperature, causing the voltage to drop in an exponential manner and approach the
steady state value.
The lamp voltage signal, as shown in figure 4.5, is used by the ICM to estimate the lamp current
required for wafer temperature tracking. Since the duration of the voltage spike (such as the one
shown in fig. 4.6) is much smaller (< 0.1 msec) than the sampling time period of the outer loop
temperature controller (1ms), this voltage spike does not pose an immediate problem to the outer
loop as the outer loop would miss seeing this very fast transient. However, this voltage spike
does pose a potential problem in the event that the outer loop frequency is increased. Moreover
this large power pulse injection into the lamp can also cause undesired effects on the wafer
temperature and the lamp electrodes.
A scenario where the outer loop (temperature controller) frequency is increased to 96kHz (same
as that of the Lamp Model in this case) and is used to set the lamp current for wafer temperature
tracking is shown in figure 4.7. As can be seen, the effect of the lamp voltage spike is a large
disturbance on the lamp current setpoint estimated by the temperature controller. As mentioned
eariler, any spike or step change in current is known to have a large negative impact on the lamp
electrode erosion and hence is unacceptable.
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Figure 4.7: Lamp voltage and current when the outer loop (ICM) is run at the same
frequency as the Lamp Model and controls the lamp current for wafer temperature
tracking
A solution to circumvent the lamp output voltage spike and the electrode erosion issue is to post-
process the signal from the temperature controller by limiting the ramp rate of the current. A rate
limiting function is implemented by forcing the current requested by the temperature
controller, 𝑖𝑙𝑎𝑚𝑝 [𝑘 + 1] to go through a current conditioning block. The function of this block is
to limit the rate of change of current to a maximum of 1000A/s and also to place a limit on the
minimum lamp current at 25A. The minimum limit will ensure that the arc current never drops
below 25A ensuring that the arc will never extinguish when the controller operates. Note that the
1000A/s limit chosen on the rate of change of current was assumed to be the same as that used
by Mattson for the top lamps and is chosen to circumvent the electrode erosion issue (the lamp
dynamics are faster and can easily keep up with this rate of change of current). This 1000A/s
value may change in future and an exact value to use will be decided by Mattson through a series
of experiments. The revised control diagram incorporating the current conditioning block is
shown in figure 4.8.
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Figure 4.8: Proposed Temperature Controller with the Current Conditioning block added.
Fig. 4.9 shows that if we limit the rate of rise on the current setpoint signal we are able to
suppress the voltage spike and also reduce the electrode erosion caused by fluctuating currents.
Figure 4.9: Lamp voltage and current when the outer loop (ICM) is engaged but the
current setpoint signal is ramp rate limited to 1000A/s.
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A consequence of limiting the rate of rise of the lamp current is that temperature tracking
accuracy is affected during the period that rate limiting is active. The inaccuracy is seen during
the very start of the temperature ramp, since the controller requests a step change in current at
this point in time. As shown in figure 4.10, the observed effect of this current ramp limitation is a
small lag in temperature tracking. This is to be expected since the step change in current
requested by the controller and required for proper tracking, cannot be delivered instantly to the
lamp. However, the overall impact on temperature tracking is only observed very briefly at the
start of the ramp and the resulting constant offset between the requested and actual temperature
is acceptable for the purpose of this project.
Figure 4.10: The impact of limiting the Lamp Current ramp rate to 1000A/s is observed as
a small lag in tracking the requested temperature.
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Transition from Open loop to Closed loop Control:
As mentioned already, the wafer temperature measurements are unavailable until the wafer
attains a value of approximately 300C. Therefore the ICM works in an open loop manner below
a temperature of 300 C to provide temperature tracking. However, when the wafer reaches or
exceeds 300C and the measurements become available, the P controller is engaged and the wafer
temperature is tracked in closed loop beyond that point.
Such a system was simulated for three different cases. Case 1 shows the simulation of the ideal
scenario, i.e. no imperfections in the models implemented in the ICM while Case 2 shows the
non-ideal scenario i.e. the existence of some deficiencies in the models used in the ICM. Case 3
shows the simulation of the ideal scenario but where the initial wafer temperature is unknown
and an artificial value of the initial temperature is given to the ICM. The value of Kp for the P
controller used in these simulations was 5.
In case 1 shown in figure 4.11, there are no imperfections in the system models used in the ICM,
therefore the ICM is able to provide perfect tracking of the requested ramp rate (ignoring the
constant offset) even before the engagement of the P controller. However, when the wafer
reaches 300C, the P controller engages and a spike in the current is observed. This spike is
caused by the temperature offset between the actual and requested profile caused by the current
ramp rate limitation. This spike is unacceptable in the current profile as it can lead to accelerated
erosion of the anode electrode.
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Figure 4.11: Case 1: Simulation of the ideal scenario where the models in the ICM are
assumed to perfectly describe the real system.
In case 2, shown in figure 4.12, an error is purposely incorporated into the ICM to replicate the
scenario of a non-ideal system model. This was done by artificially increasing the absorptivity of
the wafer model (in the ICM) by 10%. This meant that the wafer model in the ICM and the wafer
model in the simulated real wafer differed by 10%. Figure 4.12 shows that initially, in the open
loop mode, when the ICM alone is providing the temperature tracking, the real wafer
temperature deviates from the requested temperature ramp rate. This is expected since an error
had been incorporated into the ICM which results in inaccurate tracking. Once the wafer reaches
300 C, the P controller engages and there is an even worse impact on the lamp current as the P
controller corrects for the difference between the requested and measured temperature. The
larger the difference between the requested temperature ramp and the measured wafer
temperature, the larger the transient current spike observed.
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Figure 4.12: Case 2: Simulation of the non-ideal scenario where a 10% higher wafer
absorptivity was assumed for the wafer model in the ICM as compared to the real wafer.
The current spike seen in the aforementioned two cases is not acceptable in the current profile as
it is suspected of enhancing electrode erosion in the lamp. However, a control option exists to
reduce this current transient and ensure a bumpless transfer from open loop operation to closed
loop.
The control option is to reset the requested temperature 𝑇𝑟𝑒𝑞_𝑤𝑎𝑓𝑒𝑟 [𝑘] to the first measured
temperature value upon engagement of the P controller. Therefore, immediately after
engagement of the P controller, the temperature difference between the requested and the
measured temperature seen by the P controller is zero and there is no current transient. The
tracking of the temperature ramp hence forth is performed with a new requested temperature
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function, 𝑔(𝑡) which is calculated using the first measured temperature value as the initial
temperature recorded and the ramp rate requested by the user.
The performance of the control system for the same conditions as in case 2 but with the
bumpless transfer (reset) option implemented is shown in figure 4.13. Note that this method
minimizes the transient current spike significantly, but in effect does require an offset change in
the input tracking function. This offset change in input tracking function however, is acceptable
for the purpose of the project because the ramp rate of the wafer following the offset change is
the same as the one requested. Having zero error upon engagement of the P controller also means
more flexibility in the choice of the gain Kp. A Kp = 15 was used for this case and was found to
deliver adequate tracking performance, therefore these P controller values are used in the
remainder of this document and the reason for this choice will be explained shortly. Note that it
may appear from fig. 4.13 that the error in tracking after the engagement of the P controller is
zero but in reality there is a very tiny amount of error which stays more or less constant and does
not exceed 4 degrees C for this case. This tiny error will exist in general when the models used in
the ICM are not perfect.
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Figure 4.13: Case 2: The simulation of the non-ideal scenario with the reset option
implemented. As seen there is no current spike upon engagement of the P controller at
t=2.45sec.
For Case 3, the ideal scenario was simulated with the aforementioned reset option incorporated
but where the initial wafer temperature was not known. In this case the ICM is given an artificial
value for the initial wafer temperature = 25C to begin with while the real wafer was given a
value of 85C as the initial temperature. It can be seen from figure 4.14 that the artificial value
given to the ICM is used to create a requested wafer temperature profile. This simulation
however demonstrates that the lack of knowledge of the initial wafer temperature does not lead
to any current spikes in the lamp current profile and the real (or actual) wafer temperature ramp
follows the requested ramp rate unaffected.
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Figure 4.14: Case 3: Simulation of the ideal scenario where the initial wafer temperature is
unknown. The transition to closed loop is still smooth.
Choice of Kp:
The criteria for choosing Kp is purely based on controlling the shape of the current profile. The
fact that the current profile must be smooth and free of spikes plays a bigger role in determining
the Kp than the wafer temperature response. A major constraint for the choice of Kp comes from
the fact that the rate of change of current is limited to 1000A/s by the current conditioning block.
This hard limit on the rate of rise of current cannot be removed since it guarantees that the
electrode life time is not being compromised. Therefore, if the P controller requests a change in
current greater than this 1000A/s limit then its requested current will not be delivered to the lamp
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and this can cause the P controller to perform in an undesirable way and introduce spikes into the
current profile as shown in fig. 4.15.
Figure 4.15: Simulation showing the scenario where the P controller’s (Kp = 35) request
exceeds 1000A/s which results in a spike in the current profile
Therefore it is necessary to design the P controller such that its request never exceeds 1000A/s
for even the worst case possible i.e. the highest change in error encountered by the P controller.
The simulation for this worst case was done in a way which is similar to case 2, as shown in
figure 4.12, but instead of incorporating an artificial error of 10%, an error of 50% was
incorporated. A simulation of this scenario with Kp =15 is shown in fig. 4.16. Note: an error of
50% is referred as the worst case only to test the robustness of the system against such a large
error and the chances of such a scenario occurring in reality are very highly unlikely (since the
system models developed were shown in chapter 3 to be in close agreement with the
experimental results).
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Figure 4.16a: Simulation of the worst case scenario where a 50% error is incorporated in
the ICM which results in a large deviation of the wafer temperature prior to the P
controller engagement and a huge corrective action after its engagement
Figure 4.16b: Zoomed in version on Figure 4.16a
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According to figure 4.16a, before the engagement of the P controller (t < 4.3s), the presence of a
large error in the ICM causes the actual wafer temperature to deviate largely from the requested
temperature profile. While the requested ramp rate is 135C/s, the actual wafer only does 65C/s.
After the wafer reaches 300C, the temperature measurements become available and the requested
temperature is reset to the first measured value, as shown in figure 4.16b (the zoomed in version
of fig. 4.16a). At this instant the P controller would see zero error and produces zero output.
However, because of the error incorporated into the ICM, the real wafer again begins to deviate
away from this new requested profile. Since the sampling time of the temperature controller is
1ms, the error seen by the P controller in the next time step (1ms later) is 0.07C (= (135C/s-
65C/s)/1000). This change in error from 0 to 0.07C is the maximum that the P controller will
encounter during the process because, this error will cause a corrective action from the P
controller which will result in a much lower change in error in the subsequent time steps.
Therefore we can use this known, maximum change in error (for the worst case) to estimate the
maximum usable gain Kp to cause the output of the P controller to change at 1000A/s.
𝑒𝑟𝑟𝑜𝑟 = 𝑇𝑟𝑒𝑞_𝑤𝑎𝑓𝑒𝑟 − 𝑇𝑤𝑎𝑓𝑒𝑟 (4.11)
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑃 𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑒𝑟𝑟𝑜𝑟 ∗ 𝐾𝑝 (4.12)
1000𝐴/𝑠 = 0.07𝐶/𝑚𝑠 ∗ 𝐾𝑝 = 70𝐶/𝑠 ∗ 𝐾𝑝 (4.13)
𝐾𝑝 = 1000/70 = ~15 (4.14)
Therefore a Kp = 15 will never cause the change in the P controller’s output to exceed 1000A/s
for even the worst case scenario. Hence ensuring that the P controller can always deliver its
request without any obstruction from the current conditioning block. From fig. 4.16b, it can be
seen that the requested current and the lamp current overlap each other. Note that the value of Kp
=15 was chosen for a 1000A/s limit on the rate of change of current but can still be used even if
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the 1000A/s limit is increased. But if the 1000A/s limit is reduced then the value of Kp must be
recalculated using the approach shown above.
In summary, it can be seen from fig 4.16a that, after the engagement of the P controller, the
current profile is smooth and the wafer temperature follows the requested temperature profile
very closely. After the engagement of the P controller, the error remains more or less constant at
around 10 degrees and this size of error is acceptable for the project.
Although the tracking response of the P controller is sufficient for the purpose of the project, a PI
controller with Kp= Ki= 15 was also tested. The PI controller gave a slightly better tracking
performance by having a lower overall error. Therefore, either controller (P or PI) can be used as
a feedback controller in the proposed temperature controller.
4.4 Performance of the Control System:
This section will show the improvement brought about by the proposed temperature controller
relative to the existing control system used at Mattson’s Facility. The main improvement is the
smoothness of the current profile. The current profile simulated for a number of cases is free of
any oscillation. In addition, the ramp rate of the wafer temperature is steadier and follows the
requested profile very closely.
Two cases will be shown to demonstrate the improvement in the current smoothness over the
existing system: one for a 130C/sec ramp (figures 4.17a, 4.17b) and another for a 70C/sec ramp
(figures 4.18a, 4.18b). In each case, the experimentally recorded lamp current and wafer
temperature (using Mattson’s existing controller) are shown first and then the same quantities are
shown for the new proposed temperature controller.
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Case 1: 130C/s
Figure 4.17a: Experimentally Recorded Wafer Temperature and Lamp Current for a
Ramp Rate of 130C/s
Figure 4.17b: Performance of the Proposed System showing the Wafer Temperature and
Lamp Current for a Ramp Rate of 130C/s.
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Case 2: 70C/s
Figure 4.18a: Experimentally Recorded Wafer Temperature and Lamp Current for a
Ramp Rate of 70C/s
Figure 4.18b: Performance of the Proposed System Showing the Wafer Temperature and
Lamp Current for a Ramp Rate of 70C/s.
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4.5 Alternative/ Less Computationally Intense Version of Control System Architecture
The proposed temperature controller can be, from an implementation point of view,
computationally intensive because of the incorporation of the full lamp model. An alternative
system which can be used to perform temperature tracking is to use the real time voltage
measurements across the lamp (if available) to substitute using the lamp model in the ICM. This
measured voltage value can be used in equation 4.7 in place of the voltage estimated by the full
lamp model. The radiative efficiency of the lamp is stored in a lookup table as a function of input
lamp power and can be substituted in place of the radiative efficiency outputted from the lamp
model. This alternative is shown in figure 4.19.
Figure 4.19: Alternative system architecture. Real time lamp voltage measurement replaces
the lamp model.
The results of the performance of this system are not available since it requires the measurement
of the lamp voltage in real time which is only possible in an experimental setting. However, the
performance of the above system is expected to be very similar to the system which includes the
lamp model. This comes from the fact that the lamp model is sufficient (as demonstrated in
chapter 2) to predict the lamp voltage which is in very close agreement with the measured lamp
voltage.
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CHAPTER 5: CONCLUSIONS
The objective of this thesis was to develop a control system capable of controlling the
temperature ramp rate of a semiconductor wafer using two bottom lamps before flash annealing
the top surface with four lamps. A major requirement of the controller was to generate a
fluctuation free lamp current profile, since it is known that a fluctuating current profile leads to
enhanced anode electrode erosion.
The first step was to build a wafer and lamp model and validate both models using available
experimental data. Chapter 2 presented the Full Lamp Model which is capable of accurately
estimating the lamp voltage and radiated power for a given current. The modelling process
started by first describing the electrical characteristic of two electrode regions and two
constriction regions. The positive column constituting the fifth region was described using
equations obtained from previous work aside from the method used to evaluate the radiation
term. The radiation term was modeled using a net emission coefficient, initially with Rp=7 mm.
The five regions were combined together and a model describing the lamp voltage as a function
of current was produced using a numerical software package MATLAB. The positive column
was constructed using five concentric cylinders which resulted in five coupled nonlinear ordinary
differential equations describing the average arc temperature in each of the five concentric zones
and one coupled algebraic equation relating the injected current to the axial electric field in the
positive column. The positive column voltage was a post processed computation. The
comparison of the results of the model with the experimental results showed two major
discrepancies; the model describing lamp voltage as a function of current was in error and the
radiation leaving the quartz outer enclosure was not in agreement with the experimental results.
The corrected model incorporated the following changes: 1) reduced isothermal arc radius (Rp =
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5mm) by increasing the water-wall thickness and 2) employed an experimentally determined
empirical factor to account for radiation absorption by the cold argon gas, water-wall and quartz
tube. These changes resulted in very good agreement between the results from the model and the
results from experiments.
Chapter 3 presented the formulation of the wafer model, wherein an ordinary nonlinear
differential equation was developed to describe the temporal evolution of the wafer temperature.
This involved details involving the structure of the process chamber and the wafer’s reaction to
the lamp’s radiation. The wafer model and lamp model were then combined in a system that was
tested with the experimentally recorded current profiles. The resulting wafer temperature profiles
were then compared with the experimentally recorded ones and it was shown that the agreement
between theory and experiment was good.
Chapter 4 presented the controller strategy for tracking a temperature ramp for the wafer. The
control system is comprised of an inner current loop for controlling the arc lamp current and an
outer temperature loop for controlling the wafer temperature. The temperature controller operates
in open loop until a wafer temperature measurement with an acceptable level of accuracy is
available (typically at a temperature around 300 C). Below 300 C, the temperature controller
operates in open loop and the required lamp current for tracking the temperature is computed
using the wafer and full lamp models. This was referred to as the internal control model (ICM).
Once the wafer temperature exceeds 300 C, a P controller is engaged to adjust the output of the
ICM to minimize the tracking error. To also ensure a bumpless transfer from open loop to closed
loop, a reset option was also implemented. The first viable temperature measurement above 300
C is used as an initial temperature once the P controller is engaged. Finally, the full system
consisting of the controllers combined with the lamp and wafer models was tested and the
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performance of this system was compared with experimentally recorded data. It was shown that
the full system is capable of tracking a temperature ramp accurately and the resulting lamp
current profile is very smooth and free of any fluctuations.
Thesis Contributions
The thesis contributions are as follows:
Constructed a full model of the lamp consisting of the five different physical regions of
the lamp. The model is capable of providing a good estimate of the lamp voltage and
radiated power.
Produced a thermal model of the wafer to predict its temperature evolution upon
exposure to wafer radiation.
Developed the control system that provides wafer temperature control with a bumpless
transfer from open loop to closed loop wafer and no fluctuations in the arc current.
Future Work
A number of issues have arisen as a result of this research and require further study. Possible
avenues for future research include:
Simulating the cathode constriction region including the fluid flow equations in order to
correctly account for the effects of gas flow in the cathode constriction region. In
particular, the impact of gas flow on the (temperature profile and hence the) voltage drop
across the region should be clarified further and used to improve the model accuracy.
94
Studying radiation transport in the regions outside the isothermal arc in a more detailed
manner and hence improve the agreement between theory and experiment for the
radiative efficiency above the 100 kW input power level.
Investigating the physical phenomenon which leads to a discrepancy between theory and
experiment for the radiative efficiency below an input power level of 50 kW.
Attempt a slightly different controller strategy in which the inner loop regulates the
lamp’s input power rather than the lamp current. Care must be exercised as this strategy
may lead to fluctuating lamp current profiles which are not permissible.
Introducing a model order reduction technique which would allow the internal control
model to operate at a lower sampling rate.
Introducing a noise source at the output of the current to account for the chopper
switching harmonics and investigate the influence of this noise on the disturbance
rejection and on the choice of sampling frequency for the internal control model.
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transfer in high-voltage circuit breakers,” J. Phys. D: Appl. Phys., vol. 45, 2012.
[8] M. F. Modest, Radiative Heat Transfer, 2nd ed. San Diego: Academic Press, 2003, ch.13,
sec. 13.7, pp. 440-444.
[9] E. C. Beder, C. D. Bass, and W. L. Shackleford, “Transmissivity and Absorption of Fused
Quartz Between 0.22 and 3.5 micrometer from Room Temperature to 1500 degree C,”
Appl. Optics, vol. 10, no. 10, pp. 2263-2268, Oct. 1971.
[10] R.K. Endo, Y. Fujihara, M. Susa, “Calculation of density and heat capacity of silicon by
molecular dynamics simulation,”, High Temperatures - High Pressures, 35/36(5), pp. 505-
511, 2006.
[11] T. Sato, “Spectral Emissivity of Silicon,” Japanese Journal of Applied Physics, vol. 6,
pp. 339 - 347, 1967.
[12] P.J. Timans, “Emissivity of silicon at elevated temperatures,” Journal of Applied Physics,
vol. 74, pp. 6353-6364, 1993.
[13] A. A. El-Deib, “Modeling of and Driver Design for a Dielectric Barrier Discharge Lamp,”
Ph.D. thesis, Dept. Elect. And Comp. Eng., Univ. of Toronto, Toronto, ON, 2010
[14] M. F. Modest, Radiative Heat Transfer, 2nd ed. San Diego: Academic Press, 2003, ch.9,
sec. 9.6, pp. 271-274.
[15] M. S. Benilov, and M. D. Cunha, “Heating of refractory cathodes by high-pressure arc
plasmas: 2,” J. Phys. D: Appl. Phys., vol. 36, no. 6, pp. 603-614, Feb, 2003.
[16] M. F. Modest, Radiative Heat Transfer, 2nd ed. San Diego: Academic Press, 2003, ch.9,
sec. 9.5-9.10, pp. 269-279.
96
Appendix A: Ultra-fast Radiometer
The UFR (Ultra-Fast Radiometer) is a type of temperature measurement device that measures the
irradiance emitted from the body and then uses that to calculate the actual temperature of the
body. With the need to control the flash anneal process (1msec), higher sampling rates are
needed. This is the reason for it being called ultra-fast. This device provides measurements at
100 kHz.
InGaAs photo
diode
Em
itta
nce
[W
]
Low Temperature Amplifier High Temperature Amplifier
Low Temperature A/D
converter
High Temperature A/D
converter
0..
5V
0.5
V
InGaAs diode output [mA]
18
bit
Interface to the UFR FPGA
18
bit
Figure A1: Figure showing the architecture of the UFR.
The architecture of the UFR is shown on figure A1. Light that falls onto the photo diode is
previously filtered (using band gap filters) to the narrow band around the wavelength 1450nm.
97
The reason for choosing this wavelength is because the water windows filter the radiation from
the lamps at this wavelength and prevent it from entering the process chamber. This ensures that
the radience seen by the UFR is purely from the wafer. For an irradience power of 1W, the photo
diode will generate a current signal of 0.9A. This signal is then fed into the two amplifiers with
different gains, one for low- and second for high- temperature measurements. Output from the
both amplifiers is 0-5V.
Voltage signal from the amplifiers is fed to two separate 18-bit SAR A/D converters. The UFR
firmware controls the A/D converters that are producing 100K 18-bit samples per second.
Depending of the temperature either the signal from the low temperature channel, combination of
both or just high temperature channel is used for the temperature calculation.
In this section the details of the following topics are discussed:
Temperature calculation using the UFR data
Wafer top side emissivity estimation
Wafer bottom side emissivity measurement
Temperature Calculation using the UFR data:
The temperature of each side of the wafer is calculated from the spectral radiance, L (watts)
measured by the UFR by using the Planck’s Equation (the emissivity must be known):
15.273
1),(
),(2ln
1
5
2
TLS
Thck
hcT
nuationFilterAtte
B
(A1)
where:
98
T is in degrees Celsius
h = 6.6260755E-34 J*s
c = 2.99792458E8 m/s
= 1450E-9 m
kB = 1.380658E-23 J/K
= Narrowband filter bandpass = 25E-9 m.
(,T) = Spectral Emissivity (could be derived from various sources)
L(,T) = Spectral Radiance
SFilterAttenuation = Attenuation coefficient due to extra filtering in a port of the
chamber. For a transparent window, this term equals 1, but if a Neutral
Density (ND) filter, for example, is placed in a port, this term should be set to
the attenuation factor at 1450nm. Typically, a 10x ND filter is used in the top
UFR port. High accuracy is not required, so nominal values (i.e. 10) quoted
by the manufacturer are sufficient.
Calculating (or Estimating) Emissivity of the Wafer Top side
Emissivity is calculated by re-arranging Planck’s Equation (shown in the previous section) to
solve for emissivity. In this case the wafer surface temperature must be known.
1
15.273exp
),(),( 1
2 T
a
a
TLST nuationFilterAtte
(A2)
where a1 and a2 are the following constants:
Kek
hca
B
39225425.91
25
2
2 64594111.12
mWe
hca
The estimation of wafer top side emissivity is done at the point where the wafer has reached the
intermediate temperature (around 900C) and the top side of the wafer is about to undergo the
99
millisecond flash annealing. At this point the temperature of the bottom of the wafer is known.
The top side temperature of the wafer can then be approximated to be the same as the bottom
temperature. This estimated temperature will be used with the radiance from the top UFR to
calculate the emissivity for the top of the wafer. This top emissivity (assuming that it does not
change) will then be used for subsequent top temperature measurements during the flash
annealing.
Bottom side Emissivity measurement:
Figure A2: Diagram showing the UFRs in the system and the diagnostic flash setup for the
estimation of the bottom side emissivity
The emissivity of the bottom side of the wafer is measured by firing a flash from an external
source (emissometer) onto the wafer (see fig. A2). During the flash the top and bottom UFR will
estimate the transmissivity and the reflectivity of the wafer. This will then be used to estimate the
emissivity of the wafer using
tr 1 (A3)
Where r is the reflectivity and t is the transmissivity. r and t are measured simultaneously by
coordinating 3 UFRs and the diagnostic flash. The diagnostic flash is a short-duration (~1msec)
100
light source that is brighter than the wafer’s radiation. It is directed towards the center of the
wafer from the bottom corner of the chamber opposite to the top UFR. If the wafer is partially
transparent (t > 0), the top UFR sees a portion of the diagnostic flash. Similarly, the bottom UFR
sees the reflection of the diagnostic flash off the bottom of the wafer. The signal from the top
UFR is linearly related to the transmissivity of the wafer, whereas the signal from the bottom
UFR is linearly related to the reflectivity of the bottom of the wafer. A reference radiometer will
be located outside the chamber. A portion of the diagnostic flash will be delivered to the
reference UFR (via a beam-splitter of bifurcated fiber bundle, for example). This will allow the
reference UFR to provide a measure of the absolute intensity of a given flash to correct for shot-
to-shot variations.
101
Appendix B: Derivation of the total absorbed heat flux from
incident intensity
Figure B1: Geometry for derivation of the Radiative Heat Flux Absorbed
Assume that the spectral intensity incident on a differential area dAs on the surface of dV is
given by 𝐼𝜆(𝜏𝜆, Ω) from equation 2.5. The change of this incident intensity in dV as a result of
absorption is:
𝑑𝐼𝜆 = − 𝐼𝜆(𝜏𝜆, Ω) 𝜅𝜆(𝑇) 𝑑𝐿 (𝐵1)
where 𝜅𝜆(𝑇) is the absorption coefficient of the volume element dV.
Therefore the power absorbed by the differential subvolume dAs dL from this incident radiation
is:
𝑑𝐹𝑟𝜆_𝐴𝐵𝑆𝑂𝑅𝐵𝐸𝐷 = −𝑑𝐼𝜆 𝑑𝐴𝑠 𝑑Ω = 𝐼𝜆(𝜏𝜆, Ω) 𝜅𝜆(𝑇) 𝑑𝐿 𝑑𝐴𝑠 𝑑Ω (B2)
Obtaining the total power absorbed by all of dV from this incident intensity, we get:
𝑑𝐹𝑟𝜆_𝐴𝐵𝑆𝑂𝑅𝐵𝐸𝐷 = 𝐼𝜆(𝜏𝜆, Ω) 𝜅𝜆(𝑇) 𝑑Ω ∫ 𝑑𝐴𝑠 𝑑𝐿 (𝐵3)
𝑑𝑉
102
Therefore, integrating the incident intensities over all solid angles we get:
𝐹𝑟𝜆_𝐴𝐵𝑆𝑂𝑅𝐵𝐸𝐷 = 𝜅𝜆(𝑇) 𝑑𝑉 ∫ 𝐼𝜆(𝜏𝜆, Ω) 𝑑Ω (𝐵4)4𝜋
0
103
Appendix C: Derivation of the Net Emission Coefficient
Derivation of the Net emission Coefficient, equation 2.15 [M. Bartlova, V. Aubrecht, O. Coufal,
(2010)]:
From equation 2.7:
∇ ∙ 𝐹𝑟𝜆 = 4𝜋 𝜅𝜆(𝑇)(𝐼𝑏𝜆(𝑇) − 𝐽𝜆) (𝐶1)
where: 𝐽𝜆 = 1
4𝜋∫ 𝐼𝜆(𝜏𝜆, Ω) 𝑑Ω (𝐶2)
4𝜋
0
is the mean radiation intensity of the incident radiation from all directions. For an isothermal
cylinder of radius Rp, the average spectral intensity 𝐽𝜆 is approximately the same as an isothermal
sphere of radius Rp, and is given by (Liebermann et al. (1976)):
𝐽𝜆 = 𝐼𝑏𝜆(𝑇)[1 − 𝑒(−𝜅𝜆 (𝑇)𝑅𝑝)] (𝐶3)
Substituting 𝐽𝜆 from equation C3 in C1, the net emission at the arc center of an isothermal arc of
radius Rp is:
∇ ∙ 𝐹𝑟𝜆 = 4𝜋 𝜅𝜆(𝑇) (𝐼𝑏𝜆(𝑇) − 𝐼𝑏𝜆(𝑇)[1 − 𝑒(−𝜅𝜆 (𝑇)𝑅𝑝)]) (𝐶4)
∇ ∙ 𝐹𝑟 = 4𝜋 ∫ 𝐼𝑏𝜆(𝑇) 𝜅𝜆(𝑇) 𝑒(−𝜅𝜆 (𝑇)𝑅𝑝) 𝑑𝜆 = 4𝜋𝜀𝑁 ∞
0
(𝐶5)
𝜀𝑁 = ∫ 𝐼𝑏𝜆(𝑇) 𝜅𝜆(𝑇)𝑒−(𝜅𝜆(𝑇)𝑅𝑝)𝑑𝜆∞
0
[𝑊
𝑠𝑟. 𝑚3] (𝐶6)
104
Appendix D: Calculation of attenuation of incident intensity in the
outer regions of the lamp
Equation 2.18 which is repeated below is the governing equation describing the attenuation of
incident intensity along a given path in a cold medium, which is not radiating itself:
𝐼𝜆(𝜏𝜆, Ω) = 𝐼𝜆(0, Ω) exp[− ∫ 𝜅𝜆
𝑆
0
(𝑇) 𝑑𝑟] (𝐷1)
This expression can also be evaluated along a specific direction which in our case is in a radial
direction. The symbol 𝜅𝜆(𝑇) represents the absorption coefficient as a function of wavelength
and temperature where temperature is a function of position. S represents the total length over
which the integration takes place which is from the isothermal core radius to the outer quartz
tube radius.
There are three cold regions outside the isothermal arc to consider: the cold argon, the water-
wall, the quartz tube. The absorption coefficient of argon gas (98% argon & 2% water) as a
function of wavelength is shown in figure D1 for various temperatures.
105
Figure D1: Absorption coefficients of argon vs wavelength for various temperatures
The cold argon gas has a very high absorption coefficient in the 70-80nm range, as shown in Fig.
D1. Since the precise amount of power emitted by the isothermal arc within the 70-80nm range
is unknown (but is known for <200nm), an exact estimate of the transmissivity factor is not
possible. However, as will be shown, the water-wall by itself completely attenuates the radiation
with a wavelength less than 200nm, therefore, the exact estimate of the transmissivity by the cold
argon gas is not required. In addition, the lamp tube made of quartz is transparent to radiation in
the infrared and visible range but is not very transparent to wavelengths less than 200nm and
greater than 3000nm [9]. This pattern of spectral absorption is similar to the absorption pattern of
water, therefore, spectral absorption of radiation overlaps in all three regions i.e. all three regions
absorb the same wavelengths of radiation. Hence to a first approximation, we only need to
consider the water-wall and can ignore the cold argon and quartz tube, since water has the
106
highest absorption coefficients and also has a much greater thickness in the lamp in comparison
to the other two layers.
The absorption coefficient of water at room temperature is shown in figure D2 as a function of
wavelength.
Figure D2: Absorption coefficient of water at room temperature
The water-wall temperature was assumed fixed at 25C and the thickness of the water-wall was
assumed to be 5mm. Therefore S in equation D1 was taken to be 5mm.
The total transmissivity factor for water corresponding to the contribution of all spectral bands is
given by:
𝑃𝑂𝑈𝑇
𝑃𝐵= ∑
𝑃𝜆𝑖_𝑂𝑈𝑇(𝜏𝜆)
𝑃𝐵
𝑁
𝑖=1= ∑
𝑃𝜆𝑖_𝐼𝑁(0)
𝑃𝐵 exp[− ∫ 𝜅𝜆𝑖
𝑆
0
(𝑇(𝑠)) 𝑑𝑠𝑁
𝑖=1 ] (𝐷2)
107
The transmissivity factor for a particular band i is given by:
exp [− ∫ 𝜅𝜆𝑖
𝑆
0
(𝑇(𝑠))𝑑𝑠] (𝐷3)
where 𝜅𝜆𝑖(𝑇(𝑠)) represents a form of mean absorption coefficient as a function of radius for a
particular spectral band i. This expression is only an estimate of the maximum amount of
transmission expected. Therefore, the largest value expected of the individual transmissivity
factors (eq. D3) for the 8 bands is:
REST (<200nm) = 0%
UVC (200-280nm) = 100%
UVB (280-315nm) = 100%
UVA (315-400nm) = 100%
VIS2 (400-780nm) =100%
IRA (780-1400nm) = 95%
IRB (1400-3000nm) = 0%
IRC (>3000nm) = 0%
Therefore, the incoming radiation with wavelengths below 200nm and above 1400nm, is
completely attenuated inside the lamp while the other bands may have partial transmission which
cannot be predicted using equation D2. The total transmissivity factor 𝑃𝑂𝑈𝑇
𝑃𝐵 can then be estimated
using the above individual transmissivity factors and the temperature dependent data for 𝑃𝜆𝑖_𝐼𝑁(0)
𝑃𝐵
from figure 2.18. This yields a total transmissivity factor varying between 80-70% for an
isothermal arc temperature range of 10,000-13000K.
108
Temperature dependency of the Arc’s spectral power output:
As mentioned earlier, the experimental radiative efficiency of the lamp is almost independent of
the lamp input power but the estimated full lamp model predicts an increase in radiative
efficiency with input power, as shown in figure 2.19.
One possible theory explaining this difference between the model results and experimental data
is the temperature dependency of the arc radiation spectrum as shown in figure 2.18. During the
simulations of the estimated full lamp model it was seen that the arc temperature increases for an
increase in current i.e. the arc temperature is a function of the arc current and hence the input
power as per fig. 2.14. This increase in arc temperature causes the radiation spectrum of the arc
to change, shifting the output radiation more towards the shorter wavelengths.
Figure 2.18 shows that a change in arc temperature from 10,000K to 13,000K causes the
percentage of output radiation in the infrared and visible bands to reduce, along with a
simultaneous percentage increase in the power in the shorter wavelengths (<200nm). This has a
significant impact on the profile of the lamp’s radiative efficiency since the cold gas, water and
quartz tube absorb radiation having wavelengths less than 200nm. This self-absorption of shorter
wavelengths by the lamp and the fact that the lamp radiation shifts towards shorter wavelengths
at higher temperatures could explain why the radiative efficiency of the lamp as seen in the
experimental data does not increase much with an increase in input current/power.
109
Appendix E: Details of the Blocks used in the Temperature
Controller
Wafer Model:
Figure E1: the wafer model block showing the inputs and outputs
Discretized equation used in the Block:
𝑖𝑙𝑎𝑚𝑝𝐼𝐶𝑀[𝑘 + 1] =
𝑎1 ∙ 𝛽(𝑇[𝑘]) + 2𝜀(𝑇[𝑘])𝜎𝑇[𝑘]4 + 𝜅𝑁 ·( 𝑇[𝑘] − 𝑇𝑐[𝑘])
𝑙𝑔𝑎𝑝
𝑣𝑙𝑎𝑚𝑝[𝑘] 𝜂𝑙𝑎𝑚𝑝[𝑘]𝜂𝑜𝑝𝑡 𝛼 (1
𝑆𝑤)
(𝐸1)
Equation Parameters:
a1 : Requested temperature ramp rate of wafer
𝑇[𝑘] : measured or estimated wafer temperature as follows:
𝐼𝑓 𝑇[𝑘] < 300𝐶, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑇[𝑘] 𝑖𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑏𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:
𝑑𝑇[𝑘] = [−Κ𝑁 ∙𝑇[𝑘] − 𝑇𝐶[𝑘]
𝑙𝑔𝑎𝑝𝑡ℎ𝑆𝑖𝐶𝑝(𝑇[𝑘])𝜌(𝑇[𝑘])−
2𝜀𝑆𝑖(𝑇[𝑘])𝜎𝑇4
𝑡ℎ𝑆𝑖𝐶𝑝(𝑇[𝑘])𝜌(𝑇[𝑘])+
2 𝜂𝑙𝑎𝑚𝑝[𝑘]𝑣𝑙𝑎𝑚𝑝[𝑘]𝑖𝑙𝑎𝑚𝑝 [𝑘]𝛼 𝜂𝑜𝑝𝑡
𝑆𝑤𝑡ℎ𝑆𝑖𝐶𝑝(𝑇[𝑘])𝜌(𝑇[𝑘])] ∙ ∆𝑡 (𝐸2𝑎)
𝑇[𝑘 + 1] = 𝑇[𝑘] + 𝑑𝑇[𝑘] (𝐸2𝑏)
𝐼𝑓 𝑇[𝑘] > 300𝐶, 𝑡ℎ𝑒𝑛 𝑇[𝑘] = 𝑇𝑤𝑎𝑓𝑒𝑟 [𝑘] (𝑖. 𝑒. 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑏𝑦 𝑠𝑒𝑛𝑠𝑜𝑟) (𝐸2𝑐)
β(T[k]) = 𝑡ℎ𝑆𝑖 · 𝜌𝑆𝑖(𝑇[𝑘]) · 𝐶𝑝𝑆𝑖(𝑇[𝑘]) = thSi (1.95*106+ 2.33* 102T − 2.66 * 10−2T 2
+ 2.28 * 10−7T 3) (𝐸3)
ε(T[k]) = 1.53 · 10−3𝑇[𝑘] − 0.3 𝑓𝑜𝑟 280𝐶 < 𝑇[k] < 660𝐶
0.7 𝑓𝑜𝑟 𝑇[k] > 660𝐶 (𝐸4)
α : wafer absorptivity coefficient = 0.65
110
σ = 5.67×10−8 W m−2 K−4 is the Stefan–Boltzmann constant
𝜅N = 0.03 W/(m · K)
Tc[k]: temperature of the quartz substrate calculated by the following equations:
𝑑𝑇𝐶[𝑘] = [κ𝑁 ∙𝑇[𝑘] − 𝑇𝐶[𝑘]
𝑙𝑔𝑎𝑝𝑡ℎ𝐶𝐶𝑝𝐶(𝑇𝐶[𝑘])𝜌𝐶(𝑇𝐶[𝑘])+
𝛼𝐶𝜀𝑆𝑖(𝑇[𝑘])𝜎𝑇4
𝑡ℎ𝐶𝐶𝑝𝐶(𝑇𝐶[𝑘])𝜌𝐶(𝑇𝐶[𝑘])−
2 𝜀𝐶(𝑇[𝑘])𝜎𝑇𝐶[𝑘]4
𝑡ℎ𝐶𝐶𝑝𝐶(𝑇𝐶[𝑘])𝜌𝐶(𝑇𝐶[𝑘])] ∙ ∆𝑡 (𝐸5𝑎)
𝑇𝐶[𝑘 + 1] = 𝑇𝐶[𝑘] + 𝑑𝑇𝐶[𝑘] (𝐸5𝑏)
𝑙𝑔𝑎𝑝 : The space between the wafer and the quartz substrate = 1.4mm
𝑣lamp[k], ηlamp[k] : Voltage and Radiative efficiency of the lamp
𝑆𝑤= 3.14 * 0.15 * 0.15 m^2
thSi : wafer thickness = 0.775mm
𝜂𝑜𝑝𝑡: optical efficiency of the process chamber = 0.23
LAMP MODEL:
Figure E2: The lamp model block showing the inputs and outputs
Discretized equation(s) used in the Block:
𝐸[𝑘] =𝑖𝑙𝑎𝑚𝑝[𝑘]
𝐺𝑎𝑟𝑐[𝑘]=
𝑖𝑙𝑎𝑚𝑝[𝑘]
2𝜋 ∑ (𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙𝐴𝑟𝑒𝑎(𝑛)∙𝜎𝑎𝑟𝑐(𝑛,𝑇𝑎𝑟𝑐)[𝑘])50
(𝐸6)
for n=1 to 5:
111
∆𝑇𝑎𝑟𝑐(𝑛)[𝑘]
= 𝜎𝑎𝑟𝑐(𝑇𝑎𝑟𝑐(𝑛))[𝑘] · 𝐸[𝑘]2 − 4𝜋𝜀𝑁(𝑇𝑎𝑟𝑐(𝑛))[𝑘] + ((𝑇𝑎𝑟𝑐(𝑛)[𝑘] − 𝑇𝑎𝑟𝑐(𝑛 − 1)[𝑘])/𝜅(𝑇𝑎𝑟𝑐(𝑛))[𝑘])
𝐶𝑝𝑎𝑟𝑐(𝑇𝑎𝑟𝑐(𝑛))[𝑘] · 𝜌𝑎𝑟𝑐(𝑇𝑎𝑟𝑐(𝑛))[𝑘] · ∆𝑡 (𝐸7)
n++;
end
Total lamp voltage, 𝑣lamp[k] = [𝐿𝑒𝑛𝑔𝑡ℎ_𝑎𝑟𝑐 × 𝐸[𝑘]] + Voltage drop across the two
electrode interfacial boundaries and two constriction regions (50V). (𝐸8)
Positive Column Resistance = [𝐿𝑒𝑛𝑔𝑡ℎ_𝑎𝑟𝑐 × 𝐸[𝑘]] / 𝑖𝑙𝑎𝑚𝑝[𝑘] (𝐸9)
Total Power Radiated (from the isothermal core)= 𝑣𝑙𝑎𝑚𝑝[𝑘] × 𝑖𝑙𝑎𝑚𝑝[𝑘] × 𝜂𝑙𝑎𝑚𝑝[𝑘] (𝐸10)
Total Input Power = 𝑣𝑙𝑎𝑚𝑝[𝑘] × 𝑖𝑙𝑎𝑚𝑝[𝑘] (𝐸11)
Equation Parameters:
𝜂𝑙𝑎𝑚𝑝[𝑘] lookup table :
Table E1: Radiative efficiency of the lamp at a given input power
Lamp input power in Watts Radiative efficiency ( ηlamp)
0 0
6,000 0.1
9,108 0.26
21,169 0.37
34,886 0.47
60,000 0.475
140,000 0.487
705,721 0.487
Tarc(n)[k] : Temperature of the nth differential region of the arc
differential_Area(n) : area of the nth differential region
𝜀𝑁(𝑇𝑎𝑟𝑐(𝑛))[𝑘], Cparc(n, Tarc(n))[k], ρarc(n, Tarc(n))[k] : are the Net Emission
Coefficient, specific heat capacity and density characteristics of the argon and water
mixture, which are implemented in lookup tables shown below.
κ(n, Tarc(n))[k] : is the thermal conductivity parameter of the argon and water mixture.
This parameter is also stored as a lookup table shown below.
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σarc(n, Tarc(n))[k] : is the electrical conductivity parameter of the argon and water
mixture. This parameter is also stored in a lookup table shown below.
E[k] : Electric field across the positive column
Length_arc = 0.280 m
∆t = is the discrete time step size (=0.1msec)
The following figures E3-E7 are the arc transport coefficients: specific heat capacity, net
emission coefficient, thermal conductivity, electrical conductivity and density respectively,
shown as a function of arc temperature for a plasma composition of 98% Argon and 2% Water at
5 bar pressure:
Figure E3: Specific Heat Capacity (Cp) of the plasma versus temperature
Figure E4: Net Emission Coefficient (𝛆𝐍) for Rp = 5mm of the plasma versus temperature
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Figure E5: Thermal Conductivity (𝛋) of the plasma versus temperature
Figure E6: Electrical Conductivity (𝛔𝐚𝐫𝐜)of the plasma versus temperature
Figure E7: Density (𝛒𝐚𝐫𝐜)of the plasma versus temperature
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Feedback Controller (for wafer temperature):
Figure E8: The feedback controller with embedded reset option
A proportional (P) controller (with enable), shown in fig. E8 is only enabled as a feedback
controller when the temperature measurements are available. Initially when enabled, the
requested wafer temperature is reset to the first measured temperature value.
P controller equations:
𝐾𝑝 = 15 (𝐸12)
𝑒𝑟𝑟𝑜𝑟[𝑘] = 𝑇𝑟𝑒𝑞_ 𝑤𝑎𝑓𝑒𝑟[𝑘] − 𝑇𝑤𝑎𝑓𝑒𝑟[𝑘] (𝐸13)
𝑃𝑜𝑢𝑡[𝑘] = 𝐾𝑝𝑒𝑟𝑟𝑜𝑟[𝑘] (𝐸14)
Embedded code segment for temperature reset:
𝑟𝑒𝑠𝑒𝑡_𝑑𝑜𝑛𝑒 = 0; %indicator that the reset is done or not
𝑒𝑟𝑟𝑜𝑟[𝑘] = 0; % the difference between the measured and the requested temperature
𝐼𝑓 (𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒_𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 == 𝑔𝑜𝑜𝑑 && 𝑟𝑒𝑠𝑒𝑡_𝑑𝑜𝑛𝑒 == 0) % if measurements are
available and reset has not already been done
𝑒𝑟𝑟𝑜𝑟[𝑘] = 𝑇𝑤𝑎𝑓𝑒𝑟[𝑘] − 𝑇𝑟𝑒𝑞_𝑤𝑎𝑓𝑒𝑟[𝑘]; % calculate the difference
𝑟𝑒𝑠𝑒𝑡_𝑑𝑜𝑛𝑒 = 1; %set the indicator so the reset is not done again
𝑇𝑟𝑒𝑞_𝑤𝑎𝑓𝑒𝑟[𝑘] = 𝑇𝑟𝑒𝑞_𝑤𝑎𝑓𝑒𝑟[𝑘] + 𝑒𝑟𝑟𝑜𝑟[𝑘]; %add the difference to the original
requested temperature
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Current Controller and N-phase Chopper:
Figure E9: The inner current control loop
The discretized equation used for the inner loop shown in fig. E9 is derived below:
𝜏𝑑𝑖𝑙𝑎𝑚𝑝
𝑑𝑡= 𝑖𝑟𝑒𝑞 − 𝑖𝑙𝑎𝑚𝑝 (𝐸16)
Discretizing:
𝜏𝑖𝑙𝑎𝑚𝑝[𝑘] − 𝑖𝑙𝑎𝑚𝑝[𝑘 − 1]
∆𝑡= 𝑖𝑟𝑒𝑞[𝑘] − 𝑖𝑙𝑎𝑚𝑝[𝑘] (𝐸17)
Rearranging:
𝑖𝑙𝑎𝑚𝑝[𝑘] = 𝑖𝑟𝑒𝑞[𝑘] (∆𝑡
𝜏 + ∆𝑡) + 𝑖𝑙𝑎𝑚𝑝[𝑘 − 1] (
𝜏
𝜏 + ∆𝑡) (𝐸18)
Where ∆𝑡 is the discrete time step size (=0.125msec)
Current Conditioning:
Figure E10: The current conditioning block
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The blocks shown in fig. E10 when combined together, limit the rate of change and the max/min
value of the current that is imposed on the current controller.
The minimum value of current is set to not decrease below: 25A
The maximum value of current is set to not exceed: 1800A
Example operational code segment of the Ramp Saturation block, limiting current to
1000amps/sec:
𝑖𝑓((𝑖𝑟𝑒𝑞_𝑖𝑛𝑖𝑡𝑖𝑎𝑙[𝑘] − 𝑖𝑟𝑒𝑞_𝑖𝑛𝑖𝑡𝑖𝑎𝑙[𝑘 − 1]) > 1000 ∙ ∆𝑡)
%if current is changing faster than 1000amps/sec
𝑖𝑟𝑒𝑞[𝑘] = 𝑖𝑟𝑒𝑞[𝑘 − 1] + (1000 ∙ ∆𝑡); %set ramp rate to 1000amps/sec
𝑒𝑙𝑠𝑒
𝑖𝑟𝑒𝑞[𝑘] = 𝑖𝑟𝑒𝑞_𝑖𝑛𝑖𝑡𝑖𝑎𝑙[𝑘]; %otherwise follow the requested value
𝑖𝑓(𝑖𝑟𝑒𝑞_𝑖𝑛𝑖𝑡𝑖𝑎𝑙[𝑘] > 1800) %if current requested is greater than 1800amps
𝑖𝑟𝑒𝑞[𝑘] = 1800; %set the current to 1800amps
where :
𝑖𝑟𝑒𝑞_𝑖𝑛𝑖𝑡𝑖𝑎𝑙[𝑘]: lamp current requested by the controller
𝑖𝑟𝑒𝑞[𝑘] : current after saturation i.e. the final current value which is requested to the
current controller