From Fermi acceleration to collisionless discharge heating

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998 955

From Fermi Acceleration to CollisionlessDischarge Heating

Michael A. Lieberman,Fellow, IEEE, and Valery A. Godyak,Fellow, IEEE

(Invited Review)

Abstract—The heating of electrons by time-varying fields isfundamental to the operation of radio frequency (RF) and mi-crowave discharges. Ohmic heating, in which the phase of theelectron oscillation motion in the field is randomized locally byinterparticle collisions, can dominate at high pressures. Phaserandomization can also occur due to electron thermal motionin spatially inhomogeneous RF fields, even in the absence ofcollisions, leading to collisionless or stochastic heating, whichcan dominate at low pressures. Generally, electrons are heatedcollisionlessly by repeated interaction with fields that are localizedwithin a sheath, skin depth layer, or resonance layer insidethe discharge. This suggests the simple heating model of aball bouncing elastically back and forth between a fixed andan oscillating wall. Such a model was proposed originally byFermi to explain the origin of cosmic rays. In this review, Fermiacceleration is used as a paradigm to describe collisionless heatingand phase randomization in capacitive, inductive, and electron cy-clotron resonance (ECR) discharges. Mapping models for Fermiacceleration are introduced, and the Fokker–Planck descriptionof the heating and the effects of phase correlations are described.The collisionless heating rates are determined in capacitive andinductive discharges and compared with self-consistent (kinetic)calculations where available. Experimental measurements andcomputer simulations are reviewed and compared to theoreticalcalculations. Recent measurements and calculations of nonlocalheating effects, such as negative electron power absorption, aredescribed. Incomplete phase randomization and adiabatic bar-riers are shown to modify the heating in low pressure ECRdischarges.

Index Terms—Anomalous heating, collisionless heating, Fermiacceleration, plasma discharges, stochastic heating.

I. INTRODUCTION

T HE heating of electrons by time-varying fields is fun-damental to the operation of radio frequency (RF) and

microwave plasma discharges. In auniformoscillating electricfield, Re a single electron has a coherentvelocity of motion that lags the phase of the electric field force

by 90 . Hence, the time-average power transfered fromthe field to the electron is zero. Electron collisions with other

Manuscript received September 16, 1997; revised February 11, 1998. Thework of M. A. Lieberman was supported in part by the NSF under GrantsECS-9529658 and INT-9602544, by the Lam Research Corporation, and bythe Space Plasma and Plasma Processing Group at The Australian NationalUniversity, Canberra, Australia.

M. A. Lieberman is with the Department of Electrical Engineering andComputer Sciences, and the Electronics Research Laboratory, University ofCalifornia, Berkeley, CA 94720 USA.

V. A. Godyak is with the OSRAM SYLVANIA Development, Inc., Beverly,MA 01915 USA.

Publisher Item Identifier S 0093-3813(98)04407-5.

particles destroy the phase coherence of the motion, leadingto a net transfer of power. For an ensemble ofelectronsper unit volume, it is usual to introduce the macroscopiccurrent density with the macroscopic electronvelocity, and to relate the amplitudes of and through alocal conductivity: whereis the plasma conductivity and is the electron collisionfrequency for momentum transfer. In this “fluid” approach,the average electron velocity still oscillates coherently butlags the electric field by less than 90leading to an ohmicpower transfer per unit volume

Re Re Re

Although the average velocity is coherent with the field, thefundamental mechanism that converts electric field energy tothermal energy is the breaking of the phase-coherent motionof individual electrons by collisions: the total force (electricfield force plus that due to collisions) acting on an individualelectron becomes spatially nonuniform and nonperiodic intime.

These observations suggest that a spatiallynonuniformelectric field by itself might lead to electron heating, evenin the absence of interparticle collisions, provided that theelectrons have thermal velocities sufficient to sample the fieldinhomogeneity. This phenomenon has been well-known inplasma physics since Landau [1], [2] demonstrated the col-lisionless damping of an electrostatic wave in a warm plasma,and is variously referred to in the literature ascollisionless,noncollisional, stochastic, transit time, or anomalousheatingor dissipation. Since that time, collisionless dissipation hasbeen studied extensively in fusion and space plasma physics.However, within the last decade and with the increased empha-sis on industrial applications of low-pressure gas discharges, ithas become evident that collisionless dissipation phenomenaare fundamental to RF and microwave discharges.

As will be seen in the following review, stochastic colli-sionless interactions leading to electron heating can be a basicfeature of warm plasmas having space dispersion. The electronresponse at some point in the plasma is defined not onlyby the field at that point, but by an integrated effectover the neighboring space. Due to the spatial variation, thetime-varying field seen by an individual “thermal” electron isnonperiodic. The electron can lose phase coherence with thefield (which is strictly periodic), resulting in stochastic interac-tion with the field and collisionless heating. Two fundamental

0093–3813/98$10.00 1998 IEEE

956 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 3, JUNE 1998

issues arise: 1) what is the rate of energy transfer to electronsassuming that phase coherence is lost and 2) what are theconditions for loss of phase coherence? These issues form theprincipal topic of our review.

In almost all discharges, the spatial variation of the time-varying field is strongly nonuniform, with a low field in thebulk of the plasma and one or more highly localized fieldregions (RF sheath, skin depth layer, etc.), usually near theplasma boundaries. An electron, being confined for hundredsto thousands of bounce times by the dc ambipolar and bound-ary sheath potential in the discharge, interacts repeatedly withthe high field regions, but interacts only weakly during its driftthrough the plasma bulk. This suggests a dynamical model toinvestigate the energy transfer and loss of phase coherence: aball bounces elastically back and forth between a fixed andan oscillating wall. This model was first introduced by Fermi[3] to explain the origin of cosmic rays. The process in whichthe ball repeatedly interacts with the oscillating wall, resultingin phase randomization and stochastic heating, is known asFermi acceleration. This process has been studied extensivelyas a paradigm in dynamics. In this review, we adapt theFermi acceleration model as our fundamental approach forunderstanding collisionless heating in weakly ionized gasdischarges.

In the usual model of Fermi acceleration, the wall os-cillation motion is specified and the motion of the ball isthen determined. From the structure of the motion in thevelocity-position phase space of the ball, the conditions forphase randomization and the heating rates can be determined.However, the corresponding problem of collisionless electronheating in discharges has additional complexity, because thespatially nonuniform RF or microwave heating fields mustbe determined self-consistently with the electron motions.This self-consistent problem has been treated within conven-tional (warm plasma) kinetic theory for RF inductive andcapacitive discharges assuming that all electron phases arerandomized. The issue of partial phase randomization and fullyself-consistent treatments of other types of discharges is anactive area of research.

In Section II, we describe the model of Fermi acceler-ation. We motivate its introduction to explain the originof cosmic rays and introduce mapping models to describethe dynamics. We introduce a Fokker–Planck formalism todescribe the collisionless heating in the presence of completephase randomization, and we describe the effects of partialphase randomization. In Section III, we review collisionlessheating in capacitive RF discharges. We describe the earlystudies and introduce a simple Fermi acceleration model fora homogeneous sheath to determine the collisionless heatingrate. We describe modifications to the heating rate due toself-consistent sheath models and make comparisons to ex-periments and to fluid and particle-in-cell (PIC) simulations.We also describe other collisionless heating models, such asfor magnetized capacitive discharges, and we introduce someother approaches to determine the collisionless heating. InSection IV, we review collisionless heating in inductive RFdischarges. We introduce the classical and anomalous skineffects and a Fermi acceleration model of the collisionless

(anomalous) heating. We compare this to a self-consistentkinetic model and describe recent experiments that identifyeffects due to collisionless heating, such as the existenceof regions of negative electron power absorption within thedischarge bulk due to space dispersion caused by electronthermal motion. Negative absorption has also been observed incomputer simulations of capacitive discharges. In Section V,we review briefly some features of collisionless heating inelectron cyclotron resonance (ECR) discharges. We introducea Fermi acceleration heating model and show some compar-isons to experiments. Both the model and the experimentssuggest that incomplete phase randomization can reduce theheating rate and lead to an adiabatic barrier to the heating. InSection VI, we summarize our conclusions and suggest somefurther issues that need to be resolved.

II. FERMI ACCELERATION

A. Cosmic Rays: Discovery and Properties

In the morning of August 7, 1912, Austrian physicist V.Hess ascended to over 5 km in a balloon gondola as “anobserver for atmospheric electricity” [4], [5]. During the jour-ney, he made careful measurements of the rate of dischargingof three electroscopes, and he noted a several-fold increasein the rate of discharging as the balloon rose in altitude. Inhis publication inPhysikalische Zeitschriftin November 1912,Hess suggested that the results of his observations were bestexplained “by a radiation of great penetrating power enteringour atmosphere from above.” Further flights confirmed thesefindings, and the American physicist R. Millikan, althoughinitially skeptical of the extraterrestrial origin, introduced thenamecosmic rays.

It is now generally agreed that the majority of cosmic rayshave a galactic origin. The cosmic ray flux is isotropic andof order 1 cm 2 s 1, the energy density is approximately1 eV/cm3, and the lifetime is approximately 107 years. Cos-mic rays are mostly protons, but are rich in heavy nucleicompared to solar abundences. The particle energy rangesfrom – eV, with a power law distribution

Cosmic rays are believed to originate from supernovas suchas the well-studied Crab nebula, which is the remnant of asupernova in 1054 A.D. With one galactic supernova everyfifty years within a galactic disk volume of 1067 cm3 creating1043 J of fast particles, the energy balance is

Jyrs

eV/cm cmyrs

Measurements of radiation from supernova remnants clearlyshow the presence of synchrotron radiation, demonstratingthe existence of high-energy (1011 eV) electrons. Exactlyhow the fast particles are formed and accelerated is not wellunderstood. Early theories emphasized acceleration across highvoltages or by means of shock waves.

LIEBERMAN AND GODYAK: FROM FERMI ACCLERATION TO COLLISIONLESS DISCHARGE HEATING 957

Fig. 1. Fermi acceleration in which a particle bounces between a fixed andan oscillating wall.

B. Fermi’s Proposal

In 1949, Fermi put forth the idea that “cosmic rays areoriginated and accelerated primarily in the interstellar space ofthe galaxy by collisions against moving magnetic fields.” Hewent on to assert the basic acceleration mechanism as follows:

It may happen that a region of high field intensity movestoward the cosmic-ray particle which collides againstit. In this case, the particle will gain energy in thecollision. Conversely, it may happen that the regionof high field intensity moves away from the particle.Since the particle is much faster, it will overtake theirregularity of the field and be reflected backward, in thiscase with loss of energy. The net result will be averagegain, primarily for the reason that head-on collisionsare more frequent than overtaking collisions because therelative velocity is larger in the former case.

Fermi noted that this idea naturally leads to a powerlaw energy distribution, but that it failed to explain in astraightforward way the heavy nuclei observed in the primarycosmic radiation. It is now believed that cosmic rays below1015 eV are produced within our galaxy by Fermi accelerationof particles within the shock waves of supernova remnants [6].

C. Fermi Maps and Dynamical Chaos

The Fermi problem of a particle bouncing between a fixedand an oscillating wall is a classical model of Hamiltoniandynamics [7] and is illustrated in Fig. 1. This model ofenergy gain by repeated collisions of a particle with anoscillating wall was examined numerically by Ulam andassociates [8], who found that the particle motion appeared tobe stochastic, but did not increase its energy on the average.Ulam’s result was explained using a combination of analyticaland numerical work by subsequent authors. The Fermi problemwas treated using an exact area-preserving dynamical mappingfor a sawtooth wall velocity by Zaslavskii and Chirikov [9].Similar studies were performed by Brahic [10]. A “simplified”mapping, in which the oscillating wall imparts momentum tothe particle but occupies a fixed position, was introduced byLieberman and Lichtenberg [11] and studied for arbitrary wallvelocities [12], [7].

To find the exact mapping dynamics for this system, weintroduce a fixed surface of section as some const.

Defining to be the normalized velocity,to be the phase of the moving wall at theth collision withthe fixed surface at then a difference equation forthe motion of the particle can be determined in terms of awall motion where is an even periodicfunction of the phase with period and with

We obtain, in implicit form the equationsof motion

(2.1a)

(2.1b)

(2.1c)

Here is the phase at the next collision with the movingwall, after the th collision with the fixed surface

with the distance between the walls, andis the velocity impulse given to the ball. It is well known[13] that measuring the position from the fixed wall asconjugate to the velocity then the phase is a time-likevariable conjugate to the energy-like variable That is,in the extended phase space for this Hamiltoniansystem, the choice of a surface gives an area-preservingmapping for the remaining pair As we show inSection II-D, this implies that a stochastic orbit has a uniforminvariant distribution over the accessible phase space.Hence, assuming all phases are accessible, the energyhasa uniform invariant distribution.

Because of its complicated form, (2.1) is not convenient foranalytical study. Substituting assuming a sinusoidalwall motion in (2.1), and expanding to first order in (and

), we obtain

(2.2a)

(2.2b)

A still simpler, nonimplicit form can be constructed if thesinusoidally oscillating wall imparts momentum to the ball,according to the wall velocity, without the wall changing itsposition in space. The problem defined in this manner hasmany of the features of the more physical problem. In thissimplified form the mapping is

(2.3a)

mod (2.3b)

The mapping in (2.3) serves as an approximation (with suitablydefined variables) to many physical systems in which thetransit time between kicks is inversely proportional to avelocity. The absolute-value signs in (2.3) correspond to thevelocity reversal, at low velocities which appears in theexact equations (2.1). The absolute value has no effect on theregion which is the primary region of interest. For thesimplified problem, a proper canonical set of variables are theball velocity and phase just before theth impact with themoving wall. The normalized velocity then has a uniforminvariant distribution, as will be seen in Section II-D.


Fig. 2. Surface of section for the Fermi problem, showing occupation ofphase space cells for 623 000 iterations of a single initial condition. Dashedcurves are calculated from secular perturbation theory (after [7]).

Transformations of the type (2.1)–(2.3) can be examinednumerically for many thousands of iterations, thus allowingboth detailed knowledge of the structural behavior and sta-tistical properties of the dynamical system to be determined.Fig. 2 shows the - surface for the simplified Fermi map(2.3) with for 623 000 wall collisions of a singletrajectory, with an initial condition at low velocityThe surface has been divided into 200200 cells, with ablank indicating no occupation of that cell. We find that thephase plane consists of three regions:

1) a region for large in which in-variant adiabatic curves predominate and isolate narrowlayers of stochasticity near the separatrices of the variousresonances;

2) an interconnected stochastic region for intermediate val-ues of in which adiabatic islandsnear linearly stable periodic solutions are embedded ina stochastic sea; and

3) a predominantly stochastic region for smallin which all primary periodic solutions appear to beunstable.

Both regions (2) and (3) exhibitstrong or global stochas-ticity of the motion. In the latter region, although somecorrelation exists between successive iterations, over mostof the region it is possible to approximate the dynamicsby assuming arandom phase approximationfor the phasecoordinate, thus describing the momentum coordinate by a

diffusion equation. We explore this question more fully in thenext subsection.

D. The Fokker–Planck Equation

In regions of the phase space that are stochastic or mostlystochastic with small isolated adiabatic islands, it may bepossible to describe the evolution of the distribution functionin action space (or velocity space) alone. This is, in fact, theproblem of most practical interest. In the Fermi accelerationproblem, for example, the motivation was to find a possiblemechanism for heating of cosmic rays. The variations in thephases of the particles with respect to their accelerating fieldsare of little interest except as they are required for determiningthe heating rates and the final energy distribution.

Let us consider in what sense the evolution of the distribu-tion function can be described by a stochastic processin the action alone. Clearly we must confine our attentionto a globally stochastic region of the phase space in whichadiabatic islands do not exist or occupy negligible phase spacevolume. In such a region, it may be possible to express theevolution of the distribution in alone, in terms ofa Markov process in [14]

(2.4)

where the transition probability, is theprobability that an ensemble of phase points having an action

at a “time” suffers an increment in action aftera “time” . If we make the additional assumption thatthere exists an intermediate time scale such that

then we can expand the first argumentof the integrand in (2.4) to second order in to obtainthe Fokker–Planck equation

(2.5)

For Hamiltonian systems, the friction coefficient and thediffusion coefficient are related [15] as

(2.6)

allowing (2.5) to be written in the form of a diffusion equation

(2.7)

Assuming that phase randomization occurs on the time scalethen we can average over a uniform distribution of

phases to obtain the so-calledquasilinear diffusion coefficient

(2.8)

is then obtained directly from (2.6).For the simplified Fermi map (2.3) with sinusoidal velocity,

for which we obtain and Hencethe Fokker–Planck equation for the velocity distribution is

(2.9)


Fig. 3. Comparison of velocity distributionf(u) [here P (u)] for the simplified Fermi map (2.3) [solid line] and the exact Fermi map (2.1) [dashedline] (after [7]).

Similarly, for the Fermi map (2.2), we obtain

(2.10)

and the Fokker–Planck equation for the energy distributionis, from (2.7)

(2.11)

To obtain a steady-state solution to the Fokker–Planckequation, we assume perfectly reflecting barriers atand Setting in (2.9) and taking the netflux to be zero, we obtain a uniform invariant distribution invelocity const. for the simplified map. For the map(2.2), we obtain similarly a uniform invariant distribution inenergy const. Introducing the velocity distribution

for (2.2) through

(2.12)

and using we see that for (2.2).In Fig. 3 we compare the numerically calculated distributionsfor and 5 106 interactions with these predictions.In the region below the predictionsare verified. Above the distributions both fall off due tothe presence of islands and higher order correlations in thephase space, with the dips near the island centers.

We can also solve the transient Fokker–Planck equation. Forthe simplified Fermi mapping (2.3), with initial conditions ofa -function at we can solve (2.9) to obtain

(2.13)

which yields the distribution function for the transient heatingof the particles. This time development only holds, of course,until the particles begin to penetrate into the region withislands,

In real discharges there is always a nonzero flux in ac-tion space due to particle generation and loss processes.For example, electrons might be born by ionization at lowenergies and lost to the walls or by inelastic collisions at highenergies. Adding these generation and loss terms to (2.11)and solving yields nonconstant energy or velocity distributionsthat typically decrease with increasing energy [16]. Classical(electron-electron) collisions, which are always present inreal discharges, also tend to produce Maxwellian electrondistributions. The interplay among these different processesdetermines the distribution function in a way that can be verydifficult to determine analytically.

E. The Effects of Correlations

The complete dynamics, including the transition regionwith adiabatic islands embedded in a stochastic sea, is verycomplicated and can only be solved numerically. To gain someunderstanding of the diffusion in the phase space region wherecorrelations are important, it is convenient to first transformthe Fermi map to a local map near a resonance. Taking thesimplified Fermi map of (2.3), we obtain the so-calledstandardmappingby linearization in action space near a given period-1fixed point. These fixed points are located at

integer (2.14)

Putting and shifting the angle

then the mapping equations take the standard form

(2.15a)

(2.15b)

where

(2.16)


Fig. 4. Local approximation of the Fermi mapping by the standard mapping. (a) Linearization aboutu1a leading toK small and local stochasticity and(b) linearization aboutu1b leading toK large and global stochasticity (after [7]).

is the new action and

(2.17)

is thestochasticity parameter. We have thus related to theold action The conversion from Fermi to standard mappingis illustrated in Fig. 4 for two different values of leadingto two different values of

The dynamics of the standard mapping (2.15) can be con-sidered to evolve on a two-torus, with bothand takenmodulo The periodicity of the mapping in gives riseto a special type of periodic orbit (period-1 fixed point) in

which advances by every iteration of the mapping.The condition for these so-calledaccelerator modesis that

and and integers, withThe accelerator modes are stable providedwhich implies that stability windows for period-1 fixed pointsexist for successively higher values of as increases

decreases). Remnants of these accelerator modes,calledquasiaccelerator modes, can exist in the Fermi mapping,leading to enhanced diffusion.

The quasilinear transport coefficients for the standard map-ping (2.15) are and Since this mappinglocally approximates the Fermi mapping, we can relateto


Fig. 5. Plot ofD=DQL versus stochasticity parameterK. The dots are the numerically computed values and the solid line is the theoretical resultin the largeK limit (after [17]).

for the Fermi mapping. Using we find thatthe diffusion coefficients are related by

(2.18)

The island structure embedded in the Fermi stochastic seais exceeding complex, and, in fact, has fractal properties. Wemight expect this structure to lead to long time correlationof stochastic orbits in the neighborhood of adiabatic orbits,and this is in fact what happens. The quasilinear transportcoefficients are determined using the random phase assumptionapplied to a single step jump in the actionHowever, as pointed out in Section II-D, the Fokker–Planckdescription of the motion is valid only in the limitwhere is the number of steps for phase randomization tooccur. We should therefore consider the jumpwhere This was first done using Fourier techniques forthe standard mapping in the limit of large by Rechester andWhite [17] and for any by Rechesteret al. [18]. To order

the result is [7]:

J J J J

(2.19)

where and the J’s are Bessel func-tions. A numerical calculation of using 3000 particles iscompared with (2.19) in Fig. 5 [17]. There is good agreement,except near the first few peaks of which are due to thepresence ofaccelerator modes. For near but greater than

the critical value one finds numerically that

(2.20)

For an adiabatic (invariant) barrier exists and thereis no long-time diffusion.

For the Fermi map in which the phase is randomizedwithin a region of the velocity space for which the localapproximation gives a near-constant stochasticity parameter

, it is possible to derive a local (in velocity) diffusioncoefficient [19]. In this regime, for which

, the diffusion coefficient becomes, using (2.18)

(2.21)

Let us note that if we are interested in using the Fermi mapto model a heating mechanism, then particles will generallystart at low velocities, where the stable islands have negligiblysmall area. As the particles are heated they enter regions ofphase space within which large islands exist. Without extrinsicstochasticity, the particles will not penetrate these islands.Hence, although the equilibrium distribution is uniform inthe ergodic phase space surrounding the islands, the phase-averaged distribution will not be uniform, as is seenin Fig. 3. To correct for this effect, one must divide bythe fraction of phase space occupied by stochastic orbits. Thedetails of the calculation are described in [19] and [7]. Theenhanced diffusion due to quasiaccelerator modes is treatedby Lichtenberget al. [20].

Deterministic Fermi acceleration mappings are useful toolsfor understanding the purely dynamical aspects of the phase


Fig. 6. Sheath-plasma-sheath sandwich structure of a capacitive RF dis-charge.

randomization and heating of particles by periodic fields.However, let us note that for the heating of electrons in weaklyionized gas discharges, theextrinsic stochasticityassociatedwith electron-electron, electron-ion, and electron-neutral col-lisions can play a critical, and in many cases dominating role.The interplay between the intrinsic dynamical chaos and theextrinsic chaos due to collisions has received much attentionboth in the dynamics community [7] and in the dischargeheating community [21]–[23].

III. CAPACITIVE RF DISCHARGES

We begin with the application of Fermi acceleration toelectron heating in capacitive discharges where collisionlessdissipation of RF power has been intensively studied overthe last few decades, due to the wide application of thesedischarges in plasma processing [21]. A sandwich-like (sheath-plasma-sheath) structure of the capacitive RF discharge, firstproposed by Schneider [24] and now widely accepted inanalyzing the basic properties, is shown in Fig. 6. A typicaldischarge consists of a vacuum chamber containing two planarelectrodes separated by a spacing of order 2–10 cm and drivenby an RF power source. The substrates are placed on oneelectrode, feedstock gases are admitted to flow through thedischarge, and effluent gases are removed by the vacuumpump. The typical RF driving voltage is –V, and for etching of thin films pressures are in the rangefrom 10 to 100 mtorr, power densities are 0.1–1 W/cm2, andthe driving frequency is usually 13.56 MHz. Plasma densitiesare relatively low, 109–1010 cm 3, and electron temperaturesare of order 3 V. For deposition of films, pressures tend to behigher, and frequencies can be lower than 13.56 MHz.

The operation of capacitively driven discharges is reason-ably well understood. The mobile plasma electrons, respondingto the instantaneous electric fields produced by the RF drivingvoltage, oscillate back-and-forth within the positive spacecharge cloud of the ions. The massive ions respond onlyto the time-averaged electric fields. Electron thermal motionand oscillation of the electron cloud create sheath regionsnear each electrode that contain net positive charge whenaveraged over an oscillation period, i.e., the positive chargeexceeds the negative charge in the system, with the excessappearing within the sheaths. This excess produces a strongtime-averaged electric field within each sheath directed fromthe plasma to the electrode. Ions flowing out of the bulkplasma near the center of the discharge can be accelerated

by the sheath fields to energies of order of as they flowto the substrate, leading to energetic-ion enhanced processes.The positive ions continuously bombard the electrode over anRF cycle, whereas electrons are lost to the electrode only whenthe oscillating cloud closely approaches the electrode. Duringthat time, the instantaneous sheath potential collapses to near-zero, allowing sufficient electrons to escape to balance theion charge delivered to the electrode. Except for such briefmoments, the instantaneous potential of the discharge mustalways be positive with respect to any large electrode or wallsurface; otherwise the mobile electrons would quickly leakout. The sheath impedance is generally much larger than thatof the plasma and plays the essential role in limiting the RFdischarge current.

A. Early Work

Because the sheath width oscillates, electrons reflectingfrom the sheath are velocity dispersed. This idea was proposedby Gaboret al. [25] in their attempt to resolve the “Langmuirparadox” of a Maxwellian electron distribution in the positivecolumn of a dc glow discharge having negligible electron-electron Coulomb collisions. They found self-maintained RFoscillations (about 120 MHz) in the wall sheath of the positivecolumn. They believed that the electron interaction with RFfluctuations (oscillations) in the sheath was without energyexchange and was maintained solely by the energy flux of lowenergy electrons, thus, generating a high energy tail of thedistribution.

Later, Pavkovich and Kino [26] and Gould [27], analyzingthe wall sheath impedance at frequencies close to the plasmafrequency, showed that electron reflection in the oscillatingsheath is accompanied by RF energy absorption, due to a typeof transit time heating.

An explicit application of Fermi acceleration to electronheating in RF discharges was by Godyak [28]:

In an oscillating double sheath, the potential distribution,and thus the coordinate of the electron-reflection pointdepend on the time, and the electron reflection is anal-ogous to that of solid particles from a vibrating wall.On the average particles acquire energy in this case (theFermi acceleration mechanism).

Godyak went on to determine the electron power depositionfor a dc sheath with a small sinusoidally vibrating fluctuation,and put forward the idea that Fermi acceleration might bea major mechanism to sustain a capacitive discharge at lowgas pressures. These ideas were further developed by Godyak[29], [30] and by Akhiezer and Bakai [31]. The latter authorsused a simplified Fermi model (2.3) to determine the heatingrate, and noted that if there was a velocity barrier for heating,then the steady state distribution in velocity was uniform forvelocities below the barrier, in agreement with the calculationin Section II-D. Goeddeet al. [16] later considered the casein which electrons are continuously injected into a capacitivedischarge at low velocity and are lost by inelastic collisionsor escape to the walls at higher velocity, and determined thesteady state distribution, finding a power law electron energydistribution.


B. Homogeneous Discharge Model

Let us consider the collisionless power absorption in thesimplest model of a capacitive discharge with a homogeneousion background in the electrode gap and with a high RFsheath voltage. Such a simplified discharge model qualitativelydescribes the main features of capacitive discharges in practice.

Electrons reflecting from the large decelerating fields of amoving high voltage sheath can be approximated by assumingthe reflected velocity is that which occurs in an elastic collision(in the moving reference frame) of a ball with a moving wall

(3.1)

where and are the incident and reflected electron veloc-ities parallel to the time varying electron sheath velocity.If the parallel electron velocity distribution at the sheath edgeis , then in a time interval and for a speed interval

the number of electrons per unit area that collide with thesheath is given by This results in apower transfer per unit area

(3.2)

Using and integrating over all incidentvelocities, we obtain

(3.3)

In the physical problem varies with time, as the sheathoscillates, and the problem becomes quite complicated. Forthe uniform density model we note that

(3.4)

Furthermore, for the purpose of understanding the heatingmechanism we make the simplifying approximations that

can be approximated by a Maxwellian, ignoring theplasma drift, and that the meanelectron speed. These approximations simplify the calculation.Consistent with our approximation, we can set the lowerlimit in (3.3) to zero. Before performing the average over thedistribution function, we substitute

(3.5)

in (3.3) and average over time. Only the term insurvives giving

(3.6)

Now, consistent with our approximation that isMaxwellian, we note that the integral gives the usualrandom flux and (3.6) becomes

(3.7)

Inside the plasma the RF current is almost entirely conduc-tion current, such that

(3.8)

where is the cross-sectional area. Substituting (3.8) into (3.7)yields the stochastic electron power in terms of the (assumed)known current. Since we are calculating the power per unitarea, we use the current density, to obtain, for a single sheath

(3.9)

Within the homogeneous model, the time-average electronheating per unit area due to ohmic heating in the dischargebulk is

Re (3.10)

where is the length of the bulk region containing elec-trons and is the plasma conductivity.Substituting into (3.10), we find

(3.11)

where is the electron-neutral momentum transfer fre-quency.

Adding (3.9) (for two sheaths) and (3.11), the total timeaverage electron power per unit area is

(3.12)

A useful interpretation of this result is to introduce an effectivecollision frequency

(3.13)

Then we can consider that the stochastic heating introducesan additional, gas pressure-independent collision frequency,the electron bounce frequency into the expression forohmic heating of the discharge.

Although the preceding calculation illustrates simply themechanism of collisionless power absorption, in a real dis-charge one has to account for the oscillatory electron driftvelocity. If this velocity is independent of position and exactlyequals the velocity of the plasma-sheath edge, as given in (3.5),then transforming to an accelerated frame moving with theplasma electrons, we see a stationary electron sheath edge.Therefore, no energy is transfered to electrons that collidewith the sheath, and there is no stochastic heating [32]–[34].As pointed out in the latter two references, analytical RFsheath heating models are very sensitive to assumptions aboutthe form of the electron velocity distribution at the movingplasma-sheath edge. In particular, if the electron velocitydistribution there is assumed to be symmetric about the time-varying sheath velocity, then there is no heating. However,simulations for an inhomogeneous sheath indicate that thedistribution is not symmetric and that sheath heating doesoccur [35]–[38].

C. Self-Consistent Sheath

We describe now the stochastic heating for a collisionlessnonuniform high voltage RF sheath [32]. The structure of theRF sheath is shown in Fig. 7. Ions crossing the ion sheathboundary at accelerate within the sheath and strike


Fig. 7. Schematic plot of the densities in a high voltage capacitive RF sheath.

the electrode at with high energies. The ion motionis collisionless. Since the ion flux is conserved andincreases as ions transit the sheath,drops. This is sketchedas the heavy solid line in the figure. The electron sheath edgeoscillates from to as shown, where is thedistance from the ion sheath boundary at to the electronsheath edge. Time averaging this motion over an RF cycle,we obtain the time-average electron density within thesheath, sketched as the dashed line in Fig. 7. The time-averageelectron space charge leads to a modified Child law for the dcion current density

(3.14)

where is the dc (time-average) sheath potential andThis has the same scaling with and dc

sheath thickness as the normal Child law without electronshielding, which has For a fixed currentdensity and sheath voltage, the self-consistent RF ion sheaththickness is larger than the Child law sheath thickness bythe factor This increase is produced by thereduction in space charge within the sheath due to the nonzero,time-average electron density.

The solution for the electron sheath motion is sketchedin Fig. 8 [21]. The electron sheath motion is periodic butnot sinusoidal. The sheath moves faster when it is near theelectrode than when it is near the plasma because the iondensity is smaller near the electrode. This, along with theincrease in the sheath length over that for the homogeneousmodel, leads to an increase in the stochastic heating over thatfound for the homogeneous model. The power transferred tothe electrons by the sheath is found from (3.3), but nowis not a fixed Maxwellian, but is a time varying functionwith a time varying density at the electron sheathedge . To determine , we first note that the sheathis oscillating because the electrons in the bulk plasma areoscillating in response to a time-varying electric field. If thevelocity distribution function within the plasma at the ionsheath edge in the absence of the electric field is aMaxwellian having density then the distributionwithin the plasma at the ion sheath edge is

where is the time-varying oscillationvelocity of the plasma electrons. At the moving electron sheath

Fig. 8. Sketch of the electron sheath thicknesss versus!t:

edge, because not all electrons having atcollide with the sheath at Many electrons are reflected withinthe region where the ion density drops from to

This reflection is produced by an ambipolar electric fieldwhose value maintains quasineutrality at all times.The transformation of across this region to obtain iscomplicated because the rapid changes in the sheath positiondistort the electron distribution. A self-consistent model ofthis effect must incorporate the spatial variation of the electricfield. An analytical approach is described by Kaganovich andTsendin [39]. However, the essential features to determine thestochastic heating are seen if we approximate

(3.15)

Inserting (3.15) into (3.3) and transforming to a new variablewe obtain

(3.16)

Averaging over an oscillation period and integrating overyields

If the assumption is made that the sheath motion is muchslower than the electron thermal velocity, as for the homo-geneous model calculation, then one obtains [32], [21] for

that

(3.17)

where

(3.18)

Using (3.18), we can compare (3.9) and (3.17) at the samedriving voltage. Substituting the first equality in (3.18) into(3.9) and (3.17) yields

homogeneous model (3.19a)

self-consistent model (3.19b)

We see that, at a fixed driving voltage, both models yield astochastic heating proportional to , however, the stochastic


Fig. 9. Effective collision frequency�e� versus pressurep for a mercury discharge driven at 40.8 MHz. The solid line shows the collision frequencydue to ohmic dissipation alone (after [30]).

heating for the self-consistent model is a factor oflarger than for the homogeneous model. As an example, for

V, V, we findResults similar to this were also obtained for a collisional

sheath [40]. However, let us note that collisional (ohmic)heating within the sheath region can be large and can dominateover collisionless heating [41], [39], [21]. Ohmic heating perunit volume within a collisional sheath is larger than that in thebulk plasma due to the lower electron density, and thereforeincreased resistivity, of the sheath as compared to the bulk.

D. Experimental Results

Early experiments to investigate stochastic heating are de-scribed in [29], [30], [42], and [43] and are summarized in[44]. In these works electrical and plasma parameters werestudied in a parallel-plate capacitive RF discharge symmetri-cally driven at 40–110 MHz in mercury vapor. The current-voltage characteristic, the RF power, the plasma density andthe electron temperature were simultaneously measured in themercury pressure range between 210 4 and 1 10 1 torr.

The effective collision frequency versus pressure wasevaluated from the shape of the measured discharge current-voltage characteristic [30], [42] and directly by measuring theRF power absorbed by the discharge [43]. In the last casewas obtained from the relationship for the power absorbed per

unit area

(3.20)

where is the cross-section averaged discharge currentdensity. The measurements were done at relatively low RFvoltages, and the power absorption due to ion accelerationin the RF sheaths was neglected. The effective collisionfrequency found from experiment as a function of mercurypressure is shown in Fig. 9. Both the asymptotic leveling offof at low pressure characteristic of stochastic heatingwhich is independent of and the linear increase of with

at high characteristic of ohmic heating, are clearly visible.The good agreement of the measurements withcalculatedfrom the stochastic heating formula is somewhat fortuitous,however, as a uniform sheath rather than a self-consistentsheath was used in the calculation, and the ion power loss

was neglected in determining from the measurements.In these early studies, it was shown that the presence of

stochastic heating at the plasma boundaries reduces the RFelectric field and electron oscillatory velocity in the plasmabulk [29], [45]. This happens due to nonlocal electron energybalance in a low pressure RF discharge when RF powerabsorbed in the RF sheaths compensates the electron energy


losses over the entire plasma volume, so that there is no needfor a large bulk RF field to maintain the discharge.

A comprehensive experimental study of symmetric RFdischarge characteristics in argon at 13.56 MHz has beenperformed by Godyak and Piejak [46] and Godyaket al.[47], [48]. The discharge length and diameter were 6.7 and14.3 cm, respectively, approximating a uniform plane parallelconfiguration. Measurements were made of RF voltage, RFcurrent, total power absorbed, the central plasma densitymean electron energy and electron energy distributionfunction (EEDF) . The RF power was determined by av-eraging over an RF cycle, and andwere determined using Langmuir probes. Also, the ion currentto the RF electrodes and dc bias voltage in the RF sheath weremeasured to determine the ion power loss. Measurements wereperformed over a wide range of pressures from 3 mtorr to 3 torrand for powers up to 100 W. The corresponding RF voltageamplitudes were up to 1500 V, and the RF current amplitudeswere up to 2 A.

Having measured the discharge RF power, the dischargecurrent, and the EEDF, the plasma density was found by inte-gration over the EEDF, and the collisional power absorption

in the plasma bulk was then estimated using the plasmaconductivity formula. This was compared to the total RF power

transferred to the plasma electrons. The latter was foundas the difference between the total measured discharge powerand that corresponding to the ion loss. The result is shownin Fig. 10. As is seen, at relatively high gas pressure

torr), the power absorption is entirely due to collisionaldissipation, At low pressureand the ratio reaches three orders of magnitude atthe lowest pressure of 3 mtorr. Accounting for collisionalelectron heating in the sheath at this pressure reduces thisratio to two orders of magnitude. Such an enormous differencebetween and in the collisionless heating regimeis due to the stochastic heating in the RF sheaths, whichproduces a population of high energy electrons that maintainsthe discharge ionization, along with a simultaneous drop incollisional electron heating in the plasma because of bulkelectron cooling and the associated decrease indue to theRamsauer effect.

The electron cooling occurs during the heating mode tran-sition when the discharge switches from a collisionally toa stochastically dominated mode [46]. The heating modetransition is seen from a different perspective in Fig. 11 in theevolution with gas pressure of the electron energy probabilityfunction (EEPF) The EEPF is defined as

(3.21)

where is the electron kinetic energy; the EEPFis defined such that a plot of versus is a straight linefor a Maxwellian distribution. In Fig. 11, we see a transitionfrom a Druyvesteyn-like distribution,which is typical for collisional electron heating in a Ramsauergas at (high argon pressure) to a bi-Maxwelliandistribution at low argon pressure when stochastic heatingdominates. This transition is accompanied by a corresponding

Fig. 10. The ratio between total and collisional RF power transferred to thebulk plasma electrons versus argon pressure;l = 6:7 cm andJ = 1mA/cm2

(after [48]).

Fig. 11. Evolution of the electron energy probability function (EEPF)gp(E)with pressure in argon;l = 6:7 cm andJ = 1mA/cm2 (after [48]).

sharp change in plasma density and mean electron energy [46].In the stochastic heating regime the majority of electrons havea very low energy, and being trapped by the ambipolar dcfield they are not able to reach the RF sheaths where stochasticheating takes place. Having their energy close to the Ramsauerminimum of the argon cross section, the low energy electronshave a very low electron-atom collision frequency and thus,they collisionlessly oscillate in a weak RF field unable to gainenergy.


On the other hand, the high energy electrons overcome theambipolar potential and effectively interact with the oscillatingRF sheaths, bouncing between them. Phase randomizationmust occur for these electrons to be heated. The randomizationcan arise directly from the dynamics or can be induced byexogenous stochastic forces (e.g., interparticle collisions). Thecondition for dynamical stochasticity is (see Section II-C)

(3.22)

Unnormalizing (3.22) using andyields for the electron velocity For typical ex-perimental parameters cm, cm, and

s we obtain cm/s, correspondingto a longitudinal energy of V. Since the flux-averagedlongitudinal electron energy is V, the conditionfor dynamical phase randomization is marginally satisfied forthe majority of electrons. However, at lower driving frequen-cies and/or short electrode gaps, the condition for dynamicalphase randomization is not met. In this case, high frequencyplasma or sheath fluctuations can effectively randomize theelectron phase even when the electron mean free path islarger than the plasma width. This randomization mechanismmay be responsible for the effective collisionlesselectron heating by RF sheaths in experiments where thecondition of dynamical phase randomization is not satisfied.Phase randomization due to rare collisions of fastelectrons with atoms can also introduce phase randomization,since the average lifetime of electrons is alwaysmuch larger than the bounce time Here

is the Bohm (ion sound) velocity. Thus, overtheir lifetime, the bouncing electrons have enough chance tobe randomized via electron-atom collisions. As was shownrecently by Kaganovich et al [22], there are many scenarios forelectron phase randomization which depend on the relationshipbetween the bounce frequency, the driving frequency, and theelectron-neutral collision frequency.

Another kind of EEDF transition due to change in RF powerwas obtained by Godyak [49] and was studied by Buddemeieret al. [50] through experiment and numerical simulation. InFig. 12, we see evolution of the measured electron probabilityfunction with discharge current (and voltage) at a fixed lowargon pressure, from a Druyvesteyn-like distribution at lowdischarge voltage to a two-temperature distribution at highervoltage. This transition is associated with the nonlinear natureof electron heating in the RF sheath. At small dischargevoltage, the sheath heating is small or comparable to the bulkcollisional electron heating. With increasing RF current, thestochastic heating begins to dominate with a correspondingrestructuring of the electron energy distribution. These resultsare consistent with the scaling laws for sheath and bulk heating(e.g., see [21]).

E. Fluid and Particle Simulations

Monte Carlo and particle-in-cell (PIC) simulations of capac-itively coupled discharges at low pressure performed in the lastdecade have confirmed the existence of collisionless electronheating produced by oscillating electrode sheaths [51]–[54],

[36], [37], [50]. Stochastic heating due to reflection from os-cillating sheaths was observed by Kushner [51] using a MonteCarlo calculation, and the subsequent PIC calculations alsofound EEPF’s with high energy tails attributed to stochasticheating. Electron energy loss across the oscillating sheath wasinvestigated in several of these works, and a number of authorsalso show the transition from weak to strong sheath heating.

Nitschke and Graves [55] performed fluid simulations at lowpressure using a helium-like model gas and compared theseto particle-in-cell (PIC) simulations of the same system. Thefluid simulations do not incorporate the physics of stochasticheating, while the PIC simulations do. Below 100 mtorr, dis-agreement in the electron heating predicted by each simulationleads to significant differences in the discharge properties. Atthe same applied frequency (12 MHz) and voltage (500 V),the PIC simulations at 50 mtorr and 120 mm gap spacingyield roughly twice the electron power deposition as the fluidsimulations. From power balance, this yields twice the densityfor the PIC simulation compared with the fluid simulations. Byadding into the fluid model the appropriate analytic expressionfor stochastic heating as an “additional” power source, the fluidand PIC simulations were brought into closer correspondence.

Comparisons were also made between the measurementsof Godyak and Piejak [46] and PIC simulations by Vahediet al. [56]. The comparisons are in argon for an electrodediameter of 14.3 cm, a discharge length of 2 cm, and anexternal current source of 2.56 mA/cm2 at 13.56 MHz. The gaspressure was varied between 70 and 500 mtorr to observe thetransition from stochastically to ohmically dominated electronheating. Except for the normalization, the’s obtained fromthe simulations agree well with the measured’s, showingthe transition from a two temperature distribution at 70 mtorrto a Druyvesteyn-like distribution at 500 mtorr.

Another simulation of discharge behavior [35], [38], [7]was performed at mtorr (argon) with a spacing of 10cm between parallel plates, and over a range of RF voltagesbetween 100 and 1000 V. A two temperature distribution wasfound, as in the experiments, and the distribution varied inboth space and time. It is clear that a deeper understanding ofthe discharge behavior involves the space and time variationsof Fig. 13 shows the one dimensional electron distributionfunction versus at 15 positions near the sheathregion – cm) and at eight different times during the RFcycle. Each plot covers 1/32 of a cycle temporally, and eachline in a plot covers a 2 mm thick region spatially. The unitson the vertical axis are proportional to. At time 0/32, thesheath is fully expanded, and the two-temperature nature of thedischarge near the sheath can be seen as the wide “base” andnarrow “peak” of the distribution. As the RF cycle progressesto time 8/32, the distributions within the sheath region at eachposition display a drift toward the electrode (negative velocity)that is approximately equal to the collapsing sheath velocity.By time 12/32, fast electrons have arrived from the oppositeelectrode, moving at a velocity of about 4 106 m/s (smallpeak at extreme left of figure). At time 16/32, the sheath isfully collapsed, the drift in the sheath has disappeared, andthe fast electron group moving toward the electrode shows alower velocity as slower electrons arrive from the opposite


Fig. 12. Electron energy distribution functions in the midplane of a capacitive RF discharge (6 cm gap, 15 cm electrode diameter) in argon for constantpressure and varying RF current densities given in mA/cm2 (RF discharge voltages in volts) and (after [50]).

electrode. As the sheath begins to expand, as shown hereat times 18/32 and 20/32, the electrons in the sheath regionare strongly heated, and the beginning of an electron beamproduced by this expansion can be seen moving away at apositive velocity. As the sheath continues to expand, the driftof the distribution within the sheath away from the electrodecan be seen to initially match the sheath velocity (time 22/32)but then decays (time 24/32) to a velocity much slower thanwhen the sheath was collapsing. One consequence of thesecomplicated dynamics near the sheath edge is thatis notsymmetric about the velocity of the moving sheath edge duringthe entire RF cycle.

The existence of more energetic electrons near the plasmaedge due to local electron heating increases the ionization thereat higher pressures tending to flatten the plasmaprofile. Furthermore, the ionization is not constant, but followsthe density variations in space and time of the more energeticelectrons. This is shown for a PIC simulation by Vender andBoswell [57] in the plot of Fig. 14, in which the darkness ofeach square is proportional to the number of ionizing collisionswithin that square of position and time intervals. Most of the

ionization is seen to occur along a path of fastest electrons thatare reflected off of the sheath at the phase at which it is mostrapidly expanding. There is also somewhat more ionizationnear the sheaths, an effect that becomes more pronouncedat higher pressures where the ionization mean-free-path isshorter, which has been observed in various experiments.

The spatial distribution of the electron power absorptionhas been examined in several PIC simulations. While theabsorption is large and positive near and within the RF sheaths,it can become negative within certain regions in the dischargebulk under conditions of strong stochastic heating [58], [59].This is particularly apparent in simulations at low pressures ina Ramsauer gas, where the ohmic dissipation (which is alwayspositive) is small. Negative power absorption occurs where thephase of the electron current (transfered from the stochasticheating at the RF sheath edge by the electron thermal motion)differs from the phase of the local electric field by morethan 90. There have been no experimental measurementsconfirming the existence of negative power absorption inthe bulk region of capacitive RF discharges, although someexperimental results have hinted at the existence of this


(a) (b)

(c) (d)

Fig. 13. One-dimensional electron velocity distribution functionfe(x; vx; t) for a 10 cm electrode spacing in a 3 mtorr argon discharge; each plot coversa time window of 1/32 of an RF cycle. Each line on a plot represents a spatial window of 2 mm (after [35]).

effect (e.g., see [60]). Negative power absorption in inductivedischarges is treated extensively in Section IV.

F. Other Collisionless Heating Models

The effect of a weak dc magnetic field on the stochasticelectron heating by oscillating RF sheaths has been studiedby Liebermanet al. [61], Okuno et al. [62], Hutchinsonetal. [63], Turner et al. [64], and Park and Kang [65]. It wasshown in the first work that there can be stochastic heatingenhancement due to multiple correlated collisions of electronswith the moving sheath. A gyrating electron that collides oncewith the moving sheath collides again in a time interval ofapproximately half a gyroperiod. The electron trajectory can

be coherent over many such sheath collisions, leading to largeenergy gains. Okunoet al. [62] reported measurements ofsuch an effect of electron acceleration resonant with the sheathmotion in a cylindrical, magnetized RF-driven discharge, andPark and Kang [65] showed a reasonable agreement betweenmeasurements and a Child law sheath model incorporatingsuch stochastic heating. However, the same issues arise aboutthe form of the electron velocity distribution at the oscil-lating plasma-sheath edge, as for the nonmagnetized case;a distribution that is symmetric about the sheath oscillationvelocity yields no stochastic heating. Also, the cross-fieldresistivity of the bulk plasma increases due to the magneticfield, leading to an increased bulk ohmic heating. Hutchinson


(e) (f)

(g) (h)

Fig. 13. (Continued.)One-dimensional electron velocity distribution functionfe(x; vx; t) for a 10 cm electrode spacing in a 3 mtorr argon discharge; eachplot covers a time window of 1/32 of an RF cycle. Each line on a plot represents a spatial window of 2 mm (after [35]).

et al. [63] studied the competition between sheath and bulkheating experimentally and with PIC simulations, and Turneret al. [64] compared these results to a fluid (pressure heating)model. The PIC simulations did not show an enhanced heatingdue to resonant electrons, although an increase in averageelectron energy was observed within the sheaths.

Another way of looking at collisionless electron heatingassociated with RF sheath oscillations in capacitive RF dis-charges has been discussed by Surendra and Dalvie [34] andby Turner [66]. They showed that an approximation to theheating can be obtained within a macroscopic (fluid) theory byincorporating pressure effects that arise during the expansionand contraction of the sheath in a nonhomogeneous plasma

model. The pressure effect is caused by the difference inplasma density and electron energy between the bulk plasmaand the near-sheath plasma. When the sheath expands, elec-trons flow into the adjacent bulk plasma and are compressed.At the same time, electrons are rarefied as they flow into theopposite, collapsing sheath. Turner showed that using finiteelectron thermal conductivity, these simultaneous rarefactionsand compressions of the electron gas produce nonequilibriumthermal disturbances, and the net work done is not zero. Inthis way, kinetic effects are incorporated approximately intoa fluid model. Solving the fluid equations with the electronenergy balance equation, Turner verified this approach usingPIC simulations, and Turneret al. [64] showed experimentally


Fig. 14. Spatiotemporal distribution of ionizing collisions collected over 20RF cycles, for a 10 MHz, 20 mtorr hydrogen discharge [57].

and using PIC simulations that a weak transverse dc magneticfield can induce a transition between the pressure heating andohmic heating.

Collisionless, as well as collisional electron heating canbe effectively treated on the kinetic level in the frameworkof the nonlocal approachto the solution of the Boltzmannequation [67]–[71]. In this approach a complicated time-spacevariable and multidimensional problem is reduced to a zero-dimensional Boltzmann equation for the symmetric part of theEEDF

(3.23)

where is the total electron energy (kinetic plus potential),is the time-space averaged energy diffusion coefficient,

is the inelastic collision integral, is the electronloss frequency, and is the electron source. The spacevariation of and its integrals are defined by the spacevariation of , i.e., by the ambipolar potential distribution,and the electron heating and the shape of the EEDF are definedby . The electron heating power density is

(3.24)

Thus, contains all information about the electron heat-ing process. The particular shape of the EEDF andthe power deposition depend on the specific mechanism of

electron interaction with the RF field, on the electron phaserandomization mechanism [16], [22], and on the generationand loss mechanisms [72]. For stochastically heated electrons,

is the product of the square of the random walk step inenergy space and the frequencyof such an event

(3.25)

This approach has been used for low pressure capacitivelycoupled RF discharges in the stochastically heated regime byKaganovich and Tsendin [39].

Wang et al. [72], Lichtenberg [73], and Kaganovich [23]have recently considered the nonlinear effects of the dynamicson the determination of the energy diffusion coefficientand the resultant discharge equilibrium. Depending on thepressure, driving frequency, discharge length, and other dis-charge parameters, the existence of strong phase correlationsand adiabatic limitations to the chaotic dynamics can playimportant roles in the theoretical description of the heat-ing. However, for many capacitive discharges of commercialinterest, interparticle collisions play the dominant role inphase randomization. The general approach incorporating bothdynamics and collisions is to obtain and solve a space- andtime-averaged kinetic equation. The various regimes describ-ing the interplay between collisional and dynamical effectshave been classified by Kaganovichet al. [22].

In another approach Alievet al. [74] presented an analysisof collisionless electron heating in RF discharges based on thequasilinear theory of waves in warm plasma given for example,in Ichimaru [75] and Alexandrovet al. [76]. In applicationto RF discharges excited by an external RF power sourcewith frequency much lower than the cutoff frequency, theinteraction of the electromagnetic field with a bounded plasmais considered as a superposition of decaying (evanescent)waves having a wide spectrum of wavenumberswhere is the wave phase velocity. When the scale of theelectromagnetic field (sheath width or skin depth) becomessmaller than the electron mean free path then collisionlesselectron heating occurs as a result of the resonant interaction(acceleration or deceleration) of electrons with waves havingtheir phase velocity close to the electron velocity. This processis very similar to Landau damping, but occurs at frequenciesmuch lower than the plasma frequency Separatingthe space and time scales of the electromagnetic fields andlinearizing the kinetic equation, one can divide the EEDF intoa large part averaged over the length scale and asmall part accounting for small deviationsfrom on a length scale smaller than Using Fouriermethods, the structure of the electromagnetic fields and thediffusion coefficient are found for different kinds ofRF discharges (capacitive, inductive, and surface wave). Alievet al. [74] find that the diffusion coefficient that governsthe EEDF shape and the electron heating power density ismainly defined by resonance electrons with velocities

with . Let us note that thisapproach involves integration over the unperturbed motionsof the electrons, and therefore does not incorporate phaserandomization due to dynamical stochasticity.


(a) (b)

(c)

Fig. 15. Schematic of inductively driven discharge in (a) cylindrical, (b) planar, and (c) reentrant geometries, where (a) and (b) are used for materialsprocessing and (c) is used for lighting.

IV. I NDUCTIVE RF DISCHARGES

Plasma in an inductive discharge is maintained by applica-tion of RF power to an inductive coil, resulting in electronenergy absorption due to the induced RF electric field near thecoil. The driving frequency is usually 13.56 MHz, althoughlower (and higher) frequencies are sometimes used. As shownin Fig. 15(a) and (b), planar or cylindrical coils in a low aspectratio (length/diameter) discharge are generally used for lowpressure materials processing. The planar coil is a flat helixwound from near the axis to near the outer radius at one endof the discharge chamber (“electric stovetop” coil shape). Fora typical 30-cm chamber diameter, the RF power is typically100–1000 W.

Another kind of inductive discharge is an RF lamp with aninternal coil, as shown in Fig. 15(c). The internal coil, usuallywith a ferrite core, is inserted into a reentrant cavity inside aglass bulb coated inside with fluorescent powder (phosphor).The bulb is filled with a mixture of inert gases such as argonor krypton at a pressure of hundreds of millitorr, and mercuryvapor at a few millitorr. The inductive plasma excited insidethe bulb has a very high RF power conversion to the mercuryresonance ultraviolet (UV) radiation (60–70%, mainly at 253nm), and the UV radiation excites the phosphor to emit visible

light. The absence of electrodes provides a highly efficient(4–5 times more than an incandescent bulb) and durable (upto 100 000 h) light source.

Because the voltage across the exciting coil of an inductivedischarge can be as large as several kilovolts, a discharge canalso be capacitively driven by the coil. There is generally acapacitively driven discharge at low plasma densities, witha transition to an inductive discharge at high densities. Anelectrostatic shield placed between the coil and the plasma canreduce the capacitive coupling if desired, while allowing theinductive field to couple unhindered to the plasma. Inductivedischarges for materials processing are sometimes referred toas inductively coupled plasmas (ICP’s), transformer coupledplasmas (TCP’s), or RF inductive plasmas (RFI’s).

The inductive electric field is nonpropagatingand typically penetrates into the discharge a distance on theorder of a plasma skin depth, which is typically 1–3 cm. Henceplasma heating occurs near the dielectric window surface. Thedc plasma potential in these discharges is typically of orderfrom 30 to 40 V with respect to the walls, and the plasmadensity is typically in the range from 1011 to 1012 cm 3.Hence, the sheath thickness is of order from 0.1 to 1 mm(a few Debye lengths).


A. Classical and Anomalous Skin Effect

In an inductively coupled plasma, power is transferredfrom the electric fields to the plasma electrons within askin depth layer of thickness near the plasma surface bycollisional (ohmic) dissipation and by a collisionless heatingprocess in which bulk plasma electrons “collide” with theoscillating inductive electric fields within the skin layer. Inthe latter situation, electrons are accelerated (and decelerated)and subsequently thermalized much like stochastic heating incapacitive RF sheaths, which we treated in Section III.

We first consider the so-calledclassical or normal skineffectaccompanied by ohmic (collisional) electron heating in asemi-infinite spatially uniform plasma. The normal skin effectoccurs when the electron thermal motion is negligible andthere is a local coupling between the RF current densityand the RF electric field within the skin layer given by

where

(4.1)

is the complex conductivity of a cold plasma. We considerthe case when which is always true for induc-tively coupled plasmas. We also assume a Maxwellian EEDFand an energy-independent electron-atom collision frequency

const. Otherwise, and in (4.1) must be replacedby some effective and both being integrals overthe energy of the particular EEDF and the anddependence [77].

According to Maxwell’s equations, the penetration of thetransverse electric field into the plasma is described by thecomplex wave equation

(4.2)

having solution

(4.3)

where

Re (4.4)

is the inverse skin depth and

(4.5)

is the propagation constant. Substituting (4.1) into (4.4), oneobtains the general expression for the classical (normal) skindepth (see for example [78])

(4.6)

where

and (4.7)

In the collisional limit typical for nonsupercon-ducting metals and high pressure plasmas, and

(4.8)

and the RF energy collisionally dissipates within the skin layer.

In the high frequency limit called the nondissi-pative or high frequency skin effect and

(4.9)

In this case the electrons collisionlessly oscillate within theskin layer with no net energy gain. For an electromagneticwave incident on the plasma boundary this case corresponds tothe total reflection of the wave from the plasma. For dischargemaintenance in this case, the wave reflection is not perfect, anda small fraction of the incident wave power is locally and/ornonlocally deposited within the skin layer.

There is a third situation (anomalous skin effect) for whichelectrons incident on a skin layer of thicknesssatisfy thecondition where is determined below. Inthis case the interaction time of the electrons with the skinlayer is short compared to the RF period or the collision time.The RF field penetration in this regime was first estimatedby Pippard [79] with application to the high frequency skineffect in metals at low temperatures, and was determined self-consistently by Reuter and Sondheimer [80] for metals andby Weibel [78] for a homogeneous plasma half-space with aMaxwellian EEDF. To see the essential scalings, followingPippard, we consider the ordering and divide theelectrons into two groups, those moving at small angles tothe surface which spend most of their time between collisionswithin the skin layer, and the rest, whose chance of collision issmall. We ignore the latter group of ineffective electrons. Thevelocities of the effective electrons form angles less thanand their relative number is of order where is theelectron-neutral mean free path for momentum transfer. Thisresults in an effective plasma density andan effective plasma frequencywithin the skin layer, where is a constant of orderunity. Substituting these effective quantities into (4.1) and(4.8) yields an expression for the effective collision frequency

, analogous to that introduced in (3.13) for capacitive RFdischarges, and for the anomalous skin depth

A more careful averaging based on the kinetic theory of theanomalous skin effect [78], [75], [76] gives and

and

(4.10)

According to (4.10), for a strong anomalous skin effect whereand the penetration of the RF electric

field into the plasma is deeper than for the nondissipative skineffect in the high frequency limit:

A generalnonlocality parameterfor the nonlocal interactionof electrons with the electromagnetic field has been given byFried and Conte [81] and used in the analysis of the anomalousskin effect by Weibel [78] and Sayasov [82]

(4.11)


Fig. 16. Plot of ln (�e=�0) versus ln (!=�m); showing the regimes ofcollisional, high frequency, and anomalous skin effect in a semi-infiniteplasma; the solid line, corresponding to nonlocality parameter� = 1, isthe boundary of the anomalous skin effect (after [91]).

Formulae for the classical skin effect are applicable when; for , the anomalous skin effect takes

place. Note that is small for both very low and very highfrequency and reaches it maximum at where

which is close to its high frequencylimit. In the opposite case

and the normal (collisional) skineffect occurs. Fig. 16 shows the boundary dividing the clas-sical and the anomalous skin effect in the space of

andLet us describe some essential features of the anomalous

skin effect which are the result of the nonlocal interaction ofelectrons with the electromagnetic field due to their thermalmotion.

1) The spatial decay of the electromagnetic field into aplasma for the anomalous skin effect is not exponential,as it is in the classical skin effect. Moreover, the decaymay be nonmonotonic and may exhibit local maximaand minima in the plasma, as shown by Weibel [78] inhis kinetic analysis (see Fig. 17), and in the experimentsof Demirkhanovet al. [83] and Joye and Schneider [84](see Fig. 18).

2) As has been shown by Kondratenko [85], for a stronganomalous skin effect the penetration lengthof the RF electric field and that of the RF magneticfield are essentially different, while for thenormal skin effect The less rapid spatial decayof the RF magnetic field (and accompanying RF currentdensity) is due to the ballistic transport of RF currentcaused by the electron thermal motion. Different formu-lae for can be seen in the literature (which differ byfactors of order unity) due to different definitions of theelectron thermal velocity or

and more importantly, due to different

Fig. 17. Normalized electric field amplitude versus normalized distancealong the propagation direction for�m=! = 1 and for various nonlocalityparameters� (after [78]).

definitions of by different authors. The valuecan be defined as or as

or as a distance where the RF electricor magnetic field (or RF power) decays by one (or two)e-foldings. Different values can be obtained fororusing these definitions, while for the normal skin effectall these definitions give the same formula (4.6).

3) In the regime of the strong anomalous skin effect neitherthe skin depth nor the energy dissipation in the skinlayer depend on the collision frequency . In thiscase the RF energy dissipation process occurs even inthe limit The noncollisional dissipation hasa simple explanation. For a relatively thin skin layerwhen and the electrons reflecting fromthe space charge sheath at the plasma-wall boundarycross the skin layer in a time less than the RF fieldperiod. Hence, on the average the electrons gain energywithin the skin layer as in a dc field. This differs fromcollisionless electron motion in a homogeneous RF fieldwhen corresponding to the high frequencylimit of the normal skin effect in which electrons gainenergy from the field during one quarter cycle and returnthe energy back to the field during the next quarter cycle.

For both the normal and the anomalous skin effect, there isno electron heating unless some phase mixing (randomization)mechanism breaks the regularity of the electron motion. Forcollisional heating corresponding to the normal skin effect,randomization occurs locally within the skin layer due toelectron-atom (and/or electron-ion) collisions. For the anoma-lous skin effect the randomization is provided by the electronthermal motion which moves electrons out of the skin layer


Fig. 18. The radial distributions of the amplitude and phase of the RF magnetic field in a cylindrical argon plasma. Dotted lines show experimental datafor p = 10 mtorr, average plasma densityn = 4:2 � 10

12 cm�3; and Te = 2:1 V. Solid lines are calculations according to the theory of Sayasov[82] for n = 3 � 10

12 cm�3 and �m = 4 � 107 s�1 (after [84]).

into the neighboring plasma having no RF field, thus pre-venting the electrons from returning the energy acquired inthe skin layer back to the RF field. For a bounded plasmasuch as an inductive discharge where electrons can repeatedlyinteract with the RF field in the skin layer, some randomizationmechanism must be present in the bulk plasma to provide aneffective electron heating in the skin layer.

Let us note that the kinetic treatment of the anomalous skineffect (e.g., [78]) is a linear theory in which the dissipation isdetermined by integration over theunperturbedmotion (orbits)of the electrons. Low velocity electrons can have stronglyperturbed orbits, leading to wave trapping and ponderomotiveforce effects. But if the average energy gain is much lessthan the thermal energy, then this class of electrons contributelittle to the overall energy absorption. The repeated interactionof high energy electrons with the skin layer in a boundedsystem can lead to strong phase correlations and adiabaticinteractions of electrons with the skin layer in the absence ofexternal stochastic forces such as electron-neutral collisions.These mechanisms are similar for inductive and capacitivedischarges (see Section III) and have recently been analyzedby Kaganovichet al. [22].

B. Fermi Acceleration Model of Collisionless Heating

To determine the heating at low pressures using a Fermiacceleration model, we consider an electron from the bulkplasma incident on the RF electric field within a skin depth

layer in slab geometry. We assume a simple model in which thetransverse electric field within the slab decays exponentiallywith distance from the edge into the slab

(4.12)

We also assume that the force due to the RF magnetic fieldis negligible in determining the power dissipation, and weassume that the collisionality is weak, hencethere are no electron collisions within the skin layer. A similarmodel was first introduced by Holstein [86] to describe the lowtemperature optical and infrared reflectivity of metals. Becausethere are no -directed forces, we can write

(4.13)

where the electron reflects from the surface at Sub-stituting (4.13) into (4.12) yields the transverse electric fieldseen by the electron

Re

Re (4.14)

The transverse velocity impulse

(4.15)


is calculated by substituting (4.14) into (4.15) and integratingto obtain

(4.16)

The energy change averaged over a uniform distributionof initial electron phases is then

(4.17)

which can be integrated over the particle flux to obtain thestochastic heating power

(4.18)

For a Maxwellian electron distribution the integrals overand are easily done, and the integral can be evaluated interms of the exponential integral For the regime of largenonlocality

(4.19)

we obtain [21]

(4.20)

We can introduce an effective collision frequency byequating the stochastic heating (4.20) to an effective collisionalheating power flux

(4.21)

For we have

and equating this to (4.20), we determine

(4.22)

With we find in agreement with thesimple estimate (4.10).

Although Fermi acceleration models of the velocity impulselead to simple estimates of the effective collision frequencydue to stochastic heating, they are not self-consistent becausethe form of the spatial variation of the electric field has beenassumed. The heating power and effective collision frequencyhave been determined over the entire range of collisional-ity from a Fermi acceleration model with an exponentiallydecaying electric field profile by Vahediet al. [87]. Wesummarize their results and compare them to a self-consistentmodel in the next subsection. The different heating rates forreal and exponential electric field profiles are described inAliev et al. [74].

C. Self-Consistent Collisionless Heating

The self-consistent analysis of Reuter and Sondheimer [80],using a Fermi–Dirac electron distribution to determine theanomalous skin resistance of a low temperature metal, wasfirst applied by Weibel [78] to a classical plasma having aMaxwellian electron distribution. The RF power absorptiondue to the skin effect can be characterized by a complexsurface impedance

(4.23)

The time-average power absorbed by the plasma per unit areacan be written in terms of as

Re Re (4.24)

For the classical (normal) skin effect, we find

(4.25)

where For corresponding tothe normal skin effect dominated by collisional heating, wefind

(4.26)

with given by (4.8). For the anomalous skin effect (see[78], [75]), we obtain

(4.27)

with given by (4.10).The self-consistent analysis of the anomalous skin effect

of Weibel [78] and Blevinet al. [88] was applied to heat-ing in inductive discharges of finite extent by Blevinet al.[89] and Turner [90], who also compared the results to akinetic simulation. Following the Blevinet al. [88], [89]analysis, Kolobov and Economou [91] calculated analyticallythe surface impedance for a plasma slab model over a widerange of collisionality, The results, whichclosely coincide to the simulations of Turner [90], are shownin Fig. 19. In the collisional regime where , theresults of kinetic (warm plasma) theory practically coincidewith hydrodynamic (cold plasma) theory, corresponding to theclassical skin effect. In the collisionless limit whenthe real part of the surface impedance disappears in theclassical (cold plasma) skin effect theory (there is no heating inthe high frequency limit), but remains a constant, independentof the collision frequency, in the kinetic theory of the skineffect acounting for the electron thermal motion.

Typical behaviors for the anomalous skin depth, the surfaceimpedance, the effective stochastic collision frequency and thenormalized RF power dissipation in the skin layer are shownin Figs. 20–23, respectively, as calculated by Vahediet al.[87]. In these figures the calculated parameters are given asfunctions of the normalized RF frequency for

where is the nonlocality parameterfor The comparisons with self-consistent results


Fig. 19. The real and imaginary parts of the surface impedanceZs fora plasma slab, shown as functions of�m=!: The plasma parameters arene = 1011 cm�3;TV = 5 V, andL = 4 cm. The dashed lines are the coldplasma results (after [91]).

Fig. 20. The real (solid line) and imaginary (dashed line) parts ofthe normalized surface impedanceZs=Z0 versus the normalized fre-quency w = (!c)=(ve!pe) � ��1=2 for a near-collisionless case of�m=! = 0:008;Z0 = (ve=�c)(�0=�0)1=2 (after [87]).

for the normalized RF power absorption (Fig. 23) and withthe formula for the anomalous skin depth (Fig. 22 forshow good agreement and justify the simplifications assumedby Vahediet al. [87] for calculating the absorbed power.

Both collisional and collisionless results are modified whenthe system length becomes comparable to the skin depthThis was first noted by Blevinet al. [88], [89] and recently byTurner [90], who obtained from his finite length simulation areduced collisionless heating, as compared to the semi-infiniteslab result (4.27). This effect has been treated analytically byKondratenko [85], Shaing and Aydemir [92], and Kolobov andEconomou [91]. With rare electron collisions in the plasmaslab, the surface impedance and electron heating can beaffected by the bounce resonance electrons for which

Fig. 21. The normalized stochastic frequency�stoc=! versus the normalizedfrequencyw = (!c)=(ve!pe) for a near-collisionless case of�m=! = 0:008(after [87]).

Fig. 22. The normalized effective skin depth�=�p versus the normalized fre-quencyw = (!c)=(ve!pe) for a near-collisionless case of�m=! = 0:008:The dashed line shows�a=�p obtained from (4.10) for comparison (after [87]).

Fig. 23. Normalized input power versus normalized frequencyw = (!c)=(ve!pe): The dashed line shows the result obtained fromthe (non self-consistent) Fermi model for a near-collisionless case of�m=! = 0:008; line), and the solid line shows the result of the nonlocaltheory of Weibel [78] (Reuter and Sondheimer [80] theory using a Maxwellianelectron distribution) and after [87].


where and is the electron velocity componentalong .

One might think that the stochastically heated electronsabsorb energy along the direction of the wave electric field, i.e.,along the -direction in the slab model. That this is not the casecan be seen by considering the canonical angular momentum

where is obtained from thevector potential as Because the Hamiltonianfor the motion is independent of is conserved. Sincevanishes a few skin depths into the plasma, it follows that

within the plasma is a constant of the motion. Hence, thetransverse acceleration of the electron alongwithin the skinlayer is converted by the wave magnetic field into a longitu-dinal acceleration along as the electron exits the layer [93],[94]. Therefore the stochastic heating is along the directionof motion, a classical Fermi acceleration model. Because themagnetic field does no work on the electron, the kick in energyis the same; only the direction of the kick is altered.

The wave magnetic field can have other important effects,especially at low frequency and pressure when the electrongyration frequency can exceed the driving and collisionfrequencies. In this case one observes enhanced penetration ofthe wave into the plasma [95]. This effect has been modeledin terms of an effective conductivity in the presence of theRF magnetic field that is time-averaged over an RF period.This leads to an effective collision frequencywhich, when inserted into (4.8), yields an effective magneticskin depth

(4.28)

Other works that treat modifications of the skin depth andthe field profiles in terms of ponderomotive forces have beengiven by Helmer and Feinstein [96], DiPesoet al. [97],and Cohen and Rognlien [93], [94]. The role of this effectbecomes important for low frequency inductive dischargeswhere maintaining the discharge RF field requires a largevalue of because const.

Recently, extensive modeling of low pressure inductivedischarges [98]–[100], [91], [101] has demonstrated the impor-tance of nonlocal electrodynamics and electron heating effects.Turner [99] has showed that the introduction of an appropriateviscosity into fluid equations can adequately account forsome kinetic effects including the anomalous skin effect andelectron stochastic heating. An analytical solution for a one-dimensional (1-D) plasma corresponding to a planar coilexcited inductive RF discharge with finite length and atarbitrary collisionality has been obtained by Kondratenko [85]and recently by Shaing and Aydemir [92] and by Yoonetal. [100]. As shown in Fig. 24, the real part of the surfaceimpedance exhibits a slight maximum at a certain chamberlength. This maximum is a result of the finite size coupledwith the resonance condition for bouncing electrons within theplasma slab As is shown by Kolobov[102], these resonance conditions do not result in a pronouncedresonance behavior for the discharge characteristics due tothe wide ranges of electron velocity angles and magnitudesthat are present in a warm (Maxwellian) distribution. The

(a)

(b)

Fig. 24. Real (a) and imaginary (b) parts of the surface impedanceZs versusgap sizeL for various electron densities, for a collisionless plasma(�m = 0)with Te = 5 V (after [100]).

finite plasma dimension is also important for electron heatingwhen [22]. In the absence of collisions there is noheating if the RF magnetic field influence on the electronmotion is neglected. This situation corresponds to a constantbounce frequency, for which there is no intrinsically dynamicalphase randomization during the motion. Thus there is norandomization without electron-atom collisions. On the otherhand, resonance electrons can effectively gain energy due torare collisions during multiple bouncing, since the average


electron lifetime is much greater than the bounce time

Kolobov et al. [101] have modeled a 1-D cylindrical RFdischarge excited by an azimuthal induction electric fieldproduced by an infinitely long cylindrical coil. The electronheating was treated in terms of the energy diffusion coeffi-cient as in (3.24). Sharp peaks of were foundfor resonance electrons in a weakly collisional regime, butaveraging over angles in velocity space diminishes theresonance behavior. A similar effect may be expected for aninductive discharge excited by a planar coil. The EEDF inthis work was found from the linearized Boltzmann equationand from a Monte Carlo simulation taking into account theRF magnetic field and finite size effects over a large rangeof gas pressure and RF frequency. Yoonet al. [103] haverecently developed a two-dimensional (2-D) nonlocal heatingtheory for planar inductively coupled discharges and obtainedthe effective plasma resistance and the ponderomotive forcepotential.

D. Experiments

Demirkhanov et al. [83] were the first to observe theanomalous skin effect in a plasma. Measuring the RF magneticfield distribution in a toroidal inductive RF discharge with amagnetic probe, they found an anomalous penetration of theRF field into the plasma at some combination of the externaldischarge parameters (RF power, gas pressure, and frequency),typical for the anomalous skin effect. They also found aminimum in the RF magnetic field between the dischargeaxis and the wall, similar to that shown in Fig. 18, whichthey explained as resulting from RF current dispersion dueto electron thermal motion. According to the classical skineffect, the minimum should be at the discharge axis. Thisexperimental work inspired extensive theoretical studies ofthe anomalous skin effect using kinetic models during thefollowing decade. Although the anomalous RF field pene-tration and noncollisional electron heating are coupled, earlyexperimental works focused on the field penetration rather thanpower absorption.

Recently, the wide application of ICP’s and TCP’s inplasma processing and in lighting technology have revitalizedinterest in ICP physics and, particularly, in the anomalousskin effect and noncollisional electron heating. Noncollisionalheating is now recognized as being essential to ICP operationin the millitorr pressure range, but quantitative experimentalevidence that this process exists has not been easy to obtain.The main problem is to distinguish (separate) the collisionaland collisionless parts of the measured RF power absorbedby the plasma, which are parametrized by the correspondingcollision frequencies and To identify collisionlesseffects by comparing skin depths measured in experimentswith those calculated for the classical collisional or high fre-quency skin effect, or the anomalous skin effect, is practicallyimpossible since all these skin depths are nearly the samefor the usual experimental regimes. Also, plasma and RFfield inhomogeneities, together with a non-Maxwellian EEDF,make it difficult to compare the experimental observations withthe theories, which assume a homogeneous plasma having a

Maxwellian EEDF and a nonvariable . Moreover, all theo-ries describe a 1-D (flat) skin effect, whereas in experimentsthe plasma and the configuration of the RF fields are at leasttwo dimensional.

In a typical ICP experiment in a short metal cylindricalchamber with a dielectric window and a planar coil, theaxial distribution of the RF field even without plasma isvery inhomogeneous. Therefore, the RF field decay duringits propagation into the plasma is a combined effect of thechamber geometry and the plasma screening due to the skineffect. As was mentioned in Godyaket al. [104], collisionlessheating in ICP’s might occur even without the skin effectoriginated by the plasma conductivity, due to the vacuum RFfield inhomogeneity alone, which always exists in an inductivedischarge. This situation corresponds to an ICP with a lowplasma density (power) where the skin depthis greater thanthe characteristic length of the vacuum field inhomogeneity.

For the classical skin effect in a uniform plasma ex-cited by a planar coil in a metallic cylindrical chamberof radius for the lowest order radial mode, the electricfield profile next to the window has the form

where is the first-order Besselfunction and the plasma skin depth is (see, for example, [87])

(4.29)

where

(4.30)

and For the anomalous skin effect, canapproximately be replaced by an effective collision frequency

as in (4.10). It follows from (4.29) that for(low plasma density), and for (high plasmadensity), is the classical skin depth given by (4.6).

A method has been developed to evaluate experimentally theeffective electron collision frequency accounting for bothcollisional and stochastic heating, by considering the primaryinduction coil as a transformer with the plasma being thesecondary winding [105]. This method is based on calculationof the plasma -factor from the measuredchanges in the primary induction coil impedance induced bythe plasma load [104]

(4.31)

where and are, respectively, the change in the coilreactance and resistance, is the unloaded coilreactance, and is the coupling coefficient betweenthe coil and the plasma. There are a number of ways tomeasure However, in experiments the second term in(4.30) is usually small compared to the first. In this way,Godyak et al. [104] obtained s in anargon RF discharge operating at 10 mtorr and 13.56 MHzin the RF power range between 20–150 W. This value ofwas found to be much larger than calculated as

Consequently, Godyaket al. [104] stated that in


TABLE ICOMPARISON OF COLLISIONAL AND COLLISIONLESS HEATING FREQUENCIES

(a) (b)

Fig. 25. Absorbed RF power flux versus distance for (a)p = 0:3 mtorr and (b)p = 1 mtorr (after [106]).

their experiment a collisionless mechanism was dominating theelectron heating. Later, a more accurate calculation offorconditions close to this experiment based on integration of themeasured EEDF showed that the measuredwas actuallyvery close to [106]. Also, Kortshagen [107] modeled thisexperiment accounting only for collisional heating, and hisresults were in reasonable agreement with the experiment.

An experimental study of a low pressure inductive dischargeover a wide range of argon gas pressure, RF power andfrequency has been recently performed at OSRAM SYLVA-NIA and reported by Godyak [106]. The EEDF and RF fieldand current density distribution were measured in a near-collisionless regime in a metal chamber with a planar ICPexcitation coil using Langmuir and magnetic probes. Thechamber diameter was 20 cm and the length was 10.5 cm.From the magnitudes and phases of the measured componentsof RF magnetic field and within the skin layer and theplasma, the RF electric field and current density weredetermined from Maxwell’s equations. The absolute values ofthe power density absorbed and the effective electron collisionfrequency were then directly found according to the relations

where is the measured phase shift betweenand andis the local plasma density measured with a Langmuir

probe. In Table I are shown results of calculations for 6.78MHz based on measurements within the skin layer at the max-imum of the RF current density distribution (approximately1 cm from the glass window) at a radial position of 4 cm,corresponding to maximum of the RF electric field radial

distribution. As is seen in Table I, at mtorr,such that collisionless electron heating dominates. Atmtorr, and power absorption in the skin layer ispredominantly collisional. It is interesting to compare valuesof found experimentally at 1 mtorr wherewith a theoretical expression for the collisionless (stochastic)frequency given by Vahediet al. [87] fora 1-D model with an exponential RF profile. The valueswere determined as the distance from the glass window wherethe measured RF electric field decays by one e-folding. Thecalculated values of given in Table I appear to be closeto the measured values of

The RF power absorption density integrated along thedirection of electromagnetic field propagation,

is shown in Fig. 25 as a function of thedistance from the window. Here the measured total absorbedpower flux is compared with that calculated from themeasured and distributions assuming collisional powerabsorption

Re

and

Re

with given in (4.1). The comparison shows that the colli-sionless process dominates the RF power absorption in theseparticular cases.

The spatial distribution of the absorbed RF power densityis shown in Fig. 26. The power density was calculated as


(a) (b)

Fig. 26. Spatial distribution of absorbed RF power density for an argon discharge driven at 6.78 MHz; (a) 1 mtorr pressure, with absorbed plasma powervarying from 25 to 200 W, and (b) 50 W absorbed plasma power, with pressure varying from 0.3 to 100 mtorr (after [106]).

As expected, practically all power is dissipatedwithin the skin layer near the window. However, far fromthe window the power absorption for mtorr oscillateswith distance and becomes negative in certain regions. Thenegative power absorption occurs where the phase of theelectron current (transfered from the skin layer due to electronthermal motion) differs from the local RF field phase by morethan 90. Here, the electrons arriving from the skin layertransfer the energy they acquired from within the skin layerback to the RF field.

For the normal skin effect, the RF current is defined by theproduct of the RF electric field and the local value of the coldplasma conductivity, such that is always positive.For the anomalous skin effect the current far away from theskin layer is that which is translated from the skin layer by theelectron thermal motion, and its phase is defined by the transittime while the electric field phase is defined by adelay due to the phase velocity, which depends on frequencyand plasma density, The different mechanismsof phase delay for current and electric field can result invarious phase combinations including those corresponding toa negative power absorption. Apparently, the negative powerabsorption can exist only in the collisionless regimeotherwise, electron-atom collisions destroy the translationalmotion of the current, yielding a local coupling between RFcurrent and electric field, as shown in the high pressure casesin Fig. 26(b). The RF power absorption along the plasma isshown in Fig. 27 at various frequencies for mtorr(where and a discharge power of 100 W. A transittime effect is seen for the appearance of the first negativepower absorption region. The distancebetween the middleof the skin layer and the first zero crossing of the power isinversely proportional to the RF frequency, . Note thatapparently there is no negative power absorption at the lowestfrequency of 3.39 MHz, where the anomalous skin effect (large

) and the electron ballistic phenomena are expected to begreatest. This has a simple explanation: for 3.39 MHz, thechamber length is too small.

An analytical calculation of RF power absorption for pa-rameters of the Godyak [106] experiment has been performed

by Kolobov [102] within the framework of the existing theoryof the anomalous skin effect for a homogeneous plasma [85],modified to account for the inhomogeneity of the vacuum RFfield, as in (4.29). The result shown in Fig. 27(b) demonstratesa good agreement with the experiment in Fig. 27(a).

V. ELECTRON CYCLOTRON RESONANCE DISCHARGES

Waves generated near a plasma surface can propagate intothe plasma or along the surface where they can be subsequentlyabsorbed, leading to heating of plasma electrons and excitationof a discharge [21]. The classical example is an electron cy-clotron resonance (ECR) discharge, in which a right circularlypolarized electromagnetic wave propagates along dc magneticfield lines to a resonance zone, where the wave energy isabsorbed by collisionless heating. This is a type of stochasticheating in which electrons receive “kicks” in energy at eachpassage through the resonance zone, i.e., it is a type of Fermiacceleration.

ECR discharges are generally excited at microwave fre-quencies (e.g., 2450 MHz), and the wave absorption requiresapplication of a strong dc magnetic field (875 G at reso-nance). The power is usually coupled through a dielectricend-window into a cylindrical metal source chamber. One orseveral concentric magnetic field coils are used to generatea nonuniform, axial magnetic field within the chamberto achieve the ECR condition, where isthe axial resonance position. When a low pressure gas isintroduced, the gas breaks down and the discharge forms insidethe chamber. For materials processing applications, the plasmadiffuses along the magnetic field lines into a process chambertoward a wafer holder. The source diameter is 15–30 cm, andthe microwave power is 1–5 kW.

Two separated magnet coils generate amagnetic mirrorfield configuration having a high field underneath the twocoils and a weaker field in the midplane between the coils.Electrons can be axially trapped between the high field regionsby the axial magnetic field gradients and repeatedly bouncebetween the mirror coils (e.g., see [21]). By proper choiceof the field strength and profile, there can be two resonancezones symmetrically located with respect to the midplane.


(a) (b)

Fig. 27. Effect of frequency variation on the spatial distribution of the RF power density absorbed by the plasma: (a) measured distribution for 10 mtorrand 100 W absorbed plasma power and (b) distribution calculated from a 1-D model by Kolobov (after [106]).

This configuration can yield high ionization efficiencies, dueto enhanced confinement of hot (superthermal) electrons thatare magnetically trapped between the two mirror (high-field)positions.

Because the gas pressure in these discharges can be as lowas 0.1–0.01 mtorr and the field strengths can be large (largekicks), phase randomization due to nonlinear dynamical effectscan be very important. Dynamical effects such as the influenceof phase correlations in slowing the quasilinear heating rate,and the existance of adiabatic barriers to heating, have beenobserved experimentally in these discharges.

To determine the collisionless heating power from a Fermiacceleration model, the nonuniformity in the magnetic fieldprofile must be considered. For an electrondoes not continuously gain energy, but rather its energyoscillates at the difference frequency As an electronmoving along passes through resonance, its energy oscillatesas shown in Fig. 28, leading to a transverse energy gained(or lost) in one pass. For low power absorption, where theelectric field at the resonance zone is known, the heating canbe determined as follows. We expand the magnetic field nearresonance as

(5.1)

where is the distance from exact resonance,is proportional to the gradient in near the

Fig. 28. Energy change in one pass through an ECR resonance zone (after[21]).

resonant zone, and we approximate whereis the parallel speed at resonance.

The complex force equation for the right hand componentof the transverse velocity, can be written inthe form

(5.2)

where is the amplitude of the RHP wave with

Re (5.3)


and and unit vectors along the and directions. Using(5.1) and substituting into (5.2), we obtain

(5.4)

Multiplying by the integrating factor and integrating(5.4) from to we obtain

(5.5)

where

(5.6)

In the limit the integral in (5.5) is theintegral of a Gaussian of complex argument, which has thestandard form

(5.7)

Substituting (5.7) into (5.5), multiplying (5.5) by its complexconjugate, and averaging over the initial “random” phase

we obtain

(5.8)

The average energy gain per pass is thus

(5.9)

This can also be written as

(5.10)

where and

(5.11)

is the effective time in resonance. The effective resonancezone width (see Fig. 28) is

(5.12)

which, for typical ECR parameters, gives cm.The absorbed power per unit area is found by integrating

(5.9) over the flux of electrons incident on the zone,yielding

(5.13)

We can understand the form of as follows: an electronpassing through the zone coherently gains energy for a time

such that

(5.14)

Inserting (5.1) into (5.14) and solving for we obtain(5.11). A more careful derivation of the absorbed power,

including the effect of nonconstant during passage throughresonance, is presented by Jaegeret al. [108], giving similarresults.

At high power absorption, the electric field is not knownbut must be determined self-consistently with the energyabsorption, in the same manner as for inductive discharges.The propagation and absorption of microwave power in ECRsources is an active area of research and is not fully un-derstood. However, the essence of the wave coupling, andtransformation and absorption at the resonance zone, can beseen by considering the one dimensional problem of a righ-hand polarized wave propagating strictly along the magneticfield in a plasma that varies only along the axial direction

This problem was originally studied in connection withwave propagation in the ionosphere, and the solution wasobtained analytically by Budden [109] for the approximationof constant density and linear magnetic field variation. For awave traveling into a decreasing magnetic field, he obtainedthe solution

(5.15)

(5.16)

(5.17)

where Hence, some of the incident wavepower is absorbed at the resonance while some tunnels throughto the other side, but no power is reflected. Taking a typicalcase for which cm and cm we findthat corresponds to Thus at 2450 MHzwe expect most of the incident power will be absorbed for adensity cm

For high electric field strengths and low pressures, theenergy gain per passage through resonance is large, and thenonlinear dynamical aspects of the problem come into play.For electrons heated to high energies by repeated interactionwith the resonance zones, strong phase correlations can reducethe heating rate below that obtained from a random phaseinteraction, and an adiabatic barrier to heating can exist. Suchphenomena were observed experimentally in a high field (50kG) magnetic mirror compression experiment [110], in whicha short pulse (0.25 s) of high power (maximum 250 kW)microwaves was used to heat plasma electrons early duringthe compression, which were subsequently further heated bythe increasing magnetic field. With increasing microwavepower (field strength), a transition from a high to a lowerheating rate was found that agreed well with analytical andnumerical mapping estimates of the expected transition froma regime of random phase interaction to a regime of highphase correlations.

Another experiment was performed by Shoyamaet al. [111]in a magnetic mirror field configuration, ECR discharge at lowpressures (3– torr) using up to 4 kW of 2.45 GHzcontinuous wave power. A steady state plasma having a typicalelectron temperature of 6 V and density of cm wasformed, having a high energy electron tail of order 10 kV asdetermined by X-ray bremsstrahlung measurements. From thetheory of Fermi acceleration and as confirmed by numericaliteration of the appropriate Fermi mapping for this system,


the maximum possible energy of the heated electronswas found to scale with the input microwave power as

due to the existance of an adiabatic barrierto heating. The experimental data confirmed this scaling.

VI. CONCLUDING DISCUSSION

Collisionless (stochastic) heating of electrons by time-varying fields has been shown to be fundamental to theoperation of radio frequency (RF) and microwave discharges.Such heating is due to spatial variation of the fields, whichlead to randomization of the electron phase during its thermalmotion, even in the absence of collisions. Generally, electronsare heated collisionlessly by repeated interaction with fieldsthat are localized within a sheath or skin depth layer insidethe discharge. Consequently, the Fermi acceleration model ofa ball bouncing elastically back and forth between a fixedand an oscillating wall is a paradigm to describe collisionlessheating and phase randomization in capacitive, inductive, andECR discharges. We introduced several mapping models forFermi acceleration and showed how to use the Fokker–Planckformalism to determine the heating rate and the effects of phasecorrelations. We reviewed the role of collisionless heatingin capacitive and inductive discharges, using simple Fermimodels to determine the heating rates and comparing thesewith self-consistent (kinetic) calculations where available.We reviewed experimental measurements and computersimulations and compared these to theoretical calculationsof the heating. We described recent measurements andcalculations of nonlocal heating effects, such as negativeelectron power absorption. The effects of partial phaserandomization in reducing the heating rates were most clearlyseen in low pressure ECR discharges. We described the useof Fermi acceleration models to determine the collisionlessheating rates for these discharges, and showed that incompletephase randomization could reduce the heating rate and lead tothe existence of adiabatic barriers to heating.

A number of outstanding issues remain that either are notwell understood theoretically or lack experimental validation.Let us first mention that there is little understanding of therole of the RF (or microwave) magnetic field in collisionlessheating phenomena. The magnetic field can alter the skin depthfor the field decay and also the electron dynamics that leadsto phase randomization. In a cylindrical inductive dischargeexcited by a planar coil at one end, for example, the kick invelocity within the skin layer is in the direction, but thisis converted by the RF magnetic field into a kick along thedirection within the bulk plasma. This suggests the existence of

and components of the electron current densitywithin thebulk plasma, and these may in turn induce electric fields withthese same components. Hence the usual assumption that onlyazimuthal RF electric fields and currents exist in cylindricalinductive discharges may not be valid. Both theoretical modelsand experimental results obtained using RF magnetic probesmay need to be modified to account for these effects.

In RF capacitive discharges, PIC simulations suggest theexistence of a small negative electron power absorption inthe bulk of the discharge. There is no definitive experimental

confirmation of this effect. On the other hand, both theory andexperimental observations of RF inductive discharges showthat such negative power absorption regions exist within thedischarge bulk, but there are no PIC simulations demonstratingthis effect.

There is scanty experimental or PIC simulation evidencethat incomplete randomization of the electron phase plays asignificant role in determining the collisionless heating ratesin capacitive or inductive RF discharges. There is considerableevidence that these effects can be important in very lowpressure ECR discharges.

The Fermi acceleration model has been shown to be aneffective tool for describing collisionless heating in gas dis-charge plasmas having strong RF fields on the boundary, forwhich the traditional quasilinear approach in plasma elec-trodynamics may some times be questionable. Amazingly,collisionless electron heating, which is usually associated withhigh temperature space and fusion plasmas, appears to be afundamental process in the warm plasmas of low pressuredischarges that are used in today’s technology.

ACKNOWLEDGMENT

The authors gratefully acknowledge helpful discussions withA. J. Lichtenberg, I. D. Kaganovich, and V. I. Kolobov. Theircomments and suggested revisions have greatly improved thispaper.

REFERENCES

[1] L. D. Landau and E. M. Lifshitz,Electrodynamics of Continuous Media.Oxford, U.K.: Pergamon, 1960.

[2] L. D. Landau, “Oscillations of an electron plasma,”J. Phys. (USSR),vol. 10, p. 25, 1946.

[3] E. Fermi, “On the origin of the cosmic radiation,”Phys. Rev., vol. 75,p. 1169, 1949.

[4] M. Harwit, Cosmic Discovery—The Search, Scope, and Heritage ofAstronomy. Cambridge, MA: The M.I.T. Press, 1984.

[5] M. W. Friedlander,Cosmic Rays. Cambridge, MA: Harvard UniversityPress, 1989.

[6] C. J. Cesarsky, “Cosmic-ray confinement in the galaxy,”Ann. Rev.Astron. Astrophys., vol. 18, p. 289, 1980.

[7] A. J. Lichtenberg and M. A. Lieberman,Regular and Chaotic Dynamics.New York: Springer-Verlag, 1992.

[8] S. M. Ulam, “On some statistical properties of dynamical systems,”in Proc. 4th Berkeley Symp. Mathematical Statistics Probability, Univ.California, vol. 3, p. 315, 1961.

[9] G. M. Zaslavskii and B. V. Chirikov, “Mechanism for Fermi accelerationin a one-dimensional case,”Soviet Phys. Dokl., vol. 9, p. 989, 1965.

[10] A. Brahic, “Numerical study of a simple dynamical system,”Astron.Astrophys., vol. 12, p. 98, 1971.

[11] M. A. Lieberman and A. J. Lichtenberg, “Stochastic and adiabaticbehavior of particles accelerated by periodic forces,”Phys. Rev. A, vol.5, p. 1852, 1972.

[12] A. J. Lichtenberg, M. A. Lieberman, and R. H. Cohen, “Fermi acceler-ation revisited,”Physica, vol. 1D, p. 291, 1980.

[13] H. Goldstein,Classical Mechanics. Reading, MA: Addison-Wesley,1951.

[14] M. C. Wang and Uhlenbeck, G. E., “On the theory of the Brownianmotion II,” Revs. Mod. Phys., vol. 17, p. 323, 1945.

[15] L. D. Landau, “Kinetic equation for the Coulomb interaction,”Zh.Eksper. Theor. Fiz., vol. 7, p. 203, 1937.

[16] C. G. Goedde, A. J. Lichtenberg, and M. A. Lieberman, “Self-consistentstochastic electron heating in radio frequency discharges,”J. Appl. Phys.,vol. 64, p. 4375, 1988.

[17] A. B. Rechester and R. B. White, “Calculation of turbulent diffusion forthe Chirikov-Taylor model,”Phys. Rev. Lett., vol. 44, p. 1586, 1980.

[18] A. B. Rechester, M. N. Rosenbluth, and R. B. White, “Fourier-spacepaths applied to the calculation of diffusion for the Chirikov-Taylormodel,” Phys. Rev. A, vol. 23, p. 2664, 1981.


[19] N. W. Murray, M. A. Lieberman, and A. J. Lichtenberg, “Correctionsto quasilinear diffusion in area-preserving maps,”Phys. Rev. A, vol. 32,p. 2413, 1985.

[20] A. J. Lichtenberg, M. A. Lieberman, and N. W. Murray, “The effectof quasiaccelerator modes on diffusion,”Physica, vol. 28D, p. 371,1987.

[21] M. A. Lieberman and A. J. Lichtenberg,Principles of Plasma Dischargesand Materials Processing. New York: Wiley, 1994.

[22] I. D. Kaganovich, V. I. Kolobov, and L. D. Tsendin, “Stochastic electronheating in bounded radio-frequency plasmas,”Appl. Phys. Lett., vol. 69,p. 3818, 1996.

[23] I. D. Kaganovich, “On the role of collisions and non linear effects inso-called collisionless electron heating,” inElectron Kinetics in GlowDischarges, U. Korthagen and L. D. Tsendin, Eds. New York: Plenum,1998.

[24] F. Schneider, “Zum Mechanisms der Hochfrequenzentladung zwischenebenen Platten,”A. Angew. Phys., vol. 6, p. 456, 1954.

[25] D. Gabor, E. A. Ash, and D. Dracott, “Langmuir’s paradox,”Nature,vol. 176, p. 916, 1955.

[26] J. Pavkovich and G. S. Kino, “R.F. theory of the plasma sheath,” inProc. VI-th Conf. Ionization Phenomena Gases, 1963, vol. 3, p. 39.

[27] R. W. Gould, “Radio frequency characteristics of the plasma sheath,”Phys. Lett., vol. 11, p. 236, 1964.

[28] V. A. Godyak, “The statistical heating of electrons by oscillatingboundaries of the plasma,”Zhurnal Teknicheskoi Fiziki, vol. 41, p. 1364,1971; see also,Sov. Phys.—Tech. Phys., vol. 16, p. 1073, 1972.

[29] , “Stationary low-pressure high-frequency discharge,”Fiz.Plasmy, vol. 2, p. 141, 1976; see alsoSov. J. Plasma Phys., vol. 2,p. 79, 1976.

[30] V. A. Godyak, O. A. Popov, and A. H. Khanna, “Effective collisionfrequency of the electrons in RF discharge,”Fiz. Plasmy, vol. 2, p.1010, 1976; see alsoSov. J. Plasma Phys., vol. 2, p. 560, 1976.

[31] A. I. Akhiezer and A. S. Bakai, “On stochastic plasma heating by high-frequency fields,’Fiz. Plasmy, vol. 2, p. 654, 1976; see alsoSov. J.Plasma Phys., vol. 2, p. 359, 1976.

[32] M. A. Lieberman, “Analytical solution for capacitive RF sheath,”IEEETrans. Plasma Sci., vol. 16, p. 638, 1988.

[33] A. S. Smirnov and L. D. Tsendin, “The space-time-averaging procedureand modeling of the RF discharge,”IEEE Trans. Plasma Sci., vol. 19,p. 130, 1991.

[34] M. Surendra and M. Dalvie, “Moment analysis of RF parallel-plate-discharge simulations using the particle-in-cell with Monte Carlo colli-sions technique,”Phys. Rev. E, vol. 48, p. 3914, 1993.

[35] B. P. Wood, “Sheath heating in low pressure capacitive radio frequencydischarges,” Ph.D. dissertation, Univ. California, Berkeley, 1991.

[36] A. E. Wendt and W. N. G. Hitchon, “Electron heating by sheaths inradio frequency discharges,”J. Appl. Phys., vol. 71, p. 4718, 1992.

[37] M. Surendra and D. Vender, “Collisionless electron heating by radio-frequency plasma sheaths,”Appl. Phys. Lett., vol. 65, p. 153, 1994.

[38] B. P. Wood, M. A. Lieberman, and A. J. Lichtenberg, “Stochasticelectron heating in a capacitive RF discharge with non-Maxwellian andtime-varying distributions,”IEEE Trans. Plasma Sci., vol. 23, p. 89,1995.

[39] I. D. Kaganovich and L. D. Tsendin, “Low-pressure RF discharge in thefree-flight regime,”IEEE Trans. Plasma Sci., vol. 20, p. 86, 1992.

[40] M. A. Lieberman, “Dynamics of a collisional, capacitive RF sheath,”IEEE Trans. Plasma Sci., vol. 17, p. 338, 1989.

[41] G. R. Misium, A. J. Lichtenberg, and M. A. Lieberman, “Macroscopicmodeling of radio-frequency plasma discharges,”J. Vac. Sci. Technol.,vol. 7, p. 1007, 1989.

[42] V. A. Godyak and O. A. Popov, “Experimental investigation of resonantRF discharge,”Fiz. Plasmy, vol. 5, p. 400, 1979; see alsoSov. J. PlasmaPhys., vol. 5, p. 227, 1979.

[43] O. A. Popov and V. A. Godyak, “Power dissipated in low-pressureradi-frequency discharge plasmas,”J. Appl. Phys., vol. 57, p. 53, 1985.

[44] V. A. Godyak, Soviet Radio Frequency Discharge Research. FallsChurch, VA: Delphic Associates, 1986.

[45] O. A. Popov and V. A. Godyak, “Electron oscillation velocity andelectric field in collisionless radio-frequency discharge plasmas,”J.Appl. Phys., vol. 59, p. 1759, 1986.

[46] V. A. Godyak and R. B. Piejak, “Abnormally low electron energy andheating-mode transition in a low-pressure argon RF discharge at 13.56MHz,” Phys. Rev. Lett., vol. 65, p. 996, 1990.

[47] V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, “Electricalcharacteristics of parallel-plate RF discharges in argon,”IEEE Trans.Plasma Sci., vol. 19, p. 660, 1991.

[48] , “Measurement of electron enegy distribution in low-pressure RFdischarges,”Plasma Sources Sci. Technol., vol. 1, p. 36, 1992.

[49] V. A. Godyak, private communication (result reproduced in [21]), 1990.

[50] U. Buddemeier, U. Kortshagen, and I. Pukropski, “On the efficiencyof the electron sheath heating in capacitively coupled radio frequencydischarges in the weakly collisional regime,”Appl. Phys. Lett., vol. 67,p. 191, 1995.

[51] M. J. Kushner, “Mechanism for power deposition in Ar/SiH4 capac-itively coupled RF discharges,”IEEE Trans. Plasma. Sci., vol. 14, p.188, 1986.

[52] M. Surendra, D. B. Graves, and I. J. Morey, “Electron heating inlow-pressure RF glow discharges,”Appl. Phys. Lett., vol. 56, p. 1022,1990.

[53] M. Surendra and D. B. Graves, “Particle simulations of radio-frequencyglow discharges,”IEEE Trans. Plasma Sci., vol. 19, p. 144, 1991.

[54] D. Vender and R. W. Boswell, “Electron-sheath interaction in capacitiveradiofrequency plasmas,”J. Vac. Sci. Technol. A, vol. 10, p. 1331, 1992.

[55] T. E. Nitschke and D. B. Graves, “A comparison of particle in cell andfluid model simulations of low-pressur radio frequency discharges,”J.Appl. Phys., vol. 76, p. 5646, 1994.

[56] V. Vahedi, C. K. Birdsall, M. A. Lieberman, G. DiPeso, and T. D.Rognlien, “Capacitive RF discharges modeled by particle-in-cell MonteCarlo simulation. II. Comparisons with laboratory measurements ofelectron energy distribution function,”Plasma Sources Sci. Technol.,vol. 2, p. 273, 1993.

[57] D. Vender and R. W. Boswell, “Numerical modeling of low-pressureRF plasmas,”IEEE Trans. Plasma Sci., vol. 18, p. 725, 1990.

[58] D. Vender, “Numerical studies of the low pressure RF plasmas,” Ph.D.dissertation, The Australian National University, Canberra, 1990.

[59] M. Surendra and D. B. Graves, “Electron acoustic waves in capacitycoupled, low-pressure RF glow discharges,”Phys. Rev. Lett., vol. 66,p. 1469, 1991.

[60] A. H. Sato and M. A. Lieberman, “Electron-beam probe measurementsof electric fields in RF discharges,”J. Appl. Phys., vol. 68, p. 6117,1990.

[61] M. A. Lieberman, A. J. Lichtenberg, and S. E. Savas, “Model ofmagnetically enhanced, capacitive RF discharges,”IEEE Trans. PlasmaSci., vol. 19, p. 189, 1991.

[62] Y. Okuno, Y. Ohtsu, and H. Fujita, “Electron acceleration resonant withsheath motion in a low-pressure radio frequency discharge,”Appl. Phys.Lett., vol. 64, p. 1623, 1994.

[63] D. A. W. Hutchinson, M. M. Turner, R. A. Doyle, and M. B. Hopkins,“The effects of a small transverse magnetic field upon a capacitivelycoupled RF discharge,”IEEE Trans. Plasma Sci., vol. 23, p. 636, 1995.

[64] M. M. Turner, D. A. W. Hutchinson, R. A. Doyle, and M. B. Hopkins,“Heating mode transition induced by a magnetic field in a capacitive rfdischarge,”Phys. Rev. Lett., vol. 76, p. 2069, 1996.

[65] J.-C. Park and B. Kang, “Reactor modeling of magnetically enhancedcapacitive RF discharge,”IEEE Trans. Plasma Sci., vol. 25, p. 499, 1997.

[66] M. M. Turner, “Pressure heating of electrons in capacitively coupled RFdischarges,”Phys. Rev. Lett., vol. 75, p. 1312, 1995.

[67] I. B. Bernstein and T. Holstein, “Electron energy distributions instationary discharges,”Phys. Rev., vol. 94, p. 1475, 1954.

[68] L. D. Tsendin, “Energy distribution of electrons in a weakly ionizedcurrent-carrying plasma with transverse inhomogeneity,”Sov. Phys.JETP, vol. 39, p. 805, 1975.

[69] , “Electron kinetics in nonuniform glow discharge plasmas,”Plasma Sources Sci. Technol., vol. 4, p. 200, 1995.

[70] V. I. Kolobov and V. A. Godyak, “Nonlocal electron kinetics incollisional gas discharge plasmas,”IEEE Trans. Plasma Sci., vol. 23,p. 503, 1995.

[71] U. Kortshagen, C. Busch, and L. D. Tsendin, “On simplifying ap-proaches to the solution of the Boltzmann equation in spatially in-homogeneous plasmas,”Plasma Sources Sci. Technol., vol. 5, p. 1,1996.

[72] Z. D. Wang, A. J. Lichtenberg, and R. H. Cohen, “Kinetic theory ofstochastically heated RF capacitive discharges,”IEEE Trans. PlasmaSci., vol. 26, p. 59, 1998.

[73] A. J. Lichtenberg, “Application of mapping dynamics and kinetic theoryto analysis of a capacitive r.f. discharge,” inElectron Kinetics in GlowDischarges, U. Korthagen and L. D. Tsendin, Eds. New York: Plenum,1998.

[74] Yu. M. Aliev, I. D. Kaganovich, and H. Schl¨ulter, “Quasilinear theory ofcollisionless electron heating in radio frequency gas discharges,”Phys.Plasmas, vol. 4, p. 2413, 1997.

[75] I. Ichimaru,Basic Principles of Plasma Physics: A Statistical Approach.New York: W. A. Benjamin, 1973.

[76] A. F. Alexandrov, L. S. Bogdankevich, and A. A. Rukhadze,Principlesof Plasma Electrodynamics. New York: Springer, 1984.

[77] G. G. Lister, Y.-M. Li, and V. A. Godyak, “Electrical conductivity inhigh-frequency plasmas,”J. Appl. Phys., vol. 79, p. 8993, 1996.


[78] E. S. Weibel, “Anomalous skin effect in a plasma,”Phys. Fluids, vol.10, p. 741, 1967.

[79] A. B. Pippard, “The high frequency skin resistance of metals at lowtemperatures,”Physica, vol. 15, p. 45, 1949.

[80] G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalousskin effect in metals,” inProc. Roy. Soc., vol. A195, p. 336, 1949.

[81] B. D. Fried and S. D. Conte,The Plasma Dispersion Function. NewYork: Academic, 1961.

[82] Yu. S. Sayasov, “An analytical theory of the anomalous skin-effect inbounded plasmas,”Helvetica Physica Acta, vol. 52, p. 288, 1979.

[83] R. A. Demirkhanov, I. Ya. Kadysh, and Yu. S. Khodyrev, “Skin effect ina high frequency ring discharge,”Sov. Phys. JETP, vol. 19, p. 791, 1964.

[84] B. Joyce and H. Schneider, “Effect of abnormal skin in an argonplasma,”Helvetica Physica Acta, vol. 51, p. 804, 1979.

[85] A. N. Kondratenko, RF Field Penetration into Plasma. Moscow,Russia: Atomizdat, 1979 (in Russian).

[86] T. Holstein, “Optical and infrared reflectivity of metals at low temper-atures,”Phys. Rev., vol. 88, p. 1427, 1952.

[87] V. Vahedi, M. A. Lieberman, G. DiPeso, T. D. Rognlien, and D. Hewett,“Analytic model of power deposition in inductively coupled plasmasources,”J. Appl. Phys., vol. 78, p. 1446, 1995.

[88] H. A. Blevin, J. A. Reynolds, and P. C. Thonemann, “Penetration ofan electromagnetic wave into a hot plasma slab,”Phys. Fluids, vol. 13,p. 1259, 1970.

[89] , “Measurement of the attenuation of an electromagnetic wave ina bounded hot electron plasma,”Phys. Fluids, vol. 16, p. 82, 1973.

[90] M. M. Turner, “Collisionless electron heating in an inductively coupleddischarge,”Phys. Rev. Lett., vol. 71, p. 1844, 1993.

[91] V. I. Kolobov and D. J. Economou, “The anomalous skin effect in gasdischarge plasmas,”Plasma Sources Sci. Technol., vol. 6, p. 1, 1997.

[92] K. C. Shaing and A. Y. Aydemir, “Collisionless electron heating ininductively coupled discharges,”Phys. Plasmas, vol. 4, p. 3163, 1997.

[93] R. H. Cohen and T. D. Rognlien, “Electron kinetics in radio-frequencymagnetic fields of inductive plasma sources,”Plasma Sources Sci.Technol., vol. 5, p. 442, 1996.

[94] , “Induced magnetic-field effects in inductively coupled plasmas,”Phys. Plasmas, vol. 3, p. 1839, 1996.

[95] M. Tuszewski, “Enhanced radio frequency field penetration in aninductively coupled plasma,”Phys. Rev. Lett., vol. 77, p. 1966, 1996.

[96] J. C. Helmer and J. Feinstein, “Ponderomotive transport of charge inthe induction plasma,”J. Vac. Sci. Technol. B, vol. 12, p. 507, 1994.

[97] G. DiPeso, T. D. Rognlien, V. Vahedi, and D. W. Hewett, “Equilibriumprofiles for RF-plasma sources with ponderomotive forces,”IEEE Trans.Plasma Sci., vol. 23, p. 550, 1995.

[98] M. R. Gibbons and D. W. Hewett, “The Darwin direct implicit particle-in-cell (DADIPIC) method for simulation of low frequency plasmaphenomena,”J. Comp. Phys., vol. 120, p. 231, 1995.

[99] M. M. Turner, “Simulation of kinetic effects in inductive discharges,”Plasma Sources Sci. Technol., vol. 5, p. 159, 1996.

[100] N. S. Yoon, S. S. Kim, C. S. Chang, and D.-I. Choi, “One-dimensionalsolution for electron heating in an inductively coupled plasma dis-charge,”Phys. Rev. E, vol. 54, p. 757, 1996.

[101] V. I. Kolobov, D. P. Lymberopoulos, and D. J. Economou, “Electronkinetics and non-Joule heating in near-collisionless inductively coupledplasmas,”Phys. Rev. E, vol. 55, p. 3408, 1997.

[102] V. I. Kolobov, “Anomalous skin effect in bounded plasmas,” inElectronKinetics in Glow Discharges, U. Korthagen and L. D. Tsendin, Eds.New York: Plenum, 1998.

[103] N. S. Yoon, S. M. Hwang, and D.-I. Choi, “Two-dimensional nonlocalheating theory of planar-type inductively coupled plasma discharge,”Phys. Rev. E, vol. 55, p. 7536, 1997.

[104] V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, “Electricalcharacteristics and electron heating mechanism of an inductively coupledargon discharge,”Plasma Sources Sci. Technol., vol. 3, p. 169, 1994.

[105] R. B. Piejak, V. A. Godyak, and B. M. Alexandrovich, “A simpleanalysis of an inductive RF discharge,”Plasma Sources Sci. Technol.,vol. 1, p. 179, 1992.

[106] V. A. Godyak, “ICP kinetics and electromagntic fields in stochasticallyheated-regime,” inElectron Kinetics in Glow Discharges, U. Korthagenand L. D. Tsendin, Eds. New York: Plenum, 1998.

[107] U. Kortschagen, private communication, 1996.[108] F. Jaeger, A. J. Lichtenberg, and M. A. Lieberman, “Theory of electron

cyclotron resonance heating I. Short time and adiabatic effects,”PlasmaPhys., vol. 14, p. 1073, 1972.

[109] K. G. Budden, Radio Waves in the Ionosphere. Cambridge, UK:Cambridge University Press, 1966.

[110] N. C. Wyeth, A. J. Lichtenberg, and M. A. Lieberman, “Electroncyclotron resonance heating in a pulsed mirror experiment,”PlasmaPhys., vol. 17, p. 679, 1975.

[111] H. Shoyama, M. Tanaka, S. Higashi, Y. Kawai, and M. Kono, “Sto-chastic electron acceleration by an electron cyclotron wave propagatingin an inhomogeneous magnetic field,”J. Phys. Soc. Japan, vol. 65, p.2860, 1996.

Michael A. Lieberman (M’84–SM’91–F’94) re-ceived the B.S. and M.S. degrees in 1962 andthe Ph.D. degree in 1966 from the MassachusettsInstitute of Technology, Cambridge.

He joined the Department of Electrical Engineer-ing and Computer Sciences at the University ofCalifornia, Berkeley, in 1966. From 1990 to 1995,he served as Director of the Electronics ResearchLaboratory at the University of California. He haspublished over 140 refereed journal articles. HismonographRegular and Stochastic Motion(New

York: Springer-Verlag, 1983) and a second edition,Regular and ChaoticDynamics(New York: Springer-Verlag, 1992). A second monograph was alsopublished,Principles of Plasma Discharges and Materials Processing(NewYork: Wiley, 1994). His research interests are plasma science and engineeringand nonlinear dynamics.

Dr. Lieberman is a Fellow of the American Physical Society and theAmerican Association for the Advancement of Science. He received theDistinguished Teaching Award from the University of California, Berkeley,in 1971. He also received the Plasma Sciences and Applications Award fromthe IEEE in 1995.

Valery A. Godyak (M’91–SM’95–F’98) was bornJune 8, 1941, in Czernowitz, Russia. He receivedthe Engineer–Physicist Diploma (an equivalent ofthe M.S. degree) in physical electronics from theLeningrad Politechnical University, Leningrad, Rus-sia, and the Ph.D. degree in plasma physics fromMoscow State University, Moscow, Russia, in 1964and 1968, respectively.

In 1964, he became an Assistant Professor ofPhysics at the Ryazan Radiotechnical University,Ryazan, Russia. In 1968, he joined the Laboratory of

Fusion Engineering Problems at the Institute of Electro-Physical Apparatus,Leningrad, where he conducted research and development on high-currentrelativistic electron accelerators, particularly on field emission, electron optics,and the diagnostics of relativistic electron beams. In 1972, he returned tothe Physics Department at Moscow State University, where he was GroupLeader involved in the basic research of RF discharges up to his expulsionfor political reasons in 1981. In 1984, he emigrated to the United Statesand joined GTE Corporation (currently, OSRAM SYLVANIA DevelopmentInc., Beverly, MA), where he holds the position of Corporate Scientist. Hisresearch concerns RF discharges, including basic research and application tolight source technology.

Dr. Godyak is a winner of “OSRAM STAR” and “SIEMENS InternationalInnovation Competition” awards. He is a Fellow of the American PhysicalSociety.

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