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University of California, Berkeley PLASMA

PLASMA, SHEATHS AND SURFACES —

THE DISCHARGE SCIENCE OF IRVING LANGMUIR

M.A. Lieberman

Department of Electrical Engineering and Computer SciencesUniversity of California

Berkeley, CA 94720

Download this talk:

http://www.eecs.berkeley.edu/∼lieber

LiebermanGEC05 1


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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INTRODUCTION

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DISCHARGE STATE-OF-THE-ARTPRIOR TO LANGMUIR

(after J.D. Cobine, The Collected Works of Irving Langmuir, 4, xxii, Perg-amon Press, New York, 1961–62)

• High pressure arcs and low pressure mercury-arc lamps had longbeen used for illumination; electric power circuits were being in-terrupted in air and oil with electric arcs, and ac power was beingconverted to dc for battery charging and electrochemical applica-tions by mercury arc rectifiers.

• The scientific literature was largely devoted to studies of V-I char-acteristics, physical appearances, spark breakdown studies, the mo-bility 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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THE CHILD-LANGMUIR LAW(1913–24)

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ELECTRON EMISSION FROM SURFACES

• Prior to Langmuir“At that time, the discovery by Richardson of thermionic emis-sion was not many years old. A number of experimentershad found results so divergent and apparently inconsistent thata sceptical feeling was growing that there was no such thing. . .Langmuir introduced order into the solution and saved therm-ionic emission for physics. . . .The most important new consid-eration that he introduced was . . . the space charge effect.(P.W. Bridgman, The Collected Works . . . , 12, 440)

Je =49ε0

(2e

M

)1/2V 3/2

d2V+

–

Anode

Hot cathode

–Je d

• But . . .Langmuir was scooped!

(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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LANGMUIR PROBES(1924–26)

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PROBING ELECTRODES

• Prior to Langmuir“Probing electrodes, or sounds, had long been used to examinethe voltage distribution along discharges . . .The nature of theaction of a probing electrode in an ionized gas had not beenunderstood.”(J.D. Cobine, The Collected Works . . . , 4, xix)

• Langmuir and Mott-Smith“In a series of articles the authors have given an account of anew method of studying electrical discharges through gases atrather low pressures. The method consists in the determinationof the complete volt-ampere characteristic of a small auxiliaryelectrode or collector of standard shape placed in the path ofthe discharge, and in the interpretation of this characteristicaccording 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)

H - Planar probe; B - Cylindrical probePlanar probe curve

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NON-MAXWELLIAN ELECTRONS(M.J. Druyvesteyn, Phys. Z. 64, 781, 1930)

• In (almost) an aside (Section V),Druyvesteyn shows electron energyprobability function gp(V ) to beproportional to the second deriva-tive of the electron probe current

gp(V ) ∝ d2Ie

dV 2

• This has led to beautiful measure-ments of electron energy distribu-tions 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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USE OF THE WORD “PLASMA”(1928–29)

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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 containingvery few electrons, the ionized gas contains ions and electrons inabout equal numbers, so that the resultant space charge is verysmall. We shall use the name plasma to describe this regioncontaining 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—. . .5The word ‘plasma’ will be used to designate that portion

of an arc-type discharge in which the densities of ions and elec-trons are high but substantially equal. It embraces the wholespace not occupied by ‘sheaths’ ”

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FIRST FIGURE WITH THE WORD “PLASMA”

(from A.W. Hull and I. Langmuir, “Control of an Arc Discharge By MeansOf a Grid,” Proc. Nat. Acad. Sci. 51, 218, 1929)

Fig. 1. Positive ion sheaths around

grid wires in a thermionic tube con-

taining gas

I. Langmuir and A.W. Hull

“Figure 1 shows graphically the condition that exists insuch a tube containing mercury vapor. The space between fila-ment and plate is filled with a mixture of electrons and positiveions, in nearly equal numbers, to which has been given the name‘plasma’. ”

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ORIGIN OF THE WORD “PLASMA”?

(from G.L. Rogoff, IEEE Trans. Plasma Sci. 19, 989, 1991)

“While [Langmuir’s] relating theterm to blood plasma has been acknowl-edged by colleagues who worked withhim at the General Electric ResearchLaboratory, the basis for that connec-tion is unclear. One version of the storyhas it that the similarity was in car-rying particles, while another accountspeculated that it was in the Greekorigin of the term, meaning ‘to mold,’since the glowing discharge usually moldeditself to the shape of its container.”

Irving Langmuir

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JOINING PLASMA TO SHEATH(1929–32)

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“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 over46 pages of Physical Review.”

J.D. Cobine, The Collected Works. . . , 5, xix

“It is . . . reasonable to assume that each ion starts from rest andsubsequently possesses only the velocity which it acquires byfalling through a static electric field which is itself maintainedby the balance of electron and ion charges . . .The resulting in-tegral equations . . .have been set up for plane, cylindrical, andspherical plasmas, for long, short and intermediate length ionfree paths, and for both constant rate of ionization throughoutthe plasma and rate proportional to electron density . . .The so-lution of the general plasma-sheath equation has been extendedinto 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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THE PLASMA SOLUTION IS SINGULAR!

Planar, collisionless ion case

• Φ = 0.854 Te and ui ≈ 1.3 uB atthe singularity at x = s0.

uB =(

eTe

M

)1/2

“Bohm velocity”

• “The limit of validity of the plasmaequation.—The approximation whichgives the plasma equation fails atsome value of s less than s0 be-cause of the fact that d2Φ/dx2 be-comes infinite at s0.”

• To match plasma and sheath so-lutions, they choose

ρ = −ε0d2Φdx2

= f · eni

with (arbitrary) f = 0.05LiebermanGEC05 17


DISCHARGE MODEL WITH FLUID IONS

Φ

x0 l

Power

Gas

Simplest model:isothermal electrons andions with Ti → 0

• Particle balanced(niui)

dx= νizne

• Ion momentum balance

Mniuidui

dx= eniE − Mniνmiui

• Boltzmann electronsne = ne0 eΦ/Te

• Poisson equationd2Φdx2

= − e

ε0(ni − ne)

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THE PLASMA APPROXIMATION

• Use “quasi-neutrality”d2Φdx2

= − e

ε0(ni − ne) ≈ 0

• Solve to obtain

u′i ≡

dui

dx=

νizu2B + νmiu

2i

u2B − u2

i

where ui

uB

x

Plasma

0

uB =(

eTe

M

)1/2

is the “Bohm velocity”• u′

i ( and n′i and Φ′) are singular at ui = uB

Singular behavior is very general for any collisionalityPlasma solution exists only if ui < uB

(K.-B. Persson, Phys. Fluids 5, 1625, 1962)

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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, NewYork, 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

dx→ ∞ as ui → uB in the plasma

dui

dx→ 0 as ui → uB in the sheath

ui

uB

x

Plasma

Bohm

0

Actual

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THE CASE OF HIGHLY COLLISIONAL IONS

(from Tonks and Langmuir, cont’d)

“Only the low pressure case will be analyzed. . . and that com-paratively roughly because the plasma-sheath transition is in-herently more complicated than either plasma or sheath alone.The high pressure case might be covered by the assumptionthat the ion velocity was proportional to the electric field. . . ”

• Consider mobility-dominated ion momentum balance

ui = µiEwith µi = e/Mνmi = const.

x0 l

Power

Gas

ni

⇒ Cosine profile for ni

(W. Schottky, Phys. Z. 25,635, 1924)

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HIGHLY COLLISIONAL IONS (CONT’D)

• Diffusion profileni = n0 cos(πx/2l)

• Boltzmann relationui = µiE = µiTe tan(πx/2l)

• Maxwell equation ρ = ε0∇ · Eρ = ε0

( π

2l

)2

Te sec2(πx/2l)• Langmuir’s procedure ρ = f · eni

s ≈ λDs/√

fwith λDs the Debye length at thesheath edge

Es ≈√

f Te/λDs

uis ∼ uB (uB/νmiλDs)︸︷︷︸ √f

0 l

0

0 l0

ni

ρ/e

ui

–l

–l

s

uB

–uB

n0

f=1

uis

small

Collisional plasma-sheath transition at ui < uB

(J.L. Blank, Phys. Fluids 11, 1686, 1968)

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JOINING PLASMA TO SHEATH AFTER LANGMUIR

“Patching”

V.A. Godyak (1982+)N. Sternberg (1990+)

“Matching”

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 approximationthat 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 plasmaapproximation is valid except for a thin surface layer, and thatthe sheath solution is (by construction!) consistently adaptedto the corresponding plasma solution. . .These characteristicsfollow . . . from the simple sheath model in textbooks!”(K.-U. Riemann, IEEE Trans. Plasma Sci. 32, 2265, 2004)

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SURFACE CHEMISTRY(1912–32)

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THE NOBEL PRIZE IN CHEMISTRY — 1932

“For his discoveries and inventions in surface chemistry”

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ADSORPTION PHENOMENA PRIOR TO LANGMUIR

(Presentation Speech by Professor H.G. Soderbaum, Chairman of the No-bel Committee for Chemistry of the Royal Swedish Academy of Sci-ences, on December 10, 1932)

“The power of certain solid bodies to retain or, as we say, tocondense gases on their surface has long been known and madeuse of for all manner of practical purposes. But it has notbeen known how this adsorption really takes place. As a rulethe conception will probably have been that the gas nearest tothe adsorbing solid body, appears in a more or less condensedstate, and that the density decreases continuously outwards inproportion as the solid surface is left, in about the same way asthe density of the atmosphere of the earth decreases upwards inproportion as we move away from the solid crust of the earth.This year’s Nobel Prize winner in Chemistry has advanced anentirely conflicting theory and one which at first sight seems tobe particularly bold.”

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THE DISCOVERY OF SURFACE CHEMISTRY

“. . . there is an abrupt changein properties in passingthrough the surface . . .Whengas molecules impinge againstany surface they . . . condenseon the surface being heldby the field of force ofthe surface atoms. Thesemolecules may subsequentlyevaporate from the sur-face.”(from I. Langmuir, “The Ad-

sorption of Gases on Plane

Surfaces of Glass, Mica, and

Platinum,”J. Am. Chem. Soc.

40, 1361, 1918)

(from I. Langmuir, “A Chemically Active

Modification of Hydrogen,” J. Am. Chem.

Soc. 34, 1310, 1912)

(see also I. Langmuir, “The Constitution and Fundamental Properties ofSolids and Liquids,” J. Am. Chem. Soc. 38, 2221, 1916, Pt. I; 39,1848, 1917, Pt. II)

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ADSORPTION

• Reaction of an atom with the surface

A + SKa−→←−Kd

A: S

• Chemisorption (due to formation of chemical bonds)

U

x

1–1.5 Å

0.4–4 V

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STICKING COEFFICIENT

• Adsorbed fluxΓads = s · ΓAS = s · 1

4nAS vA

s(θ, T ) = sticking coefficientnAS = gas density near surfacevA = mean speed of gas atoms

• Langmuir kineticss(θ, T) = s0(T) (1 − θ)

θ = fraction of surface sites covered with absorbates0 = zero-coverage sticking coefficient

s

s0

0 0 1 θ

Langmuir

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ADSORPTION-DESORPTION KINETICS

• Consider reactions

A + SKa−→←−Kd

A: S

• Adsorption flux isΓads = KanASn′

0(1 − θ)n′

0 = adsorption site area density

• The desorption flux isΓdesor = Kdn

′0θ

• Equate adsorption and desorption fluxes

θ =KanAS

KanAS + Kd

00

1

5 10

Langmuirisotherm

KanAS/Kd

Surf

ace

cove

rage

θ

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STANDARD MODEL OF ETCH KINETICS

• O atom etching of a carbon substrate

C(s) CO(s)

θ

Ka Ki Kd YiKi

θ1 –

CO CO+O

θ = fraction of surface sites covered with C : O bondsYi = 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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SURFACE COVERAGE

• ReactionsO(g) + C(s) Ka−→ C : O (O atom adsorption)

C : O Kd−→ CO(g) (CO thermal desorption)

ion + C : O YiKi−→ CO(g) (CO ion-assisted desorption)

• Equate adsorption to thermal + ion-induced desorption

KanOS(1 − θ) = Kdθ + YiKinisθ

nOS = O-atom density near the surfacenis = ion density at the plasma edge

• Solve for surface coverage

θ =KanOS

KanOS + Kd + YiKinis

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ETCH RATES

• Flux of CO leaving surface

ΓCO = (Kd + YiKinis) θ n′0 (m−2-s−1)

• Ion-enhanced etch rate

E =ΓCO

nC(m/s)

with nC the carbon atom density of substrate

• In the usual ion-enhanced regime YiKinis � Kd

1E

= nC

1Yi Kinisn

′0︸︷︷︸ +

1KanOSn′

0︸︷︷︸

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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PATHOLOGICAL SCIENCE(1953)

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“THE SCIENCE OF THINGS THAT AREN’T SO”(I. Langmuir, “Pathological Science,” Colloquium at The Oak Knolls Re-

search Laboratory, December 18, 1953; Physics Today 42, 36, Octo-ber 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 intensity2. The effect is of a magnitude that remains close to the limit

of detectability3. Claims of great accuracy4. Fantastic theories contrary to experience5. Criticisms are met by ad hoc excuses thought up on the

spur of the moment6. 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 saucersLiebermanGEC05 35


COLD FUSION

“Inside a small cell containing a pal-adium cathode immersed in heavywater at room temperature, deuteriumnuclei were fusing and producing heatat a rate four times higher than theinput power.”

(paraphrase from Press Confer-ence of M. Fleischmann andS. Pons, Salt Lake City,23 March 1989)

Unfortunately . . .

Bel

ieve

rs

1989 Now

~50%

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HOT FUSION (ITER)

↗.

Cold Fusion Size Hot Fusion Size

Person↗

• But . . . interesting low temperature plasma near reactor walls!

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CONCLUSION

All practitioners of dischargescience should be familiar withIrving Langmuir’s beautiful work

Langmuir skiing

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ACKNOWLEDGEMENTS

• I am greatly indebted to . . .

C.K. BirdsallR.N FranklinV.A. GodyakK.-U. RiemannT. Sommerer

• Download this talk:

http://www.eecs.berkeley.edu/∼lieber

LiebermanGEC05 39

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