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IMPROVED UTILITY 0F MICROWAVE ENERGY
FOR SEMICONDUCTOR PLASMA PROCESSING
THROUGH RF SYSTEM STABILITY ANALYSIS AND ENHANCEMENT

presented by
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4

Improved Utility of Microwave Energy for Semiconductor Plasma Processing
through RF System Stability Analysis and Enhancement

By

Paul Rummel

A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of
MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2000

ABSTRACT

Improved Utility of Microwave Energy for Semiconductor Plasma Processing
through RF System Stability Analysis and Enhancement

By
Paul Rummel

The bulk of today’s semiconductor plasma processing equipment utilizes RF
energies at fi'equencies fi'om 50 KHz to 60 MHz for deposition, etching, cleaning and
various other processes. One of the impediments to utilizing microwave energy for these
processes is the inherent instability ofien encountered with systems operating at
frequencies of .S to 2.45 GHz. Systems with plasma loads excited by resonant antennas,
impedance matched by resonant circuits or cavities, and powered by generators of
various source impedances are invariably unstable over some operating conditions. For
microwave systems, this instability typically manifests itselfas a propensity for the
plasma to extinguish or rapidly change to a lower density as the impedance matching
device is adjusted to minimize reflected power to the microwave generator.

This paper shows why this instability exists and how a microwave driven plasma
system can be modified to achieve better stabilization. A Matlab Simulink model and a
state-model control analysis are used to identify system parameters that affect system
stability and to predict the results of modifying those parameters towards the goal of
improving stability.

A plasma system utilizing an MPDR13 Plasma Disk Reactor is used to first
characterize the developed models, and then modified to illustrate the models’
predictions. A high correlation between predicted and measured system stability

validates the method of using a control analysis to model plasma system stability.

TABLE OF CONTENTS

TABLE OF FIGURES ....................................................................... v

INTRODUCTION ............................................................................ 1
Motivations ........................................................................... 2
Goals ................................................................................... 3

CHAPTER 1

STABILITY ANALYSIS .................................................................... 5
Definition of stability ................................................................ 5
Stability quantified ................................................................... 6
Background ........................................................................... 7
Definition of Variables .............................................................. 10
Plasma Impedance Curve ............................................................ 12
Delivered Power Surface ............................................................ 15
Criterion for Stability ................................................................ 16
HF RF vs. Microwave Stability .................................................... 19

CHAPTER 2

PLASMA SYSTEM MODEL ............................................................... 22
Plasma System Model Outline ...................................................... 22
Plasma Density Model ............................................................... 23
Plasma Impedance Model ........................................................... 24
Microwave Cavity Model ........................................................... 24
Transmission Line Model ........................................................... 26
Microwave Generator w/circulator Model ........................................ 26
Complete Simulink Plasma System Model ....................................... 28
System Stability Plots ............................................................... 28
Stability Enhancement ............................................................... 31

CHAPTER 3

ONE DIMENSIONAL STATE MODEL CONTROL ANALYSIS ................... 34
Single First Order System Ordinary Difi‘erential Equation ..................... 35
Stability Analysis of the System Differential Equation ......................... 36
Differential Equation Modification ................................................ 37

CHAPTER 4

EXPERIMENTAL VERIFICATION ...................................................... 39
Description of System Hardware ................................................... 39

iii

Standard System Performance ...................................................... 41

Standard System Analysis .......................................................... 44
Modified System ..................................................................... 45
Model Verification ................................................................... 48
CONCLUSIONS .............................................................................. 52
REFERENCES ................................................................................ 55

iv

TABLE OF FIGURES

Images in this thesis are presented in color.
Figure 1: General microwave powered plasma system ................................. 1
Figure 2: Power-loss and power-absorbed curves for three incident powers and

three cavity length. Low-pressure model u/co << 1. Excitation fi'equency
and background pressure are constant. (N) equals average plasma

density ................................................................................ 7
Figure 3: Power delivered curves .......................................................... 12
Figure 4: Standard Smith Chart ............................................................ 13
Figure 5: Plasma Impedance Curve at the plasma ....................................... 13

Figure 6: Plasma Impedance Curve at the input to the matching device or cavity.. 14

Figure 7: Plasma Impedance Curve at Generator output ................................ 15
Figure 8: Microwave/circulator Delivered Power Surface .............................. 16
Figure 9: Plasma Impedance Curve w/Delivered Power Surface at the generator
output .............................................................................. 17
Figure 10: Microwave generator Delivered Power Surface ........................... 19
Figure 11: Delivered Power Surface of solid state HF RF generator ................. 20
Figure 12: Stable Plasma System with High Frequency RF Generator ............... 21
F igure 13: Plasma System Model Block Diagram ...................................... 22
Figure 14: Plasma System Matlab Simulink Model ..................................... 28
Figure 15: Matlab ‘M’ file for exercising Simulink model ............................ 29

Figure 16: Reflection Coefficient (Stability) Circle of the modeled microwave
plasma system ................................................................... 30

Figure 17: Reflection Coefficient Circle of Microwave system with added .16
wavelength transmission line ................................................ 3O

Figure 18:
Figure 19:

Figure 20:

Figure 21:

Figure 22:
Figure 23:
Figure 24:
Figure 25:
Figure 26:
Figure 27:
Figure 28:

Figure 29:

Figure 30:
Figure 31:
Figure 32:
Figure 33:
Figure 34:
Figure 35:

Figure 36:

Figure 37:

Microwave powered plasma system with reactive offset .................. 31

Offset = +j50 Ohms microwave Delivered Power Surface ................ 32
+j50 Ohms skewed Stability Circle with .16 wavelength

transmission line ................................................................ 32
+j50 Ohms skewed Stability Plot with .41 wavelength

transmission line ................................................................. 33
Stability plot, Of = 0 Ohms, a = 5 .............................................. 36
Stability plot, Of = 0 Ohms, 0 < a < 5 ......................................... 37
Stability plot, a = 5, -80 < Of< 80 Ohms ..................................... 38
MPDR13 Microwave Cavity .................................................. 40
Standard system schematic .................................................... 41
Standard system performance with generator power perturbation ........ 42
Standard system performance with cavity height perturbation ............ 43
Standard system differential equation, resonance/density coeff

‘a’ = 8 ............................................................................. 45
Modified system schematic ................................................... 46
Modified system reactive offset .............................................. 46
Determining effective reactive offset ........................................ 47
Modified system differential equation, a = 8, Of = -110 Ohms ........... 48

Modified system performance with generator power perturbation .. . . 49
Modified system performance with cavity height perturbation 50

Modified system differential equation with 1/4 A shorter
transmission line ................................................................ 50

Modified system performance with 1/4 A shorter transmission line. 51

INTRODUCTION

A general microwave powered plasma system is represented by Figure 1 below.

 

Microwave
Generator Circulator

Transmission Line Impedance Launching *
—>
G a. 0 R Matching Device Mechanism

Figure 1: General microwave powered plasma system

 

 

 

 

 

 

 

 

 

 

A typical microwave generator is comprised of a high voltage DC power supply driving
an output device such as a cavity magnetron or traveling wave tube (TWT). A circulator
is used at the output of the generator to protect the output device from reflected energy,
(mismatch), which could cause a shortened device lifetime and unstable operation, such
as a shift in output fi'equency. A necessary component required with the use of a
circulator is a matched or ‘dummy’ load, which the circulator directs any reflected energy
to. Directioml couplers placed along the transmission line are used to measure forward
(F) power to and reflected (R) power from the load.

Since it is often physically inconvenient to locate a bulky microwave generator
directly at the plasma source, it is common practice to use a transmission line to deliver
the microwave energy to the source.

Since a plasma is not a fixed impedance energy load, an impedance matching
device is required to facilitate maximal transfer of energy fi'om the generator to the
plasma. The impedance matching device can be an integral part of the launching
mechanism as in a cavity type plasma source, or a general two or three stub tuning device

placed along the transmission line near the radiating launching mechanism.

The term launching mechanism refers to the structure that presents the
electromagnetic fields to the plasma. Some examples are: parallel plate, solenoidal coil,
resonant cavity, toroidal coil, wave guide aperture, and stub antenna. How
electromagnetic fields are presented to plasmas play important roles in process
performance parameters such as process uniformity and process rate.

This thesis will focus mainly on a microwave cavity type plasma source though
the analysis concepts will be presented in light of and are directly applicable to
electromagnetic energy generated plasma sources in general, regardless of fi'equency or

launching mechanism.

Motivations

In practice, plasma system instabilities abound, especially in plasma processing
equipment operating at low-pressure regimes or with highly coupled source designs. One
manifestation of this instability can make it impossible to adjust the impedance matching
mechanism to obtain optimum energy transfer without extinguishing the plasma [1]. In
another manifestation, especially when the output forward power of the generator is
actively controlled, the amplitudes of the RF energy and/or plasma density oscillate
periodically [2], often on the order of 103 to 106 Hz.

Though the need to control absorbed power in the plasma system to maintain
process repeatability is somewhat obvious, there are also important motivations for
minimizing reflected energy from the matching device/ launching mechanism/plasma,
(other than efliciency reasons). For high frequency RF (HF RF) plasma sources,

harmonics of applied RF energies are generated due to the nonlinear characteristics of the

plasma. This implies that the plasma itself is an effective RF generator at these harmonic
fi'equencies. How this effective RF generator is ‘loaded’ by the impedance looking back
into the launching mechanism/ impedance transformation device can affect several
parameters of the plasma itself. The most repeatable processing is accomplished when
the plasma has repeatable ‘harmonic loading’. For microwave cavity type plasma
sources, the mechanical structure of the cavity directly affects the electric and magnetic
fields incident upon the plasma, affecting the uniformity and density profiles of the
electrons, ions, radicals, etc. At any appreciable given reflected power fi'om the cavity,
there could potentially be many cavity mechanical positions, even keeping within the
same resonance. This means that a plasma process could yield different rate/uniformity
results for a given fixed amount of reflected power depending upon how the cavity is
positioned or ‘tuned’.

To summarize, an unstable RF plasma system often causes non-repeatable plasma
processing. The RF/microwave power delivered to the plasma directly affects plasma
density, and the impedance matching device usually affects other plasma parameters in
addition to providing a means for coupling energy to the plasma. If measuring forward
and reflected RF power are the only ‘diagnostic’ means for maintaining repeatable
plasma densities, it is imperative that the reflected power be brought to a minimum by the

matching device for optimum process repeatability.

Goals

The goals of this thesis are to:

0 Define stability as it applies to RF/microwave driven plasma processing systems.

Give a brief background through a summary of a previous stability analysis.
Introduce 2 and 3-dimensional Smith plots as aids for illustrating system stability.
Develop mathematical models of each system component.

Develop a Matlab Simulink model of a plasma system for stability analysis.
Using the Matlab Simulink plasma system model, show how modifying an

RF Imicrowave generator’s source impedance modifies plasma system stability.
Describe system stability for a microwave plasma system through a single first
order differential equation.

Experimentally justify the model and control analysis through the experimental

implementation of a ‘stability enhanced’ microwave plasma system.

CHAPTER 1
STABILITY ANALYSIS

Definition of Stability
For the purposes of this thesis, stability is defined as the ability to repeatably
control plasma density through control of the RF/microwave power generator used to
create the plasma. A highly stable plasma system allows the user to repeatably create and
maintain a plasma at any desired density over various operating parameters of pressure,
gas composition, etc. . . The word ‘repeatably’ implies at a minimum:
1. The delivered RF p_ower to the plasma system can be remated.
In this analysis, forward RF/microwave power is a major control input, defined as
a nominal forward RF power setpoint, measured (external or internal) at the
output of the RF generator. Delivered RF power is defined as the difference
between this forward RF power and reflected RF/microwave power due to an
impedance mismatch, (also measured at the output of the RF/microwave
generator). Delivered power encompasses the losses in the transmission line,
impedance matching device, launching mechanism, and most importantly, power
absorbed by the plasma.
2. The transformation from the plasma imMance to the desired RF/microwm

generator nominal oppratmg' immdgce (typically 50 ohms) can be repeated.
Unless the system has a means for accurately measuring the impedance

 

transformation via measuring the plasma impedance or precisely modeling the
details of the transformation hardware, this second requirement for stability is
difficult to achieve if the reflected power is an appreciable fraction of forward

power. Only at low or zero reflected power can a repeatable transformation to the

plasma impedance be achieved. If the reflected power into the impedance
matching/plasma device is high, it becomes very difficult to know to what
impedance the device is transforming to. A constant standing wave ratio (SWR)
describes a circle on a Smith Chart. This means that there are several impedance
transformation solutions, which will yield the same SWR, whereas there is only

one solution at zero reflected power.

In summary, a plasma system is stable if RF delivered power can be maintained at
continuously variable levels while also reaching and maintaining low reflected power

levels.

Stability Quantified

The degree of stability represents the degree to which the plasma system can be
perturbed before control of the system is lost. This analysis will quantify stability by
graphical means utilizing Smith Charts. The Smith Chart plots will show, for a given
plasma system, areas of stable and unstable operating conditions, where the nominal
center impedance represents zero reflected power from a point of view at the
RF/microwave energy source, looking into transmission line towards the plasma load.
The plots will also define a quantifiable measure of stability by representing the
maximum stable operating region with a maximum reflection coefficient. The second
stability analysis through system differential equation plotting will yield a quantifiable

‘Region of Attraction’ to the desired stable operating plasma density.

Background

Previous analyses of plasma system stability have utilized graphical methods to
describe and explain the mechanism of cavity type source instabilities. A plot of one of
these analyses [3] is recreated in Figure 2 below. The y-axis represents power
absorbed/lost by the plasma and the x-axis represents plasma density. Two intersecting
curves are traced on the same plot, a power loss curve and a power absorbed curve. (This

thesis will use n0 to denote plasma density and the plot below uses <N>)

 

 

 

 

 

 

 

 

Power absorbed curve
A ""'-'Powerlosscurve
II Ll M>M>h
Pin! < Pm < Pins
unstable 4—» stable
L;
Power Pin3
(Watts) ,,,,,,
Pm. — _- L -----
Pin!
<N>

Figure 2: Power-loss and power-absorbed curves for three incident powers and
three cavity lengths. Low-pressure model u/(o << 1. Excitation frequency and
background pressure are constam. (N) equals average plasma density
The power loss curve, (dotted line), represents plasma density, no , as a firnction
of power lost to the plasma, Pm. In the simplest case, n0 is approximately linearly
proportional to PM with a proportionality constant 1/ kL , where kL is the slope of the

curve in W/cm’.

The power absorbed curve, (solid line), represents power absorbed by the plasma,
Pass, as a function of plasma density no. Power absorbed is dependent upon no because
the electrical plasma impedance changes with the density of charge carriers, no, which
affects the resonance or impedance transformation of whatever launching
mechanism/matching network is utilized. The three curves labeled Piol, Pm, and PM
represent three different incident power levels. This shows the effect incident power has
upon the curve shapes. The three curves, (L1, L2, L3), shown in the above figure
representing three different cavity resonant modes of operation.

Changes in no also affects other plasma parameters, such as sheath thickness and
mobility, that can change the complex part of the electrical impedance. As the plasma
impedance changes, power delivered to the plasma is affected due to changes in
resonance of the cavity or matching device, affecting actual power transfer. The slope at
any point along any of the Poo, vs. no curves, we will assign as kA, which is also in W/cm’.
For the analysis of this thesis, it will be assumed that only one resonant mode of
operation exists.

It is stated within this previous description that, “the system has solutions where
the curves intersect and the system is stable if, at these points, the slope of the power
absorbed curve is less than the slope of the power loss curve”. So, for the plot above,
solutions along the right side of the peaks will be stable and solutions to the left are
unstable if the power absorbed curve is steeper than the power loss curve. This is
indicated in the plot above on the second ‘peak’. Notice that the line demarking stability

is slightly offset to the left, where the slope of the power absorbed curves equals that of

the power loss curve. This criterion for stability can be justified through a simple linear

control analysis, which will also be used to establish the basic criterion for stability used
for the remainder of this analysis.

We will start with a very basic expression for plasma density no (in m") where
l/kL is the linear loss constant and 1: (in sec) is an arbitrary rate constant. This rate
constant is justified because the plasma density cannot change instantaneously with
absorbed power. Various plasma diffusion mechanisms will regulate this rate and it can
be arbitrary for this analysis because it is assumed to be the slowest responding element
of the system. The general expression for no is:

(a) no=1/kL-Pooo-1:dno/dt

where kL is the slope of the power loss curve at any solution, in Watts/m3.

Then, we state that Poo, is a function of no with the agreement thatat any real solution,
power lost is the same as power absorbed. Also, since changes of plasma density affect
absorbed power much faster than the diffusion rate constant described above, the
following fimction is not considered to be time dependant:

(b) Pooo = kA . no

where to, is the slope at any point along whatever curve is traced by the power absorbed
curve at any solution, also in Watts/m3.

Substituting kA . no for P... into equation (a) above yields:

(C) no=(1/kl. *kA' no)-tdno/dt

Solving for (1an yields:

 

(d) duo/d1: flo'(kA/kL-1)
‘r

This is the first order differential equation that describes the system. The equation

does not need to be solved to determine stability. At any of the solutions, the system is

stable as long as the right side is negative with respect to the left side. For this equation,
the system is stable when kA / kL is less than 1, or kA < k1,. Put into words, when the right
side of the equation is negative, the density no will always move in the opposite direction
of any perturbation to no. If the right side of the equation is perturbed in a positive
direction (no+dno), then the mptio_n of no , which is dno/dt , will move in the ppm;
direction, regaining equilibrium. On the other hand, if the right side of the equation is
positive with kA larger than k, then the response dno/dt will be in the same direction as
the perturbation resulting in a rapid movement away fi'om equilibrium at a rate inversely
proportional to the arbitrary diffusion time constant 1.

Thus, the previous descriptive criterion for stability is justified mathematically.
The plasma system is stable if, for any real solution, the slope of the power absorbed

curve, kA, is less than the slope of the power loss curve, k1,.

Definition of Variables

While it is immediately understood that plasma density, no , is some function of
power lost to the plasma, it may not be as clear that absorbed power, Poo. , is some
function of plasma density without the auxiliary parameter of plasma impedance, Zo.
Since plasma density is directly related to charge carrier density, it is a small step to
conclude that the real part, RP, of plasma impedance, Zo, is inversely proportional to no.
Furthermore, since no affects other plasma parameters such as sheath thickness and
carrier mobility, the reactive part, Xo, of plasma impedance 2,, is also some function of
no. Now that we have plasma impedance, Zp, as some function of no, it is easily

understood that the operating resonance of the launching mechanism and thus, the

10

effective energy transfer is some function no. Thus, the amount of reflected power fiom
this matching device/launching mechanism mismatch is some fimction of no. Lastly,
these changes in reflected RF power directly equates to changes in ‘Delivered’ RF power
through some relation of the ability of the generator to deliver power as a function of
reflected RF power, Pm, and thus as a function of no. Viewing this sequence using
plasma impedance, it is easier to see how changes to no affect absorbed power which is
the same as delivered power, Pdel, if we neglect the losses outside of the plasma.

Let us now shift our flame of reference fi'om plasma density, no , to plasma
impedance, Zp , and secondly, from Pooo & Ploss to the single term, Delivered Power Pdel-
Since there are only real solutions in a real system, we don’t need to distinguish between
PoooandPlooo,andweagreethatthepowerusedbytheplasmaisthesameaspower
delivered, Poo. , to the plasma which is also the same as forward power, PM , minus
reflected power, Pm , (excluding system losses outside the plasma). Now the stability
analysis involves two different system equations: one relating how lasma im ance is
affected by delivered power, and another relating how delivered power is_affect by thp
plasma impedance. Making this change of variables makes it easier to understand the
way the previously plotted ‘power absorbed curve’ is generated and what can affect it’s
shape.

Focusing on just the real part, RP , of the plasma impedance, Zp , we can create a

similar plot to Figure 2, using our new variable, Pact, as shown in Figure 3 below.

11

 

Delivered
Power

 

 

 

4 J

 

Real part of plasma Impedance ~ 1In°

Figure 3: Power delivered curves

Plasma Impedance Curve

Because we shifted our reference to plasma impedance, Zp , we now see that the
complete picture can only be represented in 3 dimensions, where the complete plasma
impedance including the real part, RP, and reactive part lo, are plotted against the now
singly defined Delivered Power Pdel- To increase clarity, let us use the well known Smith
chart to represent plasma impedance in 2 dimensions as a bottom polar plane, with
delivered power, Poo. , as a linear vertical center axis, originating at the nominal Zo point
of the Smith Chart. For reference, Figure 4 shows the standard (2-dimensional) Smith

Chart.

12

   

Inductive

    
       

Infinite
impedance

Figure 4: Standard Smith Chart
Rotating the standard Smith Chart to lay down almost flat and adding the third dimension
of delivered power, we arrive at a 3-dimensional (3-D) Smith Chart. With this 3-D Smith
Chart, we can now plot our first system equation in the new coordinates : Zp vs. Pool
defined as a Plasma IMce Curve, with an example shown below in Figure 5, (where
Zp is inversely proportional to noand no is linearly proportional to Past).

3-D Smith Chart
Delivered Power

:3 500

C2) 375

 

  

Figure 5: Plasma Impedance Curve at the plasma

13

Shifting our point of view to the input of the impedance matching device, the
plasma impedance curve would twist because of the affects of the plasma on the
resonance of the mtching device/launching meclmnism. Also, because of the impedance
transformation, there is now a solution of the curve at the nominal impedance, (Smith
Chart center), at a given delivered power. In the example shown below in Figure 6,

reflected power is minimized at 300 Watts delivered.

3-D Smith Chan
Delivered Power

4-:- 500

Cb 375

 

 

 

 

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cg?) 125
n-£;7-
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Figure 6: Plasma Impedance Curve at the input to the matching device or cavity

Shifting our point of view again down an arbitrary length transmission line to the
output of the microwave generator/circulator, the plasma impedance curve arbitrarily

rotates around the Smith Chart as shown below in Figure 7.

i so Smith Chart

 

Figure 7: Plasma Impedance Curve at Generator output

This is the point of view from which the rest of this stability analysis will be

made.

Delivered Power Surface

The second system equation using the defined state variables is Poo. as a fimction
of 2,. This equation defines a surface which this thesis will define as a Delivered Power
§___urfa_c§. An example of such a surface viewed at the output of a typical
microwave/circulator combination is shown in Figure 8 below. This is the same point of
view mentioned above, at the output of the generator, looking into the transmission line

towards the plasma load.

3-D Smith Chart

 

Figure 8: Microwave/circulator Delivered Power Surface

This surface represents the energy delivered by a generator versus the load
impedance presented to it. For the case of a microwave generator with a circulator at it’s
output, the equation for this surface is based on delivered power equals forward (setpoint)

power minus reflected power, where the reflected power is a fimction of load impedance.

Criterion for Stability
Putting the plasma impedance curve together with the delivered power surface we
arrive at Figure 9. The arrows indicate the two possible solutions, where the plasma

impedance curve meets the delivered power surface.

3-D Smith Chart A

 

 

 

 

 

 

.m Will '

 

Figure 9: Plasma Impedance Curve w/Delivered Power Surface at the generator output

As the output power setpoint of the RF/microwave generator is varied, the height
of the delivered power surface is affected. As the matching network or cavity is ‘trmed’,
the height and shape of the plasma impedance curve is varied. As the length of the
transmission line is varied, the radial orientation of the plasma impedance curve is varied.
For Figure 9 above, radial orientation does not affect the apparent ‘solutions’ of the curve
and surface because the surface is radially symmetric about the nominal impedance. The
inset in the upper right is the same surface and curve but rotated 90 degrees CCW to
show that the plasma impedance curve is indeed outside of the surface up until it enters
the surface at point B.

Similar to the preceding 2-dimensional stability analysis, intersecting points of the
plasma impedance curve to the delivered power surface define real solutions of the
plasma system. The criterion for stability requires that the slope of the surface tangent (in

the direction of the curve mgent) is less than the slop_e of the curve ta_ngent. If; at the

intersection, the tangent slope of the delivered power surface (in the curve direction) is
steeper than the tangent slope of the plasma impedance curve then the solution is
unstable. Using Figure 9 above, the intersecting solution at point ‘A’ of the plot is stable
(surface slope is zero in direction of curve), and the intersecting point at point ‘B’ of the
figure is unstable (surface slope in direction of curve is greater (steeper) than curve
slope).

We will define equilibrium as any operating condition in which RF energy is
being delivered to a plasma, and the state variables of delivered power and the plasma
impedance are not changing with respect to time. The extent to which a system at
equilibrium can be perturbed and still return to equilibrium is quantifiable, where a high
degree of allowed perturbation about me desired operatmg' point is considered to be a
desirable attribute of a plasma processing system.

In Figure 9 above, the matching network or cavity is set to bring a solution to low
reflected power, point A. It is now clearly seen that if the system is perturbed in such a
way as to raise the plasma impedance curve slightly, a marginally stable operating point
quickly becomes the only solution and the plasma soon extinguishes. If, however, the
system is perturbed so as to lower the curve, the system remains stable for a much larger
perturbation in that direction. Thus, it is important to note that the perturbation direction
and magnitude that will maintain system equilibrium is not symmetric. In the case above,
for some directions, a very small perturbation from the desired operating point (A) will

result in loss of equilibrium.

l8

HF RF vs. Microwave Stability

A microwave generator utilizes a circulator at it’s output to both protect the power
device (such as a magnetron), and to maintain output stability (mode reduction). Because
of the circulator, a microwave generator has an effective nominal source impedance of 50
ohms. The delivered power surface characteristics of this microwave energy source
appears as indicated by Figure 10 below.

3—D Smith Chart

Delivered Power

       

I
, o.

   

 

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Intrigue:

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Figure 10: Microwave generator Delivered Power Surface

Typical high fiequency (HF) RF generators (3 to 30 MHz) usually have a source
impedance far from their nominal operating impedance of 50 ohms. The 50-ohm
nominal impedance is a load for which the generator is best suited to deliver stable and
eflicient RF energy. They are often designed as a low impedance ‘voltage source’ which
is then transformed to some other source impedance through harmonic filtering or output
impedance matching, but still far fi'om 50 ohms. This default design strategy has

inadvertently given the HF RF generator a stability advantage for use in plasma systems

over the microwave RF generator. Figure 11 below shows a typical delivered power
surface of a solid state HF RF generator.
3-D Smith Chart

Delivered Power

It

‘4??? 1;. V... 0-. -
”1111:1-

 

Figure 11: Delivered Power Surface of solid state HF RF generator

Because of the characteristic of a ‘voltage source’ or ‘current source’, the
generator delivers much more power at some other load impedance than nominal. This is
true even if there is a control loop maintaining the forward or delivered output power
because the speed of that control loop is typically much slower than the speed at which
the plasma impedance can respond. So for this analysis, we must consider any HF RF
generator to be ‘uncontrolled’. It is now easily possible to have a stable operating point
at ‘best tune’ where the plasma impedance curve meets this surface at the nominal
impedance center. At this point there is no reflected power, and a relatively large
perturbation in any direction will not cause loss of equilibrium. The same plasma
impedance curve is introduced onto the HF RF delivered power surface in Figure 12

below.

20

3-D Smith Chart

ii

Iiiiéi

 

Figure 12: Stable Plasma System with High Frequency RF Generator

Note also that the system stability is now dependant upon choosing a transmission
line length that rotates the plasma impedance curve into the position where it's slope is in
the opposite direction as the slope of the tangent plane of the delivered power surface.
Thus, the stability of this system is now transmission line length dependant because the
delivered power surface is no longer symmetric about the nominal axis.

If a microwave generator could be modified to have an asymmetric output power
surface similar to that of an HF RF generator, the same stability could be efiected. This
will be attempted and shown subsequently in the system model with the results shown on

‘stability region’ plots.

21

CHAPTER 2
PLASMA SYSTEM MODEL

We will now utilize a Simulink model to fiuther analyze the stability
characteristics of a typical microwave plasma system. The output of the model is a 2-
dimensional Smith chart of the solution of the plasma impedance curve and the delivered
power surface, given different ‘starting’ positions of impedance. The charts will show
regions of stable solutions. The impedance indicated by the Smith Chart stability plots
will be the impedance as seen from our view point of just outside of the RF generator,
looking into the transmission line.

We will also use the model to see how the system can be modified to maximize

the degree of stability as previously defined.

Plasma System Model Outline

Figure 13 below is a block diagram to represent the various pieces of the plasma
system model to be developed. Each block will be represented by the fimctions indicated
in the blocks and the functions themselves will be defined. Then the individual functions

will be ‘connected’ together as shown in the figure in the form of a Matlab Simulink

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

model.
Plasma Density Model lrnpedance Model Microwave Cavity Model
no = f (gas, press, temp. dims, Pa... t) R. +iXp = I (no) *‘ Z: +in' = I (R9 +in freq. dimS)
Pubs = Pdcl
Microwave Generator Model Transmission Line Model
Paa=f(Setp0int,Zr+in) Zr+jlx=f(Zr'+ij'. 1)

 

 

 

 

 

 

 

Figure 13: Plasma System Model Block Diagram

22

Plasma Density Model
The form of the plasma density model used is based upon energy balance
equations for electropositive plasmas in equilibrium [4]. This form is:
no = kl. Pabs(no)
where no is plasma density, Pubs is absorbed power, and kL is the plasma load line
constant which absorbs all process parameters such as gas, pressure, temperature, F
etc. This is a linear proportionality constant relating plasma density to absorbed

power. A second term is added to represent a time dependence of plasma density

 

to changes in absorbed power, with I being an arbitrary time constant:

no = kL Pabs(no) - t dno/dt

Based upon previous data taken for the MPDR13 plasma reactor [5], (to be used
to verify the model), the plasma load line constant was chosen which yielded a
plasma density of 4.4e1 7 m'3 for 200 Watts of absorbed power:

kl, = 2.2 E15 W'm"

An arbitrarily picked ionization time constant was chosen which represented a
conceivable average time between ionizations. Since this is the only, (and
therefore dominant), time constant for the whole system, it can be arbitrary. Also,
the transient response of the system is not being investigated, only the stability:
r=llnu=3.13 E-S 3

Solving the plasma density equation for dno/dt yields the ordinary differential
equation for the system:

0 dant= Pabe(no)kL/T - no/‘r

23

Assumptions: Plasma density is linearly proportional to Pogo.
Plasma density time constant is arbitrary and possibly related to
ionization rate.

Plasma Impedance Model
Plasma impedance Zp can be defined by the series representation:
Zn = RP +.I xp
As a function of plasma density no, the real part of the plasma impedance, Ro, is

inversely proportional to conductivity which therefore makes it inversely

proportional to charge carrier density, hence inversely proportional to no.

_ kr
R" - (no + Cavity Losses)

Where kr is an arbitrary proportionality constant and ‘Cavity Losses’ is a term

introduced for the purpose of eliminating the zero plasma density singularity:
Cavity Losses = 1 E16
The reactive part ofthe plasma impedance will be ignored, as it will be absorbed
into the relation of plasma density to cavity resonance, later in the model:
e Xp = 0
Assumptions: There is a linear relationship between plasma conductivity and
density, with no appreciable reactive element.
Microwave Cavity Model
At the desired operating point, the microwave cavity transforms the real part of

the plasma impedance to the nominal transmission line impedance, Zo, of 50

 

ohms:
Zr' = kt * Rp
Zr" = k1

(no + Cavity Losses)

24

where Z.’ is the real part of the impedance Z' looking into cavity from the driving
point and k1 is the proportionality constant, (absorbs k,), that represents the
impedance transformation. For this model, k1 is chosen to make Ro = 50 ohms at
the desired operating point:

k1 = 2.25 E19 at Poo, = 200W, and no = 4.4E17

An term is added, (in brackets), to represent changes to the real part, Z,’, as an
arbitrary function of cavity probe height ho. Cavity excitation probe height hp is

normalized to a value of 50 (lo). This will be used to perturb the real part Z,':

, z: = k-* [5 A «hn' 2.) IZo)l
(n, + Cavity Losses)

 

Assumptions: The approximate model is based upon observed probe height vs.
S WR data.

The reactive part (L) of the impedance looking into cavity fi'om driving point as
a firnction of real part of plasma impedance R, can be represented by:

2:" = a "' Rp

where constant ‘a’ is a linear scaling factor representing the degree to which a
changing plasma density afl‘ects cavity resonance. This term also absorbs any
reactive part of the plasma itself as previously mentioned. A value of 5 was used
for the analysis and plots above, such as in Figure 9. This value was somewhat
randomly chosen for illustration purposes and will later be adjusted to model
actual observed cavity performance.

Assumptions: Cavity resonance is inversely aflected by plasma density, possibly

due to change of sheath thickness [6] and conductive medium
@lasma) skin depth [7].

25

Including another term to account for the effects of cavity height ho on
cavity resonance, and hence, perturb the reactive part of the impedance, gives:
- z: = a * (a..- h.)
where he is normalized to Ro nominal (Zo).
Assumptions: The dependence of Z,’ on ho model is approximately based upon
observed density and cavity height versus S WR data.
Transmission Line Model
Standard lossless transmission line equations [8], transform the impedance

looking into the cavity Z' into the impedance Z = Z, + ij looking into a A-

 

wavelengths long, Zo Ohm transmission line via the following development:

I‘v = (Z; + E; - Zo) / (Z; + ij' + Zo) (Voltage Reflection coefficient)
R,( = (1"v cos(2A) — 1) / 2
Ry = I“, sin(27l.) / 2

 

eZ,=

(Z. 51)
. Z: = S!!! !& 2 _ R‘Z
shims)
Assumptions: The transmission line is lossless.

Microwave Generator w/circulator Model
Since there is a circulator connected at the output of the microwave generator, the
delivered power, Pdcl , is a function of reflected power, Pm , and is given by:
Pdel = P fwd - Prfl
where PM is the generator setpoint or incident power, often automatically

controlled, but in the experimental case later outlined, manually set.

26

Reflected Power Pm for a given power reflection coefficient 1],, at a given forward
power PM is given by:
Pm = PM " Pp
where the power reflection coefficient F p is defined as:
I“. = ((2. +12. - z.)2 / (z. +jZ. + z.)’)
where Zo is the nominal impedance and Zr + ij is the impedance looking into the
transmission line towards the load.
Writing in terms of delivered power Poo. defined above:
Pdel = wad - (wad "' Pp)
Pder=PM* (1 ‘rp)
The complete microwave generator w/circulator model relating delivered output
power Poo; as a function of Setpoint Power Poo, nominal impedance lo, and load
impedance Zr + ij (as seen by RF Generator’s circulator output) can now be
written as:
P...=P... * [1—(z.+jz.—z.)’/(z.+jz.+z.)’]
where Foot is the forward setpoint control input, taking the place of PM.

Because imaginary terms cannot be modeled directly in Matlab Simulink,
the equation is rewritten in terms of magnitudes, yielding:
. P...=P...* I 1 — ((L-L)’+L’)I«L+L)’+L’)i

Assumptions: The circulator is ideal and lossless.

27

 

Complete Simulink Plasma System Model:

The preceding model components were brought together into a single Matlab
Simulink model shown below in Figure 14. The blocks of Figure 13 are shown with
dashed lines on the Simulink model. The Transmission Line Model is shown as a
simplified single block comprised of a Matlab Subsystem for ease of readability.

Plasma Density Model Plasma Impedance Model Microwave Cavity Model

 

.3.
Emmy HelgM Input + m

 

 

fl!” dX/dR
AK.“ an: I
+ 'L_I Ftp (nom 60 ohms cm x
no K1: no to By
mutt
memo-u

Probe Height Input “"2 Z! a

 

 

 

 

 

 

 

Plot Delivered power

Renaud 013.1 0 PlotZtl l I IX lPlotDt
l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

zr «-
’-
b‘ I‘—
4.
m ,_
' fin
Transmtdon Line
[Txllne]
when"! Pm: Setpoint "Mm MIN Input
Microwave Generator wlcirculator Model Transmission Line Model

Figure 14: Plasma System Matlab Simulink Model

System Stability Plots

The Simulink model will be used to show operating parameter ‘regions of
stability’. In this case, the operating parameters that are being tracked r + jx (complex
impedance) as seen from the output of the microwave source, looking into the
transmission line towards the plasma load. The starting plasma density is preset to a

nominal value, as if the modifiable parameters of cavity height and cavity probe height

28

are at optimum positions. Then, a Matlab ‘M’ file, (Figure 15 below), is used to
manipulate cavity and probe height positions and run the system model to track the
progression of the system parameters of impedance (r + jx) over time. A stable condition
results in r + jx moving to some impedance other than purely reactive, (which would
indicate that the plasma has extinguished). If we then repeat the simulation for different
cavity and probe positions, we will build up a Smith Chart plot showing regions of
stability. The starting points for each run are plotted in either red or green, depending
upon whether the plasma was extinguished or not at the end of the run. The size of the
green area around the desired nominal ‘center’ operating point will quantitatively define
stability. This thesis will give a ‘value’ to stability by defining the largest ‘power
reflection coefiicient’ circle that can be drawn within the stability region, about the

desired nominal operating point, (center of the Smith Chart).

 

hold off
Xline = .16;
for Probe = 10:4:90
for Cavity = 65:-.5:35
sim plasmasys
if r(500) < 5
smithrx(r(l), x(1), 'r*')
elseif r(500) > 500
smithrx(r(1), x(l), 'r*')
else
smithrx(r(l), x(1), 'g*')
end
hold on
end
end

 

 

 

Figure 15: Matlab ‘M’ file for exercising Simulink model

Figure 16 below is a run utilizing an ideal microwave energy source as indicated
by the delivered power surface of Figure 10. The largest power reflection coefficient
circle that can be drawn in the stable region is .02. Thus, the larger the allowed reflection

coefficient, the more the system can be perturbed in any direction without going unstable.

29

 

Figure 16: Reflection Coefficient (Stability) Circle of the modeled microwave plasma
system

If the length of the transmission line is changed as indicated in Figure 17 below,
the maximum allowed reflection coefiicient does not change due to the symmetry of the

microwave generator delivered power surface, (Figure 10).

 

Figure 17: Reflection Coeflicient Circle of Microwave system with added .16
wavelength transmission line

30

Stability Enhancement

The plasma impedance curve is defined by the operating parameters of the plasma
and plasma source such as pressure, chemistry, geometry, etc, and the electrical
characteristics of the transmission line, matching device and launching mechanism. The
delivered power surface, up until now, has been defined by the electrical characteristics
of an ‘ideal’ microwave energy source with an output circulator. Since the plasma
impedance curve is often mostly a given, defined by the demands of a plasma process,
plasma chamber design, and launching mechanism, we are left with the delivered power
surface and transmission line to modify, in an attempt to increase a system’s stability.

If we now introduce a reactive offset element placed between the microwave
output and it’s circulator, we can skew the microwave generator’s delivered power

surface. A block diagram of this modified system is indicated below in Figure 18.

 

Microwave Reactive Offset
Generator 51

 

Transmission Line

    

 

Impedance Launching
Matdling Device Mechanism

 

   

Sensors

 

 

 

 

Figure 18: Microwave powered plasma system with reactive ofi‘set

A ‘reactive offset’ input can also be seen in the complete Simulink model of
Figure 14. The offset creates a similar delivered power surface to the HF RF powered

system described above with Figure 11, and is shown below in Figure 19.

31

~ “.‘D
. n ""'\‘.“—‘-‘3e\i‘?.$'§g
Vllliil“ \\\‘\“\\§:iu$§h
. \\\\\\\\‘\;§\_{““-““

“‘v.‘-—-———‘

 

Figure 19: Offset = +j50 Ohms microwave Delivered Power Surface

The resulting Simulink stability run now shows a greatly increased allowed reflection

coefiicient of about .3 as shown in Figure 20 below.

 

Figure 20: +j50 Ohms skewed Stability Circle with .16 wavelength transmission line
As in the HF RF power system, the new system with a skewed delivered power

surface has now become dependant upon the transmission line length for it’s stability.

32

This is illustrated in Figure 21 below. This is the exact system as indicated above in
Figure 18 with the reactive offset, except the transmission line was increased and extra

1/4 wavelength. The system is completely unstable at the desired operating point.

 

Figure 21: +j50 Ohms skewed Stability Plot with .41 wavelength transmission line

The disadvantage of using this method of adding a reactive offset is that when the
cavity is positioned for best tune and reflected power is low, the impedance looking into
the offset from the point of view at the output of the circulator is not 50 ohms. This
means that the reflected energy due to this mismatch is dumped into the dummy load
connected to the third port of the circulator. During normal operation, efiiciency is not
maximized, and the microwave generator must be chosen to be able to provide more

energy than required by the plasma process.

33

CHAPTER 3
ONE DIMENSIONAL STATE MODEL CONTROL ANALYSIS

Analyzing the stability of the plasma system can also be accomplished by
observing the zeros and plots of the single first order differential equation that describes
the system For a first order system, solving the differential equation is not required to
observe it’s exact response. It is enough to first determine the equilibrium points, and
then determine whether the points are stable or not, then determine the ‘Region of
Attraction’ for the stable points. This region indicates how much the system can be
perturbed before it loses equilibrium [9]. The equilibrium points are the conditions which
lead to zero motion of the state variable, x, which we will use to represent plasma density
no. The equilibrium points are easily determined by setting dno/dt = O (x' = 0), then
solving for x. The next step is to simply plot x' versus x to determine the region of
attraction to the stable equilibrium points. For this system, we will find 2 stable
equilibrium points and one unstable point. If the plasma density is perturbed from the
stable to the unstable equilibrium point, the system state will transit to the third
equilibrium point which we will find to be at zero density, representing an
extinguishment of the plasma.

We must first build the whole differential equation for the system using the
component models already developed. Since we wish to observe the effects of the
introduced reactive offset and the effects of plasma density on cavity resonance, we will
carry these variables all the way through to the final equation. The equations for the
transmission line will be neglected as they would greatly increase the complexity of the
differential equation. Because of this, choosing the polarity of the reactive offset

(capacitive or inductive) will lead to either a more stable or less stable system. The

34

transmission line in a real system gives added flexibility for gaining stability at either

polarity.

Single First Order System Ordinary Difi'erential Equation

Starting with the original first order differential equation of the system:

dant = Pdd(no)kL/t — no/t

Letting:

X: no

x' =dno/dt
A=kL/t= 7.04619
B =1 /1: = 3.20654
Pdcl(no) = P(x)

0 x' = AP(x) — Bx

The function P(x) is the delivered power as a function of plasma density. Starting
with P as a function of impedance:
P(z..z.)=P..r*[1—((z.—z..f+zx2)/((Z.+z.)2+zi)1

Letting :

P3,, = 200

Zr = k1 / no

Z0 = 50 (nominal impedance)

Zx=a*((k1/no)-Zo)+0r

k1 = 2.25e19
Of = Reactive Offset in Ohms

where ‘a’ represents the effect of plasma density on system resonance and Of
represents the introduced reactive offset.

 

 

Rewriting P(Z, , Zx) as P(x):
= . _ ((zr-zo)2 +zx’)
P(z, z.) P... [1 «z. + Z0). + 2.1)]
= , _ (k/x-50)2+(ak/x—50a+0f)2
”’0 PS“ [1 (kl/x + 50)1 + (a ki/x — 50a + of? ]

35

Writing P(x) into the differential equation above yields (1):

 

_[looBaOL- 250031“. 1—1301 1&- [1008k1(l-a 1+ znaklof 1x2 + L200Ak1Pset-Bk1 (1 + a )]x
x [2500(1 + e )—1ooao,+ 0, 1x1 +[1ook.(1-a 1+ 2ak10¢|x + 1k, (1 + a )1

 

 

 

Stability Analysis of the System Diflerential Equation

Setting x’ to zero yields the equilibrium points for plasma density x. Setting the
density/resonance constant ‘a’ above to 5, with no reactive offset, (Of = 0 Ohms), then
solving the 3rd order polynomial of the numerator above yields three roots or equilibrium
points: 0, 3.7el7, and 4.4e17. Plotting the curve of plasma density motion (x') versus

plasma density (x) below in Figure 22 shows these three equilibrium points.

x1021

 

 

 

 

 

 

 

 

.3 ..
Regionol‘
10 I attraction 1;
i
o 1 2 3 4 5 6 x10"

Density x
Figure 22: Stability plot, Of: 0 Ohms, a = 5
The stability of each equilibrium point is tested by observing the slope of the
trajectory at the point. A negative slope indicates an asymptotically stable point and a
positive slope is an unstable point. Thus the equilibrium points at 0 and 4.3e17 are stable

(green), and the point at 3.7e17 is unstable (red). Any point on the line will move to the

36

right if the point is positive in the x' direction (vertical axis), and will move to the left if
the point is negative in the x' direction. Thus, as long as the operating conditions yield a
plasma density M 3.7e17, the system will move to the stable operating point, 4.4e17.
This is the region of attraction for the system and is shown shaded in green. If the system
is perturbed to the point where plasma density drops below 3.7e17, the system will transit
to the other equilibrium point, zero, at a rate governed by plasma diffusion mechanisms,
and the plasma is extinguished. Thus the ‘region of attraction’ is from plasma densities
of 3.7el7 m"3 and up. By observation, this region is not symmetrical, and it would be
desirable to increase the narrow left side to allow for larger system perturbations.
Diflerential Equation Modification

The system differential equation can now be modified to observe the resultant
regions of attraction, and thus changes to stability. A family of curves is generated below

in Figure 23, by varying the density/resonance constant ‘a’ from 0 to 5.

x1021

 

 

 

Density ‘2 ' Vz/ \

motion .4 . .
x!
-6 .

L A; L

4 5 6 x10"

 

 

 

 

 

 

 

 

o '1 '2 3
Density x
Figure 23: Stability plot, Of: 0 Ohms, 0 < a < 5

It can be immediately concluded by this family of curves that decreasing the
effect plasma density has on the resonance of the cavity greatly increases the system
region of attraction to the desired stable operating point. Leaving ‘a’ set to 5, another
family of curves shown below in Figure 24 is generated by allowing the reactive offset,

0f, vary fi'om —80 to 80 Ohms.

 

 

 

x10”
I—f v r r
so
2 4.20 l
0 o \
)1
. / 20 a
2 /é/t{o />l¥\
4 _ :/// Nu
:ZMr—‘r‘e ” 1
. MN: 80 at
Densrty -6
motion \
4’ \
1. \ .
-12 I \
14 _

 

 

 

 

 

 

 

 

 

 

 

 

o 1 2 4 5 6 x10"

Densitix
Figure 24: Stability plot, a = 5, -80 < Of< 80 Ohms
It is observed by the family of curves above that the region of attraction can be
completely cutoff for the desired equilibrium point for positive reactive offsets. A
substantial gain in stability region is observed, (distance between similar colored

equilibrium points), for negative reactive offsets, along with a corresponding small shift

in the stable equilibrium point.

38

CHAPTER 4
EXPERIMENTAL VERIFICATION

Verification of the system model was obtained through the use of an MPDR13,
(Microwave Plasma Disk Reactor) plasma system [10]. The goal will be to justify the
effectiveness of using a simple first order system differential equation for the purpose of
predicting and assessing system stability. Verification will be performed in four steps:

1. Standard system performance: Measure the region of attraction for the

standard MPDR13 through monitoring of reflected power and perturbing the
system into instability.

2. Standard system analysis: Determine the ‘a’ value (resonance/density

coefficient) for the MPDR13 based upon the measured region of attraction.

3. Modijy system: Modify the MPDR13 with an appropriate reactive offset and

transmission line length to increase stability.

4. Model verification: Measure the modified system region of stability and

compare to the model predicted region of attraction.

The system will be perturbed for the verification procedure above by moving the
height, (distance Ls of Fig, 25 below), of the cavity to create a mismatch. Secondarily,
the system will be perturbed by changing the power setpoint of the microwave generator.
Percentages of power change for which the system remains stable will be compared

between the modified and unmodified systems.

Description of System Hardware
Stability analysis verification was obtained with an MPDR13 microwave cavity

type plasma system, as shown below in Figure 25, with general dimensions in cm.

39

 

 

 

Figure 25: MPDR13 Microwave Cavity

The center fed probe (red) is excited with 2.45 GHz microwave energy, delivered
via a coaxial transmission line. The cavity excitation probe (Lp) and cavity (Ls) heights
are adjustable and the plasma is contained within a quartz dome shown in blue, located in
the lower portion of the cavity.

For the stability analysis, Argon was used at a pressure of SOOmT. At this
pressure the standard system instability was less pronounced and it was possible to obtain
a somewhat accurate measure of reflected power at the point of instability. This pressure
required a flow rate of about 75 sccm, pumped by an Edwards Model 5, two stage
mechanical pump. The cavity probe length Lp was set to 2.9 cm. The following
sequence was used for all data runs to insure plasma ignition:

a. The microwave generator was set to have a magnetron current of 300 ma or an

indicated forward output power of 200W, whichever came first.

b. The cavity height was brought to 12 cm.

40

c. The cavity height was then brought up to 15 cm, followed by upward manual

incremental tuning.

Standard System Performance

Figure 26 below is a schematic of the standard system used for verification.

 

r<}——————————-42Srnunanengni-——————————{1

dnechonal
couplers

 

drculaior

mkrowave r“1

generator

 

 

 

attenuators

 

 

bolometers

dummyload
tavfiy

 

 

 

 

 

 

[ 1:: gas inlet

 

 

 

 

 

 

 

to vac
PUMP

 

 

 

 

 

 

Figure 26: Standard system schematic

The microwave generator was a Micro-Now model 42031 with a UTE Microwave model
CT-3695N circulator. The directional couplers for forward and reflected power
measurements were two Narda 2785-30’5. Power was measured with two HP model
432A w/478A bolometer type power meters connected via two 20 db attenuators.

This standard system was run to find the maximum amount of perturbation that
could be made without causing the system to go unstable. This characterization of the

standard system versus power setpoint was performed by the following sequence:

41

l. Adjusting the cavity to a best tune position at 200 Watts incident, which was a
reading of 0 to 2 Watts, (measured after the circulator).

2. Turning the generator power control to maximum power.

3. Slowly adjusting the generator’s power setpoint control downward, while
monitoring forward power out of the generator, and reflected power fi'om the
cavity. Relative forward power was monitored via the Micro-Now’s internal
power meter.

The data for reflected power versus generator relative forward power is shown below in

 

 

 

 

 

 

Figure 27.
80
70 ~ 1
+
60 P 4»
{-
L 41 * «1
Reflected * '
power 40 l- —. H—Stable range 4
(WattS)
Pointotno return
20 l- J
Besttune + 1*
10 t 1i e 4
.*
g +
o 1 4 L r L * *4 r r
050100150200250300350400450500
Relative forward power (Watts)

Figure 27: Standard system performance with generator power perturbation

The ‘Point of no return’ indicated above is the power setpoint lower limit before the
system went unstable, and as mentioned above, occurred at about 10 Watts reflected.
The stable range indicated above is the forward power range below best tune for which

the system remains stable.

42

The ‘point of no return’ could also have been found by perturbing the cavity

height. This was attempted but the resolution of the servo system driving the short was

limited to .1 cm minimum which made it difficult to determine the exact reflected power

point at which the system went unstable.

It was found that increasing the setpoint power by any amount would not cause

the system to go unstable. This is predicted in the model by observing (Figure 22) that

plasma densities above the equilibrium point are always stable.

Perturbing the system by changing the cavity height resulted in the data shown

below in Figure 28.

8

7O

Reflected 50
(Watts) 4°

20

10b

0

 

 

{retry

 

it 1

 

14.5

15.0

15.5 16.0 16.5
Cavity height (0")

Figure 28: Standard system performance with cavity height perturbation

The rapid jump in reflected power occurred somewhere between 16.0 and 16.1cm

as the height was increased. It should be noted that this data was taken with the cavity

height moving in the increasing direction, (left to right in the plot). After the abrupt

upward jump in reflected power, recovery was not accomplished until the cavity height

43

was brought all the way back to 13.7 cm. This hysteresis effect has been observed in

several other studies of cavity type driven plasmas [11]-[19].

Standard System Analysis

The coefficient of cavity reactance/plasma resistance, ‘a’, was determined
through the information found above. Since the most reflected power observed when the
jump occurred was about 10 Watts, the lower limit of plasma density for stability was
determined by linearly equating delivered power to plasma density. A power reflection
coefficient of .05 represents 10 Watts reflected for 200 Watts forward. This 5%
perturbation in delivered power (200-10) fi'om the nominal operating point delivered
power (200-0), represented about a 5% drop in plasma density.

Setting x’ of Equation (1) above to zero and solving for the numerator polynomial
using different values for the resonance/density coefficient ‘a’, it was determined that a
value of a = 8.0 yielded solutions for plasma density at 4.34e17 and 4.09e17. These are
the zeros of a plot similar to that of Figure 22 . These two solutions are different by a
factor of about 6% which is close to that of the 5% plasma density ratio determined above
to be the left side of the region of attraction of the standard system. The plot of the
standard system differential equation with the ‘a’ coefficient set to 8 is shown below in
Figure 29. As indicated below, the region of attraction is from the unstable equilibrium

point, representing a plasma density of 4.09e17, and up.

x10”

 

 

 

 

 

 

 

 

2 T
o ,
2 t Regiond
' attraction
/
.4 ',
-6 - 1
Density
motion .8 t
x!
-1o
~12 r
-14 i- h
o 1 2 3 4 5 6 x10”

' x
Figure 29: Standard system difiaeifilTunation, resonance/density coeff ‘a’ = 8

Modified System

A large reactive offset was chosen to ensure a large effect and still remain within
the power delivery capability of the microwave generator. Since the generator was
capable of delivering 500 Watts into 50 ohms, a power reflection coefficient of .55 was
chosen to ensure that at least 200 Watts could be delivered into the modified system with
the reactive offset.

The system was modified as per Figure 18 above. The modified system

schematic is shown below in Figure 30.

45

 

4.25 m total length

 

_ directional
rirculator couplers
microwave

generator

 
   
  
 

T

reactive
offset

 

 

 

     
   

 

dummy load

cavity

 

 

 

1:: gas inlet

 

to vac
PUMP

 

 

 

 

 

Figure 30: Modified system schematic

A photo of the reactive offset setup is shown below in Figure 31.

Adjustable short

‘t’ adapter
circulator

RF input
Directional couplers ,_ t . j. .
Figure 31: Modified system reactive offset
To create the offset, a General Radio 874-D20 adjustable shorted stub was connected in

parallel to the transmission line with an ‘N’ type ‘t’ adapter. To characterize the offset,

one end of the ‘t’ adapter was terminated with 50 ohms, and the impedance was observed

 

looking into the other end with an HP 8510 Network Analyzer. The impedance was
measured to be about 16 — j55 Ohms, representing a power reflection coefficient of .55.
The model used for this stability analysis assumes that the reactive offset is a series
element. Using a Smith chart, the effective series reactance with 50 ohms real was found
by plotting a point along the 50 ohm circle that intersect at line segment of .55 power
reflection coefficient away fi'om the center. The resulting effective series impedance was

50 — jl 10 Ohms. This transformation method is illustrated below in Figure 32.

 
  

Measured 16-1'55
.55 pwr reflection radius

Figure 32: Determining effective reactive offset
The transmission line length between the power sensors and the cavity was
chosen through repeated trial and error for best stability. To predict this length would
have been too difficult given that the calculated length margin of error would have been

an appreciable percentage of a wavelength at 2.45 GHz.

47

Model Verification

Figure 33 below is a plot of the modified system differential equation. The
reactive offset was set to —1 10 Ohms, with the ‘a’ coefficient set to 8. The equilibrium
points are now at about 3.75e17 and 2.95e17. Equating plasma density to power, the
second equilibrium point is about 20% below that of the first, representing a reflected
power of 20% of indicated forward. Thus, for 200 Watts forward, the system should

remain stable up to about 40 Watts reflected.

x1021

 

 

 

 

 

 

 

5 a
0 9 Q\
1%.
5 I- "x. J
"t
. ‘\
Densrty .\
motion \
x. '10 P .1
-15 " -1
o 1 2 3 4 5 6 x10"
Density x

Figure 33: Modified system differential equation, a = 8, Of = -110 Ohms

The power setpoint perturbation method was performed on the modified system
via the sequence of steps outlined for the standard system, with the following results

shown on the plot below, Figure 34:

48

 

 

 

 

 

 

 

90
P Stable ran
5° a 96 an
70 ' l
Pointofnoretum
60 L
Reflected 50 «1
power *
(Watts) 40 4’ *
30 e
I- * * .1
2° + Besttune
10 t q
0 L L L *‘H‘eeereeeL
0 100 200 300 400 500 600
RelativeforwardpowerMatts)

Figure 34: Modified system performance with generator power perturbation
Compare the modified system performance of Figure 34 above to the standard system
performance of Figure 27. The maximum reflected power of 40 Watts is much larger
than the standard system’s 10 Watts reflected power. Also, the stable range is fi'om 225
Watts to 480 Watts as compared to the standard system’s 275 to 300 Watts.

Figure 35 below is a plot of the modified system’s actual performance for cavity
height perturbations. There was a rapid movement at about 30 - 50 Watts reflected where
the plasma would still be lit but at a reduced density. More importantly, the high density
plasma state would sometimes recover when the cavity height was returned from a high
position, (up to 16.4), back down to the best tune position, (small amount hysteresis). In
either direction of timing, there was a small abrupt jump in reflected power around the

16.2 — 16.3cm position (30-50 Watts).

49

 

70’ 1

mt *1
Reflected50r 1
power
(Watts)40' .4

30r *

20r

******
101 *+* *
*e

 

 

o L l r . 4 t a A at r a
14.4 14.6 14.8 15.0 15.2 15.4 15.6 15.8 16.0 16.2 16.4

Cavity height (cm)
Figure 35: Modified system performance with cavity height perturbation

 

To show the cable length dependence upon the modified system’s stability, the
transmission line length was shortened by approximately '/4 wavelength To model this
condition, the conjugate of the offset is chosen, changing Of fi'om -1 10 to 110 Ohms.
Figure 36 below is the modeled prediction of behavior, indicating no stable equilibrium

points, (dn/dt always less than 0), other than at zero density.

 

 

 

 

x1021
0 . r
-1 \ l
-2 . \\ 4
'3 I“ \ 1
Density 4 t \
motion
X' 5 t- .1
-6 A
/
-7 s
.8 L
o 1 2 3 4 5 6 x10"

Density x
Figure 36: Modified system differential equation with 1/4 X shorter transmission line

50

Figure 37 below is a plot of the modified system’s actual performance with the
shorter transmission line length. There was no stable operating point at zero reflected

power and the recovery position for the cavity height was well below 14cm.

m I V l U V U I I T

 

30L reetreee‘H‘I'

70 *

Reflected 50 »
power
(Watts) 40 t
30 t'
20 r- 41>

1o-

 

 

0 4 . 4

14.4 14.6 14.6 15.0 15.2 15.4 15.6 15.8 16.0 16.2 16.4
Cavity height (cm)
Figure 37: Modified system performance with 1/4 )1 shorter transmission line

 

51

Conclusions

The phenomenon of plasma system stability as a function of system parameters
outside of the plasma itself is clearly shown in the preceding analysis. This analysis
treated the plasma as if it were a simple device with a conductivity linearly proportional
to power, and with a first order arbitrary time response. The microwave cavity was
treated as a lossless, single-mode resonant device, with it’s reactive component having a
linearly proportional dependence on plasma density. All other losses external to the
plasma were neglected. There were no nonlinearities or discontinuities assumed in any of
the component models. With all of these seemingly gross approximations, stability
prediction was still reasonably well achieved. With a single first order differential
equation it was predicted that system stability could be increased or decreased through
the manipulation of the transmission line length and the microwave power supply’s
effective source impedance. Implementation of such a modified system confirmed these
predictions using two different types of perturbation.

It is concluded from these results that the modeling of a plasma system from a
‘controls’ point of view is a valid approach and that many observed plasma system
instabilities may be attributed to this systematic phenomenon This does not preclude any
direct nonlinearities or hysteresis effects of the plasma and its interactions with its
physical surroundings. Discontinuities of RF/microwave driven plasma sources are quite
common, especially in environments of complex shapes, and/or changing process
chemistries. However, it may behoove the system designer to be aware of this more
fundamental system-based source of potential instability before attempting to deal with

more complex behaviors encountered in plasma processing.

52

The models used for this analysis have pointed out one major contributor to
system instability through the cavity resonance to plasma density coefficient, ‘a’. If
plasma launching devices could be designed to reduce the effect that changing plasma
density has upon resonance, these potential instabilities could be minimized.

The system models could obviously be greatly improved, enabling even more
predictive accuracy. For example, given a more accurate relationship between the cavity
height and input impedance to the cavity, the height versus reflected power plots for the
actual and calculated data could have been directly compared. The two major areas for
improvement are in the models for the plasma and for the cavity/plasma interactions.
The following are some suggestions for model enhancements:

Plasma Model potential improvements:

1. Include the major nonlinearity of plasma ignition hysteresis. A minimum
excitation energy is required to ignite a plasma. The plasma will not extinguish
until an excitation energy is reached that is lower than the ignition energy.

2. Better characterize density as a function of absorbed power.

3. Characterize the ignition and density/power functions for different gases.

Cavity Model potential improvements:

1. Characterize cavity input impedance as a function of cavity/probe heights and
plasma density through numerical analysis [20].

2. Include firnctions for other resonant cavity modes, and the relationships that
determine which modes dominate for which conditions.

3. Include cavity wall losses as functions of calculated tangential magnetic fields.

53

Other model items that could provide overall system model enhancement:
1. Transmission line losses as a function of power and mismatch.
2. Realistic circulator model with losses and nonlinear behavior.
These model improvements are certainly realizable and would go a long ways
towards building a ‘virtual’ plasma system that could be the foundation of a more in-
depth model used to predict much more complex behaviors of RF/microwave driven

semiconductor plasma processes and machines.

54

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[81

[9]

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56