Physique des plasmas radiofréquence

Pascal Chabert

LPP, Ecole Polytechnique

pascal.chabert@lpp.polytechnique.fr

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