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Plasma Physics 2
Adam Obrusnik, obrusnik@plasmasolve.com
Tomáš Hoder, hoder@plasmasolve.com
1 Fyzika plazmatu 2
Introduction and context (CZ)
Předcházející povinné a volitelné kurzy
- F5170 Úvod do fyziky plazmatu / dr. Z. Bonaventura
(definice plazmatu, pohyb částic v el-mag poli, Transportneí rovnice v plazmatu)
- F7241 Fyzika plazmatu 1 / doc. Zajíčková
(rozdělovači funkce, teorie stěnové vrstvy)
- F3180 Výboje v plynech / prof. Cernák + Dr. Krumpolec
(klasifikace výbojů a teoorie formování jednotlivých výbojů)
- F4280 Technologie depozice tenkých vrstev a povrchových úprav / prof. Vašina + doc. Zajíčková
(plazmochemie, plazmové zdroje pro PVD a PECVD)
2 Fyzika plazmatu 2
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Course requirements
- Understanding the presented topics, especially:
- Definitions of different discharge modes and types.
- Transition mechanisms between discharge modes.
- Conditions and properties of different discharge types.
- Analytical calculations, estimating plasma properties at various conditions.
- ... (see details below)
- Oral exam
Individual consultations available:
- Adam Obrusnik
- Tomáš Hoder
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Literature
- RAIZER, Y.P. Gas discharge physics, Springer, 1991
- COBINE, J.D. Gaseous conductors, Dover Publications, New York, 1958
- CHEN, Francis F. Introduction to plasma physics and controlled fusion. 2nd ed. New York: Plenum
Press, 1984. xv, 421. ISBN 0306413329.
- BITTENCOURT, J. A. Fundamentals of plasma physics. 3rd ed. Sao Jose dos Campos: National
Institute for Space Research, 2003. xxiii, 678. ISBN 85-900100-3-1.
- BAZELYAN E.M., Raizer Y.P. Spark discharge, CRC Press, Taylor&Francis, 1998
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Lecture series contents
1. Townsend breakdown theory, Paschen's law
2. Glow discharge
3. Electric arc at low and high pressures
4. Magnetized low-pressure plasmas and their role in material deposition methods.
5. Brief introduction to high-frequency discharges
6. Streamer breakdown theory, corona discharge, spark discharge
7. Barrier discharges
8. Leader discharge mechanism, ionization and discharges in planetary atmospherres
9. Discharges in liquids, complex and quantum plasmas
10. Thermonuclear fusion, Lawson criterion, magnetic confinement systems, plasma heating and intertial confinement
fusion.
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Discharges - what this Lesson covers?
6 Fyzika plazmatu 2
O:>
-10
DARK DISCHARGE
T
TOWN SEND REGIME
SATURATION
REGIME
10
GLOW DISCHARGE
i 1
r~
BREAKDOWN VOLTAGE
ARC
NORMAL
GLOW
ABNORMAL
GLOW
< >k
GLOW-TO-ARC
TRANSITION
HIGH
i INTENSITY ARCS
BACKGROUND IONIZATION
_L i
INTENSITY
I ARCS
i r 10 10' 100 10,000
CURRENT I, Amps
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Electron avalanche, Townsend criterion,
discharge breakdown
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HI
Electron avalanche
At the turn of 19th and 20th century: Townsend a Paschen formulate the basics
of gaseous discharges / gas discharges
1930s: Experimental observation of electron avalanches in vapor chambers
and later in vacuum chambers
Luminous structures between the cathode and the anode - more luminous for
higher voltages. John S.Townsend
Zeitschrift für Physik*
Die Entwicklung der Elektronenlawine
in den Funkenkanal.
(Nach Beobachtungen in der Nebelkammer.)
Von H. Raether in Jena.
Mit 8 Abbildungen. {Eingegangen am 28. Februar 1939.)
a b c
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Electron avalanche - the principle
_Q: What energy do you need to ionize an atom?
A: 10-15 eV for gases, 5-10 eV for metal vapors
Q: Is that energy higher or lower than e.g.
dissociation of C02 => CO + O?
A: Ionization is higher, chemical bond energy
typically 1 - 8 eV
Q: Where did the first electron come from?
A: There was a "plenty" - background ionization
DC
Cathode
A ->
a
-*A
-r e
-
e--
A*
• e- -
Aiiode
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Electron avalanche - empirical summary
• Plane-parallel electrodes with constant E field
• Volume filled with primary electrons (cosmic radiation, radioactivity or external UV)
• When voltage is applied, charged particles move towards the electrodes where they are
lost. Volume recombination can be neglected in the first approximation.
• Electrons are accelerated to energies over the ionization threshold
• In such an experiment, one can observe 3 scenarios:
1. After turning off the external source of ionization, current disappears = non selfsustaining
discharge, see range T0
2. Further voltage increase leads to ionization and creation of avalanches, external
ionization source not needed = Townsend discharge (range T.,), described very well
by the electron avalanche theory
3. Further voltage increase leads to range T2 = additional phenomena (e.g.
recombination), no longer described by the simple electron avalanche theory.
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7777777,
Caff- rurtn
UV
source
0 M
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Important assumption of further slides
— In the derivations below, we assume Townsend dark discharge, i.e. we maintain
plasma densities low enough that they do not affect the electric field
- This holds only before discharge ignition, so the slides below describe how
discharges are initiated but not how they operate once they are ignited.
Fig. 8.6. Distributions of clcctroiE]ectrjc
and ionic currents (a), and chai fie|d
density distribution (b) when
field in the gap is not distorted ion
, density
space charge
The first Townsend coefficient a
Q: How do we mathematically describe the growth of electron
density in an electric field?
dne = anedx
ne = ne>0eax
E
Q: What does the coefficient depend on?
- Type of gas
- Pressure
- E field magnitude
- Energy distribution of electrons
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Anoda (+
Katoda (-)
The third Townsend coefficient y
- When an ion hits the surface, so-called secondary electron
emission can occur.
- The probability of such effect is described by a factor y (or
ISEE = ion-induced secondary electron emission).
- Values of y:
0.1 - 0.2 in laboratory plasmas (ion energy 100s of eV)
0.1 - 10 by ion milling (ion energy 1-10 keV)
Anoda (+
Katoda (-)
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The third Townsend coefficient y
At voltages below 500 eV, potential emission is the main mechanism of ISEE
Electron no.1 from the conduction band of the metal tunnels through the potential
barrier and recombines with the positive ion. The energy gain can be used for
releasing electron 2
Above ca 500 eV, kinetic energy of the ion starts to make a contribution
£
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Anoda (+
Katoda (-)
The third Townsend coefficient y
In "clean metals", which are very smooth , ISEE depends on the material work
function Wf - ion energy must be at least 2Wf for ISEE to occur.
In real life, ISEE is surprisingly quite material-independent and depends more
on surface micro-roughness
[Phelps, A. V., & Petrovic, Z. L. (1999). PSST, 8(3), R21-R44.doi:10.1088/0963-0252/8/3/201]
Anoda (+
10
« 1 r
- 1 — I I I 1111 - 1 — I — I I I I 111
- 1 — I — I I I 1 1
10
_ 1
Ion or atom energy (eV)
Figure 1. Electron yields for A r +
and Ar beams incident on
various clean metal surfaces versus particle energy. The solid
symbols are for A r +
and the open symbols are for Ar. The
symbols, metals and references are: T, W, [68]; +,v, Mo, [46];
• ,H,Mo, [47]; A , M o , [70]; • . M o , [66]; i , A u , [71]; x , C u ,
[67]; H , Pt, [69] and • , Ta, [69]. The curves drawn through
representative values will be used in our model.
T
l 1
1
dirty metals .
10 102
10a
Ion or atom energy (eV)
Figure 2. Electron yields for Ar+
and Ar beams incident on
various dirty metal surfaces versus particle energy. The open
symbols are for Ar+
and the solid symbols are for Ar. The symbols,
metals and references are: • , Pt, [69]; Kl.Ta, [69]; v . Au, [49];
O, Cu, [75]; 0, • , Cu, [45]; A , A, Ta, [80]; x, W, [79]; • , brass,
[81]; • , unknown, [50] and •, CuBe, [51]. The solid curves are
plots of the analytical yield expressions for dirty surfaces for Ar*"
and Ar, while the dashed curves are the representative yield curves
for clean surfaces for Ar4
" ions and Ar atoms from figure 1.
Katoda (-)
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What about the second
- The second Townsend coefficient p was supposed
to better describe the region T2, where glow
discharge starts to form.
- It never really caught on because it was way too
simplistic a treatment.
As a plasma-physicist, you can sometimes be
wrong. The world will still appreciate that you are
trying to tackle this beast ©
16 Fyzika plazmatu 2
coefficient?
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Discharge breakdown condition
To define the breakdown condition, we will use the
concept of "current density"
je = qen
ev
drift,e = <7eMen
eE
h = ^i^iVdrift,! = qmntE
Where v d r i f t a : is the drift velocity of a particle and \ix is
the charge carrier mobility. Other symbols usual
meaning.
Thinking in terms of current density is practical because
current is always a conserved quantitiy.
Anoda (+
Katoda (-)
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Discharge breakdown condition
— Choose the cathode at x=0 and anode at v x=d
- The cathode emits electrons through ISEE and through a constant
external source (e.g UV). Furthermore, we can link the current of ions
to electron current density.
je(0) =Jo+YW)=Jo+YJemead
~ 1) i
- Electron current from the cathode is then:
1 - y(ead
- 1)
And before they reach the anode, they multiply through volume
ionization
Anoda (+
Katoda (-)
A
[ >
rvri
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Discharge breakdown condition
— We derived the total discharge current to be je(d) =
Joe ad
l-y{ead
-l)
If we turn off the external source of electrons, the current
stops => non self-sustaining Townsend discharge
However, the current grows towards infinity if
y(ead
- 1) = 1 E
This is what we call the discharge breakdown criterion: The
amount of ions created by one electron during its
passage between the electrodes has to be such, that they
create another electron by IS EE.
Anoda (+
Katoda (-)
A
>
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Discharge breakdown condition
— The ignition condition
Is often written as
y(ead
- 1) = 1
y(ead
- 1) > 1
or even
y{ead
- 1) » 1
- This is because maintaining a Townsend discharge is usually
not the goal in the applications. What we usually want is to
ignite a stable self-sustained discharge and multiply the
Anoda (+
Katoda (-)
A
[ >
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Breakdown voltage, Paschen law
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Breakdown voltage
— Practical observation: Voltage between the cathode and the anode has to be higher than a certain
value so that a discharge is ignited.
- Typically denoted Vb
Q: What do you think affects the value of Vb
A: Cathode-anode distance (the actual "constant" is break down E field more than breakdown voltage)
A: Type of gas and electrode condition => ionization energy affects a, electrode affects y
A: On the collision mean free path (so pressure) => more collisions imply higher a
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Breakdown voltage
- Generally, we can approximate the Townsend coefficient as
- Since the mean free path depends on A ~ - — we can define the reduced Townsend coefficient
p \pJ N \NJ
where N is gas density in m - 3
.
- This hints to a special role of the quantity ^ or ^ in Plasma physics.
- It turns out that most transport and ionization coefficients depends, in very good approximation,
only on - but not on gas density or electric field alone.
E E
- Note: People use either - or - . The former is closer to the physics truth while the latter kinda sorta
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E/N in plasma science
— SI unit for E/N is V • m2
. Unfortunately, this reaches values of 102 0
— 102 5
and because physicists are
not yet comfortable with saying "zettavolts" or "yottavolts", people use 1 Td = 10~2 1
V • m2
Paschen law
From the above, we can derive the analytical formula for discharge breakdown condition
- = F{-)
- = F- (v
i n v
Vp,
Combining that with y{ead
- l ) = 1 yields a = ^-ln + l )
And if Vb = E • d before the plasma is ignited, it also has to hold that
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Paschen law
- ^ p d - J W ^ l n g + l))
- Paschen law states that: For a given
gas, the breakdown voltage is a
function of the product of pressure
and distance.
- The function minimum is called the
Stoletow point.
ioJ
pd [Torr cm]
10"
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ÍO6
Paschen law
—Q: Try to interpret the Paschen law. Why
does the curve grow in both directions?
A: Low pd implies either low pressure or
distance - so there are not enough
collisions to meet the Townsend criterion.
At higher pd and fixed voltage, the value of
a decreases because E decreases and it
is difficult to meet the Townsend criterion.
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— He
— Ne
— Ar
— H2
pd [Torr cm]
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Paschen law - quantitatively
— We can obtain a by solving BKE and somehow fit - « A • exp ( - ^
10'
TO'
: i i r-
b
• 1 - — i - i
Xe
// ff
-
H
V í l
7 Ar/I II / X e
Kr/ /
t li i i
E/p {V/cm-Torr!
i
200 £00 7000 2 4 10 2 4 102
2 4 2
FIR. Ionization coefficients for a wide range of Efp values (a) in molecular gases, (b) in inert
gases. From [4.3]
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Pivu A B oblasť £|/po
[cm-l
Torr-'I [Veiíi^Torr-'] jVťin-^orT-1
]
Hfl 3 34 20 - 150
Ne 4 iOO 100 400
Ar 14 180 100 GOO
Kr 17 240 100 - 1000
Xe 20 350 200 - 800
Tödlich 15 365 100 800
H, 5 130 150 600
N2
12 • Al íoo - eoo
CO? 20 Am hm - looo
B2 Ü 13 290 150 1000
II, 20 370 200 G00
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Paschen law - quantitatively
•=• We can obtain a by solving BKE and somehow fit ^ « A • exp ( - ^ )
- By substituting into the Paschen law and differentiating, we can get
Bpd B / 1
=
UApd) - I n [ i n ( l +1)] " b . m i „ = ^ l n ( l + -
Due to various real-life phenomena (finite electrodes, recombinations, other gas-phase collisions), this
rarely corresponds to reality. But it is a decent first estimation of the discharge voltage for
different gases and gas mixtures.
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Actual discharge ignition
- Nice overview of the problematics:
[Shishpanov, A. I., Meshchanov, A. V., Kalinin, S. A., & lonikh, Y. Z. (2017). Processes of discharge ignition in long tubes at low gas
pressure. Plasma Sources Science and Technology, 26(6), 065017. doi:10.1088/1361-6595/aa6f7c]
The ionization does not happen
instantaneously, it proceeds with
a certain ionizationwave
velocity.
This causes a delay in
discharge breakdown w.r.t
voltage application.
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A short note on actual discharge ignition
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Actual discharge ignition
- Nice overview of the problematics:
[Shishpanov, A. I., Meshchanov, A. V., Kalinin, S. A., & lonikh, Y. Z. (2017). Processes of discharge ignition in long tubes at low gas
pressure. Plasma Sources Science and Technology, 26(6), 065017. doi:10.1088/1361-6595/aa6f7c]
The ionization does not happen
instantaneously, it proceeds with
a certain ionizationwave
velocity.
This causes a delay in
discharge breakdown w.r.t
voltage application.
32 Fyzika plazmatu 2
Actual discharge ignition
- The time after which the discharge is ignited is expressed as fd — fs 4. ff + t
where the symbols ts and tf correspond to random phenomena.
- Based on that, we can express the Laue distribution
number of breakdowns with
larger breakdown time than td
n(td)
•iV
total number of breakdowns
= exppd
-fa+ u
]
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Actual discharge ignition
This expression produces "Lauegrams", from
which we can read out the probability of
discharge ignition and time delay at various
voltages
n(td) r
if =
e x
p[
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tá — (ř
f + ř
w)
]
- 0 . 5 -
-1.0
-1.5
- 2 . 0 -
2 . 5 -
3 . 0 -
i 1 1 1 r 1
—i—1
—r
-•—i—•—i—•—i—•—i—
0 2 0 0 4 0 0 600 8 0 0 1000 1200 1400
Time delay, jxs
Figure 3. Lauegrams for different pulse amplitudes: Uo = 1048 V
(1), 1197 V (2), 1290 V (3). Tube T l , neon, 1 Torr.
Main take-aways
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Take-aways from lesson 1
1. Be able to formulate and derive Townsend breakdown criterion, be aware of the underlying
assumptions and limitations.
2. Be able to accurately and exactly describe Paschen law.
3. Be aware of the special role of E/N in plasma science and where it comes from.
4. General awareness of actual breakdown mechanisms - there is a time delay, Lauegrams
exist.
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