modeling and simulation of seers · analogy: pendulum lumped model: only a simple oscillation...
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Matthias Kratzer, MP TI CS SIM2000-12-18
Modeling and Simulation of SEERS
R. P. Brinkmann and Th. MussenbrockLehrstuhl für Theoretische Elektrotechnik
Ruhr-Universität Bochum
Matthias Kratzer, MP TI CS SIM2000-12-18
Overview
Plasma Modeling and SimulationPerspectives on SEERSLumped Circuit ModelsA General Mathematical ModelA Simplified Toy ModelOutlook
Th. Mussenbrock/R.P. Brinkmann: The SEERS-Diagnostic Method in
in High-Density Plasmas
Matthias Kratzer, MP TI CS SIM2000-12-18
Plasma reactors
Typical regime:• 1 kW RF power• 1 Pa gas pressure• 10-2 m3 volume• 1017 m-3 density
Source: Sentech
Matthias Kratzer, MP TI CS SIM2000-12-18
Plasma modeling
Plasma modeling: Multiple scales, multiple physics!!!
Molecular Dynamics Model
Feature Model
Sheath Model
Surfacereaction rates
Angular andenergy distribution
ei+ n
Transport andsurface chemistry
Plasma
Plasma Model
[nm]
[µm]
[mm]
[m]
Matthias Kratzer, MP TI CS SIM2000-12-18
Practitioners view on SEERS
Typical points of investigation:• How are the parameters related to the process result?• Which of the parameters is the most sensitive? • How does SEERS compare with other diagnostics?
Matthias Kratzer, MP TI CS SIM2000-12-18
Our view on SEERS
Our points of investigation:• What is the physical background of the resonance?
• What exactly do the SEERS parameter mean?• Is there more information in the SEERS signal?
Matthias Kratzer, MP TI CS SIM2000-12-18
Nature of the resonance
Bulk is “inductive”:
Sheath is capacitive
Matthias Kratzer, MP TI CS SIM2000-12-18
Lumped circuit model
• Linear bulk model to represent effective impedance• One or two instances of a non-linear sheath model• Solved either analytically of numerically
TRF
TSEERS
t
Matthias Kratzer, MP TI CS SIM2000-12-18
Parameter study: RF-voltage
Amplitude of the oscillation increases with excitation.
VRF = 50 V
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curr
ent [
a.u.
]
VRF = 20 V
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curre
nt [a
.u.]
VRF = 100 V
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curre
nt [a
.u.]
VRF = 200 V
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curr
ent [
a.u.
]
Matthias Kratzer, MP TI CS SIM2000-12-18
Parameter study: Pressure
Damping increases with neutral gas pressure.
p = 20 mTorr
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curr
ent [
a.u.
]
p = 50 mTorr
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curr
ent [
a.u.
]
p = 100 mTorr
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curr
ent [
a.u.
]
p = 200 mTorr
-40
-20
0
20
40
60
80
100
0 0.5 1 1.5 2Phase [norm.]
RF
curre
nt [a
.u.]
Matthias Kratzer, MP TI CS SIM2000-12-18
Comparing with reality
Simulation SEERS Sensor-Data
• Good “qualitative” correspondence (form, trends).• No quantitative agreement, no “fine structures”
The real signal is obviously much more complex! WHY?
-20
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0 0.5 1 1.5 2
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-10
0
10
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30
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0 0.5 1 1.5 2
Matthias Kratzer, MP TI CS SIM2000-12-18
Analogy: Pendulum
Lumped model: Only a simple oscillation pattern possible
5 10 15 20 25
-0.4
-0.2
0.2
0.4
5 10 15 20 25
-0.4
-0.2
0.2
0.4 Velocity
Position
Matthias Kratzer, MP TI CS SIM2000-12-18
Analogy: Chain
2.5 5 7.5 10 12.5 15
-3
-2
-1
1
2
3Position
Distributed model: Complex oscillation patterns possible
2.5 5 7.5 10 12.5 15
-3
-2
-1
1
2
3
Velocity
Matthias Kratzer, MP TI CS SIM2000-12-18
Analogy: Chain
Eigenmode analysis: Superposition of oscillation modes
Matthias Kratzer, MP TI CS SIM2000-12-18
General reactor model
Arbitrary geometry, arbitrary number of applied voltages ...
Plasma Bulk
Plasma Boundary Sheath
v1
v2
Full Electromagnetics
Matthias Kratzer, MP TI CS SIM2000-12-18
General model: Equations
Definition of a state space:
Dynamical equations:
Electron equation of motion
Magnetic induction law
Sheath charging
Boundary conditions:
Matthias Kratzer, MP TI CS SIM2000-12-18
General model: Analysis
The state space Z can be interpreted as a Hilbert space, with an inner product motivated by an energy functional• The concept of orthogonality applies• The tools of functional analysis can be used
Abstract formulation of the dynamics:
Conservative part: pure oscillation
Dissipative part: damping
Nonlinear harmonics generation
External excitation
Matthias Kratzer, MP TI CS SIM2000-12-18
General model: Solution
The equation can be solved by a triple series expansion(power series, Fourier series, eigenmode expansion)
The series coefficients can be found by recursion:
Algebra in the solution space of the linear problem:
Numerically tractable even for complicated situations
Matthias Kratzer, MP TI CS SIM2000-12-18
General model: Example
• Numerical scheme with consistent discretization
• Any 2D-geometry, any electron density distribution
• Tested with analytical solutions for special cases
Matthias Kratzer, MP TI CS SIM2000-12-18
Toy model: Set-Up
Bulk
Vrf
Boundary
Sheath
Simplifications to facilitate an analytical treatment:• Simple axi-symmetric cylinder geometry• Constant bulk and sheath parameters
Matthias Kratzer, MP TI CS SIM2000-12-18
First eigenmode
The eigenmode shape depends on the plasma density
Matthias Kratzer, MP TI CS SIM2000-12-18
Toy model: Results I
SEERS signal for a small amplitude of the RF excitation
Matthias Kratzer, MP TI CS SIM2000-12-18
Toy model: Results II
SEERS signal for a medium amplitude of the RF excitation
Matthias Kratzer, MP TI CS SIM2000-12-18
Toy model: Results III
SEERS signal for a large amplitude of the RF excitation
Matthias Kratzer, MP TI CS SIM2000-12-18
Toy model: Comparing with reality
Simulation SEERS Sensor-Data
• Good “qualitative” correspondence (form, trends).• No quantitative agreement yet, but “fine structures”
For a quantitative agreement: Go beyond the toy model!
-20
-10
0
10
20
30
40
0 0.5 1 1.5 2
-20
-10
0
10
20
30
40
0 0.5 1 1.5 2
Matthias Kratzer, MP TI CS SIM2000-12-18
Perspective
What have we learned?
• Consistent framework for modeling of the SEERS effect, without any restriction on geometry or homogeneity.
• The nonlinear problem can be solved by “doing algebra”with the linear solutions. (In practice, few modes suffice).
Work in progress towards a full implementation:
• Essential elements already in place (nonlinear boundarysheath model, consistent numerical discretization ...).
• Current projects aim to include more physical effects (static magnetic fields, stochastic dissipation ...).
• Implementation into a full reactor simulator planned.
• Comparison with careful experiments is underway.
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