ABSTRACT
Title of dissertation: MODEL-BASED ANALYSIS OF ATOMIC
LAYER DEPOSITION GROWTH KINETICS
AND MULTISCALE PROCESS DYNAMICS
Curtisha D. Travis, Doctor of Philosophy, 2014
Dissertation directed by: Professor Raymond A. Adomaitis
Department of Chemical and Biomolecular
Engineering
A first principles model describing the reaction kinetics and surface species
dynamics for the trimethylaluminum (TMA) and water half-reactions of alumina
atomic layer deposition (ALD) is coupled with a dynamic film growth model and
reactor-scale species transport model. The reaction kinetics model is based on re-
ported enthalpies and transition state structures from published quantum-chemical
computational studies; these data are used to determine kinetic parameters using
statistical thermodynamics and absolute reaction rate theory. Several TMA half-
reactions were modeled to account for TMA adsorption and subsequent reaction
on a range of growth surfaces spanning bare to fully hydroxylated states. Several
water reactions were also considered. By coupling the reaction rate models with
surface species conservation equations, a dynamic model is created which is useful
for examining the relative rates of competing surface reactions. To describe the con-
tinuous cyclic operation of the deposition reaction system, a numerical procedure
to discretize limit-cycle solutions is developed and used to distinguish saturating
growth per cycle from non-saturating conditions. The transition between the two
regimes is studied as a function of precursor partial pressure, exposure times, and
temperature. Finally, a cross-flow tubular ALD reactor system model is derived with
components describing the precursor thermophysical properties, precursor delivery
system, reactor-scale gas-phase dynamics, and surface reaction kinetics derived from
absolute reaction rate theory. These model components are integrated to simulate
the complete multiscale ALD process. Limit-cycle solutions defining continuous
cyclic ALD reactor operation are computed with a fixed point algorithm based on
temporal and spatial discretization within the reactor, resulting in an unambiguous
definition of film growth per cycle. The use of the simulator for assisting in process
design decisions and optimization frameworks is presented.
MODEL-BASED ANALYSIS OF ATOMIC LAYER DEPOSITION
GROWTH KINETICS AND MULTISCALE PROCESS
DYNAMICS
by
Curtisha D. Travis
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Professor Raymond A. Adomaitis, Chair
Doctor Maria K. Burka
Professor Panagiotis Dimitrakopoulos
Professor Sheryl E. Ehrman
Professor Oden Rabin
Professor Michael R. Zachariah
c? Copyright by
Curtisha D. Travis
2014
DEDICATIONS
I dedicate this thesis to my adoring mother and to my dearest angel, Granny.
Your love has sustained me more than you will ever know.
ii
ACKNOWLEDGMENTS
Wow! Where do I start?! I?ll attempt to get through this without getting
overly mushy and drown my laptop in tears, but this has been quite the journey for
me. Before I continue, allow me to clarify what I mean by the word ?journey.? Yes,
I am in a way referring to the typical ?Woe is me! I?m a grad student... and I do
research... and sometimes I like it... but sometimes I don?t... blah blah blah...? I
think you can assume that much. So I?ll leave that stuff out.
When I say ?journey,? I?m not just referring to the last 5 years of my life. I?m
referring to... well, my life. Those who truly know me know that I?ve come such
a very long way to get here; and I certainly didn?t do it alone. There have been
some very instrumental people in my life who have helped me in some way, whether
big or small, and while I?m sure my words won?t come even close to describing how
much I appreciate all of you, please know that I am deeply grateful.
First and foremost, I have to thank God. Through Him, I know, without a
doubt, that all things are possible. This experience has strengthened my faith on
so many levels, and I am forever grateful that my Heavenly Father has been here
to hold my hand every step of the way. As I said, this journey started many years
ago, perhaps even back in 80?s when I was blessed with a beautiful and wonderfully
loving mommy to be here every step of the way, too!
Mama, there are so many things that I can say about you and the selfless
support that you have given me throughout all the years of my life. I am truly
blessed to have you in my corner! When I say that you are my biggest inspiration,
iii
I really mean it. I don?t know where I?d be without you. I would have never gotten
this far without the sacrifices you made for me, without your undying love, without
your confidence in me, and without your motivation. No matter what I was going
through, you gave me perspective; and talking to you somehow always made things
better. No one has been here for me the way you have, right by my side every single
step of the way; and so I dedicate this to you. You are the reason for so many of
the successes I?ve achieved in my life, and I truly love, adore, and appreciate you!
I am also so very thankful for my Granny and her unwavering support. While
she is no longer here, I always remember our conversations and the great confidence
she had in me. Granny always believed in me, always advocated for me, and always
encouraged me to ?Go all the way!? Although she wasn?t physically here these last
years, those simple words were enough to give me that little extra push when I
needed it. So, here we are, Granny?all the way! I hope I have made you proud.
To my Grandma, thank you for also believing in me and encouraging me.
While I didn?t always have time to respond, all of the random emails with the cute
little inspirational messages helped to brighten my days, just knowing that you were
thinking of me and praying for me. When I?ve expressed aggravation with being
in school for seemingly all my life or explained how annoyed I was that I couldn?t
do all the fun things I want to do, you always found a way to make me feel better
about it. You always told me not to worry about where things are headed, but just
to find joy in each part of the journey as I experience it. So, that?s what I did, and
it certainly made me a much happier graduate student. Thanks for your love and
your advice.
iv
I?ve been so lucky to have a wonderfully huge family of supporters! I won?t
be able to name all of you here, but just know that I appreciate each and every
one of you. Thank you for always trusting in me. Even when I had my doubts,
your support never wavered, never failed. You don?t know how much that meant to
me. To my aunties Ann, Peg, Berd, Leah, Mae, Sivi, and Netta, my uncles Willie,
Alfred, Rance, Jerry, and Perry, and all of my cousins (I would name you all, but
that would take forever!), I love you all to pieces!
Through the years, I?ve also developed one heck of an extended family. Part
of the reason why I wanted to attend graduate school in the Washington, DC area
is because of the great support network that I have here; and a huge part of that
is P.R.I.V.Y., my phenomenal line sisters. I am forever grateful for being joined
with such a dynamic group of ladies. You all have been my family away from home.
Thank you for the random nights out, the dinners, the girl talk sessions, and just
for being a staple in my life. You all have been such a huge support to me during
this time, even when you didn?t know it. Sometimes I needed to talk and sometimes
I just needed a silly, giggly, fun time. Whatever it was, you were there. You?ve
motivated me, you?ve encouraged me, and you will always inspire me. I love you all!
Another hugely important factor in the happiness and success of one?s graduate
school experience (perhaps the absolute biggest factor of all) is the advisor. Trust
me, I?ve heard so many horror stories from people about experiences with their
advisors. It can get really, really bad. But, thankfully I haven?t had any of those
experiences. I always say I feel like I really lucked up. Cheers to you, Ray! Thank
you for being such a great partner; I couldn?t have asked for anyone better! I?ve
v
learned so much from you during this time, and I?m so thankful for all of your
guidance and support. You?ve been patient with me as I?ve learned to shift my
thinking from chemistry to chemical engineering, and I really like the way we were
able to mesh our two thought processes. You?ve been immensely helpful, supportive,
positive, and you?ve been a superb example of what a professional chemical engineer
should be. I hope that someday I can be as brilliant an engineer as you.
To my college advisor, Dr. Hosten, thank you for being such a great resource
for me while at Howard and even now. You encouraged me to do the NRL intern-
ship, and that?s when I was first introduced to chemical engineering research. You
took an interest in my success a very long time ago, and I?m so very glad that we
have developed the relationship we have today. You?ve always been an honest and
unbiased person, and you made me feel comfortable to share my ideas with you,
always knowing that you?d try your best to steer me in the right direction. Your
professional advice has been immensely helpful, and I am forever grateful.
I can?t go without acknowledging financial support from my ?Uncle Bill? or
more formally, the Bill and Melinda Gates Foundation. Some 11 years ago, a com-
mittee of individuals decided that I?d make a worthy recipient of the Gates Millen-
nium Scholarship, and they?ve been committed to my academic success ever since.
Your support has definitely been instrumental throughout this process, and I know
just how privileged I am to have received it. Thank you, thank you, and thank you!
Lastly, to those I may have inadvertently left out, please accept my sincerest
apology. Charge it to my head and not my heart. Thank you all, and let the next
journey begin!
vi
TABLE OF CONTENTS
List of Tables x
List of Figures xiii
1 Introduction 1
1.1 The ALD Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Principles of ALD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Self-limiting Growth . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Growth Rate Limitations . . . . . . . . . . . . . . . . . . . . . 7
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Review of ALD Kinetics Modeling . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1.1 Reactor-scale Models . . . . . . . . . . . . . . . . . . 13
1.4.2 First Principles Methods . . . . . . . . . . . . . . . . . . . . . 15
1.5 Selection of a Kinetics Model . . . . . . . . . . . . . . . . . . . . . . 16
1.5.1 Modeling Objectives . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Surface Reaction Mechanisms and Kinetics 23
2.1 General Mechanism of Oxide ALD . . . . . . . . . . . . . . . . . . . 24
2.2 Review of Oxide ALD Processes . . . . . . . . . . . . . . . . . . . . . 27
2.3 Transition State Theory . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Molecular Chemisorption . . . . . . . . . . . . . . . . . . . . . 35
2.3.3 Ligand Exchange . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.4 Precursor Dissociation . . . . . . . . . . . . . . . . . . . . . . 37
2.3.5 Hydrogen Association . . . . . . . . . . . . . . . . . . . . . . 38
2.3.6 Densification . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Alumina Case Study 41
3.1 Modeling Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Formulation of Underlying Model Assumptions . . . . . . . . 46
3.2 Chemical Reaction Mechanism . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 The Surface State . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.2 Surface State Dynamics . . . . . . . . . . . . . . . . . . . . . 52
3.3 Surface Reaction Equilibria and Reaction Rates . . . . . . . . . . . . 53
vii
3.4 Computation of Species Partition Functions . . . . . . . . . . . . . . 57
3.4.1 Half-cycle A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1.1 R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1.2 R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.2 Half-cycle B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.2.1 R6 and R9 . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.2.2 R7 and R10 . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Representative Surface Coverages and Reaction Rates . . . . . . . . . 63
3.6 Limit-cycle Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6.1 A Collocation Procedure . . . . . . . . . . . . . . . . . . . . . 66
3.7 GPC and Film Composition for Self-limiting Growth . . . . . . . . . 69
3.7.1 GPC and gpc . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.8 A map of gpc(?A, ?B) . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.8.1 Comparison to Measured Exposures . . . . . . . . . . . . . . . 72
3.8.2 Effect of Deposition Temperature . . . . . . . . . . . . . . . . 74
3.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Kinetic Mechanism Analysis 78
4.1 Surface Reaction Mechanism . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.1 TMA Reactions on Hydroxylated Alumina . . . . . . . . . . . 95
4.1.2 TMA Reactions on Bare Alumina . . . . . . . . . . . . . . . . 96
4.1.3 Water Reactions on Methylated Alumina . . . . . . . . . . . . 97
4.1.4 Water Reactions on Bare Alumina . . . . . . . . . . . . . . . 98
4.1.5 Gas-phase Side Reactions . . . . . . . . . . . . . . . . . . . . 99
4.2 Analysis of Reaction Mechanism . . . . . . . . . . . . . . . . . . . . . 100
4.2.1 Comparative Analysis of TMA Kinetics . . . . . . . . . . . . . 101
4.2.1.1 Resolution of Thermodynamics . . . . . . . . . . . . 103
4.2.2 Comparative Analysis of Water Kinetics . . . . . . . . . . . . 105
4.2.2.1 Resolution of Thermodynamics . . . . . . . . . . . . 107
4.2.3 Gas-phase Side Reaction Mechanism Analysis . . . . . . . . . 108
4.3 Thermodynamic Sensitivity . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Growth Surface State Dynamics . . . . . . . . . . . . . . . . . . . . . 110
4.4.1 Dynamic Simulation Results . . . . . . . . . . . . . . . . . . . 112
4.4.2 Growth per cycle . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Dynamic ALD Reactor Design 121
5.1 ALD Reactor Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.1.1 Precursor Delivery . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.2 Reaction Chamber . . . . . . . . . . . . . . . . . . . . . . . . 125
5.1.2.1 Batch Reactors . . . . . . . . . . . . . . . . . . . . . 126
5.2 A Continuous Cross-flow ALD Reactor . . . . . . . . . . . . . . . . . 126
5.2.1 Process Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3 Precursor Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4 Precursor Delivery System Model . . . . . . . . . . . . . . . . . . . . 134
viii
5.4.1 Universal Gas Sizing Flow Equation for Control Valves . . . . 137
5.4.2 Development of a Continuous Flow Equation . . . . . . . . . . 139
5.5 Reactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.6 Limit-Cycle Computations . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6.1 Time Discretization for Forced-periodic Systems . . . . . . . . 148
5.6.2 Newton-Raphson Procedure . . . . . . . . . . . . . . . . . . . 150
5.7 Representative Results . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.7.1 TMA Ballast Chamber Dynamics . . . . . . . . . . . . . . . . 152
5.7.2 Reactor-scale Dynamics . . . . . . . . . . . . . . . . . . . . . 153
5.7.3 Growth Surface Dynamics . . . . . . . . . . . . . . . . . . . . 154
5.8 Mapping the gpc Behavior . . . . . . . . . . . . . . . . . . . . . . . . 155
5.8.1 The Vbc-?B Plane . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6 Distributed Reactor Model 162
6.1 Continuous Cross-flow ALD Reactor Model . . . . . . . . . . . . . . . 163
6.1.1 Formulation of Underlying Model Assumptions . . . . . . . . 164
6.1.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.3 Chemical Species Conservation . . . . . . . . . . . . . . . . . 169
6.1.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . 170
6.1.4 ALD Film Growth . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Discretization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.3 Representative Limit-Cycle Results . . . . . . . . . . . . . . . . . . . 173
6.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7 Concluding Remarks 180
7.1 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 182
7.1.1 Surface Reaction Mechanism and Kinetics . . . . . . . . . . . 183
7.1.2 Reactor-scale Model Optimization . . . . . . . . . . . . . . . . 183
7.1.3 Reactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.1.4 Extension to Additional ALD Chemistries . . . . . . . . . . . 184
A Throughput Optimization 185
Bibliography 187
Publications & Presentations 203
ix
LIST OF TABLES
2.1 Elementary reactions for the ALD of MOx/2 by metal precursor MLx
and H2O as in reaction (2.1). Surface species are denoted with double
bars (?) and y ? [1, x). Gas-phase reactions are non-ideal in ALD
and are thus excluded. Based on a similar table in [36]. . . . . . . . . 26
3.1 A subset of elementary surface reactions taking place during TMA
exposure as identified in [156]. Me = CH3. d and m subscripts denote
DMA and MMA surface species. . . . . . . . . . . . . . . . . . . . . . 49
3.2 A subset of elementary surface reactions taking place during water
exposure as identified in [156]. Me = CH3. d and m subscripts denote
DMA and MMA surface species. . . . . . . . . . . . . . . . . . . . . . 50
3.3 Surface reaction model parameters. . . . . . . . . . . . . . . . . . . . 52
3.4 Reaction rate energetics data. . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Statistical mechanical formulation of equilibrium constants and rate
expressions for elementary surface reactions taking place during TMA
exposure as identified in [156]. Me = CH3. d and m subscripts denote
DMA and MMA surface species. . . . . . . . . . . . . . . . . . . . . . 55
3.6 Statistical mechanical formulation of equilibrium constants and rate
expressions for elementary surface reactions taking place during water
exposure as identified in [156]. Me = CH3. d and m subscripts denote
DMA and MMA surface species. . . . . . . . . . . . . . . . . . . . . . 56
3.7 Molecular partition function contributions for a nonlinear polyatomic
ideal gas. DOF, degrees of freedom. n, number of atoms in the
molecule. ZPE, zero point energy. . . . . . . . . . . . . . . . . . . . . 58
4.1 A subset of elementary surface reactions taking place during TMA
exposure, beginning with molecular adsorption. Me = CH3. . . . . . . 83
4.2 continued from Table 4.1. A subset of elementary surface reactions
taking place during TMA exposure, beginning with molecular adsorp-
tion. Me = CH3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 A subset of elementary surface reactions taking place during TMA
exposure, beginning with dissociative adsorption as described in [40].
Me = CH3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 A subset of elementary surface reactions taking place during water
exposure, beginning with molecular adsorption. Me = CH3. . . . . . . 85
4.5 A subset of elementary surface reactions taking place during water
exposure as described in [23]. Me = CH3. . . . . . . . . . . . . . . . . 86
x
4.6 A subset of elementary surface reactions taking place during water
exposure, beginning with dissociative adsorption as posed in [60, 61].
Me = CH3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.7 A set of elementary gas-phase reactions taking place when both TMA
and water are present [111]. Me = CH3. . . . . . . . . . . . . . . . . 87
4.8 Reaction rate energetics. Plot estimate values noted with an aster-
isk (?). In the instance of multiple values for a parameter, the value
used in the dynamic growth model is noted with a dagger (?). Alu-
mina homodeposition studies are noted with a plus sign (+), otherwise
all other studies correspond to alumina heterodeposition onto Si-OH
substrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9 (continued from Table 4.8) Reaction rate energetics. Plot estimate
values noted with an asterisk (?). In the instance of multiple values
for a parameter, the value used in the dynamic growth model is noted
with a dagger (?). Alumina homodeposition studies are noted with
a plus sign (+), otherwise all other studies correspond to alumina
heterodeposition onto Si-OH substrates. . . . . . . . . . . . . . . . . . 89
4.10 Partition function ratios and underlying assumptions. Parenthetical
terms in the ?Z columns designate degrees of freedom assumed to be
lost due to the forward reaction. Terms in the ?Z columns which are
not in parentheses designate degrees of freedom assumed to be gained. 90
4.11 (continued from Table 4.10) Partition function ratios and underlying
assumptions. Parenthetical terms in the ?Z columns designate de-
grees of freedom assumed to be lost due to the forward reaction. Terms
in the ?Z columns which are not in parentheses designate degrees of
freedom assumed to be gained. . . . . . . . . . . . . . . . . . . . . . . 91
4.12 (continued from Table 4.11) Partition function ratios and underlying
assumptions. Parenthetical terms in the ?Z columns designate de-
grees of freedom assumed to be lost due to the forward reaction. Terms
in the ?Z columns which are not in parentheses designate degrees of
freedom assumed to be gained. . . . . . . . . . . . . . . . . . . . . . . 92
4.13 (continued from Table 4.12) Partition function ratios and underlying
assumptions. Parenthetical terms in the ?Z columns designate de-
grees of freedom assumed to be lost due to the forward reaction. Terms
in the ?Z columns which are not in parentheses designate degrees of
freedom assumed to be gained. . . . . . . . . . . . . . . . . . . . . . . 93
4.14 (continued from Table 4.13) Partition function ratios and underlying
assumptions. Parenthetical terms in the ?Z columns designate de-
grees of freedom assumed to be lost due to the forward reaction. Terms
in the ?Z columns which are not in parentheses designate degrees of
freedom assumed to be gained. . . . . . . . . . . . . . . . . . . . . . . 94
5.1 Reactor dimensions and primary system component design parame-
ters of the ALD reactor system shown in Fig. 5.1. TMA, trimethyla-
luminum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
xi
5.2 Nominal ALD reactor conditions. d-TMA, dimer of TMA. . . . . . . 133
5.3 Parameters for precursor delivery model. Valve flow equations yield
flow rates in units of mol/s with pressure in bar and temperature in K.139
5.4 Selection of data for comparison of water molar flow rate predictions
(mol/s) from randomized pressure differentials. Eqns. (5.13) and
(5.14) are standard flow equations based on ANSI/ISA guidelines [9].
Eqns. (5.16) and (5.18) are asymptotic and linear approximations,
respectively, to the standard equations. . . . . . . . . . . . . . . . . . 141
xii
LIST OF FIGURES
1.1 Number of ALD publications per year from between 1981 and 2013.
Search made with Google Scholar search terms ?atomic layer deposi-
tion? and ?atomic layer epitaxy? [49]. . . . . . . . . . . . . . . . . . . 2
1.2 Schematic illustration of one ALD reaction cycle. . . . . . . . . . . . 4
1.3 Self-limiting ALD and non-ideal growth rate behavior with possible
factors causing the observed non-ideality. (left) Growth per cycle as a
function of growth temperature. (right) Growth per cycle as a function
of precursor exposure time. Reproduced from [140]. . . . . . . . . . . 6
1.4 Elements of a complete ALD reactor system mathematical model.
Chapters 2-4 of this thesis will specifically address the surface reac-
tion kinetics and growth surface state dynamics model components.
Chapters 5 and 6 will introduce model components describing the pre-
cursor source thermodynamics, precursor delivery system dynamics,
and reactor-scale transport dynamics. Chapters 5 and 6 will also
incorporate the model components of Chapters 2-4 to make up the
complete multiscale ALD model. . . . . . . . . . . . . . . . . . . . . . 20
2.1 An idealized view of the MOx/2 ALD process cycle with metal precur-
sor
?
M and oxygen precursor
?
O. . . . . . . . . . . . . . . . . . . . . . 25
2.2 Schematic reaction path profiles for generic elementary ALD reaction
steps. Gas-phase and surface species are noted with Latin and Greek
alphabet, respectively. a) Adsorption/ligand-exchange reaction. b)
Association reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 A snapshot of a 35 nm2 portion of the ALD growth surface corre-
sponding to ?O = 0.6 and ?Me = 0.5. . . . . . . . . . . . . . . . . . . . 51
3.2 Predictions of the maximum adduct and transition state species sur-
face coverages for T = 500 K and a range of precursor partial pressures. 63
3.3 Predictions of the reaction rates for T = 500 K and a range of precur-
sor partial pressures. Gas-phase precursor wall collision frequencies,
wA and wB, are shown for reference. . . . . . . . . . . . . . . . . . . 64
3.4 Surface CH3 and OH limit-cycle coverage dynamics for PA = PB = 1
Torr and T = 500 K. The red and blue curves denote surface Me
and OH group coverage, respectively. Dots indicate collocation point
locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Surface CH3 and OH limit-cycle coverage dynamics for PA = PB = 1
Torr and T = 500 K. The red and blue curves denote the TMA and
water doses, respectively. Dots indicate collocation point locations. . . 68
xiii
3.6 A map of gpc as a function of precursor exposure level for PA = PB =
1 Torr. The point marked 2 corresponds to ?A = ?B = 5 ? 105 and
to the conditions of Figs. 3.4 and 3.5. . . . . . . . . . . . . . . . . . . 71
3.7 Effect of temperature on gpc for PA = PB = 1 Torr. Blue curve
corresponds to ?A = ?B = 2 ? 105 L, green to ?A = ?B = 5 ? 105 L,
and red to ?A = ?B = 9? 105 L. Dashed curve segments indicate the
temperature range where TMA decomposes [129]. . . . . . . . . . . . . 74
4.1 Representative reaction rates of irreversible TMA half-cycle reactions
corresponding to pressure-dependent reactions at T = 450 K and max-
imum surface concentration values for [X], [X?], [O?], [S], [Me1], and
[Me2]. For reference, the TMA wall collision rate wA also is shown.
Data labels denote the source of energetic information used in the
rate computation; e.g. EG04 corresponds to ElliottGreer2004 (first
author, last author, and year of publication). . . . . . . . . . . . . . . 102
4.2 Representative reaction rates of irreversible TMA half-cycle reactions
corresponding to pressure-independent reactions at T = 450 K and
maximum surface concentration values for [X], [X?], [O?], [S], [Me1],
and [Me2]. Data labels denote the source of energetic information used
in the rate computation; e.g. EG04 corresponds to ElliottGreer2004
(first author, last author, and year of publication). . . . . . . . . . . . 103
4.3 Representative reaction rates of irreversible water half-cycle reactions
corresponding to pressure-dependent reactions at T = 450 K and max-
imum surface concentration values for [X], [X?], [O?], [S], [Me1], and
[Me2]. For reference, the water wall collision rate wB also is shown.
Data labels denote the source of energetic information used in the rate
computation; e.g. WM02 corresponds to WidjajaMusgrave2002 (first
author, last author, and year of publication). . . . . . . . . . . . . . . 106
4.4 Representative reaction rate of the gas-phase side reaction. (top) Rate
computed with one precursor at constant partial pressure of 0.01 Torr.
(bottom) Rate computed with one precursor at constant partial pres-
sure of 0.001 Torr. All rates correspond to reactions at T = 450 K.
For reference, the gas-phase TMA-water collision frequency zAB also
is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5 Surface species dynamics for ??X(0) = 1 (top), ?
?
X(0) = 0.75 (middle),
and ??X(0) = 0 (bottom) at T=450 K and P = 2 Pa. The initial
0.02 s of the TMA half-cycle have been emphasized to capture the fast
surface reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.6 Relative film growth rate contributions of (R3) and (R13) as a function
of initial surface OH group concentration. . . . . . . . . . . . . . . . . 116
5.1 Process and instrumentation schematic of the continuous cross-flow
ALD reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 TMA vapor pressure and degree of dissociation, ?, as a function of
temperature; data presented in [92] are shown as filled red circles. . . 135
xiv
5.3 Water molar flow rate predictions for a randomized set of pressures
using the non-choked (NC) and choked (C) flow Eqns. (5.13) and
(5.14). The asymptotic approximations (AA) of Eqn. (5.16) and lin-
ear approximations (LA) of Eqn. (5.18) for the same random set
of pressures are overlayed, illustrating the approximation boundaries
with respect to the standard flow rate formulations. . . . . . . . . . . 143
5.4 Water molar flow rate predictions for a randomized set of pressures
using the non-choked (NC) and choked (C) flow Eqns. (5.13) and
(5.14). The asymptotic approximations (AA) of Eqn. (5.16) and lin-
ear approximations (LA) of Eqn. (5.18) for the same random set
of pressures are overlayed, illustrating the approximation boundaries
with respect to the standard flow rate formulations. Non-choked and
choked flow regions are shaded blue and red, respectively. . . . . . . . 144
5.5 Representative reactor limit-cycle solution with gpc = 0.806 A?/cycle.
Reactor nominal conditions consist of ?A = 0.1 s, ?AP = ?BP = 4 s,
?B = 0.1 s and Vbc/V = 0.02%. . . . . . . . . . . . . . . . . . . . . . 152
5.6 Growth per cycle (gpc) as a function of ?B (left) and Vbc/V (right). . 156
5.7 Growth per cycle (gpc) as a function of ?B and Vbc/V . The black
curves correspond to moles of TMA/cycle supplied to the reactor. . . 158
6.1 Central time and central space finite differences mesh and solution
points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 Representative gas-phase dynamics for T=450K, and ?A = 0.5 sec,
?B = 0.1 sec, and ?AP = ?BP = 2 sec. Curves show spatial reactor
position, with inlet marked by -x- and outlet by -o-. In order from top
to bottom, the ALD cycle step for each plot is as follows: pulse A,
purge A, pulse B, purge B. . . . . . . . . . . . . . . . . . . . . . . . . 174
6.3 Representative gas-phase and surface dynamics for T=450K, and ?A =
0.5 sec, ?B = 0.1 sec, and ?AP = ?BP = 2 sec. Curves show spatial
reactor position, with inlet marked by -x- and outlet by -o-. . . . . . . 175
6.4 Exposure and growth per cycle predictions as a function of spatial
position in the ALD reactor. . . . . . . . . . . . . . . . . . . . . . . . 177
A.1 Preliminary results for alumina ALD throughput as a function of pre-
cursor exposure times. . . . . . . . . . . . . . . . . . . . . . . . . . . 186
xv
1Introduction
Atomic layer deposition (ALD), formerly known as atomic layer epitaxy (ALE),
is a thin film manufacturing process based on continuous cycling of an alternating
sequence of self-terminating heterogeneous chemical reactions. Under ideal growth,
ALD has the capability to produce highly conformal ultrathin films on a variety of
materials and complex substrate geometries all while allowing for precise control of
film thickness and composition down to a fraction of a monolayer [44].
It is indisputable the tremendous impact that the semiconductor industry has
had on ALD interest and the direction of research. ALD was developed nearly 40
years ago by T. Suntola [151], but it wasn?t until the last decade that it gained
significant research interest. The seemingly sudden growth in ALD literature coin-
cides directly with the end of the classical Moore?s Law scaling era [73, 77] which
happened around the turn of the 21st century. Dielectrics and metal electrodes for
dynamic random-access memory (DRAM) capacitors are now routinely produced via
ALD [50]. With 3D tri-gate transistors requiring conformal coatings over complex
geometry and metal circuitry pushing scaling limits, ALD emerges as an indispens-
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ions
Year
Figure 1.1: Number of ALD publications per year from between 1981 and 2013.
Search made with Google Scholar search terms ?atomic layer deposition? and
?atomic layer epitaxy? [49].
able manufacturing tool. Indeed, Intel announced in 2007 that ALD is involved in
?the biggest change to computer chips in 40 years? [15].
ALD has received particular attention in depositing high-permittivity (high-
? or ?high-k?) dielectrics, with recent interest in nanostructures, allowing for very
controlled modification of nanopores by fine-tuning the deposition parameters [3,
32, 34]. These nanoporous materials have various industrial applications such as
catalysts, sensors, membranes, filters, nanotemplates, capacitors, and transistors.
Metal oxides such as alumina (Al2O3), zirconia (ZrO2), and hafnia (HfO2) have
thus been the subject of considerable investigation.
Although the ALD industry is not yet fully matured, it has already been ap-
plied to sizable material and deposition tool industries as well as huge semiconductor
chip fabrication lines. In modern logic chips, thicknesses of 2-3 nm are often desired,
meaning that variations in thickness of merely 0.2-0.3 nm already corresponds to
2
10 % of the total film thickness [73]. A technique which can controllably deposit one
monolayer of material would be an optimal method for these applications. Hence
with the accelerating advancement of micro- and nanotechnology, ALD continues to
emerge as an ideal candidate for producing ultrathin and conformal films across dif-
ferent industries with applications in photovoltaics [160], energy devices [78,101,106],
nanofabrication [81], biological systems [149], and the environment [101].
These new thin-film application areas and the ability to deposit an increas-
ing range of new materials have rekindled interest in thin film deposition process
modeling as current development activities have pushed traditional fabrication tech-
nologies to their limits, further encouraging the use of new deposition techniques
such as ALD. In integrated circuit manufacturing, the simultaneously decreasing
feature sizes, increasing aspect ratios, complicated designs, and adaptation of new
materials call for new methods that enable highly controlled deposition of very thin
and conformal films, requirements perfectly matched with ALD [140].
1.1 The ALD Cycle
An ALD cycle typically involves a controlled sequence of self-limiting reaction
steps and purge steps as schematically illustrated in Fig. 1.2. While more steps can
be incorporated, for instance in the deposition of hafnium aluminates [113], the ALD
cycle generally comprises four separate and sequential steps involving two gaseous
precursor species introduced in an alternating fashion. The four steps of the ALD
cycle can be described as follows:
3
Purge
Purge
Pulse A
Pulse B
ALD
Figure 1.2: Schematic illustration of one ALD reaction cycle.
(i) Pulse A. The first reactant (precursor A) is introduced and proceeds to react
with surface species until self-terminating.
(ii) Purge. Gas-phase by-products and unreacted precursor gas are removed.
(iii) Pulse B. The second reactant (precursor B) is introduced and proceeds to react
in an analogous fashion with the surface species until self-terminating.
(iv) Purge. Gas-phase by-products and unreacted precursor gas are removed.
Each pulsing step is often referred to as a ?half-cycle? or ?half-reaction? in the pro-
cess. Constituting one complete ALD cycle, these four steps are repeated potentially
hundreds or thousands of times until the desired film thickness is achieved. Thus
the term ?growth per cycle? (gpc) corresponds to the growth observed for the above
sequence of steps. Note that the lower-case abbreviation gpc will be used to denote
4
undersaturating growth and the capitalized GPC will be used to denote saturating
growth rates observed during self-limiting growth.
1.2 Principles of ALD
ALD and the more widely understood chemical vapor deposition (CVD) sys-
tems share a number of common characteristics: they are inherently nonlinear
and time-dependent, and model elements describing the deposition surface reac-
tion and gas-phase precursor transport are strongly coupled. In particular, the
transport/reaction models must describe phenomena over multiple time and length
scales. Because of this phenomenological overlap, some modeling concepts and com-
putational tools developed by the CVD community can be directly put to use in
the analysis of ALD systems. However, ALD processes are typically run in a lower
temperature regime when compared to CVD, making it a more desirable method
for the low thermal budget [73].
Another important distinction can be made between the two deposition pro-
cesses: the notion of the steady-state deposition rate in CVD does not exist for
ALD. The rate of ALD depends strongly on the instantaneous state of the growth
surface and this state changes continuously through each pulse and purge interval.
The completely dynamic nature of the ALD process adds considerably to the dif-
ficulty of developing simulators because the entire process cycle must be modeled
and careful chemical and mathematical analysis is required to resolve the multiple
time scales present in this process and to identify the rate-limiting steps.
5
GP
C
GP
C
Temperature Exposure time
Self-limiting growth
Self-limiting growth
T-independent
?ALD window?
Self-limiting growth
T-dependent
Figure 1.3: Self-limiting ALD and non-ideal growth rate behavior with possible fac-
tors causing the observed non-ideality. (left) Growth per cycle as a function of
growth temperature. (right) Growth per cycle as a function of precursor exposure
time. Reproduced from [140].
1.2.1 Self-limiting Growth
What also distinguishes ALD from CVD and other deposition methods is the
self-limiting nature of the precursor chemisorption during each exposure period,
giving rise to an operation region where the GPC (saturating growth per cycle) is
insensitive to perturbations in precursor partial pressure and exposure time. For
some ALD systems (i.e. different materials to be deposited), an ?ALD window?
is observed where self-limiting growth also is insensitive to deposition temperature.
The net result is highly conformal surface coverage and atomic-level control of film
thickness, with equipment-independent GPC for fixed precursor chemistry [44,131].
The classic picture of self-limiting growth consists of a plot of GPC versus tem-
perature, where the ideal ALD region is bounded at one end by high-temperature
precursor decomposition reactions or loss of surface species, and at the other by a
non-saturating dose resulting from incomplete surface reactions or precursor con-
densation [140]. Likewise, plotting GPC versus total precursor exposure (roughly
6
defined as the product of precursor pressure and exposure time) reveals a constant
GPC for sufficiently high exposures with a lower bound defined by under-saturating
conditions. While certainly desired, the notion of an ALD-window is not a require-
ment for self-limiting ALD growth and indeed many recently studied processes for
oxide films exhibit self-limiting growth which does not conform to an ALD win-
dow [119].
1.2.2 Growth Rate Limitations
Perhaps one of the most significant limitations of ALD in high-volume manu-
facturing is the relatively low growth rate when compared to other processes such
as CVD. This can be partially attributed simply to the nature of the process, i.e.
only a monolayer of growth is to be expected per cycle. But ALD usually produces
only a fraction of a monolayer per cycle and this can be attributed to precursor
chemistry [140]. This highlights the need to understand reaction mechanisms such
that optimal chemical reaction mechanisms can be pursued by manipulating the
appropriate process parameters. Note that the low growth rate of ALD processes
becomes less of a disadvantage when depositing films of only a few nanometers, as
is required in many semiconductor applications [73]. Yet and still, low throughput
is widely considered as the main problem of ALD [140].
Throughput is directly related to the ALD growth per cycle and inversely re-
lated to the cycle time length. Thus, to optimize the effective growth rate, the GPC
should be maximized while minimizing the cycle time. However, there are limita-
tions to each approach. Precursor pulsing intervals are limited by chemistry and
7
cannot be shortened beyond a certain point, or else the surface will not fully saturate
and the self-limiting ALD growth regime will not be attained. Purge periods also
cannot be arbitrarily shortened due to the necessary isolation of precursor gases in
ALD processes. Shortened purge times could result in coexistence of both precursors
in the reactor and consequently lead to CVD-like gas-phase reactions. This is non-
ideal as the uniformity and film thickness will no longer be controlled. Therefore,
careful consideration must be given to understand the process chemistry in order
to locate self-limiting growth regions for ideal uniformity and also to optimize the
throughput.
In efforts to improve throughput, deviations from ideal ALD growth are occa-
sionally created, e.g. operating the ALD process under non-saturating conditions.
Often times, total self-limiting surface saturation is limited by kinetics (as opposed
to mass transport) and can only be achieved by allowing the surface reactions time
to occur. In this instance, increasing the partial pressure of the precursor will not
cause a significant increase in growth rate. Further, increasing saturation from 90 %
to 100 % may take more time than reaching the first 90 %. It may thus be beneficial
to accept the 90 % saturation because then the exposure time can be substantially
shortened and the subsequent purge will likewise be shortened due to a lower volume
of precursor in the reaction chamber [140]. Use of this limited optimized reactions
ALD (LORA) approach has yielded a reported ten-fold increase in throughput [145],
although it may cause additional non-uniformity issues depending on the reactor de-
sign, e.g. it may not be applicable for cross-flow reactors.
8
1.3 Motivation
With all of this, the need for physically-based dynamic ALD models has never
been greater. Consider, for example, the explosion of growth in developing high-
throughput spatial ALD systems, now under development for roll-to-roll and other
large-substrate applications. The processing speed objectives of these systems can
only be fully achieved by model-based process and design optimization methods.
While throughput is the primary objective, the level of precursor utilization also
plays a significant role in the process operating costs. An important tool for maxi-
mizing tool productivity and precursor conversion are ALD deposition kinetics mod-
els that are equipment-independent.
Consider, also, the manufacturing environment where minimization of the indi-
vidual doses of each precursor is necessary to optimize the operation of commercial,
multiwafer reactor systems [52]. Precursor dose minimization becomes even more
critical in high-throughput spatial ALD systems. The intense interest in this tech-
nology can be gauged by the recent ALD 2011 and 2012 conferences where at least
six independent designs were presented at each meeting [5, 6]. Maximum substrate
throughput in all of these designs is dictated by a lower bound of the self-limiting
growth region. Model-based analysis of the self-limiting growth boundary also is
motivated by the need to elucidate the chemical reaction mechanisms that result in
unusually slow saturation rates in some ALD systems [42]. Understanding the kinet-
ics of the deposition process is critical to distinguishing between competing surface
reactions [23] and determining which constitute the self-limiting reaction steps.
9
Considering all of these known process interactions from atomic-scale chem-
istry to reactor-scale throughput, study of the ALD process intrinsically necessitates
a multiscale problem formulation. Consider, for instance, the intrinsic time scales
of the ALD process. While precursor pulsing and purge times scale on the order of
seconds to tens of seconds, surface reaction rates which ultimately determine film
growth rates scale from fast (ps-ns) to slow (?s-ms). Moreover film growth scales at
A? or nm while reactor sizes are on orders of meters [38]. While it is impossible to
describe explicitly all of these length and time scales in a single multiscale simula-
tion, effort should be made to couple these time and length scales in such a way to
resolve ALD process interactions across the multiscale domain.
1.4 Review of ALD Kinetics Modeling
Simulation tools are important for understanding the multiscale reaction phe-
nomena at work in ALD processes and for the design, optimization, and control of
deposition reactors to reach desired film property specifications [36]. A range of ALD
models has been developed at various levels of detail and theory to accommodate
specific modeling objectives. Of these, atomic-scale models have produced insightful
results by employing ab initio methods, such as density functional theory (DFT),
to investigate energetically-favored reaction mechanisms, proton transfer pathways,
structures of intermediate reaction species, and reaction energetics corresponding
to elementary reactions along each stage of ALD growth. Results of such stud-
ies have been combined with kinetic Monte Carlo (kMC) simulations to quantify
10
nucleation [28] and growth [27,104] kinetics.
Empirical growth models have also been developed to describe ALD kinetics
during the initial stages of nucleation [4,172] and island growth [112,136,137]. Ad-
ditionally, surface reaction models have been developed to predict growth kinetics
and the effects of operating parameters (e.g., temperature, pressure, precursor ex-
posure time) during undersaturating conditions and leading to self-limiting ALD
growth [96?98,121,156].
With increased interest in nanostructured materials (e.g. for energy and envi-
ronmental applications [11]), ALD has emerged as the method of choice for pro-
ducing conformal coatings in nanoporous structures [33], employed for instance
for construction of multilayer nanocapacitors [12], fine-tuning surface properties of
nanopores to capture DNA [19], control of pore wall size and composition of catalytic
membranes [123], and fabrication of spatially-controlled coatings within nanopores
for multi-step catalysis [34]. Because of these and other promising nanoscale ALD
applications [81,101], researchers have developed models for transport and reaction
in high-aspect-ratio structures.
There are two general methodologies currently used for predicting ALD re-
action kinetics: empirical methods and ab initio methods. These two methods
utilize vastly different levels of detail in regard to the underlying ALD chemistry
and consequently provide varying levels of detail in resolution to the intrinsic kinetic
mechanism. Both approaches are reviewed in the subsections to following.
11
1.4.1 Empirical Methods
Because much of the ALD reaction mechanism relies either directly or indi-
rectly on heterogeneous surface processes, it is overwhelmingly common to describe
the kinetics in terms of Langmuir adsorption in the form of a reactive sticking coef-
ficient s(?) = so(1??) where ? denotes surface coverage of adsorbed species and the
initial sticking probability for zero coverage so is treated as an adjustable parame-
ter [150]. In these approaches, the detailed surface chemistry is often ?lumped? into
this parameter, which can be estimated, fitted to computed energetics, or derived
from film growth experiments.
In the context of ALD, the reactive sticking coefficient is loosely defined as
the probability that a precursor molecule will react upon colliding with the surface
and contribute to film growth [85], combining in one parameter the total growth
probability as a result of all surface reaction steps. For this reason, some authors
also refer to the reactive sticking coefficient as the ALD ?reaction probability,? i.e.
the reactivity of a single precursor molecule with a reactive site on the surface.
Such a kinetics modeling approach is exemplified in [25] which expands on the
classic model of ALD step coverage [51] to develop a Langmuir-type surface coverage
model relating surface dynamics to precursor impingement fluxes computed using
the kinetic theory of gases with an adjustable sticking coefficient. Variations of this
reaction kinetics method have been applied to simple array [85], differential equation-
based [3,46?48,93,150,169] and Monte Carlo [32,84] descriptions of temporal surface
coverage evolution.
12
1.4.1.1 Reactor-scale Models
A number of detailed ALD reactor simulations have been reported in the lit-
erature, generally comprised of transport equations coupled with a surface reaction
model to describe the evolution of the growth surface. The elementary surface
reaction kinetics model has also been successfully employed in many of these sim-
ulations, beginning perhaps two decades ago with a convection-diffusion transport
model coupled to a general surface dynamics model in a flow-type reactor [1]. Precur-
sor concentration profiles and surface coverage were found to be strongly dependent
on the chosen sticking coefficient. This work was thereafter extended to investi-
gate temperature effects [147] and ALD growth rates for TiO2 films, again yielding
coverage and growth predictions with strong dependence on the reactive sticking
coefficient [148].
Another early transport/reaction coupling is seen in a plug-flow reactor model
where the sticking coefficient is assumed to combine condensation and activation
energy effects to yield estimates of film growth rate and uniformity [171]. More
recently, a simple plug-flow model, based largely on empirical sticking and recom-
bination probabilities, was developed to study film conformality as a function of
aspect ratio, but the details of the mathematical model were omitted [85].
Following interest in feature-scale investigations, a dynamic reactor model for
Al2O3 ALD processing combined precursor flow modeling through the reactor and
a ballistic flux model for precursor transport and reaction in substrate trenches, at-
tributing the reactive sticking coefficient to quantum-chemical simulations of surface
13
reactions, although the connection between the two was not made clear [125]. A
similar model was later developed for a multi-wafer batch reactor where the stick-
ing coefficient for the HfO2 process was deduced from experiments in a single-wafer
reactor [86].
In more recent work, a transport/reaction model for a tubular ALD reactor
was developed [170] where the reactive sticking coefficient is based on an earlier
Monte Carlo simulation in high-aspect-ratio nanopores [32]. The Monte Carlo sim-
ulation therein utilizes ALD surface reaction rates assumed proportional to a re-
active sticking coefficient based on an empirically-determined, first-order reaction
rate coefficient from [26]. The Monte Carlo simulation work in [84] gives an excel-
lent overview of this modeling approach, listing a range of sticking coefficients for
high-aspect-ratio systems.
As an empirical alternative to the reactive sticking coefficient, a rigorous dy-
namic parameter estimation problem was formulated to resolve Arrhenius rate ex-
pressions based on ex situ film thickness measurements for ZnO ALD as part of
a fluid dynamics model [68?72]. Characteristic of the ALD reaction mechanism,
high correlations were found between Arrhenius parameter estimations for the reac-
tions considered. Improvement in precision of individual parameter estimates was
achieved after reparameterization of the Arrhenius equation to effectively decrease
these correlations.
14
1.4.2 First Principles Methods
An accurate and arguably more flexible alternative to the empirical kinet-
ics approach exists: reaction kinetics formulations for each elementary ALD reac-
tion step based on first principles (FP). First principles (or ab initio) methods are
distinguished by requiring no empirical fitting parameters, allowing for genuinely
novel situations to be investigated. Such methodologies may involve deriving rate
constants from quantum-chemical and/or quantum-statistical computations coupled
with a theory of reaction rate expressions. While these theories have been long es-
tablished [41, 45, 88, 89], few results have been reported in direct computation of
ALD reaction rates using ab initio methods, perhaps due to the perceived costly
computation of necessary input parameters [36].
Perhaps the earliest reported formulation of ab initio ALD kinetics is seen in
a transient plug-flow reactor model developed using the Rice-Ramsperger-Kassel-
Marcus (RRKM) theory to derive rate constants for elementary ZrO2 and HfO2
reactions [24]. Because of the chemical detail contained in the model, empirical
observations of temperature-induced growth rate reduction was able to be explained:
an effect of surface dehydroxylation with increased temperature.
Later mechanistic studies which use kinetic parameters derived from transition
state theory have been able to predict reaction temperature and pressure dependen-
cies of Al2O3 ALD reactions to selectively decorate and passivate the zigzag and
armchair edges of graphene nanoribbons [168]. A similar approach was also used in
the theoretical study of hafnium aluminates [113]. Through the use of an ab initio
15
kinetics model, enhanced and inhibited chemisorption was found to be due to the
dehydroxylation behavior of the chemical precursor species.
Physically-based surface reaction models using statistical mechanics and tran-
sition state theory were also developed for Al2O3 in [156, 158, 167] and applied to
a dynamic reactor-scale model [157], allowing for the analysis of substrate effects,
identification of self-saturating growth regions in temperature and pressure, kinetic
comparison of growth mechanisms, and discernment of reactor design limits.
1.5 Selection of a Kinetics Model
All things considered, the detailed surface chemistry of ALD reactions is indeed
ignored in most feature-scale and multiscale model formulations [38]. Yet interest-
ingly, despite wide use of kinetics models based on the reactive sticking coefficient,
there still remains little consistency in reported values and no general agreement as
to the effects lumped into this parameter which is typically restricted s ? [0, 1], but
may vary by orders of magnitude, even when precursor systems are identical [84].
Moreover, the experimental and computational manner in which it is determined
varies, such that absolute values are rarely reported in the literature [142]. Thus,
from the existing body of work, one can see that use of the sticking coefficients
approach for ALD kinetics modeling remains problematic.
To avoid the ambiguity of empirical modeling approaches, a more consistent
reaction kinetics model is desired. Selection of an appropriate model must be made
with the understanding that, by essence, a model is an imitation of reality at a given
16
level of theory and a balance must be found between system size and computational
cost, particularly in multiscale systems such as those intrinsically necessitated by
ALD processes. Specifically, one must choose appropriate levels of theory for atomic-
scale surface reaction dynamics, nano- to micro-scale film growth dynamics, and
macro-scale chemical species transport.
Selection of model hierarchy (e.g. levels of detail and theory, characteristic
sizes, and characteristic times) for a process system should be guided by the pre-
determined modeling objectives and application [58]. Because of their proven accu-
racy, ab initio methods (e.g. density functional theory) are currently the methods of
choice for predicting atomic structures and reaction energetics for ALD processes of
today and the future [38]. Thus it naturally follows to consider an ab initio method
to model reaction kinetics, provided it aligns with the overall modeling objectives.
1.5.1 Modeling Objectives
Because the rate of deposition for ALD processes clearly depends on the satu-
rating coverage, it is appealing to develop an analytical model of surface coverage and
film growth. Early transport/reaction studies reveal that growth is not usually lim-
ited by precursor transport, but rather by surface reaction kinetics, confirming the
importance of ALD chemistry [46?48]. Indeed, surface reactions may remain short
of saturation and fall short of self-limiting growth because of slow kinetics [140]. An
early kinetic Monte Carlo simulation, likewise, affirms that appropriate ALD model
descriptions should involve atomistic concepts and a more detailed formulation of
the mechanism in terms of surface coverage and elementary reactions [24].
17
Thus, a fundamental understanding of the chemistry and kinetics of film
growth is of primary interest and importance in order to find the best condi-
tions for obtaining high-quality films. Such knowledge drives ALD process develop-
ment, through quantitative predictions of growth rate, temperature dependence and
pulse/purge durations, and assists in precursor selection and design of processes for
new materials. This leads to the first modeling objective:
First principles understanding of ALD reaction
mechanisms and growth kinetics
In typical ALD processes, many factors come into play that can lead to in-
complete surface saturation and consequently non-ideal film growth. Therefore,
determining the set of operating conditions corresponding to self-saturating growth
has practical consequences because of the potentially desirable reactor operating
conditions found at these limits [140]. A detailed reaction kinetics model should be
able capture these effects. Thus, the second modeling objective naturally follows:
Identification of ALD operating limits
for self-saturating growth
Many properties of interest in experiment (e.g. film growth, conformality, and
uniformity) are only accessible in theory via multiscale simulation. A multiscale
model that successfully combines chemical species transport with ab initio derived
reaction kinetics yields insights that would simplify the design of new precursors
and the optimization of ALD processes. First principles results have turned out to
be most powerful when they can add atomic-scale explanation to experimental data,
18
predict outcomes, and contribute to innovations in ALD processing [36, 38]. This
leads to the third and final modeling objective:
Explanation and prediction of multiscale
ALD reaction phenomena
Returning to the model selection criteria suggested at the beginning of this
section, it is now possible to state that a multiscale model consisting of an ab initio
surface reaction kinetics and film growth model coupled with a distributed reactor-
scale model would be appropriate in meeting the above stated objectives. Such a
predictive model has the potential to add significantly to the understanding and
development of ALD processes.
1.6 Outline of Dissertation
Despite the recent increase in ALD research, there still remains much to be
resolved in determining the conditions for the desired self-limiting ALD growth, a
condition ultimately dependent on ALD chemistry. Therefore, the motivation of this
thesis is to develop a generally applicable modeling procedure that makes use of the
adsorption and reaction energetics and transition-state configuration data from ab
initio studies in computing surface reaction rate coefficients.
Using these rate coefficients in the context of Al2O3 ALD, a surface reaction
kinetics model will be developed to predict the dynamic changes in surface species
concentration as a function of time during each exposure period, introducing the
concept of using limit-cycle solutions to examine film growth as a function of process
19
Precursor delivery 
system dynamicsPrecursor source thermodynamics
Reactor-scale transport dynamics 
Surface reaction kinetics
Growth surface state dynamics
Figure 1.4: Elements of a complete ALD reactor system mathematical model. Chap-
ters 2-4 of this thesis will specifically address the surface reaction kinetics and growth
surface state dynamics model components. Chapters 5 and 6 will introduce model
components describing the precursor source thermodynamics, precursor delivery sys-
tem dynamics, and reactor-scale transport dynamics. Chapters 5 and 6 will also in-
corporate the model components of Chapters 2-4 to make up the complete multiscale
ALD model.
operating conditions. Finally, this model will be coupled to a dynamic reactor-scale
species transport model (Figure 1.4) based on a laboratory-scale reactor system
currently under testing. The primary contributions of this thesis are its use of
physically-based reaction kinetics models derived from transition state theory [156,
158] and limit-cycle computations describing the steady operation of these reactor
systems across the multiscale domain [157].
In Chapter 2, a review of the surface reaction mechanism suggested for ox-
ide ALD processes is given in the context of transition state theory. The majority
of ALD kinetics studies are based on empirical parameters and thus are limited
in novel ALD chemistries and reactor designs. An alternative surface reaction ki-
netics methodology is suggested whereby statistical formulations of thermodynamic
quantities are used to make ab initio rate predictions for ALD processes.
In Chapter 3, principles of the surface kinetics model are applied to the Al2O3
20
ALD system using trimethylaluminum and water precursor gases. Molecular parti-
tion functions are derived for chemically reacting species, the rates of each elemen-
tary reaction step are computed, and a limit-cycle film growth model is presented
for the homodeposition ALD phase. Based on the surface reactions and film growth
model, an ALD growth per cycle is predicted for under-saturating conditions leading
to the identification of process parameters (namely temperature and exposure level)
corresponding to the ideal self-limiting growth regime.
In Chapter 4, the surface reaction kinetics model is further developed to ana-
lyze a network of elementary surface reactions which are probable in the context of
Al2O3 ALD. Representative rates are determined based on the reported energetics
of several quantum-chemical computational studies and a kinetic ALD mechanism
is proposed. A dynamic growth model is presented, giving insight into the relative
contributions of competing ALD surface reactions as well as dissociative adsorp-
tion reactions as a function of varying surface hydroxyl group and methyl group
concentrations.
In Chapter 5, the surface reaction kinetics model is coupled to a dynamic
lumped parameter reactor model for diagnostic design of a laboratory-scale cross-
flow ALD reactor. Based on the reactor design, a precursor delivery model is pre-
sented to effectively model the precursor dosing sequences throughout the ALD cy-
cle. Again, limit-cycle model computations are employed to investigate the steady
operation phase of the ALD process and optimization of film growth rate is assessed
in the context of the reactor design.
In Chapter 6, the surface reaction kinetics model is coupled to a distributed
21
reactor model to investigate the spatial distribution of film growth and provide
insight into the effects of the spatially and temporally distributed exposure profile
in the tubular reactor. These effects are evaluated in the context of distributed ALD
film rate growth predictions and film uniformity.
Chapter 7 provides the conclusion to this thesis.
22
2Surface Reaction Mechanisms and
Kinetics
The success of ALD ultimately rests on chemistry. As such, ALD process de-
velopment undeniably depends on a progressive knowledge of the underlying mech-
anisms of reaction, with the aim to identify elementary chemical reactions that
promote ideal film growth as well as those that lead to undesired by-products or
impurities [38,140].
This fundamental knowledge of ALD process chemistry should enable us to
answer critical questions about a particular system, namely, What are the mecha-
nisms responsible for self-saturating surface coverage during a precursor pulse? and
By what mechanisms are these surface species consumed during the subsequent pre-
cursor pulse? Quantitative answers to these questions can provide vital information
to be used for process optimization models and for the design of new ALD processes
and reactor systems [38].
To date, much of ALD research has been motivated by the semiconductor
processing industry [73]. As such, in terms of subject matter and popularity, ALD
23
simulations follow the same trend as ALD experiments, with most recent work
devoted to the study of depositing oxides and nitrides, particularly high-permittivity
(high-?) dielectrics such as Al2O3, HfO2, and ZrO2 [37]. A predominant topic of
published atomic-scale simulations is the study of chemical reaction mechanisms,
specifically the elementary reaction steps of the ALD process which lead to film
growth.
2.1 General Mechanism of Oxide ALD
The overall reaction to deposit metal oxide film MOx/2 (e.g. M = Al, Hf,
Zr, Ti) using water as an oxygen source can be described by its stoichiometrically
equivalent CVD reaction
MLx ?g?+
x
2
H2O ?g? ? MOx/2 ?s?+ xHL ?g? (2.1)
where the number of ligands (e.g. L = CH3, Cl) x on metal precursor MLx deter-
mines the molar ratio of the precursor gases and by-product HL [36,136]. In the case
of ALD, this reaction is generally separated into two half-reactions corresponding
to the sequential precursor pulse periods and may therefore be written as
MLx ?g?+ y?OH ?s? ? ?O?yMLx?y ?s?+ yHL ?g? (2.2)
H2O ?g?+ ?MLx?y ?s? ? ?MLx?y?1?OH ?s?+ HL ?g? (2.3)
24
O O O O O O O O
MM
M
M
M
Pulse A
M
O
M
O
M
O
M
O
M
O
M
O
M
O
M
O
OO OO
O
Pulse B
O
M
O
O
M
O
O
M
O
O
M
O
O
M
O
O
M
O
O
M
O
O
M
O
Purge
M
O
M
O
M
O
M
O
M
O
M
O
M
O
M
O
Purge
Figure 2.1: An idealized view of the MOx/2 ALD process cycle with metal precursor
?
M and oxygen precursor
?
O.
where the double bars (?) are used to denote a surface-bound species and ligands
eliminated during the metal adsorption reaction impact the succeeding oxygen ad-
sorption reaction such that y ? [1, x).
A simplified view of this ALD binary reaction sequence is schematically il-
lustrated in Fig. 2.1. Generally, during pulse A, the metal precursor adsorbs onto
a hydroxylated substrate, undergoing a ligand-exchange reaction with the surface-
bound oxygen species and, thereby, becoming permanently bound to the growth
surface. After a sufficient purge period leaving no reactive species in the gas phase,
the oxygen precursor is introduced during pulse B. This initiates a subsequent re-
action, which proceeds analogously to the reaction in pulse A, whereby the oxygen
precursor adsorbs onto the surface, undergoes a ligand-exchange reaction with the
surface-bound metal species and, consequently, becomes permanently bound to the
film. This step is also followed by a purge [152].
Half-reactions (2.2) and (2.3) may be decomposed into a number of elementary
reaction steps: molecular adsorption of the precursor, dissociation of the molecule
onto neighboring surface sites, surface diffusion of hydrogen [50] and association
into a by-product molecule which subsequently desorbs [36]. These corresponding
25
Table 2.1: Elementary reactions for the ALD of MOx/2 by metal precursor MLx
and H2O as in reaction (2.1). Surface species are denoted with double bars (?) and
y ? [1, x). Gas-phase reactions are non-ideal in ALD and are thus excluded. Based
on a similar table in [36].
Molecular desorption Molecular adsorption
A. MLx + ?OH 
 ?OH?MLx
MLx + ?O 
 ?O?MLx
B. H2O + ?MLy 
 ?MLy?OH2
H2O + ?M 
 ?M?OH2
C. HL 
 ?HL
Diffusion of hydrogen and
association into molecule
Dissociation of molecule
and diffusion of hydrogen
D. ?O?MLx + ?M 
 ?O?MLx?1 + ?ML
?O?MLy + ?M 
 ?O?MLy?1 + ?ML
E. ?M?OH2 + ?O 
 ?M?OH + ?OH
?OH + ?O 
 ?O + ?OH
F. ?OH?MLy?1 + ?HL 
 ?OH2 + ?MLy
?O?MLx?1 + ?HL 
 ?OH + ?MLx
?O?MLy?1 + ?HL 
 ?OH + ?MLy
Porosification Densification
G. ?MLy + ?OH 
 ?MLy?OH
?MOx/2 ?surf? 
 ?MOx/2 ?bulk?
elementary steps for reaction (2.1) are summarized in Table 2.1. Based on the results
of experimental data and quantum-chemical computations, this complex scheme of
chemical reactions leads to the formation of a stoichiometric MOx/2 submonolayer.
Adsorption/desorption (reactions A-C) is subject to variability due to depen-
dence on surface coverage and partial pressures which, for precursors, are strategi-
cally controlled by pulsing sequences throughout the ALD cycle. These reactions are
thus not restricted to a particular ALD period, although ideally precursor partial
pressures and pulse/purge lengths would be controlled such that reaction A occurs
during the metal precursor pulse which is followed by a sufficiently long purge so
26
as to completely remove any unreacted precursor, thereby impeding further hetero-
geneous reactions. Likewise, reaction B would ideally occur during the water pulse
which is followed by a sufficiently long purge so as to completely remove any un-
reacted water. By contrast, dissociation/association (reactions D-F) and densifica-
tion/porosification (reaction G) are independent of pressure, but instead dependent
on surface coverage and kinetics. Therefore, theoretically these reactions are also
not restricted to a particular ALD period and may contribute to growth throughout
the ALD cycle, subject of course to the availability of necessary surface species.
For most oxides, the individual reaction steps may be classified as either Lewis
acid-base (reactions A, B, D, F) by transfer of electrons or Br?nsted acid-base (reac-
tions E, F, G) by transfer of hydrogen. These elementary steps may also be grouped
into reaction sequences, such as dissociative adsorption (Afwd+Dfwd) or what is
commonly referred to in literature as ?ligand exchange? (Afwd+ Frev+ Crev) during
metal precursor exposure and similarly dissociative adsorption (Bfwd+Efwd) or lig-
and exchange (Bfwd+Frev+Crev) during water exposure. Indeed, most oxide ALD
reaction sequences commonly suggested may be expressed as some combination of
these elementary reaction steps [36].
2.2 Review of Oxide ALD Processes
Chemistry is a fundamental key to the success of ALD processes. The unique
characteristics of ALD can be achieved and benefited from only when precursors
provide self-limiting growth through saturative surface reactions while minimizing
27
competing non-growth side reactions. Numerous studies have attempted to explain
the ALD mechanism using an array of in situ and ex situ surface characterization
techniques (e.g. spectroscopy and microscopy methods), but there is yet much to
be understood [44]. To examine mechanisms in more detail, many researchers now
turn to quantum-chemical computational tools based on DFT [38].
Together, empirical and simulation studies provide evidence that oxide ALD
processes generally follow a similar reaction path [37]. Molecular chemisorption
of the precursor is the first stage in the chemical reactions. Upon adsorption, a
stable Lewis acid-base adduct is formed through a covalent interaction between
an empty orbital of the metal atom and a lone pair on the oxygen atom. After
adduct formation, the precursor may either desorb or undergo a ligand exchange
(also termed ?ligand elimination?) reaction, in which case it becomes incorporated
into the bulk film. Al2O3 [23,37,39,40,65,100,129,130,130,162,167], HfO2 [24,107?
109, 164], and ZrO2 [24, 59, 163, 165] are common examples of ALD systems known
to follow this adsorption/reaction path.
One of the earliest mechanisms to be computed with DFT was that of the pro-
totypical Al2O3 system grown by Al(CH3)3 and H2O. Computations yield similar
energy profiles for both precursors [162,167] and dissociative adsorption of Al(CH3)3
onto bridged oxygen sites for low surface hydroxylation [40]. For the various path-
ways investigated, hydrogen transfer was found to occur most readily from a neigh-
boring hydroxyl group during water half-reactions, increasing the Al?O coordina-
tion number and local density of the growing film [23].
Similar DFT study of the HfO2 mechanism (HfCl4 and H2O precursors) yields a
28
comparable energy profile for molecular adsorption and metal precursor dissociative
adsorption onto bridged oxygen sites [164]. In this study, large entropic differences
between the reactants and adduct were found to cause high desorption rates which
only increased with temperature. Since precursor desorption inhibits film growth,
this temperature effect is of obvious importance. It was reasoned that a higher
temperature causes loss of surface hydroxyl groups which in term decreases precursor
reactivity on the surface, a finding which has been corroborated [108]. ZrO2 (ZrCl4
and H2O precursors) has been found to follow a nearly identical ALD reaction path
with the same competitive desorption effects [59,163,165].
Collectively, these studies outline a critical role for ligand exchange in the
mechanism of oxide deposition. Because ALD depends on self-limiting reactions in
the metal precursor pulse, it can be shown that the growth rate depends linearly
on the degree to which ligands are eliminated during this phase [35]. Also impor-
tant to film growth are densification reactions in which oxygen and hydroxyl groups
facilitate incorporation of highly-coordinated metal atoms into the bulk film [116].
Considering that the ALD product is a dense solid film, precursor adsorption and
film growth must involve coordination increase of the constituent atoms. While
many of the studies cited herein suggest reaction mechanisms consistent with in-
creases in coordination number, the importance of densification as an element of
ALD is not yet widely recognized [37].
All things considered, there exists a relatively decent library of thermodynamic
data for oxide ALD processes. Several DFT studies have produced estimates of
reaction energetics, and indeed much of what is understood about the detailed oxide
29
reaction mechanism is based on these data. Yet, it is commonly stated that ALD
is driven by kinetics, rather than thermodynamics [37]. While one could argue
that certain reaction steps are reversible and thermodynamically controlled, the
competition between slower side reactions and faster growth reactions is crucial
for successful ALD. Kinetic rates determine the competition between reactions and
hence the time needed for the surface to become saturated and its chemical makeup.
However, far too little attention has been paid to rigorously predicting kinetics for
such reactions.
2.3 Transition State Theory
Transition state theory (TST)1 has been previously applied in only a few ALD
studies [24,113,167,168]. It has been applied more recently as a first principles kinet-
ics approach in complete ALD surface reaction models [156, 158] and incorporated
into a reactor-scale model for deposition of Al2O3 films [157]. Latest model develop-
ments incorporate the explicit chemical kinetics derived from TST into a distributed
transport/reaction model for process optimization and analysis of multiscale process
interactions [159].
Recall that the overall ALD reaction of Eqn. (2.1) may be decomposed into
a series of elementary reaction steps as outlined in Table 2.1. In each elementary
reaction, TST asserts the existence of an activated complex, or transition state,
which prevails as an intermediate species along the path from reactants to products.
1Transition state theory is also referred to as ?absolute rate theory,? ?absolute reaction rate
theory,? ?the theory of absolute reaction rates,? and ?activated complex theory.?
30
Formed by the union of two or more reactant molecules, this transition state is
assumed to be in equilibrium with the reactants [88,89]; e.g. in the generic reaction
A + B ? P, the theory states
A + B
K?

 (AB)?
v
? P (2.4)
In order for the products P to be formed, the transition state species (AB)?
must cross a potential energy barrier; the height of this barrier gives the reaction
activation energy. The rate of the forward reaction is thus governed by the con-
centration of the activated complex and the frequency of its decomposition along
the reaction coordinate. While the concentration of the activated complex can-
not be readily measured, the chemical equilibrium relationship based on molecular
concentration can be stated as
K? =
C(AB)?
CACB
= V exp
(
??G?
kBT
)
(2.5)
where the concentrations are expressed in units of molecules per volume for gas-
phase species and V is the volume of the gas phase. For surface-bound species,
the concentrations would be per unit area and the equilibrium constant for purely
surface reactions (i.e. not adsorption or desorption) would have units of area per
molecule.
31
As described by TST, the reaction rate v may therefore be stated as
v = ?C(AB)? (2.6)
= ?K?CACB (2.7)
= ? exp
(
??G?
kBT
)
V CACB (2.8)
= ? exp
(
?S?
kB
)
exp
(
??H?
kBT
)
V CACB (2.9)
where ? = kBT/h is the standard decomposition frequency, ?G? is the Gibbs free-
energy difference between the transition state and reactants, ?S? is the entropy
difference, ?H? is the enthalpy difference, kB is Boltzmann?s constant, T is absolute
temperature, and h is Planck?s constant [45].
2.3.1 Statistical Mechanics
The chemical equilibrium constant can also be derived from the interrela-
tionship between statistical and classical thermodynamics in the form of a system
partition function Z? which is comprised of molecular partition functions Z [143].
The molecular partition function gives a measure of how molecules are distributed
(partitioned) among energy states. It gives an indication of the average number of
states that are accessible at a particular temperature for noninteracting molecules.
These molecular partition functions typically include translational, rotational, vi-
brational, and electronic energy contributions which collectively define the chemical
species.
The statistical formulation of Helmholtz free energy for a system of nonin-
32
teracting molecules A = ?kBT ln Z? is related to the classical description of Gibbs
free energy by G = A + PV where P and V are the system pressure and volume,
respectively. For a gas of N independent molecules, PV can be replaced by NkBT
and thus
G = ?kBT ln Z? +NkBT (2.10)
where Z? = ZN/N ! corresponds to the total system partition function (all molecules)
and Z is the molecular partition function for each individual molecule. Substituting
this into Eqn. (2.10) and applying Stirling?s approximation (lnN ! = N lnN ?N),
G = ?NkBT ln
(
Z
N
)
(2.11)
which of course reduces to G = ?kBT lnZ for a single particle. Thus for the TST
reaction (2.4),
?
?G?
kBT
= ln
Z(AB)?
ZAZB
(2.12)
Applying this to the rate equation for reaction (2.4) and extracting the zero-point
energy contribution from the molecular partition functions, we have
v =
kBT
h
z(AB)?
zAzB
exp
(
?E?
kBT
)
CACB (2.13)
where E? is the activation energy for the reaction and z is the molecular partition
function per unit volume of the gas undergoing adsorption or per unit area of the
adsorption site, i.e. z = Z/(area or volume). Note that the activation energy is the
33
zero-point vibrational energy difference between the reactants and the transition
state.
It is reasonable to assume that E? ?= ?H? [41], and so comparison of Eqn.
(2.9) with Eqn. (2.13) implies that the partition function ratio in the rate expression
is essentially a quantitative description of the entropy difference between the tran-
sition state and reactants [45]. Hence, consideration of the translational, rotational,
and vibrational degrees of freedom associated with the atoms of each reactant and
intermediate species allows for computation of the molecular partition functions and
consequently reaction rate constants from first principles.
In theory, statistical thermodynamics may be used in computation of any
reaction rate, provided the availability of reaction energetics and relative molecular
structures of the participating species. When the activation energies are also derived
from first principles, a complete ab initio rate expression emerges. In the context
of ALD, rates for a series of heterogeneous (adsorption) and homogeneous (surface)
elementary reaction steps must be resolved. Drawing upon the existing body of
ab initio modeling work for reaction energetics and molecular structures, ab initio
reaction rate expressions are possible when statistical mechanics are applied with
TST principles. This reaction kinetics modeling approach and its implication on
the characteristic interdependence of ALD reactions will be demonstrated with a
general reaction scheme typical to ALD growth. An adsorption/ligand-exchange
reaction scheme is demonstrated in Fig. 2.2a with an association reaction scheme in
Fig. 2.2b, both of which result in bulk film production and a gas-phase by-product.
34
? + P
(A?)?
A?
A + ?
?Hads ?H? ?Hrxn
Reaction Coordinate
Pot
ent
ial E
ner
gy
a)
(??)?
? + ?
?H?
?Hrxn
? + P
b)
Reaction Coordinate
Pot
ent
ial E
ner
gy
Figure 2.2: Schematic reaction path profiles for generic elementary ALD reaction
steps. Gas-phase and surface species are noted with Latin and Greek alphabet, re-
spectively. a) Adsorption/ligand-exchange reaction. b) Association reaction.
2.3.2 Molecular Chemisorption
Table 2.1, Afwd or Bfwd
Before application of TST, the adsorption/ligand-exchange ALD reaction of
Fig. 2.2a
A + ?
Kads

 A?
K?

 (A?)?
v
? ?+ P (2.14)
must be separated into elementary steps where A and ? represent a gas-phase pre-
cursor and adsorption site, respectively. Consider first the barrierless adsorption of
gas-phase species A onto surface site ? and formation of stable surface adduct A?
A + ?
Kads

 A? (2.15)
35
This equilibrium relationship can be expressed as
Kads =
CA?
CAC??
=
ZA?
(ZA/V )Z?
exp
(
?Eads
kBT
)
(2.16)
where V is the gas-phase volume, Eads is the negative adsorption energy [88], and
C?? = C? ? CA? is the effective concentration of adsorption sites. Solving for CA?
and considering all thermodynamic quantities to be on a per-molecule basis, the
Langmuir isotherm is recovered.
CA? =
KadsCA
1 +KadsCA
C? =
KadsPA
kBT +KadsPA
C? (2.17)
2.3.3 Ligand Exchange
Table 2.1, Frev + Crev
After adsorption, the surface-bound A? species may undergo a forward ligand-
exchange reaction
A?
K?

 (A?)?
v
? ?+ P (2.18)
where ? is a surface product of the bulk film and P is a gas-phase by-product. With
E? activation energy, the above equilibrium may be expressed as
K? =
C?A?
CA?
=
Z(A?)?
ZA?
exp
(
?E?
kBT
)
(2.19)
36
Following Eqns. (2.16) and (2.19), the forward reaction rate is derived.
v =
kBT
h
C(A?)? (2.20)
=
kBT
h
K?CA? (2.21)
=
kBT
h
K?
KadsPA
kBT +KadsPA
C? (2.22)
Substituting in all ab initio quantities, the rate of the ligand-exchange reaction is
found to be
v =
kBT
h
Z(A?)?
(ZA/V )Z?
e
[
?
(
Eads + E
?
)
/kBT
]
kBT +
ZA?
(ZA/V )Z?
e(?Eads/kBT )PA
PAC? (2.23)
Unsurprisingly, the final rate expression includes partition functions from the reac-
tants, stable adduct, and transition state. This results from the double equilibrium
relationship of Eqn. (2.14). Following this conclusion, additional elementary reac-
tion rates can be derived.
2.3.4 Precursor Dissociation
Table 2.1, Dfwd or Efwd
Consider the ALD reaction where surface-bound precursor molecule ? loses a
ligand to neighboring adsorption site ?.
? + ?
Kdis

 ??L + ?+L (2.24)
The (?L) and (+L) superscripts respectively represent the loss and gain of a ligand.
37
Because the elementary dissociation step is a reversible process, no forward reaction
rate is considered. However, the equilibrium relationship is established as
Kdis =
C??LC?+L
C?C?
=
Z??LZ?+L
Z?Z?
exp
(
?Edis
kBT
)
(2.25)
2.3.5 Hydrogen Association
Table 2.1, Frev + Crev
Consider now the ALD association reaction of Fig. 2.2b
? + ?
K?

 (??)?
v
? ?+ P (2.26)
where ? and ? both represent available adsorption sites of different kinds. Following
similar derivation as shown in previous reactions, the equilibrium constant can be
expressed as
K? =
C(??)?
C?C?
=
Z(??)?
(Z?/S)Z?
exp
(
?E?
kBT
)
(2.27)
where S is the reactive surface area. The forward reaction rate is thus
v =
kBT
h
Z(??)?
(Z?/S)Z?
exp
(
?E?
kBT
)
C?C? (2.28)
38
2.3.6 Densification
Table 2.1, Gfwd
Lastly, densification can be represented by the reaction
? + ?
Kden

 ?? (2.29)
where surface species ? and ? form a coordinative bond. A similarity may be noted
with the previously described association reaction. While the association reaction
includes a densification step in formation of the transition state, ultimately two
ligands are removed from the surface in the forward reaction. By contrast, a pure
densification reaction restructures bonds of the surface and bulk film, but does not
yield a by-product. For this reason, it may also be referred to as a ?reorganization?
reaction.
Densification is important to ALD because it increases the coordination num-
ber of each reactant, leading to bulk film growth with equilibrium constant
Kden =
C??
C?C?
=
Z??
(Z?/S)Z?
exp
(
?Eden
kBT
)
(2.30)
where S is again the reactive surface area.
39
2.4 Concluding Remarks
A large and growing body of literature has investigated the mechanism of oxide
ALD processes, both empirically and theoretically. There is a consensus among
researchers that these processes follow a largely similar reaction path which may be
decomposed into a series of interdependent elementary reactions. Modeling research
to date has tended to focus on computing ab intio based thermodynamics for these
reaction steps, rather than kinetics. As a result, most growth models poorly predict
ALD outcomes except for the use of empirical fitting parameters.
In addition to being highly variable with limited applicability across reactor
systems, empirical methods are also limited by their inability to make predictions
for novel ALD processes. By contrast, ab initio simulations yield more flexibility
and also give a more detailed description of reaction chemistry. Indeed, much of the
chemical reaction mechanism described herein stems from experimentally-validated
ab initio simulations. The predictive power of such models has promising implica-
tions for the future direction of ALD applications.
An ab initio reaction kinetics method was thus suggested through the use of
transition state theory and statistical mechanics. This method was applied to several
representative elementary reactions, combinations of which are known to describe
real oxide ALD processes. In the next chapter, these mechanisms and reaction
kinetics formulations will be used to derive equilibrium and rate expressions for the
Al2O3 deposition system.
40
3Alumina Case Study
Exhibiting a relatively well-behaved growth cycle, alumina ALD grown from
trimethylaluminum (TMA) and water precursors is the prototypical experimental
and modeling system. While TMA is a pyrophoric material and thus requires special
care in handling, this high reactivity is a key factor in the success of the alumina
deposition process. Additionally, liquid TMA has a sufficient vapor pressure for
use in a pressure-drawn precursor delivery system, has high thermal stability, and
the methane by-product released in reactions with water is benign. While TMA
in large quantities could become expensive, it is only needed in small quantities at
the research-scale and water is relatively inexpensive. Considering these benefits,
the alumina ALD system has been perhaps the most intensely studied [44,130] and
consequently the best ALD process developed to date [140].
Yet, despite the attention paid to the alumina ALD system, much remains
unknown regarding the specific reactions that take place on the growth surface, and
no consistent picture has yet emerged in terms of the reaction kinetics of the overall
film growth process. As a result of the significant experimental (e.g., [117]) and
41
quantum-computational (e.g., DFT reaction modeling, [40, 162]) studies, elements
of the complete deposition reaction mechanism are beginning to emerge [36].
The model-based methodology presented in Chapter 2 was thus applied ex-
clusively to a case study of sequential surface chemical reactions for the controlled
deposition of alumina films using TMA and water precursors [156]. The overall
alumina ALD reaction can be represented by its stoichiometrically equivalent CVD
reaction as follows.
2Al(CH3)3 + 3H2O ? Al2O3 + 6CH4 (3.1)
The corresponding ALD growth cycle has been the subject of several empirical
investigations [26,32,75,90,102,117,118,136,138,150,174] in which the mechanisms
for each half-cycle are suggested as follows.
Al(CH3)3 + y?OH ? ?O?yAl(CH3)3?y + yCH4 y ? [1, 3] (3.2)
H2O + ?Al(CH3)y(OH)2?y ? ?Al(CH3)y?1(OH)3?y + CH4 y ? [1, 2] (3.3)
Each alumina half-reaction is further broken down into elemental adsorption
and surface reaction steps consistent with those outlined in Table 2.1. Many of
the empirical studies cited above suggest molecular precursor adsorption followed
by a ligand-exchange reaction. Subsequent studies also pose dissociative adsorption
reactions for both TMA [23, 40, 91, 100, 130, 155] and water [35, 39, 60, 61, 75, 91,
118, 138, 155], although the relative importance of these reactions requires further
42
investigation.
Because of the near impossibility of experimentally measuring the dynamic
ALD surface state in real time [117], researchers have turned to quantum chemi-
cal computations [37] to evaluate reaction mechanisms, bond configurations, and
energetics. Included in the above cited studies are a number of first principles
models developed to determine the mechanistic reaction pathways that are thermo-
dynamically favored in ALD. The earliest of such studies reported for alumina was
performed using DFT cluster models to predict reaction energetics and transition-
state structures for adsorption and a single ligand exchange during both the TMA
and water half-reactions [162].
Both reaction paths are initiated by barrierless precursor adsorption in which
the oxygen lone-pair electrons interact with the empty p-orbital of the aluminum
atom to form a stable Lewis acid-base adduct. If able to overcome an activation
energy barrier, this adduct then proceeds to a methane-producing ligand-exchange
reaction. As this methane-elimination reaction is rate-limiting and releases a CH4
molecule to the gas-phase, it may be assumed irreversible [156]. A similar model
was later developed that allows up to two ligand-exchange reactions during TMA
exposure and goes on to predict growth rates in alumina ALD reactor simulations,
although the modeling details were omitted [146].
Ab initio studies of the TMA half-cycle were further advanced in [39,40], where
careful consideration was given to the sequence of TMA adsorption and ligand-
exchange reactions occurring on bare (i.e. dehydroxylated) and fully hydroxylated
substrates. On bare alumina, TMA was found to adsorb dissociatively onto bridged
43
oxygen sites. While film growth was predicted for bare alumina, high ALD growth
rates were only possible when surface decomposition occurred in order to sufficiently
prevent TMA desorption and maximize Al deposition. On fully hydroxylated alu-
mina, TMA was found to adsorb molecularly and proceed through a ligand exchange,
one of three possible methane-elimination reactions. This reaction is considered irre-
versible because of the removal of methane by the ALD process exhaust and because
of the large energy barrier posed by the reverse reaction. The remaining AlMe2 (Me
= CH3) fragment was found to then undergo a highly energetically-favored densi-
fication reaction with a neighboring hydroxyl group. Similar energetic favorability
for densification reactions was also later reported in [23,79,80].
Ab initio studies of the water half-cycle were likewise advanced in [22,23], where
the relative activation energies associated with the (1-2) H-transfer transition states
were compared to the (1-4) H-transfer reactions resulting from adduct interactions
with neighboring Me groups. The latter was found to be more energetically favored.
Following early mechanistic studies, a quantitative model was developed to
predict saturating growth per cycle (GPC) for a subset of binary ALD reactions
[128]. The model depends only on the stoichiometry of the metal precursor ligand-
exchange reactions and assumes saturation is due to close-packing of the metal
precursor ligands on the growth surface. This was used to theoretically validate that
y in half-reaction (3.2) can vary from unity and that steric hindrance of the methyl
groups is responsible for self-saturation during TMA exposure [129]. DFT results
later confirm this theory, finding repulsion between methyl groups as the primary
reason for weaker TMA adsorption with increasing methyl group coverage [100].
44
Expanding the scope of these studies beyond mechanistic analysis, a kinetic
Monte Carlo simulation was developed where the primary focus was the definition of
the growth-surface lattice and species accounting mechanisms [104]. Reaction tran-
sition probabilities were determined from DFT methods but the simulation results
stopped short of predicting the saturation times for TMA and water exposures.
3.1 Modeling Objective
The study presented in this thesis chapter builds directly on the body of
previously-cited quantum-chemical mechanistic studies, employing reported ener-
getics and transition state structures to compute surface reaction kinetic parame-
ters used in a model of surface coverage dynamics corresponding to alumina ALD
growth. The goal is to understand the nonlinear connection between process pa-
rameters and the boundaries of the self-limiting growth region. At a minimum, the
process parameters that must be investigated are deposition temperature, precur-
sor partial pressures, the length of time corresponding to each exposure, and the
thermochemical properties of the ALD reactions.
ALD process development is driven by first understanding the steady ALD
growth phase (termed ?homodeposition?) in which a constant growth rate is achieved
over cycles of product-on-product deposition [38]. In introducing a novel ALD reac-
tion kinetics model, it is thus practical to limit initial analyses to this steady growth
phase, providing a conceptual theoretical framework for modeling homodeposition
and introducing the concept of limit-cycle dynamic solution computations.
45
3.1.1 Formulation of Underlying Model Assumptions
A model encompassing TMA (precursor A) and water (precursor B) adsorption
and subsequent surface reactions consists of two main components: a representation
of the instantaneous state (chemical composition) of the growth surface and the
chemical reaction material balances that govern the temporal surface state evolution.
Following the generally accepted mechanism in empirical [26, 32, 75, 90, 102,
117,118,136,138,150,174] and theoretical [35,55,99,129,130,136,146,162,167] alu-
mina studies, a relatively simple ALD reaction mechanism is assumed. Inherent in
the reaction mechanism suggested are several simplifications to the alumina ALD
process and assumptions in regard to the surface coverage model:
(i) Only immobile surface species are considered, i.e. surface diffusion is neglected.
(ii) All adsorption sites of the same type are assumed to be identical.
(iii) The ALD system can be accurately represented with a Langmuir adsorption
model, i.e. lateral adsorbate-adsorbate interactions are neglected.
(iv) Only molecular adsorption is considered, i.e. dissociative adsorption is ne-
glected.
(v) The TMA ligand-exchange transition state is allowed to interact with up to
two additional hydroxyl groups, leading to a monomethylaluminum or bare
aluminum surface site.
(vi) The TMA ligand-exchange reaction occurs with the (1-2) H-transfer mecha-
46
nism and energetics proposed in [162], and as such, elementary densification
steps are neglected.
3.2 Chemical Reaction Mechanism
The sequence of elementary reaction steps considered in this model is given in
the listings of Table 3.1 for the subset of TMA half-cycle reactions and Table 3.2
for the subset of water half-cycle reactions. The reaction energies for each reaction
and the sources of those values are given in Table 3.4.
During the TMA half-cycle, TMA (species A) is reversibly adsorbed on surface
hydroxyl groups X, forming surface adduct AX (R1). The adduct AX can undergo
a (1-2) H transfer reaction (R3), passing through transition state AX? (R2). In
this reaction sequence, the rate-determining step is the final irreversible reaction
that adds -Al(CH3)2 to the growth surface and produces one methane molecule
that desorbs to the gas phase. The two equilibrium reactions define the surface
concentrations of the adduct, AX, and the transition state, AX?, between the adduct
and the final reaction products. The transition state can also interact with up to
two additional hydroxyl groups to increase ligand elimination (R4 and R5).
During the water half-cycle, water (species B) is reversibly adsorbed on dimethy-
laluminum (DMA, Sd) or monomethylaluminum (MMA, Sm), forming surface adduct
BSd (R6) or BSm (R9), respectively. In the case of the DMA adsorption site, the
adduct BSd can undergo a (1-2) H transfer reaction (R7), passing through transition
state BSd? and undergoing an irreversible ligand-exchange reaction. Similarly, the
47
BSm adduct can undergo a (1-2) H transfer by proceeding through transition state
BSm?. Both of the irreversible reactions of half-cycle B (R8 and R11) involve a 1:1
methyl to hydroxyl group loss and gain, respectively. In other words, an adsorbed
water molecule can undergo only one methane-elimination reaction. This is in con-
trast to the TMA half-cycle where an adsorbed TMA molecule may react with up
to three hydroxyl sites.
3.2.1 The Surface State
The main features of the alumina growth surface are the aluminum and oxy-
gen atoms onto which water (B) and TMA (A) can adsorb. To characterize the
growth surface, [X], [O], [S], [AX], [BS] and [Me] are noted as the hydroxyl, oxygen,
aluminum, TMA-OH adduct, water-aluminum adduct and methyl group surface
concentrations (species/nm2), respectively. Despite the amorphous nature of the
alumina film, the instantaneous state of the surface is represented in the manner
shown in Fig. 3.1, where the X,O,S checkerboard pattern corresponds to a grid of
0.295 nm ? 0.295 nm squares and the Me group radius of 0.2 nm [156].
Oxygen sites consist of surface oxygen bridges and surface hydroxyl groups,
while methyl ligands can be bound to the surface aluminum atoms. Using brackets []
to denote species (surface or gas-phase) number densities and the ?hat? (?) notation
to denote a maximum value, surface coverages are defined as follows
?X =
[X]
[X?]
, ?O =
[O]
[O?]
, ?Me =
[Me]
[M?e]
(3.4)
48
R1 AlMe3 + OH
K1

 OH
AlMe3
K1 =
[AX]
[A] ([X]? [AX])
A + X AX
R2 OH
AlMe3 K2

 O
H
Me
Al
Me
Me K2 =
[(AX)?]
[AX]
AX (AX)?
R3 O
H
Me
Al
Me
Me
v3
? O
Al
Me Me
+ CH4 v3 =
kBT
h
[(AX)?]
(AX)? Sd + O + C
R4 OH + O
H
Me
Al
Me
Me
v4
? O
Al
Me
O + 2CH4 v4 = v3?X
X + (AX)? Sm + 2O + 2C
R5 2 OH + O
H
Me
Al
Me
Me
v5
? O
Al
O O + 3CH4 v5 = v3?
2
X
2X + (AX)? S + 3O + 3C
Table 3.1: A subset of elementary surface reactions taking place during TMA expo-
sure as identified in [156]. Me = CH3. d and m subscripts denote DMA and MMA
surface species.
49
R6 H2O + O
Al
Me Me
K6

 O
Al
Me Me
OH2
K6 =
[BSd]
[B]([S?]? [BSd])
B + Sd BSd
R7 O
Al
Me Me
OH2 K7

 O
Al
MeH Me
OH
K7 =
[(BSd)
?
]
[BSd]
BSd (BSd)?
R8 O
Al
MeH Me
OH
v8
? O
Al
OH Me
+ CH4 v8 =
kBT
h
[(BSd)
?
]
(BSd)? Sm + X + C
R9 H2O + O
Al
OH Me
K9

 O
Al
OH Me
OH2
K9 =
[BSm]
[B]([S?]? [BSm])
B + Sm BSm
R10 O
Al
OH Me
OH2 K10

 O
Al
OH Me H
OH
K10 =
[(BSm)?]
[BSm]
BSm (BSm)?
R11 O
Al
OH Me H
OH v11
? O
Al
OH OH
+ CH4 v11 =
kBT
h
[(BSm)?]
(BSm)? S + X + C
Table 3.2: A subset of elementary surface reactions taking place during water expo-
sure as identified in [156]. Me = CH3. d and m subscripts denote DMA and MMA
surface species.
50
Figure 3.1: A snapshot of a 35 nm2 portion of the ALD growth surface corresponding
to ?O = 0.6 and ?Me = 0.5.
where site conservation requires [O] + [X] = [X?] or ?O + ?X = 1. The fraction of
the growth surface free of methyl groups ?Me is defined as (1? ?Me). Likewise, the
fraction of the growth surface free of hydroxyl groups ?X is defined as (1? ?X).
The limiting (maximum) concentrations of surface species are computed using
the number density of the AlO3/2 film molecular unit, a value equivalent to the
number density of Al atoms in the Al2O3 film.
%N =
?NA
mAlO3/2
molecules/cm3 (3.5)
In this definition, NA is Avogadro?s number, ? = 3.3 g/cm3 is the amorphous alu-
mina film density [53], and mAlO3/2 is the molecular mass of the film minimal molecu-
lar unit AlO3/2. Using these values, %N = 39.0 nm?3 and the alumina film monolayer
thickness
?z = (%N)
1/3 = 0.295 nm (3.6)
51
Table 3.3: Surface reaction model parameters.
Parameter Description Value
[M?e] close-packing limit, surface Me 7.2 nm?2
[O?], [X?] alumina surface max O, -OH concentration 17.2 nm?2
[S?] alumina surface max Al concentration 11.5 nm?2
P ? standard pressure 101325 Pa
rAlC Al-C bond length 1.98 ? 10?10 m
rAlO Al-O bond length 2 ? 10?10 m
? amorphous alumina density 3.3 g cm?3
%N amorphous alumina number density 39.0 nm?3
?z alumina film monolayer thickness 0.295 nm
These computations are similar to those presented in other literature [118,128]; the
above computations give [S?] = 11.5 nm?2 and [O?] = [X?] = 17.2 nm?2 as the maximum
surface Al, O, and X number concentrations. The maximum surface methyl group
concentration [M?e] = 7.2 nm?2 is set as the close packing limit [128,130].
3.2.2 Surface State Dynamics
The essential self-limiting behavior of the alumina ALD process results from
surface Me saturation ([Me] ? [M?e]) shutting down TMA adsorption. During the
TMA half-cycle, the growth surface saturates with Me groups and during the water
half-cycle, these groups are replaced with OH. Combining these effects and keeping
in mind the reaction mechanism shown in Tables 3.1 and 3.2, the material balances
over surface methyl and hydroxyl groups are given by
d[Me]
dt
= v3?Me
(
2? ?X ? ?
2
X
)
? (v8 + v11)?X (3.7)
d[X]
dt
= ?v3?Me
(
1 + ?X + ?
2
X
)
+ (v8 + v11)?X (3.8)
52
subject to initial conditions [Me](t=0) and [X](t=0) at the start of each half-cycle.
During the TMA half-cycle, the surface is depleted of available OH groups by
the series of reactions (R1-R5). Reaction (R3) yields a DMA surface fragment which
may undergo reactions (R4) and (R5), both resulting in an additional methyl group
elimination subject to the availability of surface hydroxyl groups. During the water
half-cycle, two irreversible reactions take place (R8, R11), each removing one surface
methyl group and replacing it with a hydroxyl group. One can see that the overall
effects of the irreversible reactions (R3-R5, R8, R11) on d[Me]/dt and d[X]/dt are
nearly equal in magnitude but opposite in sign.
3.3 Surface Reaction Equilibria and Reaction Rates
Following the principles of transition state theory as derived in Chapter 2, the
equilibrium and rate expressions for each elementary reaction step have been for-
mulated for the TMA and water half-cycle and can be found in Tables 3.1 and 3.2,
respectively. Recall that to solve these equations numerically, they must be formu-
lated in terms of statistical partition functions considering the relative translational,
rotational, vibrational, and electronic energy states of the molecules.
Ab initio energetics are extracted from DFT studies while ab initio compu-
tation of the entropic contributions to the equilibrium relationships are the main
focus of this modeling section. Reaction energetics used in this study are listed in
Table 3.4 while the statistical mechanical formulation of the equilibrium and rate
expressions for reactions (R1-R11) are given in Tables 3.5 and 3.6.
53
Table 3.4: Reaction rate energetics data.
Parameter Description Value (eV) Source
1 R1 adsorption energy -0.61 [162]
2 R2 activation energy 0.52 [162]
6 R6 adsorption energy -0.57 [162]
7 R7 activation energy 0.70 [162]
9 R9 adsorption energy -0.74 [162]
10 R10 activation energy 0.91 [162]
54
R
1
A
+
X
K
1


A
X
K
1
=
Z
A
X
(Z
A
/V
)Z
X
e(
?
 1
/k
B
T
)
R
2
A
X
K
2


(A
X
)?
K
2
=
Z
(A
X
)?
Z
A
X
e(
?
 2
/k
B
T
)
R
3
(A
X
)?
v 3 ?
Sd
+
C
H
4
v 3
=
k B
T h
Z
(A
X
)?
(Z
A
/V
)Z
X
e[
?
(
1
+
 2
)/
k B
T
]
k B
T
+
Z
A
X
(Z
A
/V
)Z
X
e(
?
 1
/k
B
T
) P
A
P
A
[X
]
R
4
X
+
(A
X
)?
v 4 ?
Sm
+
2C
H
4
v 4
=
k B
T h
Z
(A
X
)?
(Z
A
/V
)Z
X
e[
?
(
1
+
 2
)/
k B
T
]
k B
T
+
Z
A
X
(Z
A
/V
)Z
X
e(
?
 1
/k
B
T
) P
A
P
A
[X
]2
[X?
]
R
5
2X
+
(A
X
)?
v 5 ?
S
+
3C
H
4
v 5
=
k B
T h
Z
(A
X
)?
(Z
A
/V
)Z
X
e[
?
(
1
+
 2
)/
k B
T
]
k B
T
+
Z
A
X
(Z
A
/V
)Z
X
e(
?
 1
/k
B
T
) P
A
P
A
[X
]3
[X?
]2
T
ab
le
3.
5:
St
at
is
ti
ca
l
m
ec
ha
ni
ca
l
fo
rm
ul
at
io
n
of
eq
ui
lib
ri
um
co
ns
ta
nt
s
an
d
ra
te
ex
pr
es
si
on
s
fo
r
el
em
en
ta
ry
su
rf
ac
e
re
ac
ti
on
s
ta
ki
ng
pl
ac
e
du
ri
ng
T
M
A
ex
po
su
re
as
id
en
ti
fie
d
in
[1
56
].
M
e
=
C
H
3
.
d
an
d
m
su
bs
cr
ip
ts
de
no
te
D
M
A
an
d
M
M
A
su
rf
ac
e
sp
ec
ie
s.
55
R
6
B
+
Sd
K
6


B
Sd
K
6
=
Z
B
S
d
(Z
B
/V
)Z
S
d
e(
?
 6
/k
B
T
)
R
7
B
Sd
K
7


(B
Sd
)?
K
7
=
Z
(B
S
d
)?
Z
B
S
d
e(
?
 7
/k
B
T
)
R
8
(B
Sd
)?
v 8 ?
Sm
+
X
+
C
H
4
v 8
=
k B
T h
Z
(B
S
d
)?
(Z
B
/V
)Z
S
d
e[
?
(
6
+
 7
)/
k B
T
]
k B
T
+
Z
B
S
d
(Z
B
/V
)Z
S
d
e(
?
 6
/k
B
T
) P
B
P
B
[S?
]
R
9
B
+
Sm
K
9


B
Sm
K
9
=
Z
B
S
m
(Z
B
/V
)Z
S
m
e(
?
 9
/k
B
T
)
R
10
B
Sm
K
10


(B
Sm
)?
K
10
=
Z
(B
S
m
)?
Z
B
S
m
e(
?
 1
0
/k
B
T
)
R
11
(B
Sm
)?
v 1
1
?
X
+
C
H
4
v 1
1
=
k B
T h
Z
(B
S
m
)?
(Z
B
/V
)Z
S
m
e[
?
(
9
+
 1
0
)/
k B
T
]
k B
T
+
Z
B
S
m
(Z
B
/V
)Z
S
m
e(
?
 9
/k
B
T
) P
B
P
B
[S?
]
T
ab
le
3.
6:
St
at
is
ti
ca
l
m
ec
ha
ni
ca
l
fo
rm
ul
at
io
n
of
eq
ui
lib
ri
um
co
ns
ta
nt
s
an
d
ra
te
ex
pr
es
si
on
s
fo
r
el
em
en
ta
ry
su
rf
ac
e
re
ac
ti
on
s
ta
ki
ng
pl
ac
e
du
ri
ng
w
at
er
ex
po
su
re
as
id
en
ti
fie
d
in
[1
56
].
M
e
=
C
H
3
.
d
an
d
m
su
bs
cr
ip
ts
de
no
te
D
M
A
an
d
M
M
A
su
rf
ac
e
sp
ec
ie
s.
56
3.4 Computation of Species Partition Functions
With complete derivations of the equilibrium and rate expressions from sta-
tistical mechanics and TST, computation of molecular partition functions are the
necessary next step to numerically solve the expressions. While much of this dis-
cussion has centered around molecular interactions, it is sometimes more useful to
look at a system in terms of energy storage modes, rather than of molecules.
Energy storage modes can be readily used to define a system unless they
are truly interdependent. For example, a molecule which can store translational,
rotational, vibrational, and electronic energy will have independent energy storage
modes when the following applies.
 = trans + rot + vib + elec (3.9)
Note that this equation neglects nuclear energy which is a constant background un-
less nuclear reactions occur. To handle the complexities of polyatomic molecules and
to extract the necessary partition functions for the different energy storage modes,
approximate methods must be used based upon a key assumption that the modes
of energy storage are essentially independent. If the storage modes are truly inde-
pendent, vibrational modes do not impact rotational, translational, and electronic
modes and so forth. This assumption of energy storage mode independence is almost
always a good approximation [66]. Implications of this assumption are as follows:
(i) Molecule translation does not influence rotational or vibrational energy modes.
57
Table 3.7: Molecular partition function contributions for a nonlinear polyatomic
ideal gas. DOF, degrees of freedom. n, number of atoms in the molecule. ZPE, zero
point energy.
Mode DOF Equation
translation 3 Z trans =
(
2pimkBT
h2
)3/2
V
rotation 3 Zrot =
1
?
?
pi
(
8pi2kBT
h2
)3
IxIyIz
vibration 3n? 6 Zvib =
3n?6?
n=1
e?h?n/(2kBT )
1? e?h?n/(kBT )
(with ZPE)
electronic Zelec =
?
i
?ie
?i/(kBT ) i = 0, 1, 2, ...
(ii) Electronic energy levels do not influence rotational, vibrational, or transla-
tional energy modes.
(iii) Rotational and vibrational energy modes do not influence each other.
These last two assumptions are certainly approximations, but in many cases they
work well. The assumption that energy storage modes are independent allows for
the molecular partition function to be written as follows.
Z = Z transZrotZvibZelec (3.10)
These partition function contributions for a nonlinear polyatomic ideal gas are de-
fined in Table 3.7.
Noting the structure of the equilibrium and rate expressions derived in Ta-
bles 3.5 and 3.6, computation of the absolute molecular partition functions are
58
unnecessary, as the partition function ratios (differences in energy modes between
chemical species) are what ultimately define the kinetics of each elementary reac-
tion. Thus, the motivation for not explicitly computing the internal contributions
to the partition functions is that many of these components approximately cancel
in the equilibrium relationships they ultimately define. Considering this, the par-
tition function ratios will be derived for the equilibrium relationships in each ALD
half-cycle.
3.4.1 Half-cycle A
3.4.1.1 R1
The partition function ratio for the reaction (R1) equilibrium relationship is
developed according to the following:
(i) TMA loses all translational freedom upon adsorption.
(ii) TMA loses all but z-axis (?helicopter?) rotation upon adsorption [40], but
rotational freedom of the bare adsorption site is retained. A new and substan-
tial rotational mode is created by the angular momentum IAX of the surface
adduct [156] with IAX =
?
3/4mAr2AlO where mA is the mass of the adsorbed
TMA and rAlO is the length of the Al-O bond [40].
(iii) All vibrational modes from the gas-phase TMA and bare adsorption site are
retained. The Al-O bond made in formation of the adduct results in an ad-
ditional stretching vibrational mode with frequency ?AlO. Following [95], the
vibrational frequency of the bond formed during adsorption is approximated
59
as
?AlO =
1
2pi
?
2pi%N1rAlO
mA
= 1.1? 1012 s?1 (3.11)
which falls well within the expected range of frequency values.
(iv) The electronic energy level difference can be extracted as the zero point energy
difference and described by the adsorption energy 1 of the TMA molecule.
The partition function ratio for reaction (R1) can thus be expressed as
ZAX
(ZA/V )ZX
=
1/
(
1? e?h?AlO/kBT
)
(
2pimAkBT
h2
)3/2 ?AX
?A
?
pi
(
8pi2kBT
h2
)
Ix,AIy,A/IAX
(3.12)
3.4.1.2 R2
The partition function ratio for the reaction (R2) equilibrium relationship is
developed according to the following:
(i) While the combined rotation of the adduct with moment of inertia IAX is
retained, the z-axis TMA rotational mode and an internal rotational mode
with moment IMe is lost due to formation of the methane-elimination bond.
(ii) All vibrational modes of the adduct are retained in the transition state.
(iii) The electronic energy level difference can be extracted as the zero point energy
difference and described by the reaction activation energy 2.
60
The partition function ratio for reaction (R2) can thus be expressed as
Z?AX
ZAX
=
1
1
?Me
?
pi
(
8pi2kBT
h2
)2
Iz,AIMe
(3.13)
These two partition function ratios complete the equilibrium expressions of
half-cycle A and consequently also complete the reaction rate expressions for reac-
tions (R3-R5).
3.4.2 Half-cycle B
3.4.2.1 R6 and R9
Reactions (R6) and (R9) are structurally similar and thus can be modeled in
much the same way. The partition function ratios for these equilibrium relationships
are developed according to the following:
(i) Water loses all translational freedom upon adsorption.
(ii) Water loses all but z-axis rotation upon adsorption, but rotational freedom of
the bare adsorption site is retained. A new rotational mode is gained upon
formation of the adduct with moment of inertia IBSm,d = mBSm,dr2AlO where
mBSm,d is the mass of the respective adduct in reaction (R6) or (R9).
(iii) All vibrational modes from the gas-phase water and bare adsorption sites are
retained. A new vibrational mode with frequency ?AlO = 1/2pi
?
2pi%N6,9rAlO/mB
is gained due to formation of the Al-O bond. In this formula, 6,9 is the ad-
sorption energy of water in each respective reaction (R6 or R9).
61
(iv) The electronic energy level difference can be extracted as the zero point en-
ergy difference and described by the adsorption energy (6 or 9) of the water
molecule.
The partition function ratio for reaction (R6) can thus be expressed as
ZBSd
(ZB/V )ZSd
=
1/
(
1? e?h?AlO/kBT
)
(
2pimBkBT
h2
)3/2 ?BSd
?B
?
pi
(
8pi2kBT
h2
)
Ix,BIy,B/IBSd
(3.14)
and the partition function ratio for reaction (R9) can be likewise expressed as
ZBSm
(ZB/V )ZSm
=
1/
(
1? e?h?AlO/kBT
)
(
2pimBkBT
h2
)3/2 ?BSm
?B
?
pi
(
8pi2kBT
h2
)
Ix,BIy,B/IBSm
(3.15)
3.4.2.2 R7 and R10
Reactions (R7) and (R10) are also structurally similar and thus can be mod-
eled in much the same way. The partition function ratios for these equilibrium
relationships are developed according to the following:
(i) While the combined rotation of the adduct with moment of inertia IBSm,d is
retained, the z-axis water rotational mode and an internal rotational mode
with moment IMe is lost due to formation of the methane-elimination bond.
(ii) All vibrational modes of the adduct are retained in the transition state.
(iii) The electronic energy level difference can be extracted as the zero point energy
difference and described by the reaction activation energy (7 or 10).
62
0.0 0.5 1.0 1.5 2.0
PA ,PB , Torr
10-15
10-13
10-11
10-9
10-7
10-5
10-3
10-1
101
sur
fac
e c
ov
era
ge
, m
ole
cu
les
/nm
2
AX
AX
BSd
BSd
BSm
BSm
Figure 3.2: Predictions of the maximum adduct and transition state species surface
coverages for T = 500 K and a range of precursor partial pressures.
The partition function ratios for both reactions (R7) and (R10) can thus be expressed
as
ZBSd?
ZBSd
=
ZBSm?
ZBSm
=
1
1
?Me
?
pi
(
8pi2kBT
h2
)2
Iz,BIMe
(3.16)
These four partition function ratios complete the equilibrium expressions of
half-cycle B and consequently also complete the rate expressions for reactions (R8)
and (R11).
3.5 Representative Surface Coverages and Reaction Rates
Equipped with all elementary reaction rate expressions, the expected growth
surface coverage values for the TMA and water adduct species and their transition
states are computed and shown in Fig. 3.2. Of the model predictions, the most sig-
nificant are the small surface coverage values - that typically less than one molecule
63
0.0 0.5 1.0 1.5 2.0
PA ,PB , Torr
10-2
10-1
100
101
102
103
104
105
106
107
rat
e, 
mo
lec
ule
s/n
m2
s
wA
wB
v3
v8
v11
Figure 3.3: Predictions of the reaction rates for T = 500 K and a range of precursor
partial pressures. Gas-phase precursor wall collision frequencies, wA and wB, are
shown for reference.
occupies an adsorption site per square nanometer at any instant in time. The water
precursor adsorbs more effectively than TMA, and the adsorption step of (R9) is
favored over that of (R6), but the forward reaction rate of (R8) is faster than (R11).
Not surprisingly, the surface concentration of adducts in their transition states are
several orders of magnitude smaller than the corresponding adduct species concen-
tration. The important implication of this computation is that the adsorbed species
would be very difficult to detect experimentally, and so surface state measurements
correspond to permanently bound species [26].
While the adsorbed species concentrations are quite small, the corresponding
reaction rate values are consistent with experimental observations. In Fig. 3.3, one
can see that surface reaction rate values of >100 reactions/(nm2s) would result in
surface saturation times of less than 1 s. The reaction rate for the water/MMA
64
reaction (R11) is greater than that of (R8), and both water reaction rates are signifi-
cantly higher than that of TMA reaction (R3). All computed reaction rates, though,
are much less than the wall collision rates, suggesting that alumina ALD growth is
kinetically-limited, as opposed to transport-limited.
3.6 Limit-cycle Solutions
At this point, the surface reaction model is complete and the dynamic nature
of the growth surface can be studied as a function of surface state initial conditions
and process operating parameters. In terms of the latter, the exposure of the growth
surface to precursors A and B is characterized by the partial pressure of each pre-
cursor PA and PB and the length of each period ?A and ?B, resulting in the total
exposure values of ?A = PA?A and ?B = PB?B with units 1 Langmuir = 10?6 Torr?s.
Note that this model assumes constant precursor partial pressures over the duration
of the pulse and an instantaneous reactor purge. Of course, the nature of the ex-
posures experienced by a growth surface in a reactor would be of a time-dependent
local partial pressure during the pulse period and during a portion of each purge
period; modeling these features requires coupling the surface reaction model to a
dynamic reactor transport model. This will be demonstrated in Chapters 5 and 6.
To make possible a systematic approach to studying the deposition dynamics,
focus is placed on simulating an ABABAB... sequence of reaction steps, each cor-
responding to constant values of ?A and ?B, and computing the AB reaction cycle
that would be found under cyclic operation after reaction surface nucleation and
65
other ?unsteady? (with respect to cyclic operation) effects become unimportant.
This limit-cycle behavior thus models steady reactor operation when PA and PB are
constant during each pulse and the reactor purge is instantaneous.
3.6.1 A Collocation Procedure
Computation of the limit-cycle solutions requires a numerical procedure be-
cause of the nonlinear nature of the modeling equations (3.7, 3.8). Computing the
AB limit-cycle over the time period 0 ? t ? ?A + ?B requires one additional im-
portant criterion: that the state of the surface returns to its initial condition at the
end of the cycle t = ?A + ?B. The combination of initial conditions and the model-
ing equations (3.7, 3.8) constitute a set of ordinary differential and linear algebraic
equations; this set of equations is reduced to a larger set of nonlinear algebraic
equations using a collocation procedure, solving the algebraic equations using the
Newton-Raphson method [18].
To implement the collocation method, the modeling equations over half-cycles
A and B (3.7, 3.8) are written in vector form
dC
dt
= FA(C),
dC
dt
= FB(C) (3.17)
where the species concentrations C are subject to initial conditions for each full cycle
C(0) = C(?A + ?B) and a condition requiring C to be continuous at t = ?A. Using
a simple polynomial Lobatto collocation technique over the unit interval with nT
collocation points for each half-cycle, AnT?nT is defined to be the discrete-ordinate
66
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
time, sec
0.0
0.2
0.4
0.6
0.8
1.0
A B
Me
X
Figure 3.4: Surface CH3 and OH limit-cycle coverage dynamics for PA = PB = 1
Torr and T = 500 K. The red and blue curves denote surface Me and OH group
coverage, respectively. Dots indicate collocation point locations.
formulation of the 1st-order differentiation array and Ak
1?nT the kth row of that
array
Ci(t1) = Ci(t2nT ) Ci(tnT ) = Ci(tnT+1) (3.18)
AkCi(tA) = ?AFA (Ci(tk)) AkCi(tB) = ?BFB (Ci(tk+nT )) (3.19)
for k = 2, . . . , nT , surface species i = Me,X, and collocation points tA, tB. This is a
straightforward orthogonal polynomial collocation on finite elements discretization
procedure for forced-periodic systems.
A representative limit-cycle solution computed using the collocation procedure
is shown in Fig. 3.4 where the state of the surface during the AB exposure cycle is
plotted as a function of time. Under these conditions, the growth surface approaches,
but does not fully reach a saturated state during the TMA pulse. During the water
pulse, the surface completely saturates, replacing essentially all methyl ligands with
67
0 1 2 3 4 5 6 7
[Me], CH3 /nm2
2
3
4
5
6
7
8
9
10
11
[X]
, O
H/n
m2
R8+R11
R3+R4+R5
Figure 3.5: Surface CH3 and OH limit-cycle coverage dynamics for PA = PB = 1
Torr and T = 500 K. The red and blue curves denote the TMA and water doses,
respectively. Dots indicate collocation point locations.
hydroxyl groups.
An alternate view of this limit-cycle solution is shown in Fig. 3.5 where the
changes in surface CH3 and OH groups during the TMA and water doses are shown
as the red and blue portions of the cycle, respectively. The linear nature of the water
half-cycle dynamics is attributable to the form of the ligand-exchange reaction used
in the surface material balance, where one reacting water precursor molecule results
directly in the production of one CH4 molecule desorbing to the gas phase. The
curved nature of the TMA portion of the curve is due to the slower accumulation
of CH3 groups when the growth surface has a higher density of OH groups (recall
R4 and R5), switching to a more rapid accumulation as CH3 ligands are left by the
TMA surface reactions.
68
3.7 GPC and Film Composition for Self-limiting Growth
Considering the ALD conditions which correspond to surface saturation by Me
ligands during half-cycle A followed by a one-to-one exchange of Me ligands with OH
during the water half-cycle B, the total number of Al atoms deposited from TMA
must be NAl = 2[M?e]/3 = 4.8 Al atoms/nm2 and NO = [M?e] = 7.2 atoms/nm2. Of
course, this results in a stoichiometric Al2O3 film; more importantly, the saturating
ALD growth per cycle GPC can be computed as
GPC = 10%NNAl = 1.23 A?/cycle (3.20)
essentially a perfect match to what is observed in practice for lower temperatures,
e.g., 180o C as described in [129]. This value of GPC, computed only from the film
density and the assumption of close-packed Me groups as the saturating mechanism
must be independent of temperature, pressure, and other operating conditions. The
GPC for alumina ALD is known to decrease with increasing deposition tempera-
ture [131]. This modeling work focuses on the process-dependent dynamics of the
saturation process, the temperature dependence modeling of gpc (undersaturated
growth rate), and conditions leading to GPC.
3.7.1 GPC and gpc
An important consequence of the limit-cycle solution procedure is that one can
now definitively quantify the difference between saturating ALD and non-saturating
69
film growth conditions. To make this distinction clear, gpc(PA, PB, ?A, ?B, T ) is
introduced as the film growth rate per limit-cycle with
max
PA,PB ,?A,?B ,T
gpc = GPC (3.21)
To compute gpc, recall that the number of Al and O atoms deposited per unit area
over one deposition cycle are NAl and NO atoms/nm2, respectively. These values
can be computed by integrating the reaction rates over each half-cycle
NAl =
? ?A
0
v3 dt (3.22)
NO =
? ?A+?B
?A
(v8 + v11) dt (3.23)
using the quadrature weights of the collocation procedure [18]. The gpc then is
computed in the same manner as in Eqn. (3.20). Note that all of the results presented
in this study were found to correspond to the stoichiometric NAl/NO ratio of 2/3.
3.8 A map of gpc(?A, ?B)
To illustrate the dependency of gpc as a function of both precursor exposure
levels for a fixed precursor pressure of PA = PB = 1 Torr and fixed temperature
T = 500 K, a gpc(?A, ?B) map was generated by computing the limit-cycle solutions
over a range of ?A and ?B; the results are shown in Fig. 3.6. The location of the
representative limit-cycle solution of Figs. 3.4 and 3.5 is marked as 2 on this plot,
indicating its position outside the self-limiting growth domain, in a region where the
70
12
3
Self?limiting
growth region
b
A
 (Langmuir)
b
B
 (
 L
a
n
g
m
u
ir
)
 
 
2 4 6 8 10
x 10
5
2
4
6
8
10
x 10
5 gpc (A/cycle)
0.4
0.6
0.8
1
1.2
Figure 3.6: A map of gpc as a function of precursor exposure level for PA = PB = 1
Torr. The point marked 2 corresponds to ?A = ?B = 5? 105 and to the conditions
of Figs. 3.4 and 3.5.
water exposure is non-saturating. For the purposes of this study, the self-limiting
growth region is considered as the set of process operating conditions producing a
steady film growth rate greater than a set fraction (e.g., 98%) of the saturating
GPC.
An important and completely expected feature of the gpc map in Fig. 3.6 is
the large plateau of self-limiting growth found when both precursor doses saturate
the growth surface in the limit-cycle solution. Saturation is achieved when the
product of dose pressure and exposure time is above some critical value for both of
the precursors - for this model of alumina deposition, the dark red region of Fig. 3.6
with a boundary defined by 98% of fully saturating cyclic film growth. The self-
limiting growth region in the dosage plane is bordered by a steep drop off to zero
growth in the limits of zero dose of each precursor. Note the qualitative similarities
71
between these computations and the film deposition efficiency plots of [121].
3.8.1 Comparison to Measured Exposures
Fig. 3.6 demonstrates the lower boundary for self-limiting growth consists of a
curve in the (?A, ?B) plane; for this set of precursor partial pressures and deposition
temperature, the saturating ALD conditions correspond to ?A, ?B both > 6? 105 L.
Model predictions will now be compared to TMA and water exposure levels found
in the literature.
When reviewing the alumina ALD literature in which precursor exposure levels
can be extracted, one will find that the conditions indicated can correspond to partial
or saturating doses. In some cases, exposure values can be estimated based on the
experimental conditions reported. For example, time-resolved FTIR measurements
demonstrated that the ALD surface reactions were completed after 210 s during
the TMA exposure and 180 s during the subsequent water exposure at 500 K and
0.01 Torr precursor partial pressure for both cases, giving total A/B exposures of
2.1?106 and 1.8?106 L, respectively [26]. The authors of the cited study note that
mass transport limitations introduced by the porous substrate were not taken into
account [3].
In another study, sequences of one to five 1 s pulses of D2O were used for the
water dose; given their reactor operating pressure of 2 mbar, a rough upper estimate
of the exposures necessary for saturation of the water reactions ranges from 1.5?106
to 7.5 ? 106 L [75]. These relatively high exposure values may be contrasted to
another case where a rate constant k2 = 0.004 ? 10?6 Torr?1 s?1 corresponding
72
to the slowest water reaction mode was identified from ALD experiments taking
place at 300 K and ultralow precursor partial pressures [150]. The rate constant in
the cited work is reported as k2 = 0.004 Torr?1 s?1, a value confirmed to actually
be k2 = 0.004 ? 10?6 Torr?1 s?1 [126]. In this case, one can directly compute
the exposure ?B = ?(ln 0.001)/k2 = 1700 L which would result in 99.9% reaction
saturation.
Exposures also can be extracted from time-resolved reactor process pressure
data. For example, in [118], a TMA/water dosing procedure was illustrated along
with the measured deposition chamber pressure as a function of time. After each
pulse, the chamber pressure drops in the manner of a first-order dynamical system
with the single time constant given by the ratio of chamber volume to pumping
speed. Fitting the pressure traces to a chamber pressure dynamic model, exposures
of 1.2 ? 105 and 1.7 ? 105 L are estimated for TMA and water, respectively. A
commercial ALD reactor system is also considered, where exposure estimates of
8.4?104 L for both the TMA and water pulses in a saturating-dose recipe for alumina
ALD in the Savannah 100 reactor can be computed using this procedure [144].
The data cited above give some indication as to the variability in exposures
reported in the literature for the TMA/water alumina ALD reaction system. How-
ever, the model predictions are of the correct order with respect to the majority
of reported exposure levels, with the variability attributable to the differing pre-
cursor partial pressures and reactor-specific aspects of the individual studies cited.
Model predictions will now be examined in more detail to clarify the role played by
deposition temperature on gpc and the self-limiting growth boundary location.
73
300 400 500 600 700
0.6
0.8
1
1.2
1.4
Self?limiting
growth
1
2
3
increasing water rxn
decreasing
TMA rxn rate
T (K)
gp
c 
(A
/c
yc
le
)
Figure 3.7: Effect of temperature on gpc for PA = PB = 1 Torr. Blue curve cor-
responds to ?A = ?B = 2 ? 105 L, green to ?A = ?B = 5 ? 105 L, and red to
?A = ?B = 9 ? 105 L. Dashed curve segments indicate the temperature range where
TMA decomposes [129].
3.8.2 Effect of Deposition Temperature
Recall that the ALD GPC (3.20) is not a function of temperature or any other
processing condition. Returning to the PA = PB = 1 Torr, T = 500 K gpc map
of Fig. 3.6, the three exposure-value sets of ?A, ?B will be examined. Maintaining
those three sets of conditions as constant, the corresponding gpc(T ) as a function
of temperature can be computed in Fig. 3.7 with the T = 500 K points marked.
Operating condition 1 corresponds to an under-saturating set of exposure
values for T = 500 K. One can observe that the gpc(T ) for this set of exposures
drops significantly for T < 500 K as well as for T > 600 K ? there is no self-limiting
growth region under these conditions [140]. The gpc behavior can be traced to the
relative reaction rates of TMA and water: the TMA reacts aggressively at low (e.g.,
room) temperatures compared to water. However, the water reaction rate quickly
74
climbs with temperature, only to be modulated by the TMA reaction rate which
decreases with increasing temperature. As temperature increases, the entropy of
the gas-phase precursor molecules increases significantly, making desorption more
favorable and consequently lowering the equilibrium surface coverage, thus lowering
the limit-cycle growth rate under these non-saturating conditions.
Additional insight into the initial increase in gpc(T ) with temperature can be
gained by closer examination of the water reaction thermodynamics. As listed in
Table 3.4, the desorption energies of the water reactions are smaller in magnitude
relative to the activation energies for the forward, B-cycle deposition reactions. Thus
at low temperatures the water deposition reaction is limited by its activation energy
barrier, making it thermodynamically more feasible for adsorbed water molecules
to desorb than to proceed through the surface ligand-exchange reaction. As tem-
perature increases, the activation energy of the water surface reaction is more easily
overcome and an increase in growth is to be expected. However, when temper-
ature is further increased, gas-phase water molecules become much more favored
entropically, leading to lower adsorption and reaction.
For the nominal dosage conditions 2 , the gpc(T ) peak approaches GPC and
then the curve flattens and widens into the ideal self-limiting growth region for even
higher exposures 3 . Each curve is bounded by the gpc gradients associated with
reduced water and TMA reaction rates at low and high temperatures. As stated
earlier, it is important not to confuse the temperature dependency of gpc described
here with that of GPC described elsewhere. For example, the reduction of GPC at
high temperatures generally is attributed to the reduction of surface OH groups with
75
increasing temperature [131]. While the model findings herein do not relate to the
temperature dependency of GPC, the TMA reaction rate reduction predicted by
the model might play a role in confounding the true mechanism for GPC reduction
with temperature. Consider, for example, the work of [163, 164] who find that an
increase in temperature may also cause a decrease in observed growth rate due to
increased desorption competition of the adsorbed precursor. Further clarification of
the role of temperature in both gpc and GPC is worthy of additional study.
3.9 Concluding Remarks
In this chapter, an alumina surface reaction kinetics model was derived based
on transition state theory. Energetic and configurational data were adopted from
published quantum-chemical computations and used to define partition functions
of the reaction equilibrium relationships governing adsorbed surface species concen-
trations and reaction rates. The primary goal was to derive a tractable deposition
model from which quantitative rate information could be extracted. The resulting
rate expressions were incorporated into time-dependent reaction material balance
expressions accounting for the temporal evolution of the surface states during each
exposure period. The surface reaction models were treated as true dynamic systems,
with continuous reactor operation described by limit-cycle solutions, numerically
computed using a polynomial collocation technique.
The work described in this chapter was motivated by the predictive capabilities
of physically based models to decouple the effects of precursor pressure, exposure
76
time, reactor temperature, and the dynamics of each exposure period on gpc, a
measure of film growth per limit-cycle which is defined and distinguished from the
fully saturating GPC. As such, the model demonstrated a good match between its
predictions and observed deposition behavior in experimental alumina ALD stud-
ies. In particular, the model appears to predict the correct magnitude of TMA and
water exposures required to achieve self-saturating ALD growth, with essentially no
adjustable parameters. Further refinement of the reaction kinetics model, derivation
of additional rate expressions, and derivation of surface species conservation equa-
tions to describe a more comprehensive reaction mechanism will be demonstrated
in the following chapter.
77
4Kinetic Mechanism Analysis
The alumina ALD reaction mechanism with trimethylaluminum and water
precursors has been the subject of several experimental [26, 32, 75, 90, 102, 117, 118,
136,138,150,174] and theoretical [35,55,99,129,130,136,146,162,167] studies. The
mechanism presented in the previous chapter is consistent with the generally ac-
cepted chemisorption and ligand-exchange reaction mechanism. However, as more
atomic-scale simulations arise, additional details are gained about the hydrogen
transfer path and densification mechanisms involved in elementary ALD reaction
steps. This yields a much more complex network of reactions for both ALD half-
cycles and consequently more complex surface dyamics.
For instance, several studies pose dissociative adsorption reactions for both
TMA [23,40,91,100,130,155] and water [35,39,60,61,75,91,118,138,155], reactions
which are often not captured in kinetics model formulations. Additionally, DFT
computations have suggested H transfer mechanisms for the TMA [23,40] and water
half-cycle reactions [22, 23] along with densification mechanisms [40,79,80].
In this chapter, a more comprehensive network of elementary reactions is pre-
78
sented, incorporating suggested mechanisms from several quantum-chemical com-
putational studies. Specifically, dissociative adsorption reactions of both precursors
are included and densification reactions are assumed to occur barrierlessly and thus
immediately [23]. Surface reaction rate predictions, based on thermodynamics and
suggested reaction mechanisms from alumina homodeposition [40, 60, 61, 146, 162]
and heterodeposition [23,55,74,79,80] studies, are compared. From this analysis, a
kinetically favored reaction mechanism will be suggested.
Finally, the first principles surface reaction model will be used to examine rela-
tive rates of competing surface reactions to determine which reactions are crucial to
the deposition process by determining the rate-limiting steps of the overall reaction
as the reaction proceeds in time during both precursor pulses.
H transfer paths were comprehensively analyzed for both TMA and water
half-reactions. In all of the cited studies considered in this mechanistic analysis,
(1-2) H transfer reactions were suggested as the primary mechanism of surface re-
actions during the TMA half-cycle, i.e. the first TMA ligand-exchange reaction
occurs on the surface site onto which the TMA molecule initially adsorbs. A (1-5)
H transfer reaction was also considered in [23], but the (1-2) reaction was found
to be energetically favored. The resulting dimethyl moiety can then relax to an
energetically more-favored 4-coordinate Al through a barrierless densification reac-
tion [23, 40, 79, 80]. Overall, TMA was shown to react with up to three OH sites
to form a nearly 4-coordinated Al atom complex when OH groups were available at
sufficient proximity, although the third methane-elimination reaction was found to
be unfavored energetically.
79
Barrierless dissociative TMA adsorption was found to occur on locally bare
(i.e. dehydroxylated) alumina. In this case, TMA adsorbs onto an oxygen bridge
and distributes up to two of its Me ligands amongst neighboring surface aluminum
atoms, although the second dissociation reaction requires activation [40].
In addition to the TMA half-cycle, the detailed H transfer mechanism was
investigated for the water half-cycle in [23]. Unlike the (1-2) H transfer mechanism
suggested elsewhere [55, 146, 162] the proposed water reaction involves two MMA
species. The water molecule was found to adsorb via the known Lewis acid-base
interaction with a surface aluminum, but sharing one hydrogen with an oxygen
bound to a neighboring MMA species. From this configuration, a (1-4) H transfer
reaction was strongly favored over the (1-2) reaction, the first yielding a 3-coordinate
hydroxyl group and the latter a 2-coordinate hydroxyl group.
4.1 Surface Reaction Mechanism
Consistent with the notation introduced in Chapter 3, A and B will be used to
designate gas-phase TMA and water, respectively. X?, O?, and S represent surface
hydroxyl groups (Al-OH), oxygen bridges (Al-O-Al), and surface Al respectively.
The prime (?) notation is introduced to represent 2-coordinate oxygen (O and X)
and 3-coordinate Al to distinguish them from the 3 and 4-coordinate species repre-
sentative of the bulk film [53]. The concentration of these sites is conserved during
the reactions resulting in [X]+[X?]+[O]+[O?]=[X?] where the ?hat? (?) notation is
used to denote a maximum surface concentration as identified in the previous chap-
80
ter. Other surface species, such as TMA adsorbed on X and O will be denoted by
AX and AO, respectively; activated complexes (i.e. transition states) are designated
with a double dagger superscript (?).
The comprehensive set of elementary chemical reactions considered in this
study are given in Tables 4.1 through 4.7, beginning with the TMA half-cycle re-
actions, followed by the water half-cycle reactions, and ending with a CVD-like
gas-phase reaction. In each reaction table, equilibrium constants and reaction rate
formulations are given in terms of molecular concentrations. As with the surface
reaction model derived in Chapter 3, the equilibrium constants and reaction rates
are also derived in terms of molecular partition functions based on assumptions re-
garding the change in translational, rotational, and vibrational degrees of freedom
which arise in equilibrium relationships.
The modeling theory (i.e. TST and ab initio formulation of kinetic parame-
ters) used in this chapter is consistent with the methodology presented in Chapter 3.
Thus, derivations of the molecular partition functions have been omitted. Instead,
the assumptions used in computing the partition function ratio for each equilibrium
relationship are provided. These formulations are given in Tables 4.10 through 4.14.
The corresponding reaction energies for each equilibrium reaction and the sources
of those values are given in Tables 4.8 and 4.9.
Note that in the notation while IMe only corresponds to internal rotation of
a single Me group, IMe?2 corresponds to the total partition function contributions of
two surface Me groups along with that of the adsorption site; hence depending on
the structure of the complex, these contributions may include vibrational and other
81
rotational components in addition to the internal rotational contribution of the Me
groups themselves.
The key differences in the reaction model presented in Chapter 3 and the
model presented in this study are as follows.
(i) Densification is now considered to occur spontaneously (i.e. strongly favored
energetically) after ligand-exchange reactions [23,40,79,80]. As a consequence,
the DMA and MMA species involved in the water half-cycle reactions involve
a 4-coordinate aluminum.
(ii) Additional surface reactions beyond the initial TMA ligand exchange (e.g.
with DMA and MMA species) are modeled independent of pressure [23, 40].
(iii) Dissociative adsorption is now considered for both precursors [40,60,61].
(iv) A gas-phase side-reaction is considered during non-ideal ALD conditions when
precursor gases coexist in the reaction chamber [111].
82
R1 AlMe3 + OH
K1

 OH
AlMe3
K1 =
[AX]
[A] ([X?]? [AX])
A + X? AX
R2 OH
AlMe3 K2

 O
H
Me
Al
Me
Me K2 =
[(AX)?]
[AX]
AX (AX)?
R3 O
H
Me
Al
Me
Me
v3
? O
Al
Me Me
+ CH4 v3 =
kBT
h
[(AX)?]
(AX)? Me?2 + O
? + C
R4 O OH
Al
MeMe
K4

 O
Al
MeMe
OH K4 =
[Me2X]
[Me?2][X
?]
Me?2 + X
? Me2X
R5 O
Al
MeMe
OH
K5

 O
Al
MeMe
O
H
K5 =
[(Me2X)?]
[Me2X]
Me2X (Me2X)?
R6 O
Al
MeMe
O
H v6
? O
Al
Me
O + CH4 v6 =
kBT
h
[(Me2X)
?]
(Me2X)? Me?1 + O
? + C
R7 O OH
Al
Me
O
K7

 O
Al
O
Me
OH K7 =
[Me1X]
[Me?1][X
?]
Me?1 + X
? Me1X
Table 4.1: A subset of elementary surface reactions taking place during TMA expo-
sure, beginning with molecular adsorption. Me = CH3.
83
R8 O
Al
O
Me
OH
K8

 O
Al
O
Me
O
H
K8 =
[(Me1X)?]
[Me1X]
Me1X (Me1X)?
R9 O
Al
O
Me
O
H v9
? O
Al
O O + CH4 v9 =
kBT
h
[(Me1X)
?]
(Me1X)? S + O? + C
Table 4.2: continued from Table 4.1. A subset of elementary surface reactions taking
place during TMA exposure, beginning with molecular adsorption. Me = CH3.
R10 AlMe3 + O
K10

 O
AlMe3
K10 =
[AO]
[A] ([O?]? [AO])
A + O? AO
R11 OO
AlMe3 Al
O O
K11

 OO
Al
Me Me
Al
O
Me
O K11 =
[Me?2][Me1]
[AO][S]
AO + S Me?2 + Me1
R12 OO
Al
Me Me
Al
O O
K12

 O
Al
Me Me
O
Al
O O K12 =
[(Me?2S)
?]
[Me?2][S]
Me?2 + S (Me
?
2S)
?
R13 O
Al
Me Me
O
Al
O O
v13
? O
Al
Me
O
Al
O
Me
O v13 =
kBT
h
[(Me?2S)
?]
(Me?2S)
? Me?1 + Me1
Table 4.3: A subset of elementary surface reactions taking place during TMA expo-
sure, beginning with dissociative adsorption as described in [40]. Me = CH3.
84
R14 H2O + O
Al
MeMe
O
K14

 O
Al
Me Me
OH2
K14 =
[BMe2]
[B] ([Me2]? [BMe2])
B + Me2 BMe2
R15 O
Al
Me Me
OH2 K15

 O
Al
MeH Me
OH
K15 =
[(BMe2)?]
[BMe2]
BMe2 (BMe2)?
R16 O
Al
MeH Me
OH
v16
? O
Al
MeOH
O + CH4 v16 =
kBT
h
[(BMe2)
?]
(BMe2)? Me1 + X? + C
R17 H2O + O
Al
MeOH
O
K17

 O
Al
OH Me
OH2
K17 =
[BMe1]
[B] ([Me1]? [BMe1])
B + Me1 BMe1
R18 O
Al
OH Me
OH2 K18

 O
Al
OH Me H
OH
K18 =
[(BMe1)?]
[BMe1]
BMe1 (BMe1)?
R19 O
Al
OH Me H
OH v19
? O
Al
OHOH
O + CH4 v19 =
kBT
h
[(BMe1)
?]
(BMe1)? X + C
Table 4.4: A subset of elementary surface reactions taking place during water expo-
sure, beginning with molecular adsorption. Me = CH3.
85
R20 O O
Al
Me OH
O
Al
OH Me
OH2 K20

 O O
Al
Me
OHAlOH
Me
O
H
H
K20 =
[(B(Me1)2)?]
[BMe1][Me1]
BMe1 + Me1 (B(Me1)2)?
R21 O O
Al
Me
OHAlOH
Me
O
H
H
v21
? O O
Al
Me
OHAlOH
Me
O
H
+ CH4 v21 =
kBT
h
[(B(Me1)2)
?]
(B(Me1)2)? Me1 + X + C
Table 4.5: A subset of elementary surface reactions taking place during water expo-
sure as described in [23]. Me = CH3.
R22 H2O + O
Al
O O
K22

 O
Al
O
OH2
O K22 =
[BS]
[B] ([S]? [BS])
B + S BS
R23 O O
Al
O
OH2
O
K23

 O O
HAl
O
OH
O K23 =
[(BSO)?]
[BS][O?]
BS + O? (BSO)?
R24 O O
HAl
O
OH
O
v24
? O O
HAl
O
OH
O v24 =
kBT
h
[(BSO)?]
(BSO)? X? + X
Table 4.6: A subset of elementary surface reactions taking place during water expo-
sure, beginning with dissociative adsorption as posed in [60, 61]. Me = CH3.
86
R25 AlMe3 + H2O
K25

 Me3Al OH2 K25 =
[AB]
[A][B]
A + B AB
R26 Me3Al OH2
K26

 Al
Me
Me
Me
OH
H
K26 =
[(AB)?]
[AB]
AB (AB)?
R27 Al
Me
Me
Me
OH
H
v27
? Al
Me
Me
OH + CH4 v27 =
kBT
h
[(AB)?]
(AB)? D + C
Table 4.7: A set of elementary gas-phase reactions taking place when both TMA and
water are present [111]. Me = CH3.
87
Table 4.8: Reaction rate energetics. Plot estimate values noted with an asterisk
(?). In the instance of multiple values for a parameter, the value used in the dy-
namic growth model is noted with a dagger (?). Alumina homodeposition studies
are noted with a plus sign (+), otherwise all other studies correspond to alumina
heterodeposition onto Si-OH substrates.
Parameter Description Value (eV) Source
1 R1 adsorption energy -0.70 [40]+
-0.70 [79,80]
-0.62? [113]+
-0.61 [162]+
-0.59 [55]
-0.49 [23]
-0.46 [74]
2 R2 activation energy 1.04 [74]
0.52 [162]+
0.51 [146]+
0.50 [79,80]
0.45? [113]+
0.48 [23]
0.40 [55]
0.26? [40]+
4 R4 reorganization energy -0.62 [79,80]
-0.60? [40]+
5 R5 activation energy 1.09? [40]+
0.96 [23]
0.21 [79,80]
7 R7 reorganization energy -1.10 [40]+
8 R8 activation energy 1.60? [40]+
1.21? [23]
88
Table 4.9: (continued from Table 4.8) Reaction rate energetics. Plot estimate val-
ues noted with an asterisk (?). In the instance of multiple values for a parameter,
the value used in the dynamic growth model is noted with a dagger (?). Alumina
homodeposition studies are noted with a plus sign (+), otherwise all other studies
correspond to alumina heterodeposition onto Si-OH substrates.
Parameter Description Value (eV) Source
10 R10 adsorption energy -1.20 [40]+
11 R11 dissociation energy -0.40 [40]+
12 R12 activation energy 0.50? to 0.70?? [40]+
14 R14 adsorption energy -1.04? [113]+
-0.85 [55]
-0.57 [162]+
15 R14 activation energy 0.95? [113]+
0.81 [146]+
0.70 [162]+
0.63 [55]
17 R17 adsorption energy -1.09? [113]+
-1.00 [55]
-0.75 [23]
-0.74 [162]+
18 R18 activation energy 1.06 [23]
0.93? [113]+
0.91 [162]+
0.80 [55]
20 R20 activation energy 0.33 [23]
22 R22 adsorption energy -1.01 [60,61]+
23 R23 (1-2) activation energy 0.29? [60, 61]+
(1-4) activation energy 0.095 [60,61]+
25 R25 binding energy -0.65 [111]
26 R26 activation energy 0.66 [111]
89
T
ab
le
4.
10
:
P
ar
ti
ti
on
fu
nc
ti
on
ra
ti
os
an
d
un
de
rl
yi
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as
su
m
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P
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te
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R
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q
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or
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ac
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on
ra
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Z
tr
a
n
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?
Z
ro
t
?
Z
vi
b
R
1
K
1
=
Z
A
X
(Z
A
/V
)Z
X
?
e(
?
 1
/k
B
T
)
(l
x
,A
,
l y
,A
,
l z
,A
)
(I
x
,A
,
I y
,A
)
I A
X
? A
lO
R
2
K
2
=
Z
(A
X
)?
Z
A
X
e(
?
 2
/k
B
T
)
(I
z,
A
,
I M
e)
R
3
v 3
=
k B
T h
Z
(A
X
)?
(Z
A
/V
)Z
X
?
e[
?
(
1
+
 2
)/
k B
T
]
k B
T
+
Z
A
X
(Z
A
/V
)Z
X
?
e(
?
 1
/k
B
T
) P
A
P
A
[X
? ]
-
-
-
R
4
K
4
=
Z
M
e 2
X
Z
M
e? 2
Z
X
? [X?
]e
(?
 4
/k
B
T
)
(I
x
,A
)
? A
lO
R
5
K
5
=
Z
(M
e 2
X
)?
Z
M
e 2
X
e(
?
 5
/k
B
T
)
(I
M
e)
R
6
v 6
=
k B
T h
Z
(M
e 2
X
)?
Z
M
e 2
X
e(
?
 5
/k
B
T
) [
M
e 2
]?
X
-
-
-
R
7
K
7
=
Z
M
e 1
X
Z
M
e? 1
Z
X
? [X?
]e
(?
 7
/k
B
T
)
? A
lO
R
8
K
8
=
Z
(M
e 1
X
)?
Z
M
e 1
X
e(
?
 8
/k
B
T
)
(I
M
e)
R
9
v 9
=
k B
T h
Z
(M
e 1
X
)?
Z
M
e 1
X
e(
?
 8
/k
B
T
) [
M
e 1
]?
X
-
-
-
90
T
ab
le
4.
11
:
(c
on
ti
nu
ed
fr
om
T
ab
le
4.
10
)
P
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ti
ti
on
fu
nc
ti
on
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ti
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an
d
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de
rl
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as
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tr
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vi
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R
10
K
10
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A
O
(Z
A
/V
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O
?
e(
?
 1
0
/k
B
T
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(l
x
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,
l y
,A
,
l z
,A
)
(I
x
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,
I y
,A
)
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lO
R
11
K
11
=
Z
M
e? 2
Z
M
e 1
Z
A
O
Z
S
e(
?
 1
1
/k
B
T
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(l
z,
A
)
I M
e? 2
R
12
K
12
=
Z
(M
e? 2
S
)?
Z
M
e? 2
Z
S
[S?
]e
(?
 1
2
/k
B
T
)
(I
M
e? 2
,
I M
e)
R
13
v 1
3
=
k B
T h
Z
(M
e? 2
S
)?
Z
M
e 1
(Z
A
/V
)Z
O
? (
Z
S
)2
[S
]2
[S?
][M
e 1
]e
[?
(
10
+
 1
1
+
 1
2
)/
k B
T
]
k B
T
+
Z
A
O
(Z
A
/V
)Z
O
?
e(
?
 1
0
/k
B
T
) P
A
P
A
[O
? ]
-
-
-
91
T
ab
le
4.
12
:
(c
on
ti
nu
ed
fr
om
T
ab
le
4.
11
)
P
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ti
ti
on
fu
nc
ti
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d
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de
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tr
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R
14
K
14
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M
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(Z
B
/V
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M
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e(
?
 1
4
/k
B
T
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(l
x
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,
l y
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,
l z
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)
(I
x
,B
,
I y
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)
I B
M
e 2
? A
lO
R
15
K
15
=
Z
(B
M
e 2
)?
Z
B
M
e 2
e(
?
 1
5
/k
B
T
)
(I
z,
B
,
I M
e)
R
16
v 1
6
=
k B
T h
Z
(B
M
e 2
)?
(Z
B
/V
)Z
M
e 2
e[
?
(
14
+
 1
5
)/
k B
T
]
k B
T
+
Z
B
M
e 2
(Z
B
/V
)Z
M
e 2
e(
?
 1
4
/k
B
T
) P
B
P
B
[M
e 2
]
-
-
-
R
17
K
17
=
Z
B
M
e 1
(Z
B
/V
)Z
M
e 1
e(
?
 1
7
/k
B
T
)
(l
x
,B
,
l y
,B
,
l z
,B
)
(I
x
,B
,
I y
,B
)
I B
M
e 1
? A
lO
R
18
K
18
=
Z
(B
M
e 1
)?
Z
B
M
e 1
e(
?
 1
8
/k
B
T
)
(I
z,
B
,
I M
e)
R
19
v 1
9
=
k B
T h
Z
(B
M
e 1
)?
(Z
B
/V
)Z
M
e 1
e[
?
(
17
+
 1
8
)/
k B
T
]
k B
T
+
Z
B
M
e 1
(Z
B
/V
)Z
M
e 1
e(
?
 1
7
/k
B
T
) P
B
P
B
[M
e 1
]
-
-
-
92
T
ab
le
4.
13
:
(c
on
ti
nu
ed
fr
om
T
ab
le
4.
12
)
P
ar
ti
ti
on
fu
nc
ti
on
ra
ti
os
an
d
un
de
rl
yi
ng
as
su
m
pt
io
ns
.
P
ar
en
th
et
ic
al
te
rm
s
in
th
e
?
Z
co
lu
m
ns
de
si
gn
at
e
de
gr
ee
s
of
fr
ee
do
m
as
su
m
ed
to
be
lo
st
du
e
to
th
e
fo
rw
ar
d
re
ac
ti
on
.
T
er
m
s
in
th
e
?
Z
co
lu
m
ns
w
hi
ch
ar
e
no
t
in
pa
re
nt
he
se
s
de
si
gn
at
e
de
gr
ee
s
of
fr
ee
do
m
as
su
m
ed
to
be
ga
in
ed
.
R
ea
ct
io
n
E
q
u
il
ib
ri
u
m
co
n
st
an
t
or
re
ac
ti
on
ra
te
?
Z
tr
a
n
s
?
Z
ro
t
?
Z
vi
b
R
20
K
20
=
Z
(B
(M
e 1
) 2
)?
Z
B
M
e 1
Z
M
e 1
[M?
e]
e(
?
 2
0
/k
B
T
)
(I
z,
B
,
I M
e,
I B
M
e 1
)
R
21
v 2
1
=
k B
T h
Z
(B
(M
e 1
) 2
)?
(Z
B
/V
) (
Z
M
e
1
)2
[M?
e]
e[
?
(
17
+
 2
0
)/
k B
T
]
k B
T
+
Z
B
M
e 1
(Z
B
/V
)Z
M
e 1
e(
?
 1
7
/k
B
T
)
P
B
[M
e 1
]2
-
-
-
93
T
ab
le
4.
14
:
(c
on
ti
nu
ed
fr
om
T
ab
le
4.
13
)
P
ar
ti
ti
on
fu
nc
ti
on
ra
ti
os
an
d
un
de
rl
yi
ng
as
su
m
pt
io
ns
.
P
ar
en
th
et
ic
al
te
rm
s
in
th
e
?
Z
co
lu
m
ns
de
si
gn
at
e
de
gr
ee
s
of
fr
ee
do
m
as
su
m
ed
to
be
lo
st
du
e
to
th
e
fo
rw
ar
d
re
ac
ti
on
.
T
er
m
s
in
th
e
?
Z
co
lu
m
ns
w
hi
ch
ar
e
no
t
in
pa
re
nt
he
se
s
de
si
gn
at
e
de
gr
ee
s
of
fr
ee
do
m
as
su
m
ed
to
be
ga
in
ed
.
R
ea
ct
io
n
E
q
u
il
ib
ri
u
m
co
n
st
an
t
or
re
ac
ti
on
ra
te
?
Z
tr
a
n
s
?
Z
ro
t
?
Z
vi
b
R
22
K
22
=
Z
B
S
(Z
B
/V
)Z
S
e(
?
 2
2
/k
B
T
)
(l
x
,B
,
l y
,B
,
l z
,B
)
(I
x
,B
,
I y
,B
)
? A
lO
R
23
K
23
=
Z
(B
S
O
)?
Z
B
S
Z
O
? [O?
]e
(?
 2
3
/k
B
T
)
(l
z,
B
)
R
24
v 2
4
=
k B
T h
Z
(B
S
O
)?
(Z
B
/V
)Z
S
Z
O
?
[O
? ]
[O?
]e
[?
(
22
+
 2
3
)/
k B
T
]
k B
T
+
Z
B
S
(Z
B
/V
)Z
S
e(
?
 2
2
/k
B
T
) P
B
P
B
[S
]
-
-
-
(l
x
,A
,
l y
,A
,
l z
,A
)
(I
x
,A
,
I y
,A
)
R
25
K
25
=
Z
A
B
(Z
A
/V
)Z
B
e(
?
 2
5
/k
B
T
)
(l
x
,B
,
l y
,B
,
l z
,B
)
(I
x
,B
,
I y
,B
)
l x
,A
B
,
l y
,A
B
,
l z
,A
B
I x
,A
B
,
I y
,A
B
,
I z
,A
B
? A
lO
R
26
K
26
=
Z
(A
B
)?
Z
A
B
e(
?
 2
6
/k
B
T
)
(I
M
e)
R
27
v 2
7
=
k B
T h
Z
(A
B
)?
(Z
A
/V
)Z
B
e[
?
(
25
+
 2
6
)/
k B
T
]
P
A
P
B
(k
B
T
)2
-
-
-
94
4.1.1 TMA Reactions on Hydroxylated Alumina
The TMA half-cycle is typically initiated by barrierless chemisorption of TMA
(R1), forming a coordinative bond with the oxygen of a surface hydroxyl group.
While the adsorbed TMA has no translational freedom, it maintains a ?helicopter?
rotational mode and the combined angular momentum of the adduct. This stable
adduct may then proceed through a 4-coordinate Al transition state to produce the
DMA moiety and methane gas which immediately desorbs (R2, R3). The primary
entropic difference between the adsorbed TMA and transition state is the loss of
internal rotational freedom of one Me group due to bond formation. The trigonal
planar 3-coordinate Al of the DMA that remains after the reaction is denoted by
Me?2.
This Me?2 is then assumed to undergo immediate densification with a surface
OH, recovering the 4-coordinate Al state Me2 (R4). The primary distinction between
the two is the helicopter rotation mode of Me?2 that does not exist for Me2: the
contribution of this rotation mode is approximately equivalent to rotation about
the Al-Me bond (with rotational inertia IMe?2 =
?
3r2Al?CmMe) for gas-phase TMA.
For typical operating conditions, K4 ? O(105) indicating the favorability of finding
the relaxed Me2 species relative to Me?2. Thus, if surface hydroxyl groups are present,
reaction (R3) will produce Me2, not Me?2.
Each of the two surface Me groups left after reaction (R4) can undergo subse-
quent reactions with a surface OH site, producing methane that immediately des-
orbs to the gas phase. The 4-coordinate Me2 can undergo a reaction where H from
95
a neighboring OH site is transferred to a Me group on the 4-coordinated Al to
eliminate another methane molecule in reactions (R5) and (R6). As in the pre-
vious methane elimination, the trigonal Me?1 can immediately relax to form Me1,
which caps two surface O and one hydroxyl group X, through reaction (R7) with
K7 ? O(1013). As can be readily seen by the large energy associated with the re-
organization [40] listed in Table 4.8, the Me1 species is highly favored over Me?1 as
the result of reaction (R6). Following this sequence, the final Me group can then be
eliminated by reactions (R8) and (R9), leaving a bare surface aluminum.
4.1.2 TMA Reactions on Bare Alumina
Having considered the adsorption and methane-elimination reactions on a hy-
droxylated surface, the adsorption and dissociation reactions of TMA on locally bare
(i.e. dehydroxylated) alumina will be described by reaction sequence (R10-R13). Of
course no methane is produced in any of the reaction steps because of the lack of
surface H. In reaction (R10), TMA adsorbs barrierlessly at a bridged oxygen site
O?, forming a Lewis acid-base adduct similar to the mechanism of (R1). However,
in this case, the adduct is considered unstable (in an energetic sense with 11 < 0)
and so can undergo a barrierless dissociation reaction, forming Me?2 and Me1 surface
species (R11).
Depending on the proximity of more aluminum sites, this process may be
repeated. The Me?2 moiety may further dissociate onto a neighboring aluminum,
although this does require an activation energy and consequently formation of a
transition state (R12). The energetics for this transition state, however, have only
96
been estimated [40]. The final product of the reaction sequence is a completely
dissociated TMA molecule, distributed about the surface as three MMA groups
with a 3- or 4-coordinate aluminum (R13).
4.1.3 Water Reactions on Methylated Alumina
The water half-cycle is typically initiated by barrierless chemisorption of water
(R14), forming a coordinative bond with a surface aluminum. While the adsorbed
water has no translational freedom, it maintains a ?helicopter? rotational mode
and the combined angular momentum of the adduct. Because of the highly-favored
densification reactions of the TMA half-cycle, the surface aluminum onto which
water adsorbs is likely to be 4-coordinate. Thus, one of the shared Al-O bonds of the
adsorption site may be released in interaction with an incoming water molecule [23].
In this way, the aluminum atom maintains the 4-coordinate state.
In the case that the aluminum involved in the stable adduct contains two
methyl ligands, then the adduct may proceed through a 1-2 H transfer transition
state to produce a surface hydroxyl group X, a Me?1 moiety, and methane gas which
immediately desorbs (R15, R16). The primary entropic difference between the ad-
sorbed water and transition state is the loss of internal rotational freedom of one
Me group due to bond formation. As in the TMA half-cycle reactions, Me?1 un-
dergoes a barrierless and immediate densification, recovering the 4-coordinate Al
state Me1 [23]. Specific energetics have not been reported for this densification step.
Thus, it has not been explicitly modeled in this work.
Water may also adsorb onto a surface aluminum which contains only one
97
methyl ligand (R17). This adsorption reaction is relatively the same as with the
dimethyl aluminum reaction, but with slightly different energetics. The adduct
proceeds through a 1-2 H transfer transition state (R18) and adds a surface hydroxyl
group while releasing methane to the gas phase (R19). This product is also assumed
to immediately proceed to densification.
As an alternative to the 1-2 H transfer reaction, a 1-4 H transfer mechanism
has been suggested in [23] whereby the BMe1 adduct interacts with a neighboring
Me1 to yield a more complex transition state (R20). From this transition state, a
methane molecule is formed with the methyl group of a neighboring aluminum as
opposed to the aluminum atom onto which the water molecule originally adsorbed.
Thus, several rotational modes are lost?helicopter rotation of the water molecule,
an internal methyl group rotation, and the BMe1 rotational inertia. The forward
reaction releases the methane, forming a bridged hydroxyl group between the two
aluminum atoms (R21). This 1-4 H transfer was reported to be several orders more
favored (energetically) than the 1-2 H transfer mechanism.
4.1.4 Water Reactions on Bare Alumina
Having considered the adsorption and methane elimination reactions on a
methylated surface, the adsorption and dissociation reactions of water on locally
bare (i.e. demethylated) alumina will be described by reaction sequence (R22-R24).
As in the TMA dissociative adsorption reactions, no methane is produced in any
of the reaction steps. In reaction (R22), water adsorbs barrierlessly onto a bare
aluminum atom, forming a Lewis acid-base adduct similar to the mechanism on a
98
methylated surface. Except for the absence of an adduct rotational moment, this
adsorbed water molecule is entropically identical to the molecularly adsorbed water
onto a methylated surface.
Given the availability of an undercoordinated oxygen atom, the adduct may
proceed through a transition state, transferring a hydrogen to this atom and re-
sulting in the loss of the helicopter rotational mode. While this reaction is not
barrierless, the activation energy to form the transition state (R23) is relatively low.
Although, the 1-4 H transfer path is reported to be more energetically favored when
compared to the the 1-2 H transfer path [60,61], consistent with the findings of [23]
for the methylated surface. Nonetheless, the adduct ultimately proceeds through
this transition state, resulting in the formation of two surface hydroxyl groups (R24).
4.1.5 Gas-phase Side Reactions
While gas-phase reactions are certainly non-ideal in ALD, a sequence of el-
ementary gas-phase side reactions have been included in the ALD kinetics model
to account for CVD-like reactions that may occur in the case of insufficient purge
periods, i.e. when TMA and water precursors coexist in the reaction chamber [111].
This sequence begins when a TMA and water molecule collide to barrierlessly form
a stable gas-phase Lewis acid-base adduct (R25). The translational freedom of the
separate molecules is replaced by the three translational degrees of freedom asso-
ciated with the adduct. Likewise, the rotational modes of the adduct replace the
rotational modes of the separate molecules. Internal rotation in the axial direc-
tion remains, however, for each molecule in the adduct. All vibrational modes are
99
assumed to be retained and an additional mode arises from the Al-O bond.
This adduct may proceed to form a transition state in preparation for methane
elimination. The primary entropic difference between the adduct and transition
state is the loss of internal rotational freedom of one Me group due to bond formation
(R26). The product of the forward reaction is a methane molecule and Al(CH3)2-OH
(DMA-OH) which is noted as ?D? in the model formulation developed herein (R27).
No additional gas-phase or surface reactions involving this species are considered
beyond this point. This species, however, is a known product of ALD side reactions
and serves as a way to account for the relative significance of gas-phase side-reactions
in a dynamic ALD kinetics model. Predictions of significant DMA-OH formation
would, for instance, suggest the need for longer purge periods and/or decreased
precursor exposures.
4.2 Analysis of Reaction Mechanism
Several researchers have investigated the alumina ALD surface reaction mech-
anism using quantum-chemical computational methods and have reported values
for the energetics of relevant elementary reaction steps. These values are given in
Tables 4.8 and 4.9. Of these studies, four investigate alumina homodeposition [40,
113, 146, 162], two are alumina water adsorption studies [60, 61], and the others are
alumina heterodeposition studies on Si-OH substrates. Values are not considered for
heterodeposition on other substrates (e.g. Si-H, Ge-S, TiO2 etc.), although the gen-
eral mechanism is fairly similar [23,54,56,57,83,91,99,100,113,139,155,167,168,173].
100
In some instances, the suggested reaction energies and/or reaction mechanisms
vary widely across different studies of the alumina ALD process. Considering that
energetics are exponential terms in reaction rate formulations, wide variations in
these values will produce wide variations in the reaction rates produced. In one
instance, for different H transfer paths, the difference in the computed rate is O(106)
[23], but this is also seen in reported values for the typical 1-2 H transfer mechanism.
To resolve these differences with a kinetic analysis and choose a set of energetic
parameters most consistent with the observed ALD mechanism is thus the objective
of this section.
4.2.1 Comparative Analysis of TMA Kinetics
Having derived v3, v6, v9, and v13 the values of these rate expressions are
plotted as a function of TMA partial pressure for T = 450 K. To compare the
energetics suggested for pressure-dependent TMA half-cycle reactions vs. pressure-
independent reactions, the rate expressions for molecular adsorption and ligand-
exchange v3 have been plotted with dissociative adsorption v13 in Fig. 4.1. The rates
of the subsequent surface methane-elimination reactions v6 and v9 are likewise shown
together in Fig. 4.2. Note that these values are not rate constants as they depend
on PA, [X], [X?], [O?], [S], [Me1], and [Me2]; for the figures, rates are evaluated at
the maximum surface concentration of these species, e.g., [X] = [X?]. For this reason
and because the modulating effect of surface Me group coverage has not yet been
included, the rates portrayed correspond to the upper limits of the surface reaction
rates.
101
10-2 10-1 100
PA , Torr
10-8
10-6
10-4
10-2
100
102
104
106
rat
e, 
mo
lec
ule
s/n
m2
s
wA
v3,EG04
v3,WM02
v3,DP12
v3,JE03
v3,KK11
v3,HR04
v3,ND10
v13,EG04
v13,EG04
Figure 4.1: Representative reaction rates of irreversible TMA half-cycle reactions
corresponding to pressure-dependent reactions at T = 450 K and maximum sur-
face concentration values for [X], [X?], [O?], [S], [Me1], and [Me2]. For reference,
the TMA wall collision rate wA also is shown. Data labels denote the source of
energetic information used in the rate computation; e.g. EG04 corresponds to El-
liottGreer2004 (first author, last author, and year of publication).
Because v3 and v13 depend on the adsorption of the gas-phase precursor A,
their values are expected to increase with PA while v6 and v9 are independent of PA.
The dissociative adsorption rate v13 approaches a limiting value as PA grows, indi-
cating saturation in the Langmuir equilibrium relation; an important consequence is
that the adsorbed adduct can subsequently desorb during the purge phase between
exposures A and B, reducing the effectiveness of the overall deposition process.
For reference, the TMA wall collision rate is plotted as wA = PA/
?
2pimAkBT .
For the lower end of the 13 range (i.e. the upper limit for v13 shown in the plot),
v13 > wA indicating the system is not in equilibrium with the gas-phase TMA at
PA. An additional gas-phase transport term would have to be included to sustain
these rates, otherwise this corresponds to a mass-transfer limited situation.
102
10-2 10-1 100
PA , Torr
10-5
10-3
10-1
101
103
105
107
109
1011
rat
e, 
mo
lec
ule
s/n
m2
s
v6,EG04
v6,DP12
v6,KK11
v9,EG04
v9,DP12
Figure 4.2: Representative reaction rates of irreversible TMA half-cycle reactions
corresponding to pressure-independent reactions at T = 450 K and maximum surface
concentration values for [X], [X?], [O?], [S], [Me1], and [Me2]. Data labels denote the
source of energetic information used in the rate computation; e.g. EG04 corresponds
to ElliottGreer2004 (first author, last author, and year of publication).
The energetics associated with each elementary reaction step have been ex-
tracted from quantum-chemical computational studies. As can be seen from Ta-
bles 4.8 and 4.9, in many instances, very different values have been reported. In
these cases, reaction rates were computed based on each source and noted in the
figures. For example, v3,EG04 corresponds to a rate for reaction (R3) computed us-
ing an (R1) adsorption and (R2) activation energy from [40] which has been noted
ElliottGreer2004 (first author, last author, year of publication).
4.2.1.1 Resolution of Thermodynamics
When considering the thermodynamics reported in this collection of DFT stud-
ies, it is important to choose values which are consistent with what is known about
103
the ALD mechanism. In some instances, reaction rates are several orders larger or
smaller than the general trend of quantum-chemical computations. For instance, in
EG04 [40], the estimated activation energy for 2 is significantly lower than other
reported values and thus these energetics yield reaction rates for (R3) which are
several orders larger when compared to other predictions. On the other end of the
spectrum, JE03 [74] reports a barrier more than double than what has been sug-
gested in other studies. The corresponding rate for (R3) based on those energetics
is thus significantly lower.
Note that in some instances the reported adsorption energy may be relatively
consistent with other studies, but the ligand-exchange activation energy may not
agree, or vice versa. Because the initial adsorption and transition state formation
reaction are inherently pressure-dependent, the forward rate also has this depen-
dency. Thus, to maintain consistency in rate computations, both energy values are
taken from the same source, i.e. adsorption energy from one source will not be
combined with the ligand-exchange activation energy of another.
The energetics of WM02 [162] and ND10 [113] were both taken from homod-
eposition studies and are in fair agreement with one another. They, however, yield
slightly different reaction rates. Thermodynamics from the heterodeposition studies
of KK11 [80] and HR04 [55] also yield a range of (R3) rate predictions which appear
to be consistent with other reports.
To knowledge, there has been only one quantum-chemical computational study
reporting ALD energetics for dissociative TMA adsorption on a bare alumina sur-
face [40]. The energetics for the (R10-R13) reaction sequence thus are taken solely
104
from this study. It is interesting to note that the rate of the dissociative TMA reac-
tion is significantly higher than the comparable rate of reaction on a hydroxylated
surface. Because there are no other reported values with which to compare the
thermodynamics of this reaction, the consistency of 10 ? 12 with respect to other
reports cannot be discussed. It must also be noted that the transition state was
not able to be computed for the dissociative adsorption reaction and thus 12 is an
estimate. A range is therefore shown for v13,EG04.
Moving now to the pressure-independent reactions, one will notice that there
have been notably fewer quantum-chemical computational studies devoted to the
subsequent surface reactions following adsorption and ligand exchange. KK11 [80]
reports a very small activation energy barrier 5 yielding a very high reaction rate
which is clearly in disagreement with other reports. The homodeposition study of
EG04 [40] suggests an activation energy barrier for the second methane-elimination
reaction which yields a lower (R6) rate than what has been reported in the heterode-
position study of DP12 [23]. Substrate effects may account for this difference. In
the final methane-elimination reaction (R9), EG04 was unable to locate a transition
state and thus only an estimate was given for the activation barrier. In this instance,
energetics from DP12 may be used as an alternative.
4.2.2 Comparative Analysis of Water Kinetics
Having likewise derived v16, v19, v21, and v24, the values of these rate expres-
sions are plotted as a function of water partial pressure for T = 450 K and are
collectively shown in Fig. 4.3. Again, note that these values are not rate constants
105
10-2 10-1 100
PB , Torr
10-3
10-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
1011
rat
e, 
mo
lec
ule
s/n
m2
s
wB
v16,WM02
v16,HR04
v16,ND10
v19,WM02
v19,DP12
v19,HR04
v19,ND10
v21,DP12
v24,HA00
Figure 4.3: Representative reaction rates of irreversible water half-cycle reactions
corresponding to pressure-dependent reactions at T = 450 K and maximum surface
concentration values for [X], [X?], [O?], [S], [Me1], and [Me2]. For reference, the wa-
ter wall collision rate wB also is shown. Data labels denote the source of energetic
information used in the rate computation; e.g. WM02 corresponds to WidjajaMus-
grave2002 (first author, last author, and year of publication).
as they depend on PB, [X], [X?], [O?], [S], [Me1], and [Me2]; for this figure, rates are
evaluated at the maximum surface concentration of these species, e.g., [X] = [X?]. For
this reason and because the modulating effect of surface hydroxyl group coverage
has not yet been included, the rates portrayed correspond to the upper limits of the
surface reaction rates.
Because all water half-cycle rates depend on the adsorption of the gas-phase
precursor B, their values are expected to increase with PB. Both the dissociative
adsorption rate v24 and the MMA reaction rate v19 approach a limiting value as PB
grows, indicating saturation in the Langmuir equilibrium relation. For reference,
the water wall collision rate is plotted as wB = PB/
?
2pimBkBT . Similar to the
106
TMA dissociative adsorption reaction, v24  wB indicating a mass-transfer limited
situation.
4.2.2.1 Resolution of Thermodynamics
The water adsorption and ligand-exchange activation energies associated with
reaction sequence (R14-R16) have been reported in two homodeposition studies,
WM02 [162] and ND10 [113] and one heterodeposition study HR04 [55]. The rate
computed from HR04 involves a stronger adsorption interaction and lower ligand-
exchange barrier and is consequently several orders higher. Considering this, the
thermodynamics derived from homodeposition will be considered.
The water adsorption and ligand-exchange activation energies associated with
reaction sequence (R17-R19) have been reported in the above cited studies and in
an additional heterodeposition study DP12 [23]. However, energetics from the het-
erodeposition studies yield reaction rates significantly higher and lower than those
reported for the homodeposition studies. Specifically, an alternative 1-4 H transfer
path has been suggested with reaction energies yielding rates v21,DP12 that are sev-
eral orders faster than the typical 1-2 H transfer mechanism v19,DP12 suggested in
the same study. The 1-4 H transfer path is also much faster than reaction rate pre-
diction of all other cited studies based on molecular water adsorption. Note that the
predictions of this study are based on alumina deposition onto a Si-OH substrate.
Thus, substrate effects may account for the differences in reported energetics when
compared to alumina homodeposition results. It is thus unclear if this mechanism
is applicable to the present study.
107
Reaction rate predictions based on WM02 and DP12 are very different at low
pressures, but are of similar order at higher pressures. This can be explained by the
distinct differences in the adsorption and reaction energies reported. The suggested
adsorption energy of ND10 is nearly double that of WM02, but the reaction activa-
tion energy is significantly higher. The adsorption energies reported in ND10 [113]
and HR04 [55] are relatively high, yielding rates which appear to be independent of
pressure, indicating an almost immediate surface saturation.
Two quantum-chemical computational studies are widely referenced in the
context of water interactions with bare alumina [60,61]. The energetics for the dis-
sociative water reaction sequence (R22-R24) are taken solely from these studies. As
with the TMA dissociative adsorption reactions, it is interesting to note that the
rate of the dissociative water reaction is several orders higher than the comparable
rate of reaction on a methylated surface. It must be noted that while these studies
have been referenced in other ALD works [23], the energetics may not correspond
to the amorphous alumina film grown by ALD. The reported energetics correspond
to ?-alumina which has a significantly higher density (?? = 3.98 g/cm3) compared
to the amorphous phase deposited during ALD (?am = 3.3 g/cm3) [53]. It is there-
fore uncertain whether these thermodynamics are applicable to the alumina ALD
process.
4.2.3 Gas-phase Side Reaction Mechanism Analysis
Representative rates for the gas-phase side-reaction are shown in Fig. 4.4. In
the top plot, the partial pressure of one precursor is kept constant at 0.01 Torr
108
10-4
10-3
10-2
10-1
100
101
rat
e, 
mo
lec
ule
s/n
m3
s zAB
v27
10-2 10-1 100
PA ,PB , Torr
10-5
10-4
10-3
10-2
10-1
100
rat
e, 
mo
lec
ule
s/n
m3
s zAB
v27
Figure 4.4: Representative reaction rate of the gas-phase side reaction. (top) Rate
computed with one precursor at constant partial pressure of 0.01 Torr. (bottom)
Rate computed with one precursor at constant partial pressure of 0.001 Torr. All
rates correspond to reactions at T = 450 K. For reference, the gas-phase TMA-water
collision frequency zAB also is shown.
while the other is varied, i.e. PA = 0.01 Torr. The bottom plot corresponds to a
precursor partial pressure of 0.001 Torr for one precursor while the other is varied,
i.e. PA = 0.001 Torr. This is meant to mimic residual precursor gases in the reaction
chamber following a purge period. The rate obviously increases with pressure and
could become significant without proper separation of the pulsing intervals.
4.3 Thermodynamic Sensitivity
It is very important to note the sensitivity of computed reaction rate expres-
sions to the thermodynamics used in the computations. Because reaction energies
are placed inside an exponential term, small variations in these values could result
109
in significant variations in computed kinetics. As previously stated, energetics for
the models derived in this thesis have been adopted from reported thermodynam-
ics of quantum-chemical computational studies. As demonstrated in Figs. 4.1-4.3,
reaction energetics can cause significant differences in computed kinetics, with mag-
nitude differences of several orders for the same reaction. These discrepancies may
be attributed to differences in the level of theory used in the various DFT studies
and also by current computational challenges posed by atomic-scale simulations.
In DFT, electronic structures of chemically-reacting species are determined by
approximately solving the quantum mechanical Schro?dinger wave equation, which
gives access to atomic structures, reaction energetics at zero temperature, and dy-
namics at finite temperatures. However, ALD chemistries are often very complex
and there still exist issues relating to the accuracy of DFT models such that comput-
ing Gibbs free energies of activation to ?chemical accuracy? (< 5 kJ/mol) remains a
difficulty in ab initio simulations [38]. For this reason, a detailed survey of reported
ab initio energetics was performed herein for the alumina ALD system.
4.4 Growth Surface State Dynamics
It is generally reported that steric hinderance by surface Me groups regulates
TMA surface reactions [10,40,128,129,131,161]. This conclusion is supported by ob-
servations that TMA surface reactions self-terminate when Me group concentration
reaches an approximately constant value denoting saturation of adsorbed species,
a value which approaches the Me group close-packing limit [10, 127, 131?135, 161].
110
Steric hindrance of surface Me groups appears to cause the termination of TMA re-
actions and so TMA adsorption proceeds until the surface becomes saturated with
Me groups which can shield active bonding sites. For these reasons, it is impor-
tant to include in an alumina ALD growth model an account of surface Me group
concentration.
Total concentration [Me] of surface Me groups is found to be a combination
of chemisorbed Me1 and Me2 plus the Me groups of adsorbed adducts AO and AX.
Recalling that ?Me = [Me]/[M?e] and ?Me = 1? ?Me,
[Me] = [Me1] + 2[Me2] + 3 ([AX]
eq + [AO]eq)?Me (4.1)
=
[Me1] + 2[Me2] + 3([AX]eq + [AO]eq)
1 + 3([AX]eq + [AO]eq)/[M?e]
(4.2)
where the superscript eq indicates values computed using the equilibrium relation-
ships shown in Tables 4.1 and 4.3. Likewise, ?Xtot = ?X + ?X? = [X
tot]/[X?] and
?X = 1? ?Xtot . Thus
[Xtot] = [X] + [X?] + ([BMe2]
eq + [BMe1]
eq)?X (4.3)
=
[X] + [X?] + [BMe2]eq + [BMe1]eq
1 + ([BMe2]eq + [BMe1]eq)/[X?]
(4.4)
With ?Me representing the fraction of growth surface not blocked by Me groups
and ?X representing the fraction of growth surface not blocked by hydroxyl groups,
111
the time-rate of change of surface species Me1, Me2, and O, O?, S, X, and X? are
d[Me1]
dt
= v6 ? v9 + 3v13?Me + (v16 ? v19)?X (4.5)
d[Me2]
dt
= v3?Me ? v6 ? v16?X (4.6)
d[O]
dt
= v13?Me (4.7)
d[O?]
dt
= (v3 ? v13)?Me + v6 + v9 ? v24?X (4.8)
d[S]
dt
= v9 ? 2v13?Me ? v24?X (4.9)
d[X]
dt
= v3?Me ? v9 + v24?X (4.10)
d[X?]
dt
= ?2v3?Me ? v6 + (v16 + v19 + v24)?X (4.11)
subject to initial conditions [Me1](0), [Me2](0), . . . , [X?](0). Thus, the evolution of
the surface state can be computed by integrating Eqns. (4.5-4.11) from the specified
initial conditions. Furthermore, the instantaneous adsorbed and activated-complex
species concentrations can be recovered from the states defined by Eqns. (4.5-4.11)
and the equilibrium relationships described earlier.
4.4.1 Dynamic Simulation Results
With the surface-state dynamic modeling differential and algebraic equations
in hand, the evolution of the surface state can be examined over the complete ALD
cycle by integrating the differential equations (4.5-4.11) subject to the initial state
of the surface. For this purpose, the collocation-based technique described in the
previous chapter is employed. Three representative cases are shown in Fig. 4.5
112
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.01 0.02
0.0
0.2
0.4
0.6
0.8
1.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
time, sec
X
X'
O'
Me
Me1
Me2
Figure 4.5: Surface species dynamics for ??X(0) = 1 (top), ?
?
X(0) = 0.75 (middle),
and ??X(0) = 0 (bottom) at T=450 K and P = 2 Pa. The initial 0.02 s of the TMA
half-cycle have been emphasized to capture the fast surface reactions.
and all correspond to an initial surface devoid of Me groups. The first simulation
shown in the uppermost plot corresponds to the initial state [X?](0) = [X?], a fully
hydroxylated surface. Early in the simulation, [X?] drops quickly upon exposure to
TMA, producing surface Me2 by (R3), some of which reacts with surface X to form
Me1. During the subsequent water pulse, most but not all of the Me groups are
consumed, but the water self-saturation limit is not reached.
A set of initial conditions at the other end of the range of [X?](0) corresponds
to a bare alumina surface ([X?](0)=0) and the resulting simulation is shown in the
bottom plot of Fig. 4.5. The dynamic behavior in this situation is simple and
113
quick: TMA adsorbs onto the bare O? sites and proceeds through the sequence
of reactions (R10-R13), quickly saturating the surface with Me1, after which, no
other reactions are possible. This is completely expected as no methane-elimination
reactions are possible without the presence of hydroxyl groups. What is interesting
is the behavior of the process during the water pulse, when compared to the Me1 and
Me2 terminated surface in the top plot. The dynamics of the water exposure when
starting from the bare surface are significantly faster. The Me1 groups are quickly
consumed, after which the OH concentration remains constant. In the previous case,
however, the Me2 groups were more readily consumed, but when converted to Me1,
this reduction was much slower. This demonstrates the competition between these
two reactions at higher surface hydroxylation.
Intermediate between the fully hydroxylated and bare surfaces is the case
[X?](0) = 3[X?]/4, [O?](0) = [X?]/4 shown in the middle plot of Fig. 4.5. Despite
the significant concentration of surface OH species X?, the dynamics share more
in common with those corresponding to the bare surface. As in the latter case,
reaction (R13) proceeds to quickly saturate the surface with Me1; however, after
the saturation point, reactions (R6) and (R9) can proceed, albeit slowly, because
of the available X sites. During the water exposure, Me1 groups are much more
quickly consumed than Me2, which was not the case for the surface with higher
hydroxylation. Together, these results suggest that reaction (R19) may dominate
at lower hydroxyl group coverage, perhaps due to the greater availability of Me1
groups under those conditions. But reaction (R16) becomes more competitive at
higher hydroxyl group coverage when Me2 is more readily available.
114
Another implication of these results stem from the inability of the water reac-
tions to remove all Me groups and thus produce a saturated surface under the con-
ditions considered in the model. While the TMA dynamics also do not completely
saturate, the saturation limit is approached much more quickly than in the water
half-cycle. This would suggest that the water exposure should be higher than that of
the TMA in order to ensure self-limiting growth conditions. This is consistent with
previous modeling [156] as well as experimental observations requiring higher water
exposures (relative to TMA) to sufficiently remove all methyl ligands [75,102]. Note
that dissociative water reactions were not allowed in the dynamic growth model due
to great uncertainty as to the applicability of the available thermodynamic param-
eters. Of course, inclusion of these reactions would likely lead to higher hydroxyl
coverage.
4.4.2 Growth per cycle
Defining the number of Al atoms deposited during pulse A by reactions (R3)
and (R13) by
NAl,R3 =
? ?A+?B
0
v3?Me dt (4.12)
NAl,R13 =
? ?A+?B
0
v13?Me dt (4.13)
the growth per cycle can be determined as
gpc = 10%N (NAl,R3 +NAl,R13) A?/cycle (4.14)
115
[X ](0) =
[X]
[X ](0)=
3[X]/4[X ](0)=
00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
gp
c, 
R3
R13
Figure 4.6: Relative film growth rate contributions of (R3) and (R13) as a function
of initial surface OH group concentration.
where, as previously stated, %N is the number density of amorphous Al2O3 [156].
Results are shown in Fig. 4.6. In this chart the relative contributions of Al de-
posited by reactions (R3) and (R13) are distinguished. The majority of Al atoms
are deposited by reaction (R3) for the fully hydroxylated initial surface state, re-
sulting in an overall growth rate of 0.88 A?/cycle, a value near the commonly ac-
cepted range of 0.9-1.1 A?/cycle as nominal ALD growth under saturating condi-
tions [118, 120, 129, 131, 138, 150]. From Fig. 4.5 (top) one can see that the surface
approaches, but does not reach self-saturating conditions during the TMA and wa-
ter pulses for the fully-hydroxylated initial surface. Thus it would follow that the
gpc for this case (given the pressure and pulse times used in the model) would ap-
proach, but not fully reach the saturating GPC. For the intermediate case where
the initial surface hydroxyl concentration is 3/4 the maximum value, both reactions
116
(R3) and (R13) share significant responsibility for alumina ALD growth, yielding a
gpc of 0.71 A?/cycle.
The dominance of reaction (R13) is observed under conditions corresponding to
surfaces that are dehydroxylated at the start of the TMA exposure in the left-most
plot of Fig. 4.6. Model predictions suggest a growth rate of 0.70 A?/cycle when the
surface is completely devoid of hydroxyl groups. These results are consistent with
experimental [13, 26, 43, 129, 134, 135, 150, 161] and theoretical [40, 129, 162] studies
in which it was concluded that TMA adsorbs at bridged oxygen sites in the case of
low hydroxyl group coverage.
Experimental ALD growth data have been reported [135] and numerically
analyzed [130] concluding nonzero alumina ALD growth rates on a completely de-
hydroxylated surface and suggesting that TMA chemisorption under this condition
results in three surface methyl groups, which is consistent with the products of reac-
tion (R13). The data in those studies is reported in terms of the number of aluminum
atoms deposited per cycle, which may be translated into a growth per cycle using
the formulation in [128,129], similar to Eqn. (4.14) as described herein.
Further, an empirically-based growth model suggested in [130] predicts an
aluminum deposition rate which yields a growth per cycle (using the growth rate
model in [128, 129]) of approximately 0.5 A?/cycle on a completely dehydroxylated
surface. It must be noted, however, that the model is derived from combined alumina
growth data on both alumina and silica surfaces and from a range of temperatures
353-573 K. Due to these additional variables, only a general comparison should be
expected from the model presented herein and the empirical growth data reported
117
in [130]. Even so, the film growth rate predictions shown in Fig. 4.6 are within rea-
sonable quantitative agreement with experimental ALD growth data [130, 135] and
are consistent with experiments which suggest TMA surface reaction mechanisms at
bridged oxygen sites for low hydroxyl group coverage [13,26,43,129,134,135,150,161].
It is important to note that experimental growth rates are generally quoted as
thickness increments per ALD cycle, whereas physically-based models predict mass
increments per cycle from chemical reactions. Direct comparison between a model
and thickness measurements is therefore only possible if the mass density of the
film is known, which can vary considerably with precursor chemistry and reactor
conditions. Unfortunately, film densities are rarely measured and reported along
with thicknesses [38].
4.5 Concluding Remarks
In this chapter, a surface reaction kinetics and film growth model was presented
based on combined reaction energetics and configurational data from quantum-
chemical computational studies and kinetic parameters determined using statistical
thermodynamics and absolute reaction rate theory. The reaction mechanisms which
are responsible for ALD growth were examined at varying hydroxyl group coverage.
Note that precursor exposure (combination of precursor partial pressure and pulsing
time) may significantly impact growth rates as was shown in the previous chapter.
This study was focused to examine the kinetic reaction mechanism and the effects
of surface coverages on kinetic reaction favorability.
118
Results indicate that TMA dissociation reactions on bare alumina oxygen
bridges are kinetically competitive with TMA ligand-exchange reactions with hy-
droxyl sites. Due to the currently available approximate activation energy for disso-
ciative adsorption and the consequential uncertainty in reaction rate, determination
cannot be made as to whether molecular or dissociative adsorption reactions are
always kinetically favorable. However, dissociative adsorption does appear to domi-
nate at low surface hydroxylation, while molecular adsorption reaction activity grows
with increasing hydroxylation and appears to dominate at saturation. Maximum
ALD growth was only achieved at an initially saturating hydroxyl group density.
Together molecular and dissociative adsorption TMA reactions are more kinetically
favored than the other considered reactions and are predominantly responsible for
ALD growth.
As such, subsequent reactions to the initial ligand exchange are significantly
less favored kinetically, with complete removal of methyl groups during TMA expo-
sure being the least favored reaction even at the upper limit of its reaction rate, con-
firming that complete removal of all three Me groups during the TMA half-reactions
is both thermodynamically and kinetically unlikely. Results further indicate the rel-
ative favorability of the water reactions with varying hydroxyl site density. At
higher hydroxyl coverage, the dimethyl and monomethyl aluminum species present
fair competition for the selectivity of incoming water molecules. However, model
predictions suggest that at lower hydroxyl site density, the reaction with the Me1
species will dominate.
Building on previous mechanistic ALD studies by quantifying surface reaction
119
rate kinetic parameters, this work provides detailed insight into reaction selectivity
and surface species dynamics. The surface reaction and growth-surface dynamics
models, when combined with reactor-scale models of precursor transport, consti-
tute useful tools for quantifying the rates of the deposition reactions as well as for
dynamic optimization of commercial ALD processes. This will be described in the
following chapter.
120
5Dynamic ALD Reactor Design
Dynamic reactor-scale models with accurate surface reaction rate expressions
are necessary to identify the ALD process operating limits corresponding to satu-
rating growth conditions. One of the most important contributions of system-level
reactor models for thin-film processes is the ability to use dynamic models to accu-
rately characterize the time-dependent composition of the reactant gases to which
the growth surface is exposed. As such, these models constitute useful tools for the
design of novel reactors, quantifying the rates of deposition reactions, as well as for
optimization of deposition conditions and adaptation to commercial ALD processes
for high-throughput systems.
In this chapter, the ALD surface reaction and dynamic film growth model
developed in Chapter 4 is coupled to a dynamic reactor-scale model for a process
design application. Effects of structural and parametric changes are assessed in an
optimization framework aimed at increasing the ALD growth per cycle. A lumped
parameter dynamic model is employed to study the temporal behavior of the ALD
process and provides a useful tool for novel design diagnostics.
121
5.1 ALD Reactor Designs
ALD reactors share many similarities with their CVD counterparts, but there
are several key distinctions that reflect the unique features of ALD. One is the need
to supply precursors in an alternating fashion hundreds or thousands of times in each
deposition process. Reasonable throughput requires this cycling to be carried out
as rapidly as possible, thereby putting strict demands on both the precursor source
and reaction chamber designs. At the same time, uniform distribution of precursors
across the substrate may be necessary to ensure film growth uniformity, although
this theoretically should not be an issue for a truly self-limiting ALD process.
There are two distinct types of ALD reactors which can be differentiated by
the presence or absence of a carrier gas. The first type is a reactor where precursor
gases are delivered into the reaction chamber without the use of a carrier gas (e.g. by
throttled pumping) and thus the total pressure in the reaction chamber corresponds
to the precursor pressure [118]. After each pulse period, the reaction chamber
is opened and completely evacuated by use of a vacuum pump. So there is no
continuous flow of gases, but rather a series of exposure and evacuation steps. While
this approach leads to longer residence times and thus more efficient use of the
precursors, the evacuation step is often slow which consequently leads to longer
cycle times.
Another type of reactor, the most employed design in practice, is a continuous
flow reactor where precursors are introduced into the reaction chamber with an inert
carrier gas which flows continuously to a downstream vacuum pump. This carrier
122
gas is used for transportation of the precursor species and to facilitate purging, which
is faster than evacuating the reaction chamber between each precursor pulse [31].
Thus, the advantage of this continuous flow reactor is much shorter cycle times
relative to the previously discussed design. Most often nitrogen or argon is used
as the inert gas at flow rates yielding a reaction chamber pressure in the range
1? 10 Torr [140]. In these flow-type reactors, the residence time in the chamber is
often very short due to the high flow rate of the carrier gas, consequently making
the precursor pulse dispersion much smaller when compared to the previous design.
But theoretically, this should not be an issue due to the self-limiting nature of
ALD reactions. Because this is the more widely used design, all further reactor
components will be discussed in the context of the continuous ALD flow reactor.
5.1.1 Precursor Delivery
As previously noted, the primary feature of ALD is separate and sequential
precursor pulsing intervals providing means for self-limiting film growth. This re-
quires a separation of the precursor source and the reaction chamber by a mechanical
valve or some other rapid valving system. To avoid deposition in the source lines,
precursors are commonly transported to the reaction chamber along separate lines,
one for each precursor.
For precursors with sufficient vapor pressure, a pulsing valve separates the
source from the reaction chamber, and, when the valve is opened, precursors are
simply drawn into the carrier gas stream by a pressure differential across the control
valve. Since a pressure drop across the valve is the necessary driving force, it follows
123
that precursor gases (or liquids) must have a vapor pressure higher than the reaction
chamber pressure in order for this precursor delivery method to be effective. In
some cases, the precursor will need to be heated to reach the desired vapor pressure,
in which case the control valve and source lines must also be heated to prevent
condensation. A bubbler may be used for precursors with insufficient vapor pressure
or when a faster delivery rate is desired, e.g. for large surface area applications [140].
In some instances, there may be the desire to carefully control the amount
of precursor gas that enters the reactor in any given pulse period. Of course the
pulsing valve enables one to control dosing level by adjusting the amount of time
that the valve is allowed open. To more accurately control dosing, however, two
valves may be placed in series with a storage volume in between [63, 157]. So, the
configuration of the precursor delivery system would be
source ? control valve ? storage volume ? control valve ? reaction chamber.
First, the valve near the source is opened and the high vapor pressure of the pre-
cursor facilitates flow across the control valve, thereby filling the storage volume.
Then, the first valve is closed and the valve nearest the reaction chamber is opened,
thereby evacuating the storage volume and introducing the precursor into the re-
action chamber. Using this method and monitoring the pressure inside the storage
volume enables one to accurately compute and control the precursor dose.
124
5.1.2 Reaction Chamber
Single-wafer ALD reactors may have different configurations for gas flow?
usually either ?cross-flow? (also called flow-channel or traveling-wave) or ?show-
erhead? (also called perpendicular-flow). In a cross-flow reactor, gases flow parallel
across the substrate, whereas gases flow perpendicular to the substrate in shower-
head designs. When the gas inlet is very near to the substrate, the design may
be better described as radial-flow and shares more similarities with the cross-flow
design in terms of advantages and disadvantages [44].
Compared to the showerhead design, the cross-flow reactor has the advantage
of higher throughput. This speed benefit arises from the simple geometry of the
reaction chamber, in that the height of the channel (distance from the substrate
to the top wall of the flow channel) can be made to be very small, on the order of
millimeters. In these high-aspect-ratio systems, precursor molecules collide multiple
times with the substrate along the length of the reactor, thereby increasing the
probability that it will find an open adsorption site as well as the rate at which
the substrate becomes saturated due to irreversible surface reactions [140]. This
configuration enables increased precursor utility and shortened exposure time, when
compared to a more open reaction chamber design. The small volume of the flow
channel also allows for rapid purging of the reactor, another key feature for high-
throughput designs.
The disadvantage of the cross-flow reactor is its sensitivity to non-idealities
such as precursor decomposition and by-product readsorption, both leading to non-
125
uniformity in film thickness [94]. When precursor molecules undergo irreversible
surface reactions, a by-product is formed which immediately releases to the gas-
phase. This by-product can then compete with precursor molecules for adsorption
and can also effectively decrease the precursor partial pressure, if sufficient by-
product is present. By contrast, the showerhead reactor design theoretically ensures
that each site on the substrate receives the precursor pulse at the same time, thereby
decreasing by-product adsorption competition.
5.1.2.1 Batch Reactors
To improve the cost and time-effectiveness of commercial ALD processes, batch
reactors are often employed to deposit films on multiple substrates simultaneously
[44]. While the cycle length for these reactors is often much longer due to the larger
reaction volume, shortened cycle times can be achieved when substrates are packed
back-to-back to form flow channels similar to the single-wafer cross-flow reactor [62].
Large batch tools of 50-150 wafers (up to 30 cm in diameter) are commercially
available and commonly used in the semiconductor industry [21, 52, 115] as well as
other industries, e.g. 44 substrates measuring 40?50 cm or 82 substrates at 16?27
cm [140,141].
5.2 A Continuous Cross-flow ALD Reactor
The laboratory-scale ALD reactor considered in this thesis is a continuous
cross-flow reactor currently in testing phase and to be used in evaluating ALD for
126
a range of spacecraft-related thin-film applications [30]. As illustrated in Fig. 5.1,
the reaction chamber (RC) consists of a stainless steel process tube surrounded by a
bench-top tube furnace, containing the substrate(s) and a quartz crystal microbal-
ance (QCM) for real-time mass deposition measurements. The reactor performance
will be initially tested with the commonly used ALD precursors of trimethylalu-
minum and water; both are contained in the liquid state in temperature-controlled
laboratory bottles. External induction heating is provided by a furnace which sur-
rounds the reaction chamber, acting as a hot-wall temperature controller (TC). Each
of the precursors flows through a separate sequence of needle valves/orifices to con-
trol their flow rates and, then, through solenoid-activated control valves, CV1-5, to
regulate the precursor dosages and lengths of the purge periods.
A mass flow controller (MFC) is used to regulate the Argon purge gas flow rate,
and low-pressure manometers (P1 and P2) respectively record gas pressure at the
reactor outlet and the small ballast chamber (BC) used to regulate the TMA dose.
Residual gas analysis (RGA) is performed using a mass spectrometer; the primary
pressure controller (PC) consists of a vacuum pump located downstream of the
RGA, and a smaller pump is used to vent water vapor between water doses. Reactor
dimensions and other reactor component specifications are listed in Table 5.1.
127
Table 5.1: Reactor dimensions and primary system component design parameters of
the ALD reactor system shown in Fig. 5.1. TMA, trimethylaluminum.
Parameter Description Value
As substrate growth surface area 6.45? 10?4 m2
A total growth surface area 0.207 m2
L reactor length 0.864 m
AR aspect ratio 11.3
m?I Ar mass flow controller range 0? 500 sccm
R reactor internal radius 0.0381 m
V reactor volume 3.94? 10?3 m3
Vbc TMA ballast chamber volume 7.85? 10?7 m3 (0.02% of V )
St primary exhaust pump capacity 14.6 ft3/min
128
ne
ed
le
 v
al
ve
or
ifi
ce
 /
ne
ed
le
 v
al
ve
CV
1
CV
3
CV
4
CV
5
RC
CV
2
H 2
O
Ar
TM
A
BC
L
su
bs
tra
te
QC
M
R
RG
A
P2
P1
TC TC
PC
M
FC
pu
m
p
F
ig
ur
e
5.
1:
P
ro
ce
ss
an
d
in
st
ru
m
en
ta
ti
on
sc
he
m
at
ic
of
th
e
co
nt
in
uo
us
cr
os
s-
fl
ow
A
L
D
re
ac
to
r.
129
5.2.1 Process Recipe
The carrier gas flows continuously while each precursor is introduced sequen-
tially, separated by a purge period to prevent gas-phase reactions.
pulse A ? purge A ? pulse B ? purge B ? pulse A ? . . .
?A sec ?AP sec ?B sec ?BP sec ?A sec . . .
where the AB cycle repeats?potentially thousands of times?building the film one
submonolayer at a time. After the initial nucleation transient following a change in
the precursor system (e.g., when depositing a nanolaminate consisting of alternating
thin-film materials), the deposition behavior during each AB cycle approaches a
limit-cycle solution, the computation of which is the main focus of this simulation
study. The time length (in seconds) for the AB pulses and purge periods are denoted
as ?A, ?B, ?AP and ?BP , respectively.
Referring to Fig. 5.1, control valves CV2 and CV5, normally are open during
all pulse and purge periods and, so, will not be discussed further. During the purge
period prior to pulse A, CV1 is closed to allow TMA to fill the ballast chamber;
the TMA partial pressure in this chamber can potentially reach the vapor pressure
of TMA in the supply bottle containing liquid-phase TMA. During pulse A, CV1
is opened to allow TMA vapor to flow into the reactor, reducing the pressure in
the ballast chamber. Note that a small flow of TMA will continue through the
orifice/needle valve during pulse A. At the end of pulse A, CV1 is closed, and the
pressure rebuilds in the TMA ballast chamber.
Regardless of the position of CV4, Ar carrier gas flows continuously during
130
all pulse and purge periods. During both purge periods and during pulse A, CV3
is open between the water source and the water purge pump, as well as to CV4;
however CV4 is closed in the direction of CV3, resulting in no water flow to the
reactor. During the water pulse, CV4 is switched to the all-open position, but CV3
is closed in the direction of the water purge pump, allowing the flow of Ar and water
to the reactor. This configuration was designed to prevent condensation in the water
delivery system and to improve the reproducibility of the water dose [82].
5.3 Precursor Characteristics
Development of the ALD reactor system model begins with precursor ther-
mophysical property and precursor delivery system modeling. From the National
Institute of Standards and Technology (NIST) WebBook [114], the Antoine?s equa-
tion coefficients for TMA (between 337?400 K) and water (between 293?343 K) are
determined as
log10 P
vap
TMA = 4.67984?
1724.231
T ? 31.398
+ log10
760
1.01325
(5.1)
log10 P
vap
H2O = 6.20963?
2354.731
T + 7.559
+ log10
760
1.01325
(5.2)
where the vapor pressure units are Torr and T is in K. The vapor pressures computed
by these relationships corresponding to the TMA and water sources are given in
Table 5.2.
As pointed out in [166], at lower temperatures and higher TMA partial pres-
131
sures, the dimer of TMA (d-TMA) is favored over the monomer (m-TMA) in the
gas phase. This is important in determining the true dose values for the precur-
sor delivery system. The d-TMA dissociation extent as a function of temperature
was studied in [92], where values for the degree of dissociation, ?, were given as a
function of temperature. The degree of dimer dissociation is defined as follows
? =
moles of m-TMA
moles of m-TMA + 2(moles of d-TMA)
=
moles of m-TMA
n?A
(5.3)
where n?A is the total number of Al(CH3)3 molecules whether in monomer or dimer
form. Using a least-squares procedure, an exponential expression was fit to the data
contained in the cited source to find
lnKd = ?
13756.5425
T
+ 32.2019 (5.4)
where T again is in K. For a binary mixture of d-TMA and m-TMA, the equilibrium
expression may be written as
Kd =
(ymP
vap
TMA/P
?)2
(1? ym)P
vap
TMA/P
?
(5.5)
for the mole fraction of the monomer ym with P ? = 760 Torr. Note that the m-TMA
mole fraction is related to the degree of dissociation, ? [92], by ym = 2?/(1 + ?)
when considering a binary mixture.
In a mixture containing additional species that do not participate in the dimer-
ization reaction, defining nN as the total moles of species not participating in the
132
Table 5.2: Nominal ALD reactor conditions. d-TMA, dimer of TMA.
Parameter Description Value
m?I nominal Ar molar flow 3.72? 10?4 mol/s
mbase reactor base molar capacity 1.42? 10?4 mol
?res reactor base residence time 0.381 s
Tamb ambient temperature 300 K
Tbc ballast chamber temperature 300 K
T reactor temperature 450 K
Pbase reactor base pressure 1.01 Torr
Pbc,base ballast chamber base pressure 13.67 Torr
?Pz base pressure drop across reactor 6.8? 10?4 Torr
P vapTMA TMA vapor pressure at Tamb 13.67 Torr
P vapH2O water vapor pressure at Tamb 26.82 Torr
? d-TMA dissociation degree at Tamb 4.04? 10?3
x1 base pressure differential ratio at CV1 0.93
x2 base pressure differential ratio at CV2 0.00
x3 base pressure differential ratio at CV3 0.96
?A Pulse A 0.1 s
?AP Purge A 4 s
?B Pulse B 0.1 s
?BP Purge B 4 s
133
d-TMA/m-TMA reaction, the gas-phase m-TMA and d-TMA mole fractions, ym
and yd, can be written in terms of ?, as follows
ym =
2?
1 + ? + 2?
, yd =
1? ?
1 + ? + 2?
(5.6)
where ? = nN/n?A. If Kd corresponds to the d-TMA dissociation equilibrium coef-
ficient determined above, defining
y2m
yd
= ? =
P ?Kd
P
(5.7)
allows for the following.
? =
???+
?
?2?2 + ?(4 + ?)(1 + 2?)
4 + ?
(5.8)
Given the conditions inside the TMA source (T = 300 K, P = P vapTMA and ? = 0), the
extremely small value of ? listed in Table 5.2 shows that TMA leaves the bubbler
essentially entirely as the dimer, d-TMA. This can also be observed in Fig. 5.2,
where the TMA vapor pressure and degree of dissociation are plotted.
5.4 Precursor Delivery System Model
The objective of this model element is to predict the time-dependent precursor
molar flow rates, m?Ain(t) and m?
B
in(t), as they enter the reaction chamber. As seen in
Fig. 5.1, the precursor delivery system is designed to inject a TMA dose regulated
134
300 350 400 450 500
T, K
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(PvapTMA)
L&G 1941
PvapTMA
200
400
600
800
1000
Pva
p
TM
A, 
To
rr
Figure 5.2: TMA vapor pressure and degree of dissociation, ?, as a function of
temperature; data presented in [92] are shown as filled red circles.
by the size of a ballast chamber and the TMA pressure P2 in this chamber prior
to the opening of control valve CV1. A material balance on the ballast chamber is
thus written as
dmbc
dt
= m?CV2
1 + ?bc
1 + ?source
? ?1(t)m?CV1 (5.9)
where ?source and ?bc are the degrees of d-TMA dissociation in the TMA source and
ballast chamber, respectively, computed using Eqn. (5.8) with ? = 0. The function
?1 indicates the time-dependent position of CV1. CV2 is always open under typical
operating conditions, allowing for finite gas flow rate whether or not CV1 is open.
m?CV1 and m?CV2 are the molar flow rates through control valves CV1 and CV2,
respectively.
Following this, the molar flow rate of TMA from the ballast chamber into the
135
reaction chamber is computed by
m?Ain(t) = ?1(t)m?CV1
1 + ?rxr
1 + ?bc
(5.10)
where ?rxr is the d-TMA dissociation degree under reactor pressure and composition
conditions and is computed using Eqn. (5.8) and ?rxr.
Because of the potential for condensation of water in a ballast chamber for this
precursor, the alternative design developed by [82] is used where water evaporating in
the H2O source is vented using an auxiliary purge pump during the TMA pulse and
purge periods (a similar approach cannot be used for TMA, because of the expense
of discarding unused precursor). This configuration gives rise to a relatively simple
model for the water dose
m?Bin = ?3(t)m?CV3 (5.11)
where the function ?3 indicates the time-dependent position of CV3 and m?CV3 is
the molar flow rate of water through control valve CV3.
The molar flow rate of gases into the reactor is ultimately dependent on the
pressure differential between the reaction chamber and precursor storage vessels.
The reactor and ballast chamber pressures are computed by
P1 ? P =
mRgT
V
, P2 = Pbc =
mbcRgTbc
Vbc
(5.12)
where P is the reactor pressure, approximately measured by manometer P1.
136
5.4.1 Universal Gas Sizing Flow Equation for Control Valves
The dependence of gas molar flow rate through the control valves would be
a significant challenge to predict by a purely physically-based modeling approach.
To perform such computations would require (at a minimum) the size and shape
of the orifice, the diameter of the pipe, the pressure drop across the valve, and the
fluid density. But in addition to this, all of the valve passage dimensions and all
of the changes in size and direction of flow through the valve would be required, as
well. However, rather than doing these computations, the valve flow coefficient Cv
is employed, which combines the effects of all flow restrictions in the valve.
Valve manufacturers determine the valve flow coefficient by testing the valve
with water at several flow rates, using a standard test method developed by the
International Society of Automation (formerly the Instrument Society of America)
for control valves and now used widely for essentially all valves [153]. Following
standards set forth by the American National Standards Institute (ANSI) and the
International Society of Automation (ISA) [9], the flow rate of a compressible fluid
through a control valve can be determined by
m? = NCvPin
(
1?
x
3F?xT
)?
x
GgTinZ
(5.13)
where Pin is the upstream (or inlet) pressure and x = ?P/Pin = (Pin?Pout)/Pin is
the differential pressure between upstream and downstream pressure taps. Gg is the
gas specific gravity, the density ratio of the gas to air with both at standard condi-
137
tions, which is considered in this practice to be equal to the ratio of the molecular
weights. Tin is the absolute temperature of the inlet stream, Z is the compressibility
factor of the fluid (Z = 1 for ideal gases), and N is a numerical constant for units.
F? is a specific heat ratio factor and xT is the critical pressure differential ratio
factor at choked flow. These terms will be discussed further. Parameters for the
valve flow model are listed in Table 5.3.
The parenthetical expression in Eqn. (5.13) is known as the expansion factor
and accounts for the change in density as the fluid passes from the valve inlet to
the vena contracta (the location just downstream of the orifice where the jet stream
area is a minimum). It also accounts for the change in the vena contracta area
as the pressure differential is varied. If the inlet pressure is held constant and the
outlet pressure pressure is progressively lowered, the mass flow rate through a valve
will increase to a maximum limit, a condition referred to as choked flow. Further
reductions in outlet pressure will produce no further increase in flow rate. This limit
is reached when the pressure differential x reaches the value of F?xT .
The limiting value of x is defined as the critical differential pressure ratio.
Thus, the value of x, for computational purposes, is held to this limit. Although
the actual pressure differential ratio may be greater, x is not to exceed F?xT . If
x ? F?xT , then the x value is replaced with F?xT , reducing Eqn. (5.13) to
m? =
2
3
NCvPin
?
F?xT
GgTinZ
(5.14)
for the choked flow regime. Note that the factor xT is determined experimentally and
138
Table 5.3: Parameters for precursor delivery model. Valve flow equations yield flow
rates in units of mol/s with pressure in bar and temperature in K.
Parameter Description Value Source
N numerical units constant 4.889 [9]
Cv Swagelok valve flow coefficient 0.27 [154]
F?,A specific heat ratio factor, TMA 0.8 [9]
F?,B specific heat ratio factor, water 0.9 [9]
xT critical pressure differential ratio factor 0.55 [9]
Gg,A gas specific gravity, TMA 2.489
Gg,d-A gas specific gravity, TMA dimer 4.889
Gg,B gas specific gravity, water 0.622
Z compressibility factor 1
given for certain valve types. This value is based on air near atmospheric pressure
as the flowing fluid with a specific heat ratio of 1.40. The specific heat ratio factor
(F? = ?/1.40) is thus used to adjust xT using the appropriate specific heat ? for the
flowing fluid.
5.4.2 Development of a Continuous Flow Equation
The discontinuity introduced by the switch between non-choked and choked
flow may bring rise to numerical difficulties when implementing these flow equa-
tions into an iterative Newton-Raphson procedure. The Jacobian matrix elements
necessary for this procedure can be computed by finite differences; typically sam-
ples are selected within a specified range of a function?s input values to determine
the derivatives based on the function?s response to such sampling. A discontinuous
function or one with a discontinuous derivative could be problematic in numerical
model implementation, e.g. as in the models presented herein.
To avoid these numerical issues, an alternative valve flow equation is proposed
139
where a single continuous flow equation is used for both choked and non-choked
flow. Noting the behavior of Eqns. (5.13) and (5.14) with respect to ?P and x, a
continuous equation is formulated using maximum flow (choked flow) as a horizontal
asymptote approached as x??.
m? = m?choked
x
?
x2 + (2/3)2
(5.15)
m? =
2
3
NCv?P
?
F?xT
GgTinZ
?
1
x2 + (2/3)2
(5.16)
A continuous flow equation may also be formulated using a linear approximation as
follows.
m? = m?chokedx (5.17)
m? =
2
3
NCv?P
?
F?xT
GgTinZ
(5.18)
Comparative values of the molar flow rates predicted by these equations are
shown in Table 5.4 for a randomized set of pressures. To generate this data, the inlet
and outlet pressure was randomized with Pin ? [0, 5] bar and Pout ? [0, 5] bar and
then used to compute the pressure differential ratio x. These values were then sup-
plied to each molar flow rate equation. From the randomized pressure experiment
data shown in Table 5.4, one can see that the continuous flow equation approxi-
mations are typically within an order of magnitude of the standard ANSI/ISA flow
equations, improving in accuracy as the pressure differential increases to the choked
140
Table 5.4: Selection of data for comparison of water molar flow rate predictions
(mol/s) from randomized pressure differentials. Eqns. (5.13) and (5.14) are stan-
dard flow equations based on ANSI/ISA guidelines [9]. Eqns. (5.16) and (5.18) are
asymptotic and linear approximations, respectively, to the standard equations.
m? m? m?
?P x (5.13, 5.14) (5.16) (5.18)
0.040 0.014 0.032 0.003 0.002
0.531 0.118 0.137 0.036 0.024
0.928 0.206 0.170 0.060 0.042
1.530 0.313 0.209 0.094 0.069
0.442 0.402 0.049 0.026 0.020
2.248 0.531 0.192 0.120 0.102
1.773 0.640 0.126 0.087 0.080
2.164 0.745 0.132 0.098 0.098
3.889 0.840 0.210 0.164 0.176
3.774 0.938 0.182 0.149 0.171
flow regime, a behavior that is to be expected.
This must be considered in context of the ALD reactor to be modeled. From
the nominal reactor conditions listed in Table 5.2, one can see that the pressure in
the reaction chamber is significantly lower than pressures in the precursor source
containers. An analysis of the nominal pressure differential ratios shows choked
flow at CV1 and CV3, while no pressure differential exists at CV2, signifying initial
equilibration of the ballast chamber pressure with the TMA source. However, as
CV1 is opened and the ballast chamber pressure decreases, then flow will move from
non-choked and potentially choked flow across CV2. Because choked flow appears to
be the primary regime in the ALD reactor system modeled herein, the lower accuracy
of the approximations in non-choked flow may not be significant, except at times for
the ballast volume pressure differential with respect to the TMA source. However,
141
this does not significantly impact flow rate predictions for TMA into the reaction
chamber, since the near vacuum reactor pressure will ultimately yield choked flow
pressure differentials.
Graphical representation of the randomized pressure experiment used to gen-
erate the data in Table 5.4 is shown in Figs. 5.3 and 5.4. The scatter plot arrays
are used to demonstrate the qualitative behavior of the valve flow rate equations
with respect to ?P and x for a typical range of pressure differentials x ? [0, 1]. One
can see that, when combined, the two standard flow rate equations reveal a some-
what asymptotic relation bounded by choked flow, suggesting that a continuous
asymptotic equation would be an appropriate approximation tool.
In both figures, the standard non-choked (NC) flow equation predictions are
shown in blue with the standard choked (C) flow equation predictions shown in
red. The asymptotic approximations (AA) and linear approximations (LA) are both
shown in white overlays to the standard equations. From these plots, one can see the
boundaries of all flow equations. The standard non-choked flow equation predicts a
maximum flow rate, after which point the rate decreases with increase in pressure
differential. Of course, this does not make physical sense. On the other hand,
the standard choked flow equation predicts a flow rate only dependent on the inlet
pressure, which overshoots the flow rate prediction in the low pressure regime. One
thus switches between these two equations to yield accurate predictions depending
on the pressure differentials across the control valve. This is the standard routine
for determining volumetric valve flow rates [9].
Instead of switching between Eqns. (5.13) and (5.14), the asymptotic and
142
0.3
0.2
0.1
0.0
0.1
0.2
0.3
m?
, m
ol
/s
NC C NC + C
0.3
0.2
0.1
0.0
0.1
0.2
0.3
m?
, m
ol
/s
w/ AA w/ AA w/ AA
6 4 2 0 2 4 6
?P, bar
0.3
0.2
0.1
0.0
0.1
0.2
0.3
m?
, m
ol
/s
w/ LA
6 4 2 0 2 4 6
?P, bar
w/ LA
6 4 2 0 2 4 6
?P, bar
w/ LA
Figure 5.3: Water molar flow rate predictions for a randomized set of pressures using
the non-choked (NC) and choked (C) flow Eqns. (5.13) and (5.14). The asymptotic
approximations (AA) of Eqn. (5.16) and linear approximations (LA) of Eqn. (5.18)
for the same random set of pressures are overlayed, illustrating the approximation
boundaries with respect to the standard flow rate formulations.
linear approximation equations suggested herein are used. Specifically, Eqn. (5.16)
maintains the qualitative relationship between the molar flow rate and pressure
differentials, while also maintaining agreement within an order of magnitude to the
standard equations for the pressures of interest in the ALD reaction chamber.
Note that the additive term in the square root of Eqn. (5.16) may be adjusted
to better fit the predictions of the standard flow equations, yielding a steeper slope
near the origin, as is seen in the non-choked flow rate predictions of Figs. 5.3 and 5.4.
143
0.3
0.2
0.1
0.0
0.1
0.2
0.3
m?
, m
ol
/s
NC C NC + C
0.3
0.2
0.1
0.0
0.1
0.2
0.3
m?
, m
ol
/s
w/ AA w/ AA w/ AA
1.5 1.0 0.50.0 0.5 1.0 1.5
x
0.3
0.2
0.1
0.0
0.1
0.2
0.3
m?
, m
ol
/s
w/ LA
1.5 1.0 0.50.0 0.5 1.0 1.5
x
w/ LA
1.5 1.0 0.50.0 0.5 1.0 1.5
x
w/ LA
Figure 5.4: Water molar flow rate predictions for a randomized set of pressures using
the non-choked (NC) and choked (C) flow Eqns. (5.13) and (5.14). The asymptotic
approximations (AA) of Eqn. (5.16) and linear approximations (LA) of Eqn. (5.18)
for the same random set of pressures are overlayed, illustrating the approximation
boundaries with respect to the standard flow rate formulations. Non-choked and
choked flow regions are shaded blue and red, respectively.
However, the flow equation derivatives with respect to pressure become infinitely
large in this region, potentially causing additional convergence issues in the numeri-
cal model. For this reason, the squared 2/3 is used. This also has physical meaning:
it is the expansion factor under choked flow. When x ? 0, this term cancels with
the expansion factor in the equation, yielding an expansion factor of approximately
one, consistent with low pressure-drop in non-choked flow.
144
5.5 Reactor Model
With the precursor delivery model in place, the reactor model can now be
developed. Defining m(t) as the total moles of gas-phase species in the reaction
chamber, m?in, the total reactor feed molar flow rate, and m?out, the molar flow of
residual gas out of the reactor, an overall reactor material balance gives
dm
dt
= m?in ? m?out ?
(
?A + ?B + ?C + ?D
)
(5.19)
The instantaneous consumption rates of the TMA and water precursors ?A and ?B,
methane by-product ?C , and gas-phase side-reaction product ?D are based on the
reaction kinetics model of Chapter 4. The surface phase rate equations are coupled
to the gas phase through the precursor partial pressures PA and PB. To compute the
rate of gas-phase depletion of the precursors due to the surface (and possibly gas-
phase) reactions and the rate of by-product production, the rates of consumption
(positive quantities) of TMA, water, methane, and DMA-OH must be determined.
These are denoted as ?A, ?B, ?C , and ?D respectively:
?A = (v3 + v13)?MeA+ v27V molecules/s (5.20)
?B = (v16 + v19 + v24)?XA+ v27V (5.21)
?C = ?[v3?Me + (v16 + v19)?X ]A? v27V (5.22)
?D = ?v27V (5.23)
145
Vacuum pump capacity is rated by the volumetric pumping capacity per unit
time, St, given in Table 5.1. Over the pump?s operating range, this value is consid-
ered pressure-independent. Based on m, the reactor instantaneous total pressure is
computed using the ideal gas law Eqn. (5.12); so, the residual gas molar flow rate
can be computed by
m?out =
StP
RgT
=
mSt
V
(5.24)
The total molar feed to the reactor is the sum of the precursor and inert flow rates,
and so
m?in = m?
A
in(t) + m?
B
in(t) + m?
I
in(t)
= ?1(t)m?CV1
1 + ?rxr
1 + ?bc
+ ?3(t)m?CV3 + m?CV5 (5.25)
Under steady flow conditions with no precursor feed, such as at the end of a very long
purge period, no surface reactions take place, and so, the baseline reactor pressure
is defined using m?out = m?in = m?Iin and
Pbase =
RgT
St
m?Iin (5.26)
which for a nominal argon flow m?Iin gives Pbase listed in Table 5.2.
Defining yA as the gas-phase mole fraction of m-TMA plus d-TMA and yB, yC
and yD as, respectively, the gas-phase water, methane, and DMA-OH mole fractions,
146
the dynamic species conservation equations can be written as follows
dm
dt
= m?in ?
mSt
V
?
(
?A + ?B + ?C + ?D
)
(5.27)
m
dyA
dt
+ yA
dm
dt
= m?Ain ?
mSt
V
yA ? ?
A (5.28)
m
dyB
dt
+ yB
dm
dt
= m?Bin ?
mSt
V
yB ? ?
B (5.29)
m
dyC
dt
+ yC
dm
dt
= ?
mSt
V
yC ? ?
C (5.30)
m
dyD
dt
+ yD
dm
dt
= ?
mSt
V
yD ? ?
D (5.31)
subject to initial conditions, yA(0), yB(0), yC(0), yD(0), m(0), the time-varying state
of the growth surface and the total molar inlet flow m?in, given by Eqn. (5.25). Note
that at this time it is unclear whether d-TMA or only m-TMA can participate in
the adsorption reactions, and so, the total yA is used.
5.6 Limit-Cycle Computations
The surface reaction rate equations are coupled to the gas phase through the
precursor partial pressures, PA and PB, which are used in the surface kinetics model
as shown in Chapter 4. Likewise, the rates of precursor depletion and by-product
production are computed as functions of the surface and possible gas-phase reaction
rates. At this point, the complete model is written as
d?
dt
= f(?) (5.32)
147
and the differential equation right-hand sides, model variables and process recipe
parameters can be collected in the following table
modeling differential equations f = [(5.9), (5.27)?(5.31), (4.5)- (4.11)]T
process variables ? = [mbc,m, yA, yB, yC , yD, ?Me1 , ?Me2 , ?X ]
T
process recipe ?A, ?AP , ?B, ?BP , Vbc, T = 450 K
where yI = 1? yA ? yB ? yC ? yD and the length of the full process cycle is
?cycle = ?A + ?AP + ?B + ?BP (5.33)
All simulations are implemented in the Python programming language, making ex-
tensive use of the PyLab and Numdifftools modules. Therefore, any computationally
specific discussions that follow will be in the context of a Python implementation.
5.6.1 Time Discretization for Forced-periodic Systems
The solution strategy implemented for the dynamic ALD process is to only
consider the limit-cycle solutions that describe steady (but periodic) operation of
the reactor system during the homodeposition ALD phase; i.e. nucleation and other
events leading to transients spanning multiple exposure cycles are not considered.
Computation of limit-cycle solutions naturally lend themselves to a two-step proce-
dure, where the modeling Eqn. (5.32) is first discretized in time over each pulse and
purge period using global collocation over each of the four periods (?A, ?AP , ?B, ?BP ),
enforcing continuity between each interval, effectively resulting in periodic bound-
ary conditions between the end of purge B and the start of the next A pulse. The
148
resulting nonlinear equations, then, are solved using the Newton-Raphson method.
To implement the collocation procedure, the format of the Python array used
to represent the time-discretized vector of state variables, ? ? Rn, as ? ? Rmn,
must first be defined. For reasons advantageous to computing the Jacobian array
elements in the Newton procedure, the Python list, ?, is defined as a list of the
process variables, where ?(i, j) is state j at point i in time; defined in this manner,
?(i) is a list representing a snapshot of the states at a specific time. Given this
format for the discretized states, the discretized form of Eqn. (5.32) can be written
as
??A?? f(?) = 0 (5.34)
where the ? list is flattened to the shape appropriate for matrix multiplication using
the Python ravel method, and ??A is defined below.
With An?n corresponding to the standard Lobatto first-order differentiation
array (computed using either finite differences or using polynomial collocation tech-
niques), the discretization array suitable for vectors of discretized states, A?, is cre-
ated from diagonal m?m arrays from elements of A
A?
mn?mn
=
?
?
?
?
?
?
?
?
?
?
?
?
I1,1 0 . . . 0
... a2,2
. . .
an,1 an,n
?
?
?
?
?
?
?
?
?
?
?
?
with ai,j =
?
?
?
?
?
?
?
?
?
?
?
?
Ai,j 0
Ai,j
. . .
0 Ai,j
?
?
?
?
?
?
?
?
?
?
?
?
m?m
Note that the identity array, I1,1, has dimensions n ? n and is used to satisfy the
149
initial conditions. In this study, each of the discretized intervals uses n collocation
points (including both endpoints). Writing the vector of discrete points in time over
each pulse and purge period,
t =
?
?
?
?
?
?
?
?
?
?
?
?
tA
tAP
tB
tBP
?
?
?
?
?
?
?
?
?
?
?
?
, ??A =
?
?
?
?
?
?
?
?
?
?
?
?
A?A 0 0 P
C A?AP 0 0
0 C A?B 0
0 0 C A?BP
?
?
?
?
?
?
?
?
?
?
?
?
where the off-diagonal blocks, C, are used for continuity across the spline point, and
the off-diagonal block P is used to enforce periodicity.
5.6.2 Newton-Raphson Procedure
With the discretization complete, the Newton-Raphson procedure is written
in terms of the residual, r, update, u, and refined solution estimate, ?, at iteration
?.
r? = ??A?? ? f(??) (5.35)
u? = [J? ]?1r? (5.36)
??+1 = ?? + u? (5.37)
While perfectly standard, the Newton-Raphson procedure is presented to point out
the structure of the Jacobian array. Numerical approximation of the full Jacobian
150
array does not take advantage of its structure
Jmn?mn = ??A?
?
?
?
?
?
?
?
?
?
?
?
?
[
?f
??
]
i=0
0 ? ? ? 0
0
[
?f
??
]
i=1
? ? ? 0
...
...
. . .
...
0 0 ? ? ?
[
?f
??
]
i=n?1
?
?
?
?
?
?
?
?
?
?
?
?
(5.38)
This limits the extent to which a finite-difference procedure must be applied to
compute the block-diagonal Jacobian elements corresponding to the relatively com-
plicated nonlinear terms in the rate expressions, precursor thermodynamics descrip-
tions, and reactor material balances. The Jacobian array J is then constructed in a
block-diagonal manner, calling the Python function, numdifftools.Jacobian, for each
(collocation) point in time to define the block-diagonal elements of Eqn. (5.38).
5.7 Representative Results
A limit-cycle solution is presented in Fig. 5.5 corresponding to a base case
design, both in terms of the reactor component specifications and the process recipe.
The nominal design parameter values have been listed in Table 5.1. Four sets of plots
are presented in Fig. 5.5 in which subplots illustrating the TMA ballast chamber,
reactor gas phase, and growth surface composition dynamics are shown. One can
observe that all states conform to periodic boundary conditions over the processing
cycle; the markers indicate the locations of the temporal collocation points and the
shaded rectangles, the collocation interval endpoints.
151
2
4
6
8
10
12
14
pr
es
su
re
, T
or
r A purge B purge
Pbc
PvapTMA
0.0
0.5
1.0
1.5
2.0
pr
es
su
re
, T
or
r
P
TMA
H2O
Ar
0.000
0.001
0.002
0.003
0.004
0.005
pr
es
su
re
, T
or
r
CH4
DMA-OH
0 1 2 3 4 5 6 7 8
time, sec
0.0
0.2
0.4
0.6
0.8
1.0
, 
Me
Me1
Me2
X
bc
rxr
Figure 5.5: Representative reactor limit-cycle solution with gpc = 0.806 A?/cycle.
Reactor nominal conditions consist of ?A = 0.1 s, ?AP = ?BP = 4 s, ?B = 0.1 s and
Vbc/V = 0.02%.
5.7.1 TMA Ballast Chamber Dynamics
The full process limit-cycle is shown to begin with pulse A, where the valve be-
tween the ballast chamber and reactor is opened; the valve between the TMA source
and ballast chamber (CV2) is always open through the entire process cycle. During
pulse A, the TMA flows from the ballast chamber to the reaction chamber, reducing
the pressure of the former. At the end of pulse A, the ballast chamber/reactor valve
CV1 is closed, allowing the TMA pressure to rebuild during the subsequent purge
and B pulse intervals.
152
As seen in Fig. 5.5, top, the ballast chamber pressure behaves exactly as
expected, with a rapid initial drop in pressure, due to the small volume Vbc. However,
what is interesting to observe is the degree of d-TMA dissociation ?bc in the bottom
plot of Fig. 5.5: except for a slight upward deviation during the ballast chamber
depressurization, ?bc ? 0, indicating that virtually all of the TMA is in dimer form
while in the TMA ballast chamber.
5.7.2 Reactor-scale Dynamics
As seen in the center plots of Fig. 5.5, during pulse A, the total reactor pres-
sure, TMA partial pressure and methane partial pressure all increase as expected,
while the Ar carrier gas partial pressure remains constant. During the subsequent
purge period, the total pressure relaxes to the baseline pressure. It is now interest-
ing to observe that the TMA monomer fraction in the reactor is essentially unity
(?rxr ? 1), indicating that the d-TMA dissociates as it enters the lower-pressure,
higher-temperature reaction chamber. Note that the energy required to heat the
incoming reactant and inert gases is negligibly small compared to the rates of radia-
tive heat transfer in the reactor, and so, the thermal dynamics of the gases as they
are heated to T from Tamb or Tbc are not explicitly modeled.
During pulse B, a steeper increase in total pressure is observed, because water
is supplied to the reactor from a water vapor source held at constant pressure with
P vapwater  Pbase. Again, the system relaxes to the baseline pressure Pbase after valves
CV3 and CV4 are switched to their purge positions.
153
5.7.3 Growth Surface Dynamics
At the start of pulse A in the limit-cycle solution, ?Me ? 0 and ?X ? 0.6.
As TMA enters the reactor, a small fraction rapidly reacts with the surface OH,
leading to a reduction in ?X and the increase in ?Me shown in Fig. 5.5. With suf-
ficient TMA dosage levels, ?Me ? 1 very rapidly, indicating the aggressiveness of
the adsorption/ligand-exchange reaction as evidenced by the steep increase in ?Me2 .
As the growth surface saturates with Me groups, the rate of the adsorption/ligand-
exchange reaction slows, and the rate-controlling step switches to the second ligand-
elimination reaction. Observe that additional ligand elimination continues through-
out the purge period, reducing both the surface Me2 and X ligand density. However,
the total Me density decreases only slightly, indicating that some TMA adsorption
and ligand-exchange reactions continue throughout the purge period until all gas-
phase TMA is swept from the reactor.
When water is introduced during pulse B, ?Me rapidly drops as the surface
Me groups are replaced with OH in both the dimethyl and monomethyl reactions.
As the water partial pressure drops during purge B, all reactions, except residual
ligand elimination, come to a stop, leaving a nearly unchanging growth surface for
much of the purge period. The full length of the purge period is required, however,
to prevent remaining gas-phase H2O from reacting with surface Me and gas-phase
TMA once pulse A resumes.
154
5.8 Mapping the gpc Behavior
As described earlier, a distinguishing feature of ALD processes is the self-
limiting nature of the surface reactions, leading to a fixed gpc during steady, but
cyclic, reactor operation. The total number of Al and O atoms deposited per unit
area over one deposition cycle are denoted as NAl and NO atoms/nm2, respectively;
these values are computed by integrating the consumption rates of both precursors,
as shown in Section 4.4.2, over the limit-cycle using the quadrature weights of the
collocation procedure.
Alternatively, if ? is the extent of reaction for TMA adsorption/ligand ex-
change, and this is assumed the only reaction involving gas-phase TMA (i.e. ne-
glecting dissociative adsorption), then
NAl = ? =
[Me]f ? [Me]o + [X]o ? [X]f
3
(5.39)
where the subscripts, o and f , denote surface concentrations at the start of pulse
A and at the end of purge A, respectively. Note that this relationship holds only
when no reactions take place under CVD conditions (where both gas-phase pre-
cursors are found in the reactor, resulting in the possibility of all reactions taking
place simultaneously). Under these idealized ALD reactor operating conditions and
under fully saturating conditions when dissociative TMA adsorption is negligible,
[Me]o = [X]f = 0 and [Me]f = [X]o = [M?e], resulting in Eqn. (5.39), reducing to the
maximum gpc possible for an idealized ALD process, a value derived in Chapter 3
155
Figure 5.6: Growth per cycle (gpc) as a function of ?B (left) and Vbc/V (right).
as GPC = 1.231 A?/cycle.
The gpc = 0.806 A?/cycle of the base case design, corresponding to the limit-
cycle solution of Fig. 5.5, is much less than the maximum indicated by Eqn. (3.20)
for idealized, saturating ALD conditions. Thus, two modes of increasing the dose of
each precursor will be examined. During pulse B, the water dose is regulated by the
timing of the gas delivery system valves, CV3 and CV4. With a base case design of
?B = 0.1, observe in Fig. 5.6, left that gpc ? 0 as ?B ? 0, while keeping Vbc fixed,
exactly as expected. Note that ?B = 0 actually corresponds to a bifurcation point,
where the branch containing the limit-cycle solution shown in Fig. 5.5 meets a trivial
solution characterized by ?X = ?Me = 0 for all points in time. The physical meaning
of the multiple solutions will be explored in follow-up work. As ?B is increased
from the nominal operating conditions, the rate of gpc increase lessens; the CVD
conditions and the slower surface reactions contribute to the gradual increase in
gpc with no self-limiting regime to be found under the selected set of operating
conditions of the plot.
156
The base case ratio of the TMA ballast chamber/reaction chamber volume is
0.02%, and one expects that increasing this ratio will result in an increased TMA
dose in the reactor vessel. Keeping the H2O dose fixed, the overall results are seen in
Fig. 5.6, right. As Vbc ? 0, gpc ? 0.78 A?/cycle?not zero?because ?A is nonzero,
and the TMA bleeding through valve CV2 always results in a nonzero TMA dose.
Observe that gpc grows with Vbc, but again, while the rate of gpc increase declines
with ballast chamber volume, a plateau indicating saturating growth is not observed
under the operating conditions of the plot.
5.8.1 The Vbc-?B Plane
Because the gpc values of Fig. 5.6 neither reach a limiting value nor the pre-
dicted theoretical maximum, gpc? GPC = 1.231 A?/cycle, the gpc is presented as
a contour plot in the Vbc-?B plane in Fig. 5.7. In this figure, the base case design
corresponds to the lower-left corner of the plot.
In Fig. 5.7, gpc = GPC = 1.231 A?/cycle is marked by the white contour line
found over this range of ?B and Vbc values. The large region above and to the right
of the curve corresponds to large doses of both TMA and water, resulting in growth
exceeding what would be possible for a pure, surface reaction-limited ALD process.
Examining the limit-cycle solution in this region reveals that because the amount of
TMA and water is so large, a significant amount remains after each purge period,
resulting in each of the precursors being found in the gas phase at the start of the
other pulse period.
During pulse A, excess water in the gas phase reacts with surface Me de-
157
0.2 0.4 0.6 0.8 1.0
Vbc/Vrxr %
0.1
0.2
0.3
0.4
0.5
0.6
B 
se
c
CVD conditions
base-case design
1 . 2
e - 0
5
1 . 1
e - 0
5
9 . 0
e - 0
6
6.0
e-0
6
0.80
0.88
0.96
1.04
1.12
1.20
1.28
1.36
1.44
gp
c A
/cy
cle
?
%
?%
Figure 5.7: Growth per cycle (gpc) as a function of ?B and Vbc/V . The black curves
correspond to moles of TMA/cycle supplied to the reactor.
posited by TMA, increasing the ability of TMA to adsorb and react during pulse A.
Likewise, excess TMA present at the start of pulse B generates surface Me, which
subsequently react with gas-phase H2O, adding to the overall consumption of that
reactant. Likewise, gas-phase reactions occur more readily in this non-ideal region.
While the ALD growth model currently does not yet account for film growth as a
result of gas-phase reactions, precursor consumption and by-product increase are
noted.
These unintentional reactions are characterized as being under CVD conditions
instead of true ALD reactions. Because CVD conditions are generally undesirable in
ALD processes, due to the poor film quality and reduced conformality produced by
the resulting reactions, the curve corresponding to gpc = GPC is marked to indicate
an approximate lower limit, where reactions under CVD conditions are significant.
158
Thus, ALD reactor operation should be limited to the region below and to the left
of this curve.
The practical upshot of this computation is immediate: there is little incentive
for process designs where Vbc/V > 0.25% and that, generally, ?B > 0.2 s (given that
all other parameters are fixed, of course). The rationale for these limits is further
clarified by considering the economics of this ALD process: given the relative value of
TMA to water, plus the costs of disposing unused TMA downstream of the reactor, a
simple yet reasonable optimization objective would be to minimize TMA use alone.
To quantify TMA use, the TMA flow rate through CV1 is integrated using the m?Ain
term of Eqn. (5.25)
mAcycle =
? ?cycle
0
?1(t)m?CV1(t)
1 + ?rxr
1 + ?bc
dt. (5.40)
These curves are shown in black in Fig. 5.7, where the values indicated correspond to
moles of TMA/cycle supplied to the reaction chamber. As expected, the contours
show a reduction of TMA use as Vbc is decreased, but note that for large dose
volumes, both ?B and Vbc affect mAcycle, due to the increased time for regenerating
the TMA pressure in the ballast chamber during pulse B.
5.9 Concluding Remarks
In this chapter, a laboratory-scale atomic layer deposition (ALD) reactor sys-
tem model was developed by integrating components describing the precursor ther-
mophysical properties, precursor delivery system, reactor-scale gas-phase dynamics,
159
and surface reaction kinetics derived from absolute reaction rate theory. ALD reac-
tor operation was limited to steady cyclic operation with limit-cycle solutions com-
puted using a collocation discretization scheme in time. This study demonstrated
that a key advantage to the fixed-point approach was the resulting unambiguous
definition of growth per cycle, making possible parametric studies of film growth
rates to that expected for ideal ALD.
The utility of the resulting ALD system simulator was demonstrated by the
strong interactions found between different reactor, reaction, and process recipe
elements, interactions that would be otherwise difficult to predict without such
simulators. In particular, this work demonstrated that surface reactions normally
associated with one specific precursor exposure can take place during the purge or
even during exposures to the other of the two precursors. The ability to model the
interaction between pulse and purge periods was critical to uncovering the surface
reactions occurring under CVD conditions and identifying processing regimes where
these reactions are most likely to take place.
Physically-based models have the capabilities to decouple the effects of pre-
cursor pulsing time, reactor temperature and pressure, and the dynamics of each
exposure period from the growth per cycle, and can thus distinguish saturating from
non-saturating ALD growth conditions. The ability to predict both gpc and cyclic
precursor feed rates for a real reactor and gas delivery system made it possible to
generate simple, but physically meaningful, design rules for adjusting the precursor
doses to minimize TMA consumption and undesirable CVD conditions, while main-
taining a high value of gpc necessary for acceptable reactor throughput. One of the
160
most important contributions of reactor system-level models for thin-film processes
is the ability to use dynamic models to accurately characterize the time-dependent
composition of the reactant gases to which the growth surface is exposed. Because
direct experimental measurement of gas-phase characteristics local to the growth
surface are challenging at best, physically-based reactor models are needed to inter-
pret measured gpc levels. The utility of models of this form extend to other ALD
process chemistries, other gas delivery systems and more complex (e.g., multi-wafer)
reactor systems. The extension of this modeling work to the tubular geometry of
the ALD reaction chamber is examined in the chapter to follow.
161
6Distributed Reactor Model
In this chapter, the design model presented in Chapter 5 is extended by the
development of a distributed transport dynamics model, described by partial dif-
ferential equations, to take into account the spatial gradients inside the reaction
chamber. Although this adds considerable computational complexity to the prob-
lem, this work is motivated by a desire to understand film thickness non-uniformity
experienced in certain flow-type reactors; recall that non-uniformity is uncharacter-
istic of self-limiting ALD growth.
In cross-flow ALD reactors, different substrate areas can experience substan-
tially different exposures to by-product gases. If film growth begins only near the
leading edge of the substrate, this part is exposed to only a small amount of by-
product molecules while the trailing edge will receive a much higher amount [94,140].
Thus, because the trailing edge of the substrate is exposed to a more depleted precur-
sor flow than the leading edge, the observed growth rate across the substrate may be
inconsistent [20,52,64]. Spatial non-uniformity was likewise noted in plasma-assisted
ALD processes, but reasoned due to a change in growth regime from reaction- to
162
recombination-limited [85].
The rigorous reactor-scale transport model described in this chapter incorpo-
rates the dynamic surface reaction model governing alumina ALD growth outlined in
Chapter 3. Through this modeling description, the mass deposited by film growth
was conveniently expressed in terms of the spatial and temporal evolution of the
surface state for each half-cycle. In this way, the film growth rate per limit cycle
and its relative uniformity can be unambiguously defined. Thus, the purpose of this
study is to describe a generally-applicable model of ALD precursor reaction and
transport that allows a clear identification of the links between growth conditions,
precursor properties, and surface coverage profiles.
6.1 Continuous Cross-flow ALD Reactor Model
Reactor geometry and gas-flow conditions are vital elements in film deposition
by ALD. Variables such as precursor partial pressure, pulse/purge time, flow rate,
reactor dimensions, and reactor layout must be optimized. It is therefore useful to
develop models of the complete reaction chamber?flow of gas from the inlet, through
the reactor, across the reactive surface, and to the exhaust [36].
In this section, a physically-based model of the continuous cross-flow ALD
reactor (see Section 5.2) with temporally separated precursor pulsing is presented.
The model component describing the reactor-scale gas-phase dynamics is coupled
with the surface reaction and growth dynamics model presented in Chapter 3 and
the precursor delivery model presented in Section 5.4 to describe the spatial and
163
temporal evolution of the ALD growth surface. This model thus encompasses all
elements of a complete ALD reactor system model as schematically illustrated in
Fig. 1.4.
6.1.1 Formulation of Underlying Model Assumptions
In order to simplify the overall modeling framework and reduce the compu-
tational requirements, several assumptions have been made without sacrificing the
integrity and applicability of the model. The surface reaction model developed in
Chapter 3 has been coupled to the reactor-scale model designed here. Assumptions
in relation to the surface reaction mechanism have been previously discussed (see
Section 3.1.1). In this multiscale model several additional assumptions have been
made in relation to the ALD reactor.
(i) The reaction chamber is surrounded by a tube furnace which provides a reason-
ably homogeneous temperature field and precursors are stored in temperature-
controlled vessels. Thus, the hot-wall stainless steel reactor tube is assumed
to be isothermal [1, 69, 147, 148, 170, 171]. Moreover, because growth rates in
ALD are typically only weakly temperature-dependent (within a reasonably
small range), temperature uniformity is usually not critical [140]. Of course,
the heat equation may be employed in reactors where strong temperature gra-
dients exist.
(ii) The system is assumed to be thermodynamically ideal, i.e. there are no in-
teractions between molecules in the gas phase and the relationship between
164
pressure and density is defined by the ideal gas law. This assumption is ap-
propriate considering the low operating pressure typical in ALD reactor sys-
tems [1, 44,68?72,147,148,170,171].
(iii) Given the high aspect-ratio of the tubular reaction chamber (> 11), flow can be
considered to be approximately one-dimensional in the axial direction. Thus
the axial velocity of the carrier gas is used as an approximation, understanding
that the carrier gas flow is kept constant during all dose and purge periods in
the reactor to be modeled despite the pulsed nature of the ALD process. The
average axial velocity is found by
?uz? =
1
2
P ?
P
T
T ?
Q?
piR2
m/s (6.1)
where Q? is the volumetric flow rate of the carrier gas at T ? = 273K and
atmospheric pressure P ? = 1 atm. Reactor temperature T , pressure P , and
cross-sectional area piR2 are used with the ideal gas law to derive the axial
velocity which is divided by two in order to obtain the average value [14]. The
average axial velocity is a reasonable approximation for continuous flow ALD
systems where the carrier gas is kept at a much higher partial pressure than
the precursor species [1, 147,148,170,171].
(iv) Under the nominal reactor operating conditions, the Reynold?s number (Re)
was determined to be 3.7, well within the laminar region. From this low Re,
the entrance region was determined to be less than 2% of the length of the
165
reaction chamber and thus fully-developed flow is a reasonable approximation.
In computing Re, viscosity was determined using the Chapman-Enskog theory
for monatomic gases [76] as follows
? = 2.6693? 10?6
?
m/T
?2?
Pa s (6.2)
In the equation, m is the atomic weight and ? is the Lennard-Jones molecular
diameter as given in [124]. ? is the diffusion collision integral and is deter-
mined from the accurate empirical relation given in [110] which is a function
of temperature and the Lennard-Jones well depth, .
? =
A
T ?B
+
C
exp (DT ?)
+
E
exp (FT ?)
+
G
exp (HT ?)
(6.3)
where values for A, B, C, D, E, F, G, and H are 1.06036, 0.1561, 0.193, 0.47635,
1.03587, 1.52996, 1.76474, 3.89411, respectively [124] and temperature is re-
lated to the Lennard-Jones energy by T ? = kBT/.
(v) Considering the continuous flow ALD reactor where the mass flow rate of the
carrier gas is kept constant, vacuum pump speed is kept constant, and the
partial pressures of the precursor gases are typically much lower than that of
the carrier gas, it is reasonable to assume that no significant spatial gradient
in total pressure exists within the reactor, i.e. only small pressure variations
are encountered in the flow during precursor pulsation. As a simple test of
this assumption, the pressure drop across the reactor was computed using the
166
Hagen-Poiseuille equation [14] as follows.
?P =
8?LQ?
piR4
=
4?Luz
R2
(6.4)
Substituting the nominal reactor parameters into this equation yields a pres-
sure drop of 6.8 ? 10?4 Torr, a value which corresponds to only 0.067% of
the base pressure in the reaction chamber. This suggests that the approxi-
mation to neglect total pressure gradients (i.e. dP/dz = 0) is valid for the
reactor system in this investigation and thus gaseous species are modeled as
incompressible [1, 147,148,170,171].
6.1.2 Model Description
A model is developed which couples chemical species transport equations with
a formulation of surface species conservation corresponding to the alumina ALD
surface reaction model derived in Chapter 3 to describe the spatial and temporal
evolution of the growth surface. As such, all surface reaction modeling equations
derived previously should be assumed. Mass transport of chemical species ? through
the reaction chamber is controlled by the conservative formulation of the convection-
diffusion equation in one spatial dimension
?P?
?t
= D?I
?2P?
?z2
? ?uz?
?P?
?z
+ S? (6.5)
167
where t and z are time and the spatial coordinate along the length of the reactor,
respectively. P? is the partial pressure of species ?, D?I is the binary diffusion
coefficient of species ? in the inert carrier gas I, and S? is the source term accounting
for the consumption and production rate of ? as a result of chemical reactions. Using
the ideal gas relationship and applying the product rule,
?y?
?t
= ?
y?
m
dm
dt
+D?I
?2y?
?z2
? ?uz?
?y?
?z
+ S? (6.6)
where m is the total moles of gas-phase species in the reactor and y? is the gas-phase
mole fraction of chemical species ?.
The binary diffusion coefficients were determined from physical properties of
the gases using the Chapman-Enskog kinetic theory of dilute gases [67].
D?I = 2.6280? 10
?7
?
T 3
(M? +MI)
2M?MI
1
P?2?I??I
m2/s (6.7)
Here, M? and MI respectively represent the molecular weights of the ? species and
carrier gas in atomic mass units, T is absolute temperature, P is the total pressure
in atmospheres, and ??I in A?ngstro?m units is given by
??I =
1
2
(?? + ?I) (6.8)
where ?? and ?I are the Lennard-Jones molecular diameters of each respective
species [124] and ??I is the collision integral determined as previously described.
The self diffusion coefficient for the carrier gas was also determined by the
168
Chapman-Enskog kinetic theory by the following.
DI = 2.6280? 10
?7
?
T 3
MI
1
P?2I?I
m2/s (6.9)
6.1.3 Chemical Species Conservation
Applying the precursor delivery model described in Section 5.4 and defining
yA as the gas-phase mole fraction of TMA, and yB, yC , and yD as the gas-phase
water, methane, and DMA-OH mole fractions, the reactor material conservation
and transport equations can be written as
dm
dt
= m?in ? m?out ?
(
?A + ?B + ?C + ?D
)
(6.10)
?yA
?t
= DAI
?2yA
?z2
? ?uz?
?yA
?z
?
yA
m
dm
dt
?
?A
m
(6.11)
?yB
?t
= DBI
?2yB
?z2
? ?uz?
?yB
?z
?
yB
m
dm
dt
?
?B
m
(6.12)
?yC
?t
= DCI
?2yC
?z2
? ?uz?
?yC
?z
?
yC
m
dm
dt
?
?C
m
(6.13)
?yD
?t
= DDI
?2yD
?z2
? ?uz?
?yD
?z
?
yD
m
dm
dt
?
?D
m
(6.14)
subject to initial conditions at the start of each pulse and purge period. Initial
conditions are determined at the onset of the ALD cycle before the first exposure
period, while continuity conditions determine the initial conditions from one period
to the next. The initial conditions are prescribed as follows: m(t = 0) = mbase,
yA(z, t = 0) = 0, yB(z, t = 0) = 0, yC(z, t = 0) = 0, yD(z, t = 0) = 0, and
yI(z, t = 0) = 1, i.e. only the carrier gas flows through the reactor initially.
169
Continuity conditions defined at the initiation of purge A, pulse B, and purge B
are such that the values from the final time step in the previous period are conserved.
The inert mole fraction yI is determined by yI = 1 ?
?
y? where ? = A,B,C,D.
The set of ordinary differential equations (ODEs) derived in Section 3.2.2 to describe
the temporal evolution of the growth surface are also solved with the set of partial
differential equations above, subject to initial conditions corresponding to the fully
hydroxylated state.
6.1.3.1 Boundary Conditions
The inlet boundary conditions for Eqns. (6.11-6.14) are prescribed according
to the molar flow rates of the chemical species. As such, the flux across the inlet
boundary for each component is specified as follows.
J?|z=0 piR
2 = m??in (6.15)
(
?D?I
dy?
dz
?
?
?
?
z=0
+ ?uz? y?|z=0
)
P
RgT
piR2 = m??in (6.16)
where m??in varies according to the control valve pulsing sequence. The Neumann
condition is enforced at the reaction chamber outlet. Thus, prescribing a constant
flux and assuming the diffusive mass flux to be zero, the final boundary condition
is written as
dy?
dz
?
?
?
?
z=L
= 0 (6.17)
170
6.1.4 ALD Film Growth
As in Chapter 5, limit-cycle solutions are computed for the multiscale system,
requiring the gas-phase and surface state at the end of purge B return to their
respective initial conditions consistent with the beginning of pulse A. The surface
state is defined by Eqns. (3.7) and (3.8) and as such the film growth computation
follows Section 3.7. Recall the material deposited per cycle is found by
NAl =
? ?
0
v3 dt (6.18)
NO =
? ?
0
(v8 + v11) dt (6.19)
Note that these reaction rates correspond to the reaction mechanism presented in
Chapter 3, not to be confused with the reactions numbered in Chapter 4. The
material deposited per cycle is key to determining the growth per cycle gpc which
is computed according to the following.
gpc = 10%NNAl (6.20)
Spatial uniformity is investigated in the context of gpc and by specifically mon-
itoring the amount of material deposited [69]. The total mass gain from aluminum
and oxygen deposition is found by
ms = (NAlMAl +NOMO)
105
NAvo
ng cm?2 (6.21)
171
where MAl and MO are the molecular weights of aluminum and oxygen, respectively.
The average mass gain of the film is governed by
d ?ms?
dt
=
1
L
? L
0
ms dz (6.22)
and film uniformity is described by the following.
? = 1?
? L
0
|ms ? ?ms? | dz
? L
0
ms dz
(6.23)
In essence, ? = 1 - standard deviation/mean, where ? = 1 for a perfectly uniform
film.
6.2 Discretization Procedure
Due to the nonlinear nature and complexity of the modeling equations, com-
putation of the solutions requires a numerical technique. The set of nonlinear partial
differential Eqns. (6.11-6.14) are discretized using a central time and central space
(CTCS) finite differences technique with non-uniform intervals resulting in a large
set of nonlinear algebraic equations. The finite differences mesh is shown in Fig. 6.1
with discretization intervals initialized according to
?t
?z2
D?I,max ? 0.75 (6.24)
172
i-1 i i+1
j+1
j
j-1
z
t
?z
?t
Figure 6.1: Central time and central space finite differences mesh and solution points.
where ?t is the nominal time step and ?z = 0.1L is the nominal displacement
in space set to a tenth of the reactor tube length. The ODEs representative of the
material balance on the ballast chamber (Eqn. 5.9) and reaction chamber (Eqn. 6.10)
are discretized using the same central temporal discretization steps. All discretized
ODEs and PDEs are solved simultaneously in a fashion similar to what has been
previously discussed in Chapter 5, employing a Newton-Raphson procedure with an
Euler method utilized for the initial guess.
6.3 Representative Limit-Cycle Results
Representative limit-cycle solutions were computed with ?A = 0.5 sec, ?B = 0.1
sec, and ?AP = ?BP = 2 sec. Examination of the precursor pulsing sequence shown
in Fig. 6.2 reveals the manner in which the TMA and water precursors propagate
through the reaction chamber. From top to bottom, the four plots depict the four
steps of the ALD cycle: pulse A, purge A, pulse B, and purge B. At the start of
173
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P A
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
P A
0.0
0.5
1.0
1.5
2.0
P B
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.0
0.5
1.0
1.5
2.0
P B
, To
rr
, To
rr
, To
rr
, To
rr
z, m
Figure 6.2: Representative gas-phase dynamics for T=450K, and ?A = 0.5 sec,
?B = 0.1 sec, and ?AP = ?BP = 2 sec. Curves show spatial reactor position, with
inlet marked by -x- and outlet by -o-. In order from top to bottom, the ALD cycle
step for each plot is as follows: pulse A, purge A, pulse B, purge B.
pulse A, no TMA exists in the reaction chamber. As TMA is introduced, a gradual
increase in partial pressure is observed along the reaction channel. During purge A,
a TMA partial pressure drop is experienced more suddenly near the reactor inlet
and then the partial pressure gradually decreases throughout the reactor as residual
gases are removed. Following this, a quick and relatively large pulse is experienced
during the water exposure. Gaseous species are subsequently swept from the reactor
during the next purge period.
While the water pulse time is shorter than the TMA pulse, water precursor de-
174
0.5
0.0
0.5
1.0
1.5
2.0
P A,
P B
A purge B purge
0.005
0.000
0.005
0.010
0.015
0.020
P C
0 1 2 3 41.0
1.2
1.4
1.6
1.8
2.0
2.2
P
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Me
0 1 2 3 4t, sec
0.0
0.2
0.4
0.6
0.8
1.0
X
, To
rr
, To
rr
, To
rr
Figure 6.3: Representative gas-phase and surface dynamics for T=450K, and ?A =
0.5 sec, ?B = 0.1 sec, and ?AP = ?BP = 2 sec. Curves show spatial reactor position,
with inlet marked by -x- and outlet by -o-.
livery is actually more efficient than that of TMA due to differences in the precursor
vapor pressures. The vapor pressure of water in its source container is significantly
higher than that of TMA; and thus the pressure differential across the pulsing valve
drives more water into the reaction chamber. Yet both precursor gases are effectively
swept from the reactor in the same purge time spans, indicating the important role
of the downstream vacuum pump in facilitating precursor propagation through the
reaction chamber. This is not surprising considering that convection is significantly
more important than diffusion in this system, with relatively large Peclet numbers:
Pe ? 100 for TMA and Pe ? 40 for water.
175
Model predictions of the gas-phase and surface dynamics are shown in Fig. 6.3
with the TMA and water pulse periods shaded red and blue, respectively. In the
top plot, the precursor gas dynamics are given as a function of spatial position in
the reaction chamber. During each exposure, surface reactions produce methane
by-products according to the spatial dynamics shown in the second plot; although,
compared to the magnitude of TMA and water partial pressures, the partial pressure
of methane gas in the reactor is negligibly small. The next plot in this series is similar
to the pressure dynamics shown in the lumped parameter model of Chapter 5. The
reactor pressure is normally around 1 Torr, increasing during precursor pulse periods
and relaxing back to the nominal pressure during reactor purge periods.
An examination of the growth surface in the last two plots of Fig. 6.3 reveals
what appears to be self-limiting growth in both precursor exposure periods, but
this is not the case. Neither Me nor OH groups reach maximum values, but growth
halts. The methyl group density does not reach the close-packing limit during the
TMA exposure, suggesting perhaps a transport-limited situation. However, self-
limiting growth is achieved in the water exposure due to methyl group depletion.
During the TMA exposure, the growth surface reaches only around 70% saturation,
but all methyl groups are removed during the water exposure. This indicates that
the current water partial pressure and pulse time are sufficient to chemically react,
removing all surface methyl groups; but self-limiting growth for the ALD cycle is not
achieved due to insufficient saturation during the TMA exposure. The TMA partial
pressure and/or pulse time should therefore be increased to allow for more TMA
adsorption and sufficient surface saturation by methyl groups until self-termination.
176
100000
150000
200000
250000
300000
350000
exp
osu
re, 
Lan
gm
uir
TMA
water
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8z, m
0.810
0.815
0.820
0.825
0.830
0.835
0.840
gpc
,    
 /cy
cle
Figure 6.4: Exposure and growth per cycle predictions as a function of spatial position
in the ALD reactor.
It is interesting to note that a majority of surface reactions occur during the
pulse periods. This is interesting because these periods are typically considered
only in the context of removing by-products and unreacted chemical species. Thus,
the purge dynamics are many times omitted from ALD simulations. These model
results show the significance of modeling the complete ALD cycle, including purge
period dynamics, as this yields key insight into surface reaction dynamics which
could potentially guide more-informed optimization decisions.
Lastly, the spatially distributed growth per cycle and related film uniformity
were investigated according to Eqns. (6.20) and (6.23), respectively. The spatially-
177
distributed exposure and gpc are shown in Fig. 6.4, indicating a higher TMA expo-
sure than water due to the longer TMA pulsing interval. Both exposures, however,
were found to be relatively uniform throughout the reaction chamber. One would
then expect a relatively uniform growth rate. While the gpc appears to be non-
uniform, note that gpc variability is actually very small, hundredths of an A?ngstro?m,
with gpc = 0.815 ? 0.835 A?. Moreover, film uniformity was computed for this sim-
ulation and ? = 0.99. Thus, despite under-saturating growth conditions, the ALD
film is predicted to be essentially uniform across the length of the reaction channel.
This finding highlights a key advantage of ALD growth processes.
6.4 Concluding Remarks
In this chapter, a first principles-derived surface reaction model was coupled
to a spatially and temporally distributed reactor system model to describe ALD film
growth. Results suggest that reactor transport effects can significantly impact sur-
face coverage dynamics, potentially hindering self-limiting growth. The importance
of surface saturation during the TMA exposure was shown by the undersaturating
growth rates predicted with insufficient TMA pulsing. Limit-cycle solutions predict
a relatively uniform growth rate with gpc = 0.815 ? 0.835 A?, which falls short of
the self-saturating growth regime. Examination of the precursor pulsing dynamics
yields more efficient water precursor delivery due to the higher vapor pressure of wa-
ter. In order to achieve better TMA pulsing, the exposure time length would need
to be increased or a more efficient TMA delivery system should be investigated (e.g.
178
use of a bubbler).
The implication of this distributed model is significant due to the level of mod-
eling detail contained in each model component?from precursor source dynamics,
to precursor delivery through the control valves, to transport through the reaction
chamber, and then to reaction and film growth at the substrate surface. Carefully
coupling all of these model components reveal significant growth rate dependencies
on the various reactor and operation parameters. Models such as the one presented
in this chapter serve as very useful reactor design and optimization tools for ex-
isting and potentially novel ALD reactor systems. Further model refinement and
optimization will be the subject of follow-up work.
179
7Concluding Remarks
The work presented in this thesis was motivated by the need for model-based
ALD process optimization and greater understanding of ALD reaction dynamics.
This lead to the development of a physically-based ALD reaction kinetics and film
growth model which is used to examine growth surface dynamics in the period lead-
ing to self-saturation during each half-cycle. In coupling this model with reactor-
scale transport phenomena, additional insight is given into multiscale process inter-
actions.
Recall the modeling objectives described in the introductory chapter:
(i) First principles understanding of reaction mechanisms and growth kinetics
In Chapters 3 and 4, a physically-based ALD surface reaction kinetics model
was presented based on combined reaction energetics from quantum-chemical
computational studies studies and kinetic parameters determined using sta-
tistical thermodynamics and absolute reaction rate theory. Several reaction
paths were investigated and a kinetic reaction mechanism was suggested based
on dynamic film growth rate analysis. Model predictions were in good agree-
180
ment qualitatively and quantitatively with the experimentally observed ALD
mechanism. A key contribution of this work is that the level of modeling detail
allows for keen insight into molecular-level surface species interactions and thus
provides means to study the ALD saturation and growth mechanism in relation
to competing reactions. This analysis also allowed for comparison of suggested
thermodynamically favored mechanisms and H transfer paths, details which
lead to better understanding of the dynamic ALD growth mechanism. [158]
(ii) Identification of operating limits for self-saturating growth
The surface kinetics and limit-cycle growth model described in Chapter 2 was
motivated by the predictive capabilities of physically-based models to decou-
ple the effects of reactor input parameters (namely temperature and exposure
times) on the measured film growth rate. A key contribution of this model
is the demonstration of the transition state applicability to ALD processes,
which even for a simplified reaction mechanism, yields growth rate, exposure,
and temperature predictions consistent with experimentally-observed reaction
phenomena. This demonstrated the applicability of the first principles ki-
netics modeling approach. Another major contribution of this work is the
introduction of the limit-cycle growth model, describing the continuous cyclic
operation of the deposition reaction system. Limit-cycle solutions developed
were able to distinguish saturating GPC from non-saturating gpc conditions.
The transition between the two regimes was then able to be studied in the
context of temperature and exposure, yielding self-limiting growth boundary
181
predictions. [156]
(iii) Explanation and prediction of multiscale reaction phenomena
In Chapters 5 and 6, a laboratory-scale reactor system model was developed
by integrating components describing the precursor thermophysical properties,
precursor delivery system, reactor-scale gas-phase dynamics, and surface reac-
tion kinetics derived from absolute reaction rate theory. A key finding is that
unintended chemical vapor deposition conditions can mask regions of opera-
tion that would otherwise correspond to ideal saturating ALD operation. The
ability to model the interaction between pulse and purge periods was critical
to uncovering the surface reactions occurring under CVD conditions and iden-
tifying processing regimes where these reactions are most likely to take place.
Extension to the distributed reactor model allowed for discussion of transport
effects and reactor parameters associated in film non-uniformity [157,159].
7.1 Suggestions for Future Work
ALD modeling is a specialized niche represented by around 150 papers pub-
lished mostly over the last decade [38] and only a handful of those papers have
implemented physically-based growth models to unelucidate the kinetic reaction
mechanism [37]. The modeling work presented in this thesis was thus meant to
provide a relatively straightforward framework from which additional models can
be derived. Several paths are suggested for continuing this work.
182
7.1.1 Surface Reaction Mechanism and Kinetics
The ALD modeling efforts described herein reveal an increasingly complex
picture of the surface processes taking place even in those systems once thought to
represent ?ideal ALD,? such as alumina ALD using trimethylaluminum and water
as precursors. The scheme of 27 elementary reactions presented in Chapter 4 gives
a glimpse into how large these systems of reactions can grow. Therefore, a rational
framework must be developed for the analysis and simulation of the increasingly
complex ALD surface process models, perhaps using a model dynamic reduction
framework which can accommodate the growing number of chemical reactions and
species necessary to accurately model ALD growth.
7.1.2 Reactor-scale Model Optimization
Considering the progression of model development in this thesis, it naturally
follows to apply an optimization procedure aimed at two immediate objectives: in-
creasing reactor throughput and improving film uniformity. It would be interesting
to see the interplay between these two variables since non-uniformity is often at-
tributed to pressure gradients or precursor depletion. So, an increase in precursor
dose would be the natural solution. But this also causes an increase in either the
pulse period or the purge necessary to remove unreacted species. This is a nice
problem for ALD optimization with very practical consequences. A preliminary
throughput analysis is presented in Appendix A.
183
7.1.3 Reactor Model
The distributed model presented in this work assumed a negligible pressure
drop across the reactor and thus gases were assumed incompressible. While the
pressure drop was computed to be negligibly small, it is not completely clear if this
assumption maintains validity with varying precursor pulsing conditions. Consid-
ering that exposure level is critical in ALD processes, future work should consider
employing the Navier-Stokes equations to more rigorously account for these vari-
ables.
7.1.4 Extension to Additional ALD Chemistries
Finally, this modeling work should be applied to additional ALD processes.
Preliminary investigation of the BN ALD system has shown much similarity in the
thermodynamic mechanism and initial results have appeared to be promising. At
this point, the multiscale modeling approach presented in this thesis has only been
applied to the alumina process in entirety. Considering that the surface reaction
model is developed from first principles, it is assumed to be widely applicable. Thus,
it would be a good study to assess the relative applicability of the model by extending
study to additional ALD systems of interest.
184
AThroughput Optimization
Reactor throughput, or effective growth rate, is directly related to the growth
per cycle by
?eff =
gpc
?A + ?B + ?AP + ?BP
, A?/s
where ?eff is the effective growth rate and gpc is film growth per limit cycle. A
similar formula is used in [97]. The growth per cycle is know to be affected by
a number of factors, namely reactor temperature, precursor partial pressure, and
reactant exposure times.
Until the saturating growth per cycle is reached, an increase in exposure time
for either precursor causes an observed increase in the growth per cycle. On the other
hand, exposure time has an inverse relationship with respect to throughput. Purge
times are modulated by the reactor volume and speed of the vacuum pump which
ultimately determine the time it takes to sufficiently remove unreacted precursor
gas during purge periods. Although the effective growth rate is likely to increase
with a decrease in the purge time interval, doing so could cause non-ideal gas-phase
185
1 2 3 4 5 6
?A (s)
1
2
3
4
5
6
? B
 (s
)
Throughput ( ?A/s)
0.036
0.044
0.052
0.060
0.068
0.076
0.084
Figure A.1: Preliminary results for alumina ALD throughput as a function of pre-
cursor exposure times.
reactions. Preliminary throughput optimization results are shown in Fig A.1.
186
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PUBLICATIONS & PRESENTATIONS
Refereed Journal Publications
Travis, C. D. and R. A. Adomaitis, Modeling alumina atomic layer deposition
reaction kinetics during the trimethylaluminum exposure, Theor. Chem. Acc.
133, 1414 (2014).
Travis, C. D. and R. A. Adomaitis, Dynamic modeling for the design and cyclic
operation of an atomic layer deposition (ALD) reactor, Processes 1, 128?152
(2013).
Travis, C. D. and R. A. Adomaitis, Modeling ALD surface reaction and pro-
cess dynamics using absolute reaction rate theory, Chem. Vap. Dep. 19, 4?14
(2013).
Publications to be Submitted
Travis, C. D. and R. A. Adomaitis, Model-based analysis of atomic layer depo-
sition growth kinetics and multiscale process dynamics, (tentative submission
May 2014).
Conference Presentations
Adomaitis*, R. A. and C. D. Travis, Current modeling and simulation chal-
lenges in thin-film deposition processes, Paper 105b, AIChE Annual Meeting,
San Francisco, CA, November 3?8, (2013).
Travis*, C. D. and R. A. Adomaitis, First principles modeling of atomic layer
deposition surface reaction and process dynamics using absolute reaction rate
theory. Paper 362d, AIChE Annual Meeting, Pittsburgh, PA, October 28 ?
November 2, (2012).
Travis*, C. D. and R. A. Adomaitis, The computational challenges of simu-
lating atomic layer deposition process dynamics, Paper 595a, AIChE Annual
Meeting, Pittsburgh, PA, October 28 ? November 2, (2012).
203
Dwivedi, V., R. A. Adomaitis* and C. D. Travis, Passively controlled thermal
material deposited by atomic layer deposition, Paper 755c, AIChE Annual
Meeting, Pittsburgh, PA, October 28 ? November 2, (2012).
Travis*, C. D. and R. A. Adomaitis, Modeling ALD surface reaction and pro-
cess dynamics using absolute reaction rate theory, UMD Graduate Research
Interaction Day (GRID), College Park, MD, April 11, (2012).
Travis*, C. D. and R. A. Adomaitis, Modeling ALD surface reaction and pro-
cess dynamics using absolute reaction rate theory, UMD Research Fest, College
Park, MD, March 30, (2012).
Travis, C. D. and R. A. Adomaitis*, Coupling ballistic transport and surface
reaction models for simulation of atomic layer deposition processes, Paper
317b, AIChE Annual Meeting, Minneapolis, MN, October 16?21, (2011).
Travis*, C. D. and R. A. Adomaitis, Surface reaction modeling for atomic layer
deposition processes, UMD Research Fest, College Park, MD, March 8, (2011).
*Presenter
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