Download 3.1) Plasma physics for ion sources

3.1) Plasma physics for ion sources
To understand ion sources operation, knowledge in plasma physics is required.
The constituents of plasma, depending on the ionization degree, are
Ions, electrons and atoms (neutrals)
Different to a gas, the particles in plasma interact strongly due to Coulomb forces. The behaviour of
plasma can be influenced by electric and magnetic field.
The plasma is often referred to the „fourth state of matter! The transition from gas to the plasma state
is not a real phase transition in the thermodynamic view, but energy (latent heat) is required to ionize
atoms in the gas!
Examples for plasma:
Natural:
Sun, ionosphere, polar lights, flames, flashes, e- in metals
Artifical:
Neon tube, plasma screens, plasma etching systems,
fusions plasmas, metal-vapor discharge lamps, ion source plasmas
Due to free moving charged particles in the plasma, it has a high conductivity!
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3.1
Plasma density und plasma temperature
The plasma density of ions or electrons are determined as particle densities
ni
and
ne
 1 
 cm3  or
in
 1 
 m3 
A multi component plasma is neutral and the neutrality condition is:
q n
i
i
 ne
(3.1)
The degree of ionisation or fractional ionization is defined as
pion 
ni
ni  nneutral
pion = 1 is the case of a completely ionized plasma.
A plasma is called highly ionized if the fractional ionization is larger than 10%.
Typical plasmas in the lab: ne ~ 1014 – 1022 1/m3
For comparison: Gas density at room temperature and 10-2 Pa: n = 2.5*1018 1/m3
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3.2
The temperature of a plasma is typically given in eV:
1 eV = 11600 K
e*U = kT  U = 1 V  T = e/k
Ion temperature Ti and electron temperature Te don't have to be identical. In presence of a magnetic
field, an anisotropy is introduced, leading to different T parallel and perpendicular to the field:

Ti|| , Ti  , Te|| , Te 
The reason for that is the different mobility of the particles parallel and perpendicular to the field.
The concept of temperature is also applied to plasma, which are not in thermal equilibrium!
Typical electron temperatures:
 surface ionized plasma
 arc discharge plasma
 microwave generated plasma
Te ~ 0.2 eV
Te ~ 1 eV
Te ~ some keV
Ti < 1 eV
Plasma frequency
The electrons as well as the ions can oscillate inside the plasma. Depending on the component
different modes can be present, electron-plasma oscillation (Langmuir wave) or "ion sound".
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3.3
Continuity equation:


 div j  0
t


j    v ,   ne  me
The motion of plasma is implied by Euler's equation (hydrodynamic equation of motion for
incompressible and frictionless flow)

 v  
   v 
 t


dv 
v 
F
dt

 


with F  ne eE
Divide the charge carrier density ne into a constant part ne0 and a flow ne1 with
ne0  ne1
, in which ne1 is not neutralized!
Due to the quasi-neutrality
 el  e  ne0   qi ni 0  e  ne1  e  ne1
i
From the 1. Maxwell equation follows
Assumption:
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

v , ne1 und E
  el
e  ne1
div E 

0
0
are small numbers  linearization of equation is possible:
3.4

v
e 

E
t
me
from the Euler equation and

ne1
 ne 0  div v  0
t
from the continuity equation.
With both equations one get:
 2 ne1
e 2 ne1
 ne 0
0
2
t
me 0

oscillation equation with
2
e
 ne 0
 p2 
me   0
(3.2)
ne1 (t )  A  cos  p t
Plasma frequency
p is the characteristic time, in which the plasma reacts on a disturbance!
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3.5
1
fp 
2
e2
 ne  8,98  ne [ m13 ]Hz 
 0me
Only electromagnetic waves with f > fp are able to travel through the plasma. p or fp is called the cutoff-frequency of the plasma.
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3.6
The calculation of the electric field of a disturbance shows that the disturbance can only be small.
Assuming a disturbance of 1% in the quasi-neutrality in a local environment of 1mm size:
 
div E 
0
E 0.01  ne  e
10 V


  E  10
x
0
m
Such a field can't be preserved. Thus, quasi-neutrality is given, with only small disturbances. The
electron density at which an electro-magnetic wave is reflected is called the critical density
4 2 0 me 2
2 1 




ncritical 

f

0
.
0124

f
Hz
p
p
 m3 
e2
 
Example for cut-off frequencies:
 Free electrons in metal, density ~1028 1/m3  f ~ 1015 Hz, visible light is reflected, UV-light can
pass.
 Ionosphere, density ~109 – 1010 1/m3  f ~105 –106 Hz, medium wave is reflected, metric wave
can pass.
The Debye-length
The quasi-neutrality causes a shielding of plasma particles among each other. The region of the
shielding can be calculated via perturbation calculation:
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3.7




ne r   ne  ne1 r  ; ni r   ni  ni1 r 
Poisson equation
  



1
   e  ne  Ze  ni  e  ne1 r   Ze  ni1 r 
0
0
With the assumption of thermal equilibrium (Te=Ti and Boltzmann distribution)


e

(
r
)

ne1 r   ne  exp 

 kT 


Ze

(
r
)

ni1 r   ni  exp 

 kT 
,
(3.3)
and quasi-neutrality, one can obtain a linearization of (3.3) and insertion in Poisson equation delivers:
 e2  ne  1
Z  
0


 r  


r


 0  kTe kTi 
0

Solution:
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 r  
1
q
 e
40 r

r
D

 rˆ
for


0 r   q   ( r )
(for a single charge q)
3.8

2D 
 0  k  Te
ne  e 2
(3.4)
Debye-length
The Debye-length is the characteristic length for the shielding of constant background charge density
0. The solution of the differential equation
above is the Coulomb potential with shielding.
The function is shown by the following graph.
For r > D the potential drops with as e-function
Each charged particle is surrounded by a
Debye-cloud
Calculation of Debye-length:
D 
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 0  k  Te
ne  e 2
3.9
D  7437
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Te [eV ]
[m ]
1
ne [ m 3 ]
3.10
Plasma edge and plasma potential
A plasma appears as quasi-neutral on a scale x > D. The potential inside the plasma with respect to
the wall is constant and is called the plasma potential p. If plasma particles would be elastically
reflected by the wall, p would be equal to the wall potential. The particles recombine with a probability
of 99%.
Plasma-wall interface: a) ideally reflected, b) complete wall recombination
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3.11
The flux towards the wall follows from the Maxwell velocity dostribution:
 m 
dW  f (v )d 3v  

2

kT


average velocity:
3/ 2
 mv2  3
d v
exp  
 2kT 
v   v f ( v ) d 3v 
number of particles with
  
v  v , v  dv 
8kT
m
Flux on the wall:
1/ 2 
 m 
  n

2

kT


nv
 mv 2 
0 vx exp   2kT dvx  4
Flux is proportional to the mean velocity of the particles. Therefore
e 
mi
i
me
Given that the velocity, and thereby the flux of electrons, is higher compared to the ion flux, a current
would break the quasi-neutrality. Thus the wall becomes negatively charged with respect to the plasma
or the plasma positive against the wall. Hence the electrons are decelerated and the ions accelerated
which balances the net currents. A charge double-layer develops which can be in the order of the
Debye-length. The layer is called Debye-layer.
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3.12
The thickness of the plasma edge depends on the potential of an eventually existing electrode with
>> p.
In this case
d sheath  D 
e  Zug
kT
(3.5)
and with
ne = 1018 1/m3, Te = 1 eV, D = 10 m
and  = 20000 V
results in dsheath = 1.4 mm.
An insulated electrode inside the plasma will be charged up to a „floating potential“ f, which is about 34 times kTe. For this electrode
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e  i
holds true, i.e. there is not net current.
3.13
With a probe reaching in to the plasma,
which is connected to a power supply,
it is possible to measure the potential.
Such a probe is called
Langmuir-probe.
By varying the potential of the probe
with respect to the wall, the characteristic
current curve can be obtained.
The saturation currents are proportional to the fluxes:
I e,sat
d (ln I p )
1
mi


I i ,sat
,
dV
T( eV )
me
,
ne 
4 I e, sat
e  Aprobe T( eV )
The application of the Langmuir-probe in practice is limited by the power dissipated on the probe.
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3.14
3.2) Generation of ion source plasma
Erzeugung eines quasineutralen Plasmas durch Stoßionisation
q n
i i
 ne
Bereitstellung freier Atome im
Plasmagenerator durch:
Bereitstellung freier Elektronen
durch:
Bereitstellung der Ionisationsenergie durch:
• Einlassen eines Arbeitsgases
• Glühemission
• Beschleunigung der Elektronen
• Schmelzen und Verdampfen
• Photoionisation
• HF-Heizung
• Sputtern von Feststoffen
• Funkenentladung
• E x B-Drift
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3.15
A simple experiment to generate a plasma is the gas discharge. For the experiment a vacuum tube is
needed. The following figures show the simple experimental setup. By lowering the gas pressure
inside the tube a glow of the cathode can be recognized. The neutral gas is then partially converted to
a plasma (it became conductive).
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3.16
In the neutral gas, atoms can by ionized (e.g. by
X-rays from the environment) and thereby
electrons are released. The current between
anode and cathode depends on the amount of
charges generated on the electrons path.
Probabilit y for ionization by e
path length
first Townsend coefficient
The second Townsend coefficient  describes the
ionization of an atom by an ion (is negligible in source plasmas).
For the generated electrons the following expressions are valid:
dN e
   Ne
dx

N e (d )  N e 0ed
Ne0 = Number of generated electrons at the cathode
dNi
   Ni
dx

N i ( d )  N i 0e d
same for ions
For d >> 1  N(d) >> N0
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charge or Townsend avalanche
3.17
Below a certain gas pressure p a stationary discharge emerges, i.e. electrons are continuously
generated. Impacts of the ions onto the cathode material releases secondary electrons.
The number of secondary electrons per ion impact is given by the third Townsend coefficient with
dN e  Ni   .
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3.18
  0.01  0.1
ignition condition
  ed  1
From the ignition condition follows the ignition voltage:
With
U Ignition 
c2 p  d
ln c1 p  d   ln ln( 1 )


f = mean free path in gas, Ui = ionization potential
(Ignition- or Paschen - curve)
For high p*d the short mean free path
has to be compensated by a higher U.
For small p*d, U  ∞, because the mean
free path becomes >> d.
For the minimum of the curve, the mean
free path equals ~d.
The difference between the curves arises
from specific secondary-electron yield 
for each element.
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3.19
At low gas pressures it is called a glow discharge and for higher pressure and current a arc discharge.
The gas discharge
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3.20