PLASMA, SHEATHS AND SURFACES —
THE DISCHARGE SCIENCE OF IRVING LANGMUIR
M.A. Lieberman
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720
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PLASMA
OUTLINE
• Introduction
• Child-Langmuir law (brief)
• Langmuir probes (brief)
• Use of the word “plasma”
• Joining plasma to sheath
• Surface chemistry
• Pathological science (brief)
• Conclusion
GE Research Lab (1909–50)
Irving Langmuir (1888–1957)
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PLASMA
INTRODUCTION
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PLASMA
DISCHARGE STATE-OF-THE-ART
PRIOR TO LANGMUIR
(after J.D. Cobine, The Collected Works of Irving Langmuir, 4, xxii, Pergamon Press, New York, 1961–62)
• High pressure arcs and low pressure mercury-arc lamps had long
been used for illumination; electric power circuits were being interrupted in air and oil with electric arcs, and ac power was being
converted to dc for battery charging and electrochemical applications by mercury arc rectifiers.
• The scientific literature was largely devoted to studies of V-I characteristics, physical appearances, spark breakdown studies, the mobility and diffusion of gaseous ions, and ionization phenomena.
• The principal treatises in English were those of J.S. Townsend (1915)
and J.J. Thomson (1906)
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PLASMA
THE CHILD-LANGMUIR LAW
(1913–24)
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PLASMA
ELECTRON EMISSION FROM SURFACES
• Prior to Langmuir
“At that time, the discovery by Richardson of thermionic emission was not many years old. A number of experimenters
had found results so divergent and apparently inconsistent that
a sceptical feeling was growing that there was no such thing
. . . Langmuir introduced order into the solution and saved thermionic emission for physics. . . . The most important new consideration that he introduced was . . . the space charge effect.
(P.W. Bridgman, The Collected Works . . . , 12, 440)
4
Je = 0
9
2e
M
1/2
V 3/2
d2
• But . . . Langmuir was scooped!
+
V
–
Anode
–Je
d
Hot cathode
(C.D. Child, Phys. Rev. 32 (5), 492, 1911)
(I. Langmuir, Phys. Rev. 2 (6), 450, 1913)
(Initial velocities, cylinders and spheres with K.D. Blodgett, in 1923–24)
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PLASMA
LANGMUIR PROBES
(1924–26)
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PLASMA
PROBING ELECTRODES
• Prior to Langmuir
“Probing electrodes, or sounds, had long been used to examine
the voltage distribution along discharges . . . The nature of the
action of a probing electrode in an ionized gas had not been
understood.”
(J.D. Cobine, The Collected Works . . . , 4, xix)
• Langmuir and Mott-Smith
“In a series of articles the authors have given an account of a
new method of studying electrical discharges through gases at
rather low pressures. The method consists in the determination
of the complete volt-ampere characteristic of a small auxiliary
electrode or collector of standard shape placed in the path of
the discharge, and in the interpretation of this characteristic
according to a new theory.”
(I. Langmuir and H.M. Mott-Smith, Gen. Elect. Rev. 27, 449, 538, 616,
762, 810, 1924; Phys. Rev. 28, 727, 1926)
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LANGMUIR PROBES AND CURVES
• Measurements and theory for planar (with guard ring), cylindrical,
and spherical probes (low pressure, Maxwellian electrons)
Planar probe curve
H - Planar probe; B - Cylindrical probe
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PLASMA
NON-MAXWELLIAN ELECTRONS
(M.J. Druyvesteyn, Phys. Z. 64, 781, 1930)
• In (almost) an aside (Section V),
Druyvesteyn shows electron energy
probability function gp (V ) to be
proportional to the second derivative of the electron probe current
d2 Ie
gp (V ) ∝
dV 2
• This has led to beautiful measurements of electron energy distributions in discharges
(V.A. Godyak and R.B. Piejak, Phys.
Rev. Lett. 65, 996, 1990)
Argon capacitive discharge
13.56 MHz, 2 cm gap
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PLASMA
USE OF THE WORD “PLASMA”
(1928–29)
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PLASMA
LANGMUIR INTRODUCES THE WORD “PLASMA”
(from I. Langmuir, “Oscillations in Ionized Gases,” Proc. Nat. Acad. Sci.
14, 627, August 1928)
“Except near the electrodes, where there are sheaths containing
very few electrons, the ionized gas contains ions and electrons in
about equal numbers, so that the resultant space charge is very
small. We shall use the name plasma to describe this region
containing balanced charges of ions and electrons.”
(from L. Tonks and I. Langmuir, “Oscillations in Ionized Gases,” Phys.
Rev. 33, 195, February 1929)
“ A. Plasma-electron5 oscillations—. . .
5
The word ‘plasma’ will be used to designate that portion
of an arc-type discharge in which the densities of ions and electrons are high but substantially equal. It embraces the whole
space not occupied by ‘sheaths’ ”
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PLASMA
FIRST FIGURE WITH THE WORD “PLASMA”
(from A.W. Hull and I. Langmuir, “Control of an Arc Discharge By Means
Of a Grid,” Proc. Nat. Acad. Sci. 51, 218, 1929)
Fig. 1. Positive ion sheaths around
grid wires in a thermionic tube containing gas
I. Langmuir and A.W. Hull
“Figure 1 shows graphically the condition that exists in
such a tube containing mercury vapor. The space between filament and plate is filled with a mixture of electrons and positive
ions, in nearly equal numbers, to which has been given the name
‘plasma’. ”
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PLASMA
13
ORIGIN OF THE WORD “PLASMA”?
(from G.L. Rogoff, IEEE Trans. Plasma Sci. 19, 989, 1991)
“While [Langmuir’s] relating the
term to blood plasma has been acknowledged by colleagues who worked with
him at the General Electric Research
Laboratory, the basis for that connection is unclear. One version of the story
has it that the similarity was in carrying particles, while another account
speculated that it was in the Greek
origin of the term, meaning ‘to mold,’
since the glowing discharge usually molded
itself to the shape of its container.”
Irving Langmuir
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PLASMA
JOINING PLASMA TO SHEATH
(1929–32)
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PLASMA
“A GENERAL THEORY OF THE PLASMA OF AN ARC”
(L. Tonks and I. Langmuir, Phys. Rev. 34, 876, 1929)
. . . “one of the most important papers in the field,. . . spread over
46 pages of Physical Review.”
J.D. Cobine, The Collected Works. . . , 5, xix
“It is . . . reasonable to assume that each ion starts from rest and
subsequently possesses only the velocity which it acquires by
falling through a static electric field which is itself maintained
by the balance of electron and ion charges . . . The resulting integral equations . . . have been set up for plane, cylindrical, and
spherical plasmas, for long, short and intermediate length ion
free paths, and for both constant rate of ionization throughout
the plasma and rate proportional to electron density . . . The solution of the general plasma-sheath equation has been extended
into the sheath surrounding the plasma.”
Separate treatment of plasma and sheath physics
“Plasma approximation” or “quasi-neutrality” (ne ≈ ni )
Kinetic treatment of ions
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PLASMA
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THE PLASMA SOLUTION IS SINGULAR!
Planar, collisionless ion case
• Φ = 0.854 Te and ui ≈ 1.3 uB at
the singularity at x = s0 .
1/2
eTe
“Bohm velocity”
uB =
M
• “The limit of validity of the plasma
equation.—The approximation which
gives the plasma equation fails at
some value of s less than s0 because of the fact that d2 Φ/dx2 becomes infinite at s0 .”
• To match plasma and sheath solutions, they choose
d2 Φ
ρ = −0 2 = f · eni
dx
with (arbitrary) f = 0.05
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PLASMA
DISCHARGE MODEL WITH FLUID IONS
Power Φ
Simplest model:
isothermal electrons and
ions with Ti → 0
Gas
0
l
• Particle balance
• Ion momentum balance
x
d(ni ui )
= νiz ne
dx
dui
M ni u i
= eni E − M ni νmi ui
dx
• Boltzmann electrons
ne = ne0 eΦ/Te
• Poisson equation
d2 Φ
e
= − (ni − ne )
2
dx
0
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PLASMA
THE PLASMA APPROXIMATION
• Use “quasi-neutrality”
e
d2 Φ
= − (ni − ne ) ≈ 0
2
dx
0
• Solve to obtain
2
2
u
+
ν
u
ν
du
i
iz
mi
i
B
=
ui ≡
dx
u2B − u2i
ui
where
1/2
uB
eTe
uB =
M
is the “Bohm velocity”
• ui ( and ni and Φ ) are singular at ui = uB
Plasma
0
x
Singular behavior is very general for any collisionality
Plasma solution exists only if ui < uB
(K.-B. Persson, Phys. Fluids 5, 1625, 1962)
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PLASMA
BOHM’S COLLISIONLESS SHEATH ANALYSIS
(D. Bohm in The Characteristics of Electrical Discharges in Magnetic
Fields, ed. by A. Guthrie and R.K. Wakerling, McGraw-Hill, New
York, Chap. 3, p. 77, 1949)
• Drop all collisions (νiz , νmi ); keep the full Poisson equation;
assume E ≈ 0 at the plasma-sheath edge
• A sheath solution exists only if ui ≥ uB
(Recall a plasma solution exists only if ui < uB )
Collisionless plasma-sheath transition at ui ≈ uB
• But. . . the plasma and sheath solutions do not join smoothly!
dui
→ ∞ as ui → uB in the plasma
dx
ui
Bohm
uB
Actual
Plasma
dui
→ 0 as ui → uB in the sheath
dx
0
x
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PLASMA
THE CASE OF HIGHLY COLLISIONAL IONS
(from Tonks and Langmuir, cont’d)
“Only the low pressure case will be analyzed. . . and that comparatively roughly because the plasma-sheath transition is inherently more complicated than either plasma or sheath alone.
The high pressure case might be covered by the assumption
that the ion velocity was proportional to the electric field. . . ”
• Consider mobility-dominated ion momentum balance
with µi = e/M νmi
ui = µi E
= const.
⇒ Cosine profile for ni
Power
ni
(W. Schottky, Phys. Z. 25,
635, 1924)
Gas
0
l
x
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PLASMA
HIGHLY COLLISIONAL IONS (CONT’D)
n0
• Diffusion profile
ni = n0 cos(πx/2l)
• Boltzmann relation
ui = µi E = µi Te tan(πx/2l)
• Maxwell equation ρ = 0 ∇ · E
π 2
ρ = 0
Te sec2 (πx/2l)
0
2l
–l
• Langmuir’s procedure
ρ
=
f
·
en
i
s ≈ λDs / f
uB
with λDs the Debye length at the
uis
sheath edge 0
Es ≈ f Te /λDs –uB
uis ∼ uB (uB /νmi λDs ) f
small
–l
ni
f=1
ρ/e
s
0
l
ui
0
l
Collisional plasma-sheath transition at ui < uB
(J.L. Blank, Phys. Fluids 11, 1686, 1968)
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PLASMA
JOINING PLASMA TO SHEATH AFTER LANGMUIR
“Matching”
“Patching”
V.A. Godyak (1982+)
N. Sternberg (1990+)
S.H. Lam (1965+)
R.N. Franklin (1970+)
K.-U. Riemann (1981+)
I.D. Kaganovich (2002)
M.S. Benilov (2002+)
• But. . .
“Actually, the problem to construct a smooth approximation
that is uniformly valid both in the plasma and sheath regions,
is more a problem of aesthetics than of practical necessity.
. . . For most applications, it is sufficient to know that the plasma
approximation is valid except for a thin surface layer, and that
the sheath solution is (by construction!) consistently adapted
to the corresponding plasma solution. . . These characteristics
follow . . . from the simple sheath model in textbooks!”
(K.-U. Riemann, IEEE Trans. Plasma Sci. 32, 2265, 2004)
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PLASMA
SURFACE CHEMISTRY
(1912–32)
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PLASMA
THE NOBEL PRIZE IN CHEMISTRY — 1932
“For his discoveries and inventions in surface chemistry”
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PLASMA
ADSORPTION PHENOMENA PRIOR TO LANGMUIR
(Presentation Speech by Professor H.G. Söderbaum, Chairman of the Nobel Committee for Chemistry of the Royal Swedish Academy of Sciences, on December 10, 1932)
“The power of certain solid bodies to retain or, as we say, to
condense gases on their surface has long been known and made
use of for all manner of practical purposes. But it has not
been known how this adsorption really takes place. As a rule
the conception will probably have been that the gas nearest to
the adsorbing solid body, appears in a more or less condensed
state, and that the density decreases continuously outwards in
proportion as the solid surface is left, in about the same way as
the density of the atmosphere of the earth decreases upwards in
proportion as we move away from the solid crust of the earth.
This year’s Nobel Prize winner in Chemistry has advanced an
entirely conflicting theory and one which at first sight seems to
be particularly bold.”
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PLASMA
THE DISCOVERY OF SURFACE CHEMISTRY
“. . . there is an abrupt change
in properties in passing
through the surface . . . When
gas molecules impinge against
any surface they . . . condense
on the surface being held
by the field of force of
the surface atoms. These
molecules may subsequently
evaporate from the surface.”
(from I. Langmuir, “A Chemically Active
Modification of Hydrogen,” J. Am. Chem.
Soc. 34, 1310, 1912)
(from I. Langmuir, “The Adsorption of Gases on Plane
Surfaces of Glass, Mica, and
Platinum,”J. Am. Chem. Soc.
40, 1361, 1918)
(see also I. Langmuir, “The Constitution and Fundamental Properties of
Solids and Liquids,” J. Am. Chem. Soc. 38, 2221, 1916, Pt. I; 39,
1848, 1917, Pt. II)
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PLASMA
27
ADSORPTION
• Reaction of an atom with the surface
K
a
−→
A + S ←− A: S
Kd
• Chemisorption (due to formation of chemical bonds)
U
1–1.5 Å
x
0.4–4 V
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PLASMA
STICKING COEFFICIENT
• Adsorbed flux
Γads = s · ΓAS
1
= s · nAS v̄A
4
s(θ, T ) = sticking coefficient
nAS = gas density near surface
v̄A = mean speed of gas atoms
• Langmuir kinetics
s(θ, T) = s0 (T) (1 − θ)
θ = fraction of surface sites covered with absorbate
s0 = zero-coverage sticking coefficient
s0
Langmuir
s
0
0
1
θ
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PLASMA
ADSORPTION-DESORPTION KINETICS
• Consider reactions
Ka
−→
A + S ←− A: S
Kd
• Adsorption flux is
Γads = Ka nAS n0 (1 − θ)
n0 = adsorption site area density
• The desorption flux is
Γdesor = Kd n0 θ
• Equate adsorption and desorption fluxes
Surface coverage θ
1
0
Langmuir
isotherm
0
5
10
KanAS/Kd
Ka nAS
θ=
Ka nAS + Kd
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PLASMA
STANDARD MODEL OF ETCH KINETICS
• O atom etching of a carbon substrate
O
Ka
CO
+
Ki
Kd
1 –θ
θ
C(s)
CO(s)
CO
Y i Ki
θ = fraction of surface sites covered with C : O bonds
Yi = yield of CO molecules desorbed per incident ion
(T.M. Meyer and R.A. Barker, J. Vac. Sci. Technol. 21, 757, 1982)
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PLASMA
SURFACE COVERAGE
• Reactions
Ka
C:O
O(g) + C(s) −→
Kd
C : O −→ CO(g)
Y K
i i
ion + C : O −→
CO(g)
(O atom adsorption)
(CO thermal desorption)
(CO ion-assisted desorption)
• Equate adsorption to thermal + ion-induced desorption
Ka nOS (1 − θ) = Kd θ + Yi Ki nis θ
nOS = O-atom density near the surface
nis = ion density at the plasma edge
• Solve for surface coverage
Ka nOS
θ=
Ka nOS + Kd + Yi Ki nis
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PLASMA
ETCH RATES
• Flux of CO leaving surface
ΓCO = (Kd + Yi Ki nis ) θ n0
(m−2 -s−1 )
• Ion-enhanced etch rate
ΓCO
(m/s)
E=
nC
with nC the carbon atom density of substrate
• In the usual ion-enhanced regime Yi Ki nis Kd


1
1
1


= nC
+
E
Yi Ki nis n0
Ka nOS n0
Γis
ΓOS
• The ion and neutral fluxes and the yield (a function of ion energy)
determine the ion-assisted etch rate
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PLASMA
PATHOLOGICAL SCIENCE
(1953)
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PLASMA
“THE SCIENCE OF THINGS THAT AREN’T SO”
(I. Langmuir, “Pathological Science,” Colloquium at The Oak Knolls Research Laboratory, December 18, 1953; Physics Today 42, 36, October 1989, transcribed and edited by R.N. Hall)
• In pathological science, scientists manage to fool themselves
“Symptoms of Pathological Science:
1. The maximum effect that is observed is produced by a
causitive agent of barely detectable intensity
2. The effect is of a magnitude that remains close to the limit
of detectability
3. Claims of great accuracy
4. Fantastic theories contrary to experience
5. Criticisms are met by ad hoc excuses thought up on the
spur of the moment
6. Ratio of supporters to critics rises up to somewhere near
50% and then falls gradually to oblivion”
• Davis-Barnes effect (1929), N-rays (1904), Mitogenic rays (1923),
Allison effect (1927), ESP (1934), flying saucers
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PLASMA
COLD FUSION
“Inside a small cell containing a paladium cathode immersed in heavy
water at room temperature, deuterium
nuclei were fusing and producing heat
at a rate four times higher than the
input power.”
(paraphrase from Press Conference of M. Fleischmann and
S. Pons, Salt Lake City,
23 March 1989)
Believers
Unfortunately . . .
~50%
1989
Now
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PLASMA
HOT FUSION (ITER)
.
Cold Fusion Size
Person
Hot Fusion Size
• But . . . interesting low temperature plasma near reactor walls!
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PLASMA
CONCLUSION
All practitioners of discharge
science should be familiar with
Irving Langmuir’s beautiful work
Langmuir skiing
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PLASMA
ACKNOWLEDGEMENTS
• I am greatly indebted to . . .
C.K. Birdsall
R.N Franklin
V.A. Godyak
K.-U. Riemann
T. Sommerer
• Download this talk:
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PLASMA