physique des plasmas radiofréquence - polytechnique · physique des plasmas radiofréquence pascal...
TRANSCRIPT
Planning
trois cours : Lundi 30 Janvier: Rappels de physique des plasmas froids Lundi 6 Février: Modèle d’une décharge capacitive Lundi 13 Février: Décharges capacitives multifréquence
En parallèle: travail personnel ou en binôme
Pour le 6 Février: Lecture des chapitres 2 & 3 Pour le 20 Février : Ecrire un modèle global d’une décharge inductive utilisée comme source d’ionisation d’un propulseur à grille
Propulseur à grille excité par une décharge inductive (ICP)
Ar
Ar+
e
Ar+
e Ar Ar+ e
Ar Ar+ e
Ar Ar
Ar Ar
Ar+
Ar
Ar Ar+
Ar
Ar Ar+
Ar+
1. Temperature and density domains
2. Non equilibrium
3. Thermal equilibrium properties
4. Collisions and reactions
5. Sheaths, Debye length, Child law
Introduction to plasma discharges
Plasmas
• Plasma : ionized gas
– Fully (fusion energy program)
– partially, ionization degree:
• Three types of species :
– electrons
– Ions (positive and negative)
– neutrals (radicals or stables)
• Discharges : 54 10101 outypicallyxiz
ig
iiz
nn
nx
Plasma discharges
• Potential is maximum at the
plasma center:
– Electrons are confined
– Negative ions are confined
– Positive ions are accelerated
toward the walls
Ji
Je JN
• Positive ion flux : directed
• Electron flux : isotropic
• Neutral flux : isotropic
Plasma
n+ = ne+ n-
<V>
~
Positive space
charge: sheaths
0
Relative densities and energies
n (cm-3) n (cm-3)
Non Equilibrium
• Plasma is sustained by electrons which provide ionization
e + Ar → Ar+ + e + e
• Electrons are usually near thermal equilibrium and have a large
average energy: Te is typically 35 000 K
• Ions and neutrals (heavy) remain near room temperature: 300K
• However, in discharges, ions gain directed energy toward the
surfaces, i.e. they are not in thermal equilibrium
Distribution functions Thermal Equilibrium
Collisions, cross sections Mean free path, collision frequency
Rate coefficient
Thermal equilibrium properties (1)
see page 22
Thermal equilibrium properties (2)
see page 24
Thermal equilibrium properties (3)
see page 24-25
Averages over Maxwellian distributions (1)
see page 23-24
Averages over Maxwellian distributions (2)
Averages over Maxwellian distributions (3)
Averages over Maxwellian distributions (4)
Collisions, cross sections
Number of targets in V:
Proportion of scattered flux:
“area” of targets (cross section):
Mean free path, collision frequency
The flux decays exponentially:
The characteristic length is called the mean free path:
The collision frequency is :
The rate coefficient for the collision process is :
In plasmas
The cross section is a function of the incident electron energy
Electrons have a velocity distribution. The collision frequency is
defined as follows:
Elastic Collisions
Inelastic Collisions, e.g. ionization
f ( )
( )
Eiz
Idealized cross section:
Integral over a Maxwellian yields:
Which may be further simplified :
Rate coefficients for reactions
e + Ar → Ar+ + e + e Kiz
1. The central problem of discharge modelling
2. Fluid equations
3. Particle and energy balance
4. Electromagnetic properties – waves
5. Radiofrequency Reactors
Plasma Dynamics
Central problem of discharge modeling
• Electromagnetic fields generate forces on particles
• But, particle motion generates electromagnetic field!
To find self-consistent solution…
• Need to solve simultaneously plasma transport and Maxwell’s
equations
• Difficult problem; needs simplification
• Various level of simplification of plasma transport
• Go from EM fields to Voltage and Currents: circuit theory
Kinetic description
• Follow the motion of each particle in the field: impossible
• Define macro-particle and solve the motion of each of these
self-consistently with the fields : Particle-in Cell simulations
• Or, define a distribution function and follow the evolution using
Boltzmann or Vlasov equations: kinetic theory
All of these are complicated and simpler approaches are often possible
Fluid equations
Particle conservation equation
Momentum conservation equation
Global model: particle balance
Plasma (volume V)
Surrounded
by a surface A
Global model: energy balance
Plasma (volume V)
Surrounded
by a surface A
Electromagnetic properties
The plasma may be treated as a dielectric with the following
dielectric constant (page 47):
If one ignores displacement currents then the plasma conductivity is:
Dispersion relation of EM waves
The plasma is in
fact a conductor at
low frequency and
a dielectric at high
frequency
Skin depth
Dielectric at high frequency ( > pe) Conductor at low frequency ( < pe)
pi pe rf domain
MHz GHz
Waves are absorbed in a skin depth Propagating waves (microwave
diagnostics: interferometry, reflectometry
etc.)
Inertial (low pressure)
Capacitively-coupled plasma
~ rf
Inductively-coupled plasma
~ rf
~ rf
Typical etching reactors: CCP’s, ICP’s
• Electrons follow the rf field
• Ions follow time-averaged field pi pe
rf domain MHz GHz
13.56 MHz or higher?
Magnetic confinement
m
qBc
qB
mRL
v
Cyclotron frequency:
Larmor radius:
For typical conditions (B 50 Gauss):
• Non-magnetized ions: RL 10-20cm
• Magnetized electrons: RL 1-2 mm
Anisotropic dielectric constant
Waves in magnetized plasmas
see page 265
Helicon reactors
Water cooling
rf
13.56 MHz
Matching network
and source cooling
Helicon antenna
Wafer holder
Load lock and
cartridge transfer
Source
solenoid
400l/s
turbo
pump
150l/s
turbo
Ar, SF6
Chamber
solenoid
B0 Helicons generate high density
plasmas. Interesting for:
• Very deep etching
• Space plasma propulsion
1. DC sheaths
2. Plasma/sheath transition
3. Plasma transport
4. Plasma flux leaving the plasma and
reaching the surfaces
Bounded Plasmas
Why sheaths?
• Without sheaths, currents at the wall are :
e
ee
m
TkneJ
20
M
TkneJ i
i2
0
• Since me << M and Te >> Ti :
– Je >> Ji loss of electrons
• The positive space charge builds up an E field
directed to the walls which confines electrons and
accelerate ions to the wall
Plasma
n+ = ne+ n-
<V>
~
Positive space
charge: sheaths
0
E
Debye length (1)
Negative potential
perturbation
Field or potential screening occurs
within the Debye length
Boltzmann electrons:
Debye length (2) Space charge density:
Poisson’s equation:
Child law sheath (1)
• At high voltage, no electrons in the sheath
• No ion-neutral collisions
• Positive ion current is limited
by the charge space
Plasma Ji
s
-V0
0
x
Child law sheath (2)
Child law sheath (3)
Integrate twice over x:
Plasma Ji
s
-V0
0
x
Sheath thickness
• Positive ion current produced by the plasma (described later in the course):
• Using current continuity and Child law,
we obtain the sheath thickness:
4
3
0
eDe Tk
Ves
M
Tkneh
s
V
M
q eel
i02
23
02
1
0 2
9
4
M
TknehJ e
eli 0
Plasma
V01 V02>V01
+
-
Ii
+
+
+
+ +
+
+
+
e- +
+
+
e- e-
Plasma/sheath transition
This velocity is called the Bohm velocity
and is noted us= u B
The flux at the wall may then be written:
Plasma transport
Electric force
Pressure force
Friction force
Forces must balance !
ns
n0
Plasma transport
The transport of the plasma, and consequently the ratio hl that controls the plasma
flux at the boundary, depends upon the pressure regime:
- So-called Schottky or ambipolar diffusion at high pressure
- Godyak solution at intermediate pressure regime
Ambipolar diffusion (1) – Page 84
Ambipolar diffusion (2) – Page 86
• At low pressure the ion-neutral collision frequency, and consequently the ion mobility, becomes a function of the fluid velocity:
• This leads to the following edge-to-center density ratio:
Godyak’s solution for intermediate pressures – Page 87
To summarize, at low and intermediate pressure…
Sheath
x xs
ns
n0
0
n
Plasma
d
at high pressure…
Sheath
x xs
ns
n0
0
n
Plasma
hl vs pressure
The issue of electronegative plasmas and neutral depletion
• In the previous theories, we considered only positive ions and electrons, ne=ni, and
we considered constant neutral density.
• However, processing gases are electronegative and the plasma may contains a
large amount of negative ions, ne+nn=ni
• Moreover, contemporary reactors have high plasma densities which may lead to
neutral depletion at the reactor center
• These issues are very important ! All transport theories must be revisited (some of
these issues will be treated later in this course)
• Later in this course, we will see the effect of electronegativity on the plasma
stability