Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Implicit time integration methods and inexact Newton methods: application
to chemical vapor deposition
S. van Veldhuizen1
C.R. Kleijn2
1 Delft University of Technology
Delft Institute of Applied Mathematics
J.M. Burgerscentrum
2 Delft University of Technology
Department of Multi Scale Physics
J.M. Burgerscentrum
Kolloquium der Arbeitsgruppe Modellierung, Numerik,
Differentialgleichungen
Technical University Berlin, Germany
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
C. Vuik1
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
1
Introduction
Chemical Vapor Deposition
Transport Model
2
Numerical Methods
Properties
Positivity
Nonlinear Solvers
Linear Solvers
3
Numerical Results
4
Conclusions
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Outline
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Chemical Vapor Deposition
Transport Model
Chemical Vapor Deposition
Synthesizes thin solid film from gaseous phase by
chemical reaction on solid material
Delft Institute of Applied Mathematics
Reactions driven by thermal energy
Vuik, Van Veldhuizen and Kleijn
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Chemical Vapor Deposition
Transport Model
Chemical Vapor Deposition
Applications
Solar cells
Optical,
mechanical and
decorative
coatings
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Semiconductors
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Chemical Vapor Deposition
Transport Model
Transport Model for CVD
Mathematical Model
Continuity equation
Navier-Stokes equations
Energy equation in terms of temperature
Ideal gas law
Species Equation
Advection Diffusion Reaction Equation
∂(ρω)
= ∇ · (ρvω) + ∇ · (D∇ω) + m
∂t
Vuik, Van Veldhuizen and Kleijn
#reactions
X
k =1
νk RkG
Delft Institute of Applied Mathematics
Species equation
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Chemical Vapor Deposition
Transport Model
Transport Model for CVD
Reaction Rate
Net molar reaction rate
S
X
kνi,s kci − kk ,bw
S
X
k − νi,s kci
i=1
i=1
Ek
Modified Law of Arrhenius kk ,fw = Ak · T βk e− RT
max(kk ,fw )/ min(kk ,bw ) = 1028
Properties Mathematical Model of CVD
Consists of (#species − 1 + 3 + d ) coupled PDEs
Stiff nonlinear system of species equations
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
RkG = kk ,fw
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Numerical Methods
Properties
Positivity (= preservation of non-negativity):
Negative Species can cause blow up of the solution
Efficiency / Robustness
Method of Lines approach
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Stiff Problem → Stable Time Integration
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Positivity
Mass fractions
A natural property for mass fractions is their non-negativity
1
Model equations
2
Spatial discretization: Hybrid scheme
Introduces locally first order upwinding
3
Time integration
4
Iterative solvers:
(Non)linear solver
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Positivity of mass fractions should hold for . . .
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Positivity for ODE systems
Euler Backward
wn+1 − wn = τ F (tn+1 , wn+1 )
Theorem (Hundsdorfer, 1996)
Euler Backward is positive for any step size τ
Theorem (Bolley and Crouzeix, 1970)
Any unconditionally positive time integration is at most first
order accurate
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Unconditionally stable (A-stable/ stiffly stable)
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Nonlinear Solvers
Let x0 be given.
FOR k = 1, 2, . . . until ‘convergence’
Find some ηk ∈ [0, 1) and sk that satisfy
kF (xk ) + F ′ (xk )sk k ≤ ηk kF (xk )k.
Set xk +1 = xk + sk .
ENDFOR
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Inexact Newton to solve F (x) = 0
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Nonlinear Solvers
Inexact Newton Condition
Choices for Forcing Term
kF (xk )k−kF (xk −1 )−F ′ (xk −1 )sk −1 k
ηk =
kF (x
)k
k −1
ηk =
2
γ kFkF(x(xk )k)k2
k −1
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
kF (xk ) + F ′ (xk )sk k ≤ ηk kF (xk )k
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Properties
Huge condition numbers due to chemistry terms
12
10
11
10
10
9
10
8
10
7
10
6
10
5
10
−5
10
−4
10
−3
10
−2
10
−1
10
Time (s)
Vuik, Van Veldhuizen and Kleijn
0
10
1
10
2
10
Delft Institute of Applied Mathematics
Condition number
10
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Delft Institute of Applied Mathematics
Lexicographic ordering (left) and Alternate blocking per grid
point(right)
Vuik, Van Veldhuizen and Kleijn
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Iterative Linear Solver
Right preconditioned BiCGStab
Incomplete Factorization: ILU(0)
lexico alternate blocking
Number of
graphic
per gridpoint
F
220
197
Newton iters
124
111
Linesearch
12
7
Rej. time steps
0
0
Acc. time steps
36
36
CPU Time
400
300
linear iters
444
346
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
’Heavy’ chemistry terms → diagonal blocks
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Preconditioners: Lumping
Delft Institute of Applied Mathematics
Important: Lumping per species
Vuik, Van Veldhuizen and Kleijn
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
Preconditioners: Block Diagonal
series of uncoupled
systems → LU
factorization per
subsystem
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
‘natural’ blocking
over species
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Properties
Positivity
Nonlinear Solvers
Linear Solvers
To compute: inverse of a diagonal block → solve linear
system directly
Storage: factorization of diagonal blocks
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Preconditioners: Block D-ILU
Block version of the matrix derived from central differences
on a Cartesian product grid
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Computational Domain
fSiH4= 0.001 fHe= 0.999
inflow
solid
wall
dT/dr = 0
v=0
u=0
z
θ
r
0.1 mole% SiH4 at the
inflow
Rest is carrier gas
helium He
Susceptor does not
rotate
susceptor
u, v = 0 Ts=1000 K
0.15 m
Axisymmetric
outflow
dT/dz = 0
dv/dz = 0
0.175 m
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Tin=300 K
0.10 m
uin= 0.10 m/s
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Computational Domain
0.09
0.07
Temperature:
Inflow 300 K
Susceptor 1000 K
0.06
0.05
0.04
0.03
Uniform velocity at
inflow
0.02
0.01
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
grid sizes: 35 × 32 up
to 70 × 82 grid points
0.08
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Chemistry Model: 16 species, 26 reactions [1]
Above heated wafer SiH4 decomposes into SiH2 and H2
Including the carrier gas the gas mixture contains 17
species, of which 14 contain silicon atoms
Irreversible surface reactions at the susceptor leads to
deposition of solid silicon
[1] M.E. Coltrin, R.J. Kee and G.H. Evans, A Mathematical Model of the Fluid Mechanics and Gas-Phase Chemistry
in a Rotating Chemical Vapor Deposition Reactor, J. Electrochem. Soc., 136, (1989)
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Chain of 25 homogeneous gas phase reactions
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Numerical Results
ILU(0)
F
Newton
linesearch
Rej. time step
Acc. time step
lin iters
CPU
197
111
7
0
36
346
300
Lumped
Jac
310
185
20
3
41
3693
590
Vuik, Van Veldhuizen and Kleijn
block
DILU
210
112
13
0
36
676
380
block
diag
3181
1239
0
459
774
3315
3250
direct
solver
190
94
11
1
38
6500
Delft Institute of Applied Mathematics
Integration statistics: 35 × 32 grid
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Numerical Results
ILU(0)
Lumped
Jac
F
869
nf
Newton
476
nf
linesearch
127
nf
Rej. time step
15
nf
Acc. time step
62
nf
lin iters
8503
nf
CPU
7400
nf
Direct solver is not feasable.
Vuik, Van Veldhuizen and Kleijn
block
DILU
613
327
106
0
37
2036
5300
block
diag
nf
nf
nf
nf
nf
nf
nf
Delft Institute of Applied Mathematics
Integration statistics: 70 × 82 grid
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Validation: Species mass fraction along the symmetry axis
−2
10
4
2
−4
10
−5
Si H SiH
2 6
10
2
−6
10
solid: Kleijn’s
solutions
H SiSiH
2
2
−7
10
Si2
−8
10
circles: our
solutions
−9
10
Si
−10
10
−11
10
−12
10
0
0.002
0.004
0.006
0.008
0.01
0.012
Height above Susceptor (m)
Vuik, Van Veldhuizen and Kleijn
0.014
Delft Institute of Applied Mathematics
Species Mass Fractions (−)
SiH
H
−3
10
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Validation: Radial profiles of total steady state deposition rate
wafer
temperature
from 900 K up
to 1100 K
Deposition rate (nm/s)
6
5
1100 K
4
1050 K
3
2
1000 K
1
0
950 K
900 K
0
0.05
0.1
Radial coordinate (m)
Vuik, Van Veldhuizen and Kleijn
0.15
solid: Kleijn’s
solutions
circles: our
solutions
Delft Institute of Applied Mathematics
7
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Transient behavior of deposition rates
2
1.8
Total
Si2H2
1.4
along
symmetry axis
1.2
1
H2SiSiH2
0.8
wafer 1000 K
0.6
0.4
SiH2
0.2
0
0
1
2
3
4
5
Time (s)
6
7
Vuik, Van Veldhuizen and Kleijn
8
9
Delft Institute of Applied Mathematics
Deposition rate (nm/s)
1.6
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Kleijn’s Benchmark Problem
Transient behavior of deposition rates
2.5
Deposition rate (nm/s)
2
1050 K
along
symmetry axis
1000 K
1.5
wafer
temperatures
from 900 up to
1100 K
950 K
1
0.5
0
900 K
0
1
2
3
4
5
6
7
Time (s)
Vuik, Van Veldhuizen and Kleijn
8
9
Delft Institute of Applied Mathematics
1100 K
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Conclusions
Euler Backward is unconditionally positive, but Inexact
Newton does not preserve this property
Alternate blocking per grid point is more effective
Easy preconditioners are effective for 2D problem
Chemistry source terms should be in the preconditioner
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Conclusions and Future Research
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Conclusions and Future Research
How to preserve positivity when iterative linear solvers are
used ?
More realistic chemistry/surface chemistry models
Steady state solver
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
Future Research
3D transient simulations
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
References and Contact Information
S. VAN V ELDHUIZEN , C. V UIK AND C.R. K LEIJN, Comparison of
Numerical Methods for Transient CVD Simulations, Surface and
Coatings Technology, 201, pp. 8859-8862, (2007)
S. VAN V ELDHUIZEN , C. V UIK AND C.R. K LEIJN, Numerical
Methods for Reacting Gas Flow Simulations, International
Journal for Multiscale Computational Engineering, 5, pp. 1-10,
(2007)
S. VAN V ELDHUIZEN , C. V UIK AND C.R. K LEIJN, Numerical
Methods for Reacting Gas Flow Simulations, in: V.N. Alexandrov
et al. (Eds.): ICCS 2006, Part II, LNCS 3992, pp.10-17, (2006)
C.R. K LEIJN, Computational Modeling of Transport Phenomena
and Detailed Chemistry in Chemical Vapor Deposition- A
Benchmark Solution, Thin Solid Films, 365, pp. 294-306, (2000)
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
References
Implicit time integration methods and inexact Newton method
Chemical Vapor Deposition (CVD)
Numerical Methods
Numerical Results
Conclusions
Contact Information
Email: c.vuik@tudelft.nl
Url: http://ta.twi.tudelft.nl/users/vuik/
Vuik, Van Veldhuizen and Kleijn
Delft Institute of Applied Mathematics
References and Contact Information
Implicit time integration methods and inexact Newton method