Toward controlled ultra-high vacuum chemical vapor deposition processes
Dresscher, Martijn
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Dresscher, M. (2019). Toward controlled ultra-high vacuum chemical vapor deposition processes. Rijksuniversiteit Groningen.
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Chemical Vapor Deposition Processes
Een korte absorptie van het licht veroorzaakt vervolgens een richtingsverander-ing.
Book cover: Light passes through a vacuum window and crosses paths with an atom. A short absorption of the light subsequently causes a change in direction.
ISBN: 978-94-034-1083-8 (printed) 978-94-034-1082-1 (ebook)
Chemical Vapor Deposition Processes
Proefschrift
ter verkrijging van de graad van doctor aan de
Rijksuniversiteit Groningen
op gezag van de
rector magnificus prof. dr. E. Sterken
en volgens besluit van het College voor Promoties.
De openbare verdediging zal plaatsvinden op
vrijdag 11 januari 2019 om 12.45 uur
door
Martijn Dresscher
geboren op 21 juli 1989
Prof. dr. B. Jayawardhana Prof. dr. ir. B.J. Kooi Prof. dr. ir. J.M.A. Scherpen
Beoordelingscommissie
Prof. dr. M.R. de Baar Prof. dr. ir. R. Findeisen Prof. dr. G. Palasantzas
Samenvatting ix
Summary xi
Acknowledgments xiii
List of symbols xv
List of figures xxiv
List of abbreviations xxv
1 Introduction 1
1.1 Thin films . . . 2
1.2 Thin film deposition through vacuum evaporation processes . . . 4
1.2.1 Ultra-high vacuum chemical vapor deposition . . . 6
1.2.2 Low pressure chemical vapor deposition . . . 7
1.2.3 Molecular beam epitaxy . . . 8
1.3 UHVCVD operation recipe . . . 8
1.4 Motivation . . . 9
1.4.1 Modeling of free molecular flow dynamics, partial pressure measurement and controller implementation . . . 9
1.4.2 Controller design for deterministic systems with stochastic ini-tial conditions . . . 12
1.5 Outline & contributions . . . 15
1.6 Origins of the chapters . . . 16
1.7 Applied units . . . 17 v
I
Modeling of Free Molecular Flow Dynamics, Partial Pressure
Measurement and Controller Implementation
19
2 Modeling of fluxes and partial pressures for controller design 21
2.1 Free molecular flow preliminaries . . . 21
2.2 Modeling framework for FMF . . . 23
2.2.1 The transfer and leakage matrices . . . 25
2.2.2 The sorption function . . . 26
2.2.3 The input function . . . 26
2.2.4 The output function . . . 26
2.3 Methods for obtaining the transfer matrix . . . 27
2.3.1 Validation of MC simulation based method . . . 28
2.4 Numerical controller simulations for FMF dynamics . . . 29
2.4.1 Point-wise min-norm controller design . . . 30
2.4.2 Numerical flux control example . . . 35
2.4.3 Numerical partial pressure control example . . . 39
2.5 Concluding remarks . . . 42
Appendices 2.A Derivation of the scattering pdf . . . 43
3 Experimental Reactor Design and AAS Measurement Implementation 45 3.1 Atomic absorption spectroscopy preliminaries . . . 45
3.1.1 Motivation of AAS sensor selection . . . 45
3.1.2 AAS measurement technique . . . 47
3.2 Experimental design and scope . . . 49
3.2.1 Experimental scope . . . 49
3.2.2 Experimental setup design . . . 49
3.2.3 Other design choices, simplifications and assumptions . . . 51
3.3 Modeling for experimentation . . . 52
3.3.1 AAS signal interpretation . . . 52
3.3.2 Model implementation . . . 53
3.4 Experimental results and discussion . . . 56
3.4.1 AAS sensor assessment and calibration . . . 56
3.4.2 Comparison of theoretical vapor pressure, model and sensor performance . . . 58
3.5 Concluding remarks . . . 59
Appendices 3.A Sorption for the fluxes model . . . 61
3.B Sorption for the moles model . . . 62 vi
4 Partial pressure controller design and implementation 63
4.1 Experimental setup for partial pressure control . . . 63
4.2 Evaporation process modeling . . . 65
4.2.1 Stochastic input-output mapping . . . 65
4.2.2 Controller design choices . . . 66
4.3 PI controller with feedforward component . . . 68
4.3.1 Obtaining the feedforward component . . . 68
4.3.2 Controller design . . . 69
4.3.3 Experimental results and discussion . . . 70
4.4 Model-free PI controller . . . 71
4.4.1 Controller design . . . 71
4.4.2 Experimental results and discussion . . . 72
4.5 Controller comparison . . . 73
4.5.1 Comparison for discontinuous reference signal . . . 73
4.5.2 Comparison for continuous reference signal . . . 75
4.6 Concluding remarks . . . 77
II
Controller Design for Deterministic Systems with Stochastic
Initial Conditions
79
5 Containment Control problem 81 5.1 Containment control problem definition . . . 815.1.1 Dynamical system equations . . . 82
5.1.2 Candidate transient specifications . . . 82
5.1.3 Containment control problem formulation . . . 83
5.1.4 Containment control problem example . . . 83
5.2 CCP for linear systems . . . 85
5.3 CCP for nonlinear systems . . . 87
5.4 CCP controller simulation for a nonlinear robotic manipulator . . . 90
5.4.1 Dynamics and controller design . . . 90
5.4.2 Control design parameters . . . 92
5.4.3 Simulation results . . . 94
5.5 Concluding remarks . . . 94
Appendices 5.A Contraction preliminaries . . . 97
6 Shape Control Problem 99 6.1 Shape control problem definition . . . 99
6.1.1 Dynamical system equations . . . 99
6.1.2 Candidate transient specifications . . . 100
6.1.3 Shape control problem formulation . . . 100 vii
6.2 SCP for linearly matching pdfs . . . 101
6.3 SCP for nonlinearly matching pdfs . . . 104
6.4 Numerical evaluation of SCP controller for matching pdfs . . . 106
6.5 Concluding remarks . . . 108
7 Conclusions 109 7.1 Conclusions Part I . . . 109
7.2 Conclusions Part II . . . 111
7.3 Recommendations for future research . . . 112
Bibliography 113
E
en ultra hoog vacu ¨um chemische damp afzetting (UHVCVD, ultra-high vacuumchemical vapor deposition) reactor wordt gebruikt om zeer dunne vaste stof la-gen op gekozen oppervlakken te produceren vanuit atomaire en moleculaire precur-sors in de dampfase. Deze precurprecur-sors kunnen met elkaar reageren nabij of op het oppervlak en nieuwe verbindingen en stoffen maken met specifieke eigenschappen. Deze processen worden onder andere gebruikt om halfgeleiders te maken voor de chip en processor industrie, maar de mogelijkheden zijn divers. Dit procestype on-derscheidt zich van vergelijkbare processen doordat er zeer weinig verontreinigin-gen in de afgezette laag komen. Minder verontreiniginverontreinigin-gen zorverontreinigin-gen ervoor dat de opgedampte laag beter functioneert.Om de ultra hoge vacu ¨um condities te bereiken moet een reactor een thermische reiniging ondergaan. Dit houdt in dat de hele reactor voor meerdere uren op hoge temperatuur (ongeveer 330 graden Celsius) gehouden wordt, zodat achtergebleven atomen en moleculen verdampen en weggepompt kunnen worden. Een nadeel hier-van is dat veel elektronische meetinstrumenten een dergelijke behandeling niet kun-nen doorstaan. Hiernaast zorgt de afwezigheid van warmtegeleidende lucht ervoor dat de temperaturen in de reactor niet snel uniform zijn en dat thermische contact-metingen onnauwkeurig worden. Deze factoren zorgen er gezamenlijk voor dat er weinig (actuele) informatie beschikbaar is over relevante toestanden in de reactor, zoals dampdrukken en temperaturen.
Om een regelsysteem te ontwerpen voor een dergelijk proces hebben we actuele (zogeheten echte-tijd) informatie nodig over de relevante toestanden. Een regelsys-teem bestaat over het algemeen uit: (i) echte-tijd metingen van relevante staten voor terugkoppeling van het effect van de aansturing, (ii) een dynamisch model dat met redelijke nauwkeurigheid beschrijft hoe de aansturing de gemeten staten be¨ınvloed en (iii) regelalgoritmen, afgestemd op (ii), die aan de hand van (i) gepaste aansturin-gen bepalen.
In deze thesis leveren we bijdragen aan de ontwikkeling van regelalgoritmen voor echte-tijd deeldruk regulatie in een UHVCVD proces. We implementeren een
echte-tijd deeldruk meting die geschikt is voor gebruik door een regelalgoritme. Hi-ernaast hebben we een model ontwikkeld dat potentieel met hoge spati¨ele resolutie de gas stromen in de reactor kan beschrijven. Dit model kan daardoor de infor-matie over de reactoraansturing gebruiken om te voorspellen hoe de deeldruk ve-randert. We laten zien dat dit model accurate voorspellingen kan geven van de deel-druk wanneer we in dampdeel-druk condities opereren. Het model, als alle componen-ten ge¨ıdentificeerd zijn, en de deeldruk metingen kunnen samen gebruikt worden voor het ontwerp van regelalgoritmen. We laten de toepasbaarheid van de deel-druk metingen voor het ontwerpen van regelalgoritmen zien door twee regelaars te implementeren die een minimale modelidentificatie vergen. De resultaten laten zien dat beide regelaars geschikt zijn voor deze doeleinden, voor de reactorcondities waarin ze ge¨evalueerd zijn.
Naast de bovenstaande bijdragen, leveren we ook bijdragen aan de ontwikke-ling van theoretische regelalgoritmes voor systemen die omschreven kunnen wor-den met deterministische dynamica en stochastische initi¨ele staten. We laten zien dat onze regelalgoritmes theoretische garanties kunnen geven dat het effect van de variatie beperkt blijft voor een vooraf bepaald deel van de initi¨ele staten.
A
n ultra-high vacuum chemical vapor deposition (UHVCVD) reactor is used todeposit thin solid layers on desired surfaces from atomic and molecular precur-sors in the vapour phase. These precurprecur-sors can react near or on these surfaces to form new compounds with specific properties. One of the applications these pro-cesses are used for is to build semiconductors for the chips and processor industries, but the range of applications is wide. These processes are distinct from other depo-sition processes because they can deposit layers of extreme high purity. This high purity in turn allows the deposited layer to perform better.A UHVCVD reactor has to undergo a thermal cleaning process in order to reach ultra-high vacuum conditions. This entails keeping the entire reactor on a high tem-perature (approximately 330 degrees Celsius) for multiple hours. This allows resid-ual atoms and molecules to evaporate so that they can be pumped away from the system. A disadvantage of this procedure is that many electronic measurement de-vices cannot survive prolonged exposure to such temperature. The absence of heat conducting air furthermore causes reactor temperatures to be non-uniform, which in turn causes thermal measurements to be local. The availability of (real-time) infor-mation on the relevant reactor states, like pressures and temperatures, is accordingly limited.
We need real-time information on the relevant reactor states for the control sys-tem of such a process. The control syssys-tem typically requires three components: (i) real-time measurements of relevant reactor states to provide feedback on the effect of the actuation, (ii) a dynamical model that can, with reasonable accuracy, describe how the actuation influences the states, and (iii) a control algorithm, that is based on (ii), that calculated in real-time the required actuation, based on (i).
In this thesis we contribute to the development of real-time partial pressure con-trollers for UHVCVD processes. Firstly, we develop a real-time partial pressure mea-surement that is suitable for the implementation of model-based controllers. Sec-ondly, we have developed a mathematical model that can describe the evolution of particle fluxes inside the reactor with a high spatial resolution. This model uses
the information on the reactor inputs for this purpose and provides estimated par-tial pressures inside the reactor as output. We show that this model can provide accurate predictions of the partial pressure in vapor pressure conditions. The pro-vided model, once identified, and the partial pressure measurements can together be used for controller design. We furthermore show the efficacy of the real-time partial pressure measurement for controller design by implementing two controllers. These controllers rely on minimal modeling and identification efforts. The obtained results show that both controller designs are suitable for such purposes under the evaluated reactor conditions.
In addition to the contributions listed above, we provide contributions to the development of theoretical control algorithms for systems that can be described through deterministic dynamics and stochastic initial condition of the states. We show that our control algorithms can give theoretical guarantees that the impact of the variations remains bounded for a predetermined fraction of the initial states.
T
he majority of my time during the last four years has been spent obtaining knowledge and skills that were required for delivering and communicating the content of this thesis. This has been a journey that would have been significantly more difficult without the people close to me, on both a professional and a social level.To my doctoral advisers, dear Bayu, Bart and Jacquelien. Thank you for learning me the academic skills an independent researcher relies on. It is not given that the relation and the contact between a PhD candidate and his advisers is as smooth and pleasant as it was for us. I hence count myself lucky to have been able to rely on each of you for both solid and friendly advise throughout the years.
To my colleagues, peers and students, dear Emilie, Dmitry, Jack, T`abitha, Sanne and all the others. Thanks to all the pleasant interactions throughout these years and for helping me complete this project by letting me learn from you and work with you. Both my workplaces had a very friendly and supporting environment, allowing for the work to be mostly stress-free and enjoyable. Having been able to share the work with you has furthermore helped me to take some welcome breaks from the periods of individual work that are part of a PhD project. Working with you has provided me with a versatile skill set that I will be sure to put to good use in the future.
To my friends and family, which I need not name individually. I am lucky to have so many good contacts, even if it is sometimes infrequent. Your support and friendship these years kept me going, whether we had a meal together, did sports, had drinks, simply watched a movie or went on holiday. I could not have completed this challenging project without the great interactions with you. I dearly hope that we can maintain these contacts in the future, despite any challenging circumstances caused by geographic distance or busy schedules.
Part I
M 2-dimensional set of surfaces facing the atom cloud . . . 22
ω1, ω2 Surfaces, inM . . . 22
θdω1, θdω2 Angles with normals of dω1and dω2 . . . 23
dE(·) Euclidean distance . . . 23
φdω1, φdω2 Rotations around normals of dω1and dω2 . . . 23
ϕ Relative rotation dω1and dω2around central axis of cylinder . . . . 23
P Partial pressure of a precursor . . . 23
V Volume of considered body holding the atom cloud . . . 23
N Number of moles of a precursor in atom cloud . . . 23
R The ideal gas constant . . . 23
Ta Temperature of the atom cloud . . . .23
t Time variable . . . 23
˙ Nin Moles per second entering the volume holding the atom cloud . . 23
˙ Nout Moles per second leaving the volume holding the atom cloud . . . 23
˙ Ns(·) Function describing moles per second sorbed to surfaces . . . 23
n Number of discrete surfaces considered . . . 24
¯ x Fluxes in moles per second on a surface ω . . . 24
pA Knudsen cosine law transfer matrix . . . 24
s Sorbed moles, on a surface ω . . . 24
G(·) Input function . . . 24
u General model input . . . 24
z Time derivative of s . . . .24
A Corrected Knudsen cosine law transfer matrix, A = δpA . . . 24
δ Time scale constant . . . 24
I Identity matrix . . . 24
L Leakage matrix . . . 24
f (·) Sorption function . . . 24 xv
T Collection of temperatures of surfaces inM . . . 24
g(·) Input function, relates u to y . . . 24
m Number of inputs . . . 24
¯ y Fluxes model output in partial pressure . . . 24
h(·) Output function, relates x to y . . . 24
M Molar mass . . . 25
1 n-length column vector of ones . . . 25
Symmetric path length matrix . . . 25
v Average speed of a precursor in atom cloud . . . 25
q Directed flux matrix between surfaces . . . 27
θ Generalized coordinate, angle with normal . . . 28
φ Generalized coordinate, rotation around normal . . . 28
ϑ(·) Departing particle angle pdf . . . 28
¯ r Displacement along radial line from cylinder center . . . 28
¯ H Cylinder height . . . 28
¯ R Cylinder radius . . . 28
¯ h Displacement along height line from cylinder bottom . . . 28
P Set of safe states . . . 30
xd Desired trajectory of x . . . 30
x∗ Desired operating point . . . 30
x General model states . . . 30
k(·) Control law . . . 30
D Set of unsafe states, complement ofP . . . 30
B(·) Control barrier function . . . 31
C1 Class of functions that are once continuously differentiable . . . 31
∂D Boundary of the setD . . . .31
Int(D) Interior of the setD . . . 31
y General system output . . . 31
ˆ f (·) General nonlinear vector field function . . . .31
ˆ g(·) General nonlinear input vector field function . . . 31
ˆ h(·) General nonlinear output vector field function . . . 31
Lfˆ(·) Lie-derivative over ˆf at a point . . . 31
V(·) Control Lyapunov function . . . 31
U Set of admissible inputs . . . 31
X Set of admissible states . . . 31
c1, c2, c3 Constants > 0 . . . 31
κ1, κ2 Constants > 0 . . . 31
α1, α2 ClassK functions . . . 32
μ Feasible input . . . 32
Pclf&cbf Set of safe state where both clf and cbf constraints are active . . . 33
Pclf Set of safe state where only clf constraints are active . . . 33
γclf, γcbf Constant components of clf and cbf contraints . . . 33
kclf&cbf Control law in set of states where both clf and cbf are active . . . 33
kclf Control law in set of states where only clf is active . . . 33
F (·), H(·) Cost functions components for quadratic program . . . 33
Kclf(·) CLF admissible input set for a point in state space . . . 33
Kcbf(·) CBF admissible input set for a point in state space . . . 34
a, r, h Cylinder discretization parameters . . . 35
i1, i2, i3 Integers used for cylinder discretization . . . .35
B Matrix allocating input flux to surfaces . . . 36
C Matrix relating states to output . . . 36
H∞ H infinity norm . . . 37
ˆ u Derivative of flux input . . . 37
ˆ A, ˆB, ˆC System matrices for reduced order model . . . 37
ˆ a1, ˆa2 Constant components of ˆA . . . 37
ˆb1, ˆb2 Constant components of ˆB . . . 37
y∗ Desired output . . . 40
ϑθ(·) Departing particle angle pdf for given φdω1 . . . 43
ϑφ(·) Departing particle angle pdf for given θdω1 . . . 43
Θθ(·) Departing particle angle cdf for given φdω1 . . . 43
Θφ(·) Departing particle angle cdf for given θdω1 . . . 43
E Photon energy . . . 47
lw Wavelength of a photon . . . .47
hp Planck’s constant . . . 47
c Speed of light in a vacuum . . . 47
Tr Cold spot temperature reference signal . . . .52
ν Temperature input for experiment . . . 52
Tc Cold spot temperature . . . 52
λ Absorbance . . . 52
Qin Light intensity of bundle entering the atom cloud . . . 52
Qout Light intensity of bundle exiting the atom cloud . . . 52
l Length of absorbing path . . . 53
ka absorbing coefficient . . . 53
Qs Measured light intensity with shutter open . . . 53
Qb Measured light intensity with shutter closed . . . 53
Qref Measured baseline light intensity . . . 53
Λ Measured sodium pressure . . . 53
Ψ(·) Mapping from λ to Λ . . . 53
PN a Theoretical sodium vapor pressure . . . .53
ξ(·) Sodium vapor pressure function . . . 53
b1, b2, b3 Parameters in sorption function f (·) . . . 54
β1, β2 Parameters in sorption function ˙Ns(·) . . . 54
ˆ y Moles model output . . . 54
e Error signal, ¯y− PN a . . . 55
eΛ Absolute and normalized error of Λ w.r.t. PN a. . . 58
eyˆ Absolute and normalized error of ˆyw.r.t. PN a . . . 58
ey¯ Absolute and normalized error of ¯yw.r.t. PN a . . . 58
ζ Constant component of output function h(·) . . . 61
a1, a2 Constant components of ξ(·) . . . 61
ρ Measure of evaporation history . . . 65
ς Measure of temperature . . . 65
Υ(·) Pdf describing input-output probabilities . . . 65
Υu,λ(·) Marginal pdf of input current u and absorbance λ . . . 66
R Domain of ρ . . . 66
T Domain of ς . . . 66
Γ Set of input-output pairs (u, λ) for which Υu,λ> 0 . . . 66
ud Input required to realize λd . . . 67
λd Desired absorbance . . . 67
Φ(·) mapping of u to λ for feedforward component of controller . . . 68
A Considered domain for controller design of λ . . . 69
k Discrete time step counter . . . 69
Kp Proportional control gain . . . 69
ui Integral control input component . . . 69
efb Error between λdand λ . . . 69
Ki Integral control gain . . . 69
Δtk Discrete time step length between timestep k and k− 1 . . . 69
uff Feedforward control component . . . 69
ufb Feedback control component . . . .69
tk Time associated with timestep k . . . 70
η Online identified component in input-output relation . . . 71
Constant relating magnitude of input and output . . . 71
umf Model-free control component . . . 71
Δ Update parameter . . . 72
Part II
x Stochastic state variable . . . 82
X Domain of x . . . 82
u System input, generalized force . . . 82
U Domain of u . . . 82
t Time indicator . . . 82
f (·) System vector field . . . 82
x0 Stochastic initial condition . . . 82
X0 Domain of x0 . . . 82
φx0(·) Probability density function of x0 . . . 82
φx0,t(·) Probability density function of x(t) . . . 82
T Transient time for transient performance specification . . . 82
Ξ Subset of X, desired containment set . . . 82
ΦΞ,T(·) Cumulative density of φx0,T over the set Ξ . . . 82
ξ Integrator variable . . . 82
σ Second moment of a probability density function . . . 82
μ Mean value of probability density function . . . .82
p∗ Desired containment level . . . 83
d(·) Unspecified distance . . . 83
xd Desired asymptotic trajectory of x . . . .83
k(·) Control law . . . 83
A, B System matrices for linear system . . . 84
x∗ Constant desired asymptotic convergence value . . . .84
K Control gain matrix . . . 84
˜ x Stochastic state variable after coordinate change . . . 84
˜ x0 Stochastic initial condition after coordinate change . . . 84
N (·) Normal distribution . . . 84
˜ Ξ Desired containment set after coordinate change . . . 84
˜ xT ,low, ˜xT ,up Bounds on ˜Ξ . . . 84
˜ Ξ0 Initial time containment set after coordinate change . . . 84
˜ x0,low, ˜x0,up Bounds on ˜Ξ0 . . . 84
pmax Maximum containment level . . . .84
erf(·) The error function . . . 84
τ Time at which desired asymptotic behavior can be realized . . . 85
ud Desired input signal associated with desired trajectory xd . . . 85
dE(·) Euclidean distance between two points . . . 85
1, 2 Center of ball or distance set . . . 86
κ1, κ2 Radius of ball or distance set . . . 86
Bκ(·) Ball set defined with respect to Euclidean distance . . . 86
xr Tracking reference signal . . . .86
u∗ Feedforward control input . . . 86
z State of reference system . . . 86
ζ State of error system . . . 86
ζ0 Stochastic initial condition of error system . . . 86
λ Contraction exponential rate constant . . . 86
dF(·) Finsler distance . . . 87
fc(·) Closed-loop system vector field . . . 88
C Contraction region . . . 88
Dκ(·) Distance set defined with respect to Finsler distance . . . 88
X Domain of robot manipulator states . . . 90
Q Domain of q . . . 90
q Generalized position vector . . . 90
S Unitary circumference . . . 90
p Generalized momentum vector . . . 90
θ1, θ2 Rotational positions robot joints . . . 90
s Translational position of robot joint . . . 90
pθ1, pθ2, ps Momentum of robot joints . . . 90
M (·) Inertia matrix . . . 90
F1, F2, F3 Robot manipulator input components . . . 90
0 Zero matrix . . . 90 I Identity matrix . . . 90 H(·) Hamiltonian function . . . 90 D(·) Damping matrix . . . 90 G(·) Input matrix . . . 90 V (·) Potential energy . . . 90
m1, m2, m3 Robot manipulator link masses . . . 90
M1, M2 Inertia matrix components . . . 90
l1, l2 Robot manipulator link lengths . . . 90
q0 Stochastic initial condition of robot manipulator position . . . 90
μq Mean of q . . . 90
Σq Covariance of q . . . 90
μq,1, μq,2, μq,3 Mean values of q components . . . 91
Σ Covariance matrix . . . 91
σq,1, σq,2 Variance of q components . . . 91
p0 Initial condition of robot manipulator momentum . . . .91
gc Gravitational constant . . . 91
qd Desired asymptotic trajectory for generalized position . . . 91
pd Desired asymptotic trajectory for generalized momentum . . . 91
qr Reference trajectory for generalized position . . . 91
q Point associated to qr(T ) . . . 91
˜ q Error in q, defined as q− qd . . . 91
ω Error in p, defined as p− pr . . . 91
pr Reference trajectory for generalized momentum . . . .91
pdω Reference component for p . . . 91
Λ Controller component . . . 91
p Point associated to pr(T ) . . . 91 ueq, uat Input components for robot manipulator controller . . . 92
Kd Controller gain matrix . . . 92 VF(·) Finsler-Lyapunov function . . . 92 xv Virtual system state . . . 92 δxv Virtual system tangent vector . . . 92 δqv, δpv Virtual system tangent vector components . . . 92
Θ Constant component of VFfor robot manipulator example . . . 92 P (·) Functional component of VF for robot manipulator example . . . . 92
Γ(·) Collection of γ curves between two points . . . 92
γ(·) Differentiable curve between two points . . . 92 Υ(·) Functional component of contraction rate λ . . . 92
r Polar coordinate . . . 93 ρ(·) Angle used in figures . . . 94
1(·) Step function . . . 94
TxX Tangent space of point x . . . 97 T X Tangent bundle of all points in X . . . 97
c1, c2, c3 Constants used for defining Finslet-Lyapunov function . . . 97
F Finsler structure . . . 97 I Domain existing between 0 and 1 . . . 98
α(·) Function defining contraction properties . . . 98
B(·) The Bhattacharyya coefficient . . . 100
dh(·) The Hellinger distance . . . 100 db(·) The Bhattacharyya distance . . . 100 Desired Hellinger distance . . . 100
Y Domain of a pdf, specified to define matching property . . . 101
ϕ(·) Arbitrary pdf . . . 101
η, β Matrices describing linear matching of pdf . . . 101
ε Scaling correction for linear matching of pdf . . . 101 ˜
ε Scaling of pdf caused by system dynamics . . . 102 ˜
η, ˜β Matrices describing pdf change through system dynamics . . . 102 ˜
y, y Points after two different coordinate changes from x . . . 103
Z Domain of a pdf, specified to define matching property . . . 104
ϑ(·) Correction for elongation of pdf for nonlinear matching map . . . 104 Ψ(·) Function relating two pdfs in nonlinear matching map . . . 104
μd Desired mean point . . . 106
Σd Desired covariance matrix . . . 106 μT Realized mean . . . 107
ΣT Realized covariance matrix . . . 107
1.1 The third edition cover and the heliocentric model from Copernicus . . 2 1.2 Thin film examples . . . 3 1.3 Comparison of vacuum evaporation processes . . . 6 1.4 Conceptual control diagram for UHVCVD processes . . . 11 2.1 Knudsen cosine law coordinate illustration . . . 22 2.2 Validation of MC based method for obtaining the transfer matrix . . . 29 2.3 Illustration showing the partitioned safe and unsafe sets . . . 33 2.4 Discretization of cylinder for numerical flux control example . . . 35 2.5 Allocation of inputs and outputs for flux controller . . . 36 2.6 Phase plot of numerical flux control example . . . 39 2.7 Time trajectory of numerical flux control example . . . 39 2.8 Phase plot of numerical partial pressure control example . . . 41 2.9 Time trajectory of numerical partial pressure control example . . . 42 3.1 Wavelength spectrum for sodium HCL . . . 48 3.2 Schematic of experimental UHVCVD setup with AAS . . . 50 3.3 Detailed view of AAS components in experimental setup . . . 51 3.4 Origin of signal outputs for experimentation . . . 55 3.5 Applied and realized temperature signals for experiments. . . 57 3.6 Comparison of measured absorbances and associated pressures . . . . 57 3.7 Relation between measured absorbances, temperatures and pressures . 58 3.8 Comparison of model outputs, measured and theoretical pressures . . 59 4.1 Experimental setup configuration for controller implementation . . . . 64 4.2 Graphical example of two input-output trajectories . . . 66 4.3 Overview of observed relations between input and absorbance . . . 68 4.4 Block diagram for PI controller with feedforward component . . . 69 4.5 Performance graph of PIFF controller . . . 70
4.6 Block diagram for model-free PI controller . . . 72 4.7 Performance graph of MFPI controller . . . 73 4.8 Reference step response for MFPI controller . . . 74 4.9 Reference step response for PIFF controller . . . 75 4.10 Sinusoidal reference response for MFPI controller . . . 76 4.11 Sinusoidal reference response for PIFF controller . . . 76 4.12 Errors of the MFPI and the PIFF controller for sinusoidal reference . . 77 5.1 Containment control problem illustration . . . 83 5.2 Distance sets for CCP applied in the robot manipulator simulation . . 94 5.3 States and distance sets for robotic manipulator simulation . . . 95 5.4 Time evolution of robot manipulator states . . . 95 6.1 Shape control problem illustration . . . 101 6.2 Contour plot showing the realized and desired pdfs . . . 107 6.3 Contour plot showing the difference between pdfs . . . 107
AAS Atomic absorption spectrometry
CBF Control barrier function
CCP Containment control problem
CLF Control Lyapunov function
CVD Chemical vapor deposition
DOF Degree of freedom
EIES Electron impact emission spectrometry
ES-CBF Exponentially safe control barrier function
ES-CLBF Exponentially stable control Lyapunov-barrier function ES-CLF Exponentially stable control Lyapunov function
FMF Free molecular flow
HCL Hollow-cathode lamp
KKT Karush-Kuhn-Tucker
LPCVD Low pressure chemical vapor deposition
MBE Molecular beam epitaxy
MC Monte-carlo
MFPI Model-free proportional integral controller
MS Mass spectrometry
pdf Probability density function
PI Proportional integral
PID Proportional integral derivative
PIFF Proportional integral controller with feedforward component
PVD Physical vapor deposition
QP Quadratic program
SCARA Selective compliance assembly robot arm
SCP Shape control problem
UHV Ultra-high vacuum
UHVCVD Ultra-high vacuum chemical vapor deposition
Introduction
“Therefore I would not have it unknown to Your Holiness, that the only thing which induced me to look for another way of reckoning the movements of the heav-enly bodies was that I knew that mathematicians by no means agree in their in-vestigation thereof. For, in the first place, they are so much in doubt concerning the motion of the sun and the moon, that they can not even demonstrate and prove by observation the constant length of a complete year; and in the second place, in determining the motions both of these and of the five other planets, they fail to employ, consistently one set of first principles and hypotheses, but use methods of proof based only upon the apparent revolutions and motions.”
– Nicolaus Copernicus, 1543.
T
he publication of “De revolutionibus orbium coelestium” (On the Revolutions of the Heavenly Spheres) in 1543 by Nicolaus Copernicus is considered to have marked the beginning of the scientific revolution. The third updated and anno-tated edition of this manuscript was interestingly published in 1616 in Amsterdam by Nicolas Mulerius, professor of medicine and mathematics at the University of Groningen. The cover page and one of the models is shown in Fig. 1.1. The preface of Copernicus’ work considers the justification of his research and methods, addressed to Pope Paul III and, indirectly, to the physics and mathematics communities. His work was highly unconventional, as his theory had to be understood through math-ematics instead of physics. Now, 475 years later, we can safely say that the field of physics can no longer be understood without at least a basic understanding of math-ematics. This thesis considers the application of mathematical system and control engineering principles to a physical ultra-high vacuum chemical vapor deposition (UHVCVD) process, used for thin film deposition. As Copernicus did in his meth-ods, we aim to develop mathematical models and laws (or hypotheses) in order to test them with observations from the physical system. However, before doing so, we will introduce the physical process that we are working with and motivate our contributions. Let us accordingly begin this thesis by providing background infor-mation on thin films in Section 1.1. Following this, we will discuss vacuum evap-oration principles and some relevant techniques in Section 1.2. Subsequently, we will motivate the research directions in Section 1.4. This is followed by an outline of the remaining thesis chapters in Section 1.5. Lastly, we relate the chapters and their content to the original publications in Section 1.6.Figure 1.1: The third edition cover and the heliocentric model from Copernicus
1.1
Thin films
The name thin film suggests that thin films are specified by their thickness. The small film thickness is however a characteristic of the thin film but no specific bounds can be given. The reason for this is that we consider a body to be a thin film when certain physical anomalies appear. Accordingly, the physics branch of thin films deals with systems which have the common property that one dimension is very small, while other characteristics and the methods of investigating them may differ completely between the films (Eckertov´a 1977). Outside this branch, we are typically concerned with characterization of three dimensional bodies. For such bodies, we assume that their characteristics are volume independent. This assumption is however only valid for a small surface-to-volume ratio. For bodies that are very small in one dimension, like thin films, this surface-to-body ratio increases drastically and this assumption therefore becomes invalid. Let us shortly motivate why this is the case. In a bulk material, there are always forces acting upon given molecules or atoms. These forces are from all directions. They can be very structured or periodical, as is the case for crystalline materials, or they can lack such properties, as is typically the case for amorphous materials. However, when considering a surface area, some of these forces are cut off. The molecules or atoms at a surface are accordingly in a
differ-Figure 1.2: Thin film examples
Two examples of thin films are shown. In the left figure, a bell of soap shows different colors. The bell consists of a thin water film encapsulated by two soap films. Light is reflected from both layers of soap in the same direction. The thickness (or thinness) of the layer of water determines which wavelength of light results from the interference. The figure on the right shows camera lenses. These lenses are covered with anti-reflective coatings, which are thin films. A complex lens can have over 30 air-to-glass interfaces and the anti-reflective coating is essential in reducing the accompanied light loss caused by these transitions. Without these coatings, combining such a large number of lenses would result in unacceptable accumulated loss of light. (left: ©Brocken Inaglory, CC BY-SA 3.0 / right: ©Bill Ebbesen, CC BY 3.0)
ent state than the molecules or atoms in the bulk material. We consider this to be a surface state and the associated energy can differ substantially from the bulk state energy. For a body that is very small in one dimension, the two surfaces are so close to each other that they can have a profound influence on the internal characteristics of the body. This thin body can therefore differ substantially from a thicker version of itself. In fact, this difference can be so profound, that completely new phenom-ena can arise. This accordingly justifies why an entire branch of physics has been dedicated to thin films and the related technological branches.
Optical phenomena are some of the most apparent phenomena associated with thin films. An example is that of interference colors, which can be seen when water is covered by a thin layer of oil or with the bell of soap shown in Fig. 1.2. These phe-nomena have attracted attention since the second half of the seventeenth century and have resulted in modern applications of anti-reflective and decorative coatings (see Fig. 1.2 for the application to camera lenses). Electronic properties have been stud-ied from the beginning of the twentieth century. Phenomena that received particular interest are super- and semiconductivity. The development of electronics during and since the second world war has resulted in continuously decreasing electronic dimensions. This has been further promoted by the light-weight requirements of the space and avionic industries, as well as the development of medical electronics, which can be placed on or in the body of a patient. One of the main contributors
to this advance in recent years is the consumer electronics industry, e.g. computers, smartphones, tablets, watches, cars and many other applications. This miniaturiza-tion of electronic elements like tubes, resistors and capacitors has invoked the use of semiconductor elements, like diodes and transistors. This was followed by the introduction of small ceramic plates on which the elements are prefabricated, mostly in the form of thin films.
Associated thin film production technologies can be considered as a further sub-division of the physics of thin films branch. In most of the processes, the thin film is deposited on a substrate. These depositions can occur through a chemical reac-tion (chemisorpreac-tion), through physical forces of attracreac-tion (physisorpreac-tion) or a com-bination of both. Accordingly, the deposition processes are typically categorized as chemical vapor deposition (CVD) or physical vapor deposition (PVD) processes. Further categorization of the processes is difficult because the processes, but also the application of each process, may vary in many ways. What further complicates the matter is that the number of processes is vast. In (Vossen and Kern 1978) 33 different processes are listed and a subset is often considered in the literature.
In this thesis, we will focus on a specific realization of thin film deposition through vacuum evaporation. We will use the next section to go further into detail on this deposition technology.
1.2
Thin film deposition through vacuum evaporation
processes
Deposition through vacuum evaporation (Berry et al. 1968, Eckertov´a 1977, Vossen and Kern 1978, Ohring 2001), sometimes referred to as thermal evaporation, has been one of the most popular thin film deposition methods due to its relative simplicity and its potential to produce films of extreme high purity. The actual deposition pro-cess consists of several steps:
1. Transformation of the materials to be deposited (precursors) to the vapor phase; 2. Mass transfer of precursors from the evaporation source to the substrate; 3. Deposition of atoms or molecules on the substrate.
Vacuum evaporation deposition is generally considered to be a PVD process, since the principle mechanism binding the vapors to the substrate (or any surface) is ph-ysisorption. However, physical, chemical and hybrid binding mechanisms are all possible. There are situations where a direct gas insertion is desired instead of an evaporation or sublimation process, but we will not cover these situations in this thesis. Instead, we will assume in the sequel that evaporation sources releases pre-cursor atoms, as this is typically the case. Evaporating atoms directly furthermore
allows for the formation of films of the highest purity, which is one of the main ben-efits of the vacuum evaporation process that plays a central role in this thesis.
The depositions in a vacuum evaporation deposition process typically occur in-side a dedicated chamber, called the deposition chamber. Such a chamber allows for easy cleaning after a process run. The deposition chamber is located in a vacuum system, which is an integrated part of the vacuum evaporation reactor. Before com-mencing depositions, the vacuum system holding the deposition chamber needs to decrease the pressure many orders of magnitude from atmospheric condition so that: (i) undesired atoms or molecules are (mostly) removed from the deposition chamber, (ii) deposition chamber pressure is below the associated vapor pressure of precur-sors and (iii) the mean free path of precurprecur-sors becomes far larger than the deposition chamber dimensions. Evaporation (or sublimation) of the precursors is achieved by either resistance heating, radio-frequency induction or electron bombardment. Mass transfer subsequently occurs through the free molecular flow (FMF, also known as Knudsen flow) transport regime. The evaporation is typically performed in line-of-sight of the substrate, since desorption of the precursors from surfaces other than the evaporation source typically occurs in minimalistic degree, caused by relatively low surface temperatures and low vapor pressures of the precursor elements.
Both the physical and chemical binding mechanisms are highly sensitive to tem-perature. As a consequence, choosing appropriate operating temperatures of reactor components is very important. There are three temperature domains of interest: (i) evaporation source temperatures, (ii) reactor surface temperatures and (iii) sub-strate temperature. The evaporation source temperatures (i) are determined by the characteristics of the evaporation source and the required precursors. When con-sidering the reactor surface temperatures (ii), we can distinguish between hot-wall and cold-wall processes. In cold-wall processes, the reactor is not heated (only indi-rectly through the evaporation) or some parts may even be cooled, this will therefore cause vapor pressures to be very low. In hot-wall processes, the reactor is heated and this promotes further precursor migration. The latter can be desirable when the pre-cursors are not evaporated in line-of-sight of the substrate. Placing the evaporation sources out of the line-of-sight of the substrate can promote purity and uniformity by decreasing the chance that some contaminants reach the substrate and by causing the angle of incidence of arriving precursors to be distributed more uniformly. The choice between a hot-wall or a cold-wall process is hence dependent on the required precursors and on purity and uniformity demands. The substrate temperature (iii) influences characteristics of the layer that grows on the substrate. Requirements on the substrate temperature are therefore often dictated by the application of the thin film.
There are three particular realizations of vacuum evaporation deposition pro-cess that we highlight. The first is Ultra-High Vacuum Chemical Vapor Deposi-tion (UHVCVD) (Meyerson 1986, Greve and Racanelli 1991, Meyerson 1992, Adam et al. 2010) and is shown in Fig. 1.3a. This realization is the main subject of this
a)
UHVCVDb)
LPCVDc)
MBEFigure 1.3: Comparison of vacuum evaporation processes
A comparison between ultra-high vacuum chemical vapor deposition (UHVCVD), low pressure chemical vapor deposition (LPCVD) and molecular beam epitaxy (MBE) is shown. UHVCVD and LPCVD are hot wall processes, which is shown in red here, while MBE is a cold wall process, which is shown in blue. UHVCVD and LPCVD furthermore have an additional chamber inside the reactor, which is the deposition chamber. This chamber is replaced by a shield in the MBE process. The deposition chamber serves to reduce leakage to the pump so that some precursor partial pressures can build-up. There are three precursors, which are yellow, purple and green. The blocks in these colors are the associated evaporation sources. The purple precursor is evaporated in line-of-sight of the substrate in all three processes. In UHVCVD and LPCVD, the yellow and and green precursors are not evaporated in line-of-sight of the precursors. The green, yellow and purple dots are precursor atoms and the grey dots are residual molecules or atoms. The UHVCVD process has significantly less residual molecules or atoms in the process due to the UHV conditions. The downward pointing arrows indicate the system exhaust, this is where the vacuum pumps are connected.
thesis. The second is Low Pressure Chemical Vapor Deposition (LPCVD) (Fossum et al. 1985, Rausch and Burte 1993) and is shown in Fig. 1.3b. This realization is closely related to UHVCVD but operates in less strict pressure regions. The third is Molecular Beam Epitaxy (MBE) (Cho and Arthur 1975, Schlom et al. 2008) and is shown in Fig. 1.3c. This realization is the Physical vapor deposition (PVD) cousin to LPCVD and UHVCVD and can be considered as the state-of-the-art in vacuum evaporation deposition. We will use the remainder of this section to provide some background information on these processes and to compare them to each other.
1.2.1
Ultra-high vacuum chemical vapor deposition
The first application of UHVCVD as described by (Meyerson 1986) considers the chemisorption of a Germanium vapor to a Silicon substrate. This application utilizes purely chemical binding, but there is often a combination of physical and chemi-cal binding mechanisms in modern applications. Such a process can include line-of-sight evaporators such as effusion cells (also known as Knudsen cells), but also evaporators that are not placed in line-of-sight of the substrate. Precursors evaporat-ing from the latter migrate to the substrate via multiple deposition chamber surface
collisions. To promote repeated desorption of the precursors from these surfaces, a hot-wall process is desired. This heating of the deposition chamber surfaces then causes physisorptions to be reduced. Accordingly, binding of these precursors to the substrate requires either: (i) a chemical binding to the substrate, (ii) a chemical change in the precursor (by binding with another precursor) or (iii) a substrate that has a reduced temperature in comparison to the deposition chamber walls. Notice that (i) relies on chemisorption, (iii) on physisorption and (ii) on both. Combin-ing this insight with the observation that application of (iii) is not directly visible in the literature then justifies the general association to CVD processes. The indirect migration of precursors to the substrate causes a pressure buildup in the deposi-tion chamber. This is exacerbated with increasing surface temperatures and with decreasing deposition chamber leakage to the vacuum pumps. UHVCVD is a par-ticularly suitable processes when precursors with high vapor pressures are involved, because these precursors can be promoted to migrate with a limited heating of the deposition chamber surfaces. The indirect migration of the precursor to the substrate can have three favorable effects. The first is that any undesired elements escaping from the evaporation source can easily become stuck to the deposition chamber sur-faces when their vapor pressure is significantly lower than the vapor pressure of the precursor. The second is that the indirect migration of the precursors to the sub-strate surface allows for increased film uniformity, due to the more uniform angle of incidence in comparison with a line-of-sight evaporation. The third is that there is potentially a high degree of controllability of this process because, in particular when (ii) is utilized, chemical reactions can be activated. This in turn allows for time to settle on desired precursor pressures and temperatures before activation. UHV-CVD is accordingly an interesting candidate technology for applications that require extreme high purity of the deposited film. However, the technology is lacking in terms of controllability and repeatability. There are three main causes: Run-to-run variations in the substrate (on the nanoscale), run-to-run variations in evaporation sources and in-situ sensor placement restrictions. It is difficult to detect and correct for the variations during processing due to in-situ sensor placement restrictions. The
in-situ sensor placement restrictions are a direct consequence of a necessary reactor
bake-out, which is instrumental to reaching UHV conditions.
1.2.2
Low pressure chemical vapor deposition
The main difference between UHVCVD and LPCVD lies in the vacuum level. The LPCVD vacuum conditions can be achieved without the aforementioned bake-out of the reactor. This in turn causes LPCVD to have less restrictions on sensor place-ment inside the vacuum system. For example, the well-known quartz crystal micro-balance and some mass spectrometers can be used for certain applications in LPCVD, while this is not possible in UHVCVD. There is accordingly more flexibility to deal with run-to-run variations in LPCVD than there is in UHVCVD. The price to pay is
that the lower vacuum contains more residual atoms or molecules, which in turn can cause impurities in the deposited thin film.
1.2.3
Molecular beam epitaxy
MBE is strictly a line-of-sight evaporation process and belongs to the PVD processes. The precursors are typically evaporated in effusion cells. These cells facilitate tem-perature control and a shutter, which allows for a high degree of outgoing precursor flux control. When the cells are heated sufficiently and the shutter is open, the pre-cursors are transported to the substrate, where they are bound through physisorp-tion and where they can chemically react with other precursors. The substrate is typically rotating around its central axis to promote deposition uniformity. MBE processes are cold wall processes, as it is desired to have the Knudsen cells dictate all precursor fluxes to the substrate. The deposition chamber is furthermore replaced by a shield, intended to catch any precursor atoms that do not hit the substrate. This shield can be cooled to further promote physisorption of the precursors. MBE op-erates under similar conditions as the LPCVD process, and therefore offers more flexibility in sensor placement than a UHVCVD process. There are little restrictions on substrate and evaporator temperature. MBE processes are popular due to their relatively high degree of controllability in comparison with other thin film deposi-tion techniques. The process does however face two major disadvantages: (i) fluxes and temperatures need to be controlled very precisely since the depositions are al-most instantaneous as a consequence of the direct physisorption that occurs after evaporation and (ii) impurities in the deposited film can occur due to contaminants in the evaporation source and residual gasses in the system, as a result of the process not being performed in UHV and evaporation being line-of-sight.
1.3
UHVCVD operation recipe
Let us shortly describe a typical UHVCVD operation recipe before we continue to the motivation for the contributions in this thesis. UHVCVD operation recipes are application dependent, but a controlled batch deposition process roughly consist of the following processing steps:
1. Pre-process substrate, reactant sources and deposition chamber. 2. Build-in components and create UHV conditions.
3. Regulate to desired background pressures and temperatures. 4. Enable chemical reactions at substrate.
6. Cool down reactor.
The pre-processing of components in step 1 is typically a cleaning and mounting process. Step 2 involves preparing the reactor for deposition. The reactor needs to be heated so that residual molecules evaporate and can be pumped out of the sys-tem, in order to achieve the UHVCVD conditions. This step can furthermore involve pre-heating of the precursor evaporation sources to remove contaminants. In step 3 the reactor is prepared for the first deposition cycle. The reactor is regulated so that desired background pressures and temperatures are attained. At this stage un-desired reactions are generally avoided by carefully choosing which precursors are available and/or by temperature control. The reactions are subsequently activated in step 4. This activation can be performed by either adding a previously missing precursor (preferably as a molecular beam) and/or through the control of surface temperatures (in particular the substrate temperature). We remark that activation of the reactions is not essential, but it does employ a high degree of control over the de-positions that occur. Step 5 involves iterating steps 3 & 4, in order to build up layer thickness. Iterations should be small enough to allow for the desired chemical equi-librium to be maintained. Continuous operation is achieved when the iteration time is infinitesimally small. Lastly, when the desired thickness is achieved, the reactor is gradually cooled down in step 6.
1.4
Motivation
With this thesis, we aim to contribute to the development of control for UHVCVD reactors. To this end, we contribute to: (i) development of the physical UHVCVD process in combination with a mathematical process model and controller design and (ii) development of mathematical control algorithms. This results in the thesis being split in two parts:
• Part I: Modeling of free molecular flow dynamics, partial pressure measure-ment and controller implemeasure-mentation.
• Part II: Controller design for deterministic systems with stochastic initial con-ditions.
We use the remainder of this section to motivate our interest in the two directions listed above.
1.4.1
Modeling of free molecular flow dynamics, partial pressure
measurement and controller implementation
In the first part of the thesis we provide the first steps in facilitating controller imple-mentation for UHVCVD processes. We have motivated in Section 1.2.1 why
UHV-CVD can be the deposition technology of choice for some applications, in particu-lar when extreme high purity of the deposited thin films is required. However, we also have discussed that the controllability and reproducibility are still open issues, caused by run-to-run variations in the evaporation sources and substrates. An effec-tive method of dealing with such issues by implementing run-to-run and real-time feedback control (Edgar et al. 2000).
A conceptual control diagram for UHVCVD processes is shown in Fig. 1.4. The run-to-run controller aims to optimize the runs with knowledge on previous runs. Such a controller can consist of (a collection of) algorithms and a reference manager. The reference manager sets desired (future) values for the real-time controllers. The run-to-run controller can use both (previous) real-time and ex-situ data for its pur-pose. The real-time controllers aim to steer relevant temperatures and partial pres-sures to the desired values as dictated by the reference manager. Such controllers therefore rely on real-time (and past) measurements of these states and the availabil-ity of such measurements is thus essential.
The relevant states to measure for UHVCVD reactor controller design are: (i) rel-evant temperatures, (ii) film characteristics (like thickness) and (iii) precursor partial pressures (or fluxes). The states (i) and (iii) together affect the chemical reactions that are the depositions, which in turn affects (ii), this view is in accordance with model-ing for UHVCVD as in (Greve and Racanelli 1991). Measurmodel-ing temperatures (i) can be done through contact (thermocouple) and optically (pyrometer). Both of these methods are well established and compatible with UHV. Temperature control is typ-ically performed by placing PID-controlled heating or cooling elements at locations of interest, such as the substrate. Such a form of control is rudimentary but works well, in particular in combination with a reference manager. Measuring layer char-acteristics (ii) can be done optically through Ellipsometry, reflection or transmission measurements (as for example implemented in (Middlebrooks and Rawlings 2007)). The techniques are also well established but require application dependent models to relate measured variables to film characteristics. Also, any control action based on a real-time layer characteristic measurement will demand a change in pressure or temperature (through the reference manager, for example), since these are the vari-ables that affect the chemical reactions that are the depositions. We therefore con-sider such a controller to be of a higher hierarchical level than the partial pressure or temperature controllers in such a process. This causes its effectiveness to be de-pendent on the performance of the lower level pressure and temperature controllers. Developing such a controller is accordingly interesting once good temperature and partial pressure controllers are available. In contrast to the above, a method for mea-suring the precursor partial pressures (iii) is not clearly visible in the literature. The exposition above motivates our interest in the development of a real-time precursor partial pressure controller for a UHVCVD reactor. Such a controller can accordingly reduce the effects of the run-to-run variations in the evaporation sources.
de-Partial pressure controller precursor 1
Real-time contr
ollers
Run-to-run contr
oller
Reference manager
Algorithms
Real-time layer characteristics measurements
Real-time temperature & pressure measurements
Ex-situ
layer characteristics measurements
Partial pressure controller precursor
n
Substrate temperature controller
Reactor temperature controller
Reference signals
Input signals
Updated values
Evaporation precursor 1
Evaporation precursor
n
Substrate heating/cooling
Reactor heating
Energy / mass
UHVCVD pr
ocess
Mass transport,
thermodynamics
& chemistry
Coated substrate
Output
Figure 1.4 : Conceptual contr ol diagram for UHVCVD pr ocesses A conceptual contr ol diagram for UHVCVD pr ocesses is shown. The diagram contains two contr oller layers and UHVCVD pr ocess dynamics. The pr ocess dyna mics ar e subject to heating, cooling and evaporations thr ough the system inputs, supplied by the real-time contr ollers. These changes in ener gy and mass cause depositions to occur thr ough mass transport, thermodynamics and chemistry and ther efor e result in a coated substrate. The evaporation, heating and cooling pr oces ses can be contr olled individually by the real-time contr ollers, which in turn aim to follow a refer ence trajectory supplied by the refer ence manager . This refer ence mana ger is part of the run-to-run contr oller and is updated/dir ected thr ough algorithms. The pr ocess can accordingly impr ove performance between runs thr ough this mechanism. The real-time contr ollers requir e past and real-time measur ements of the variables that they contr ol, so that these variables can be steer ed to the supplied refer ence value. Th e run-to-run contr oller can use both the real-time data and ex-situ data for learning purposes.sign so that we obtain a system with actuation and measurements that are required for both the modeling of the system and the controller design. The second step con-siders the modeling of the systems dynamics, so that we obtain a sufficiently accu-rate model that relates the reactor input to the (indirectly) measured reactor outputs. The third step considers the model-based controller design. Such a controller design is typically model based so that tuning can be performed analytically and offline, having a model furthermore allows for usage of controllers that rely on predictions from the model online. The last step is the integration of the previously mentioned components, to obtain the controlled system.
In part I of this thesis, we will present contributions to UHVCVD reactor design, modeling of free molecular flow (FMF) dynamics and controller design for partial pressure control in UHV conditions. Our contributions to the UHVCVD reactor de-sign concerns implementation and calibration of an atomic absorption spectroscopy (AAS) based measurement that can measure precursor partial pressures with excel-lent accuracy and in very low pressure regions. We contribute to the modeling of FMF dynamics by proposing a flux dynamical model. This model allows us to con-sider the spatial dependence of fluxes and sorption explicitly and can therefore pro-vide further insights in these phenomena. We validate both our AAS-based measure-ment and the fluxes model by comparing their performances with a lumped (bench-mark) model and the theoretical partial pressure in the vapor pressure regime. Our contribution to the controller design for partial pressure control utilizes the AAS-based partial pressure measurement but not the fluxes model. We instead implement two simple controllers that require minimal identification using input-output data. The first controller is a proportional integral controller with a feedforward compo-nent. The second controller is a model-free proportional integral controller (based on results from (Fliess and Join 2013)). We show that we can achieve good performance with both controllers and compare their performances.
1.4.2
Controller design for deterministic systems with stochastic
initial conditions
In Part II of this thesis we provide controller design (methods) that can deal with un-certainties in initial conditions of the states of a deterministic dynamical system. Let us first motivate why such a controller design can be useful for UHVCVD processes. From there, we will move on a more general introduction of the subject, as we will contribute to this control problem in a general setting.
Controlling variations in UHVCVD processes
UHVCVD is a process type that can potentially benefit from controlling variations that are inherited from the initial conditions. Our contributions in Part I facilitate improved understanding of process dynamics through modeling and real-time
