Electron Kinetics and Non-Joule Heating in Near-Collisionless Inductively Coupled Plasmas

PHYSICAL REVIEW E VOLUME 55, NUMBER 3 MARCH 1997

Electron kinetics and non-Joule heating in near-collisionless inductively coupled plasmas

V. I. Kolobov Plasma Processing Laboratory, Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792

D. P. Lymberopoulos Applied Materials, 3100 Bowers Avenue, Santa Clara, California 95054

D. J. Economou Plasma Processing Laboratory, Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792 ͑Received 1 July 1996͒ Electron kinetics in an inductively coupled plasma sustained by a coaxial solenoidal coil is studied for the near-collisionless regime when the electron mean free path is large compared to the tube radius. Emphasis is placed on the influence of the oscillatory magnetic field induced by the coil current and the finite dimension of the plasma on electron heating and formation of the electron distribution function ͑EDF͒. A nonlocal approach to the solution of the Boltzmann equation is developed for the near-collisionless regime when the traditional two-term Legendre expansion for the EDF is not applicable. Dynamic Monte Carlo ͑DMC͒ simulations are performed to calculate the EDF and electron heating rate in argon in the pressure range 0.3–10 mTorr and

driving frequency range 2–40 MHz, for given distributions of electromagnetic ﬁelds. The wall potential ␾w in DMC simulations is found self-consistently with the EDF. Simulation results indicate that the EDF of trapped

electrons with total energy ␧Ͻe␾w is almost isotropic and is a function solely of ␧, while the EDF of untrapped electrons with ␧Ͼe␾w is notably anisotropic and depends on the radial position. These results are in agreement with theoretical analysis. ͓S1063-651X͑97͒05403-2͔

PACS number͑s͒: 52.80.Ϫs, 52.65.Ϫy

I. INTRODUCTION even in the absence of collisions if there is a ‘‘phase randomization’’ mechanism that is equivalent to electron momentum High-density plasmas operating at low gas pressures transfer in collisions with gas species. Such collisionless (Ͻ50 mtorr͒ have recently attracted considerable attention ͑stochastic͒ electron heating is well known for capacitively as primary candidates for the manufacturing of ultra-large- coupled plasmas ͑CCPs͒ where it occurs due to electron in- scale integrated circuits ͓1͔. Inductively coupled plasma teractions with oscillating sheath boundaries ͓6,7͔. Between ͑ICP͒ sources are particularly attractive because of their rela- the two extremes of Joule heating and collisionless heating, tively simpler design. Although ICPs have been known and there is an important regime of what we call ‘‘hybrid’’ heat- studied for more than a century ͓2͔, the low-pressure operating. In the hybrid regime, an electron ‘‘forgets’’ the field ing regime desirable for modern microelectronics applica- phase due to collisions, but in contrast to Joule heating, hy- tions is historically unusual and has not been extensively brid heating is nonlocal: the place of the electron interaction explored until recently. Nonlocal electron kinetics is a dis- with the field and the place where phase-randomizing colli- tinctive feature of this regime. Most of the recent kinetic sions occur are separated in space. Generally, the behavior of studies of low-pressure ICPs ͓3–5͔ have been limited to the electrons in this regime is governed by three frequencies: the collisional regime for which the electron mean free path ␭ is frequency of the rf field ␻, the collision frequency ␯, and the small compared to characteristic discharge dimensions. Mod- bounce frequency ⍀ defined below. Depending on the rela- ern applications, howerer, call for a plasma that is as free of tive magnitude of these frequencies, different electron dy- collisions as possible. Collisionless electron heating and namics and a variety of heating regimes can be distin- anomalous skin effect are typical of the near-collisionless guished. Both hybrid and collisionless heating belong to the operating regime, for which ␭ is larger than or comparable to category of non-Joule heating. the discharge dimensions. Interesting kinetic effects are There is one property that fundamentally separates ICPs caused by thermal motion of electrons in this regime. from CCPs. The magnetic field is a crucial factor for ICPs Electron heating is one of the key processes that deter- since its time variation induces an electric field that acceler- mine the power deposition and spatial uniformity of the ates electrons. Contrary to the electric field in the CCP, the plasma. Heating is a statistical process that transfers the di- inductive electric field in the ICP is solenoidal, i.e., nonpo- rected energy acquired from the field into random energy of tential in nature. The field lines are closed within the plasma thermal motion. At high gas pressures, electron collisions and do not form oscillating sheaths as in CCPs. However, with neutral atoms are responsible for this transfer; colli- due to the nonpotential nature of the field, charged particles sional ͑Joule͒ heating predominates. Heating at low pres- change energy in a round-trip through the field region even sures is due to a combined effect of electron interactions with in the absence of collisions ͓8͔. The existence of collisionless the fields, reflections from the plasma boundaries ͑potential heating in the ICP was suggested by Godyak et al. ͓9͔ based barriers͒, and collisions with gas species. Heating may occur on measurements of external electrical characteristics of an

1063-651X/97/55͑3͒/3408͑15͒/$10.0055 3408 © 1997 The American Physical Society 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3409 inductive discharge. Turner ͓10͔ has drawn similar conclu- sions by calculating the surface impedance of a planar plasma slab using a particle-in-cell ͑PIC͒ Monte Carlo collisions simulation of electrons. He found that the real part of the surface impedance does not vanish with a reduction of gas pressure, indicating, in an indirect way, the existence of power absorption in the collisionless limit. An analytical model of electron heating was developed in ͓11͔ for a spatially homogeneous plasma with a Maxwellian electron distribution function. It was assumed that electrons interact with inductive electric field within a skin layer and, due to collisions with plasma species, forget the phase of the field between subsequent interactions. The model assumed implic- itly that the electron mean free path does not exceed the characteristic dimension of the plasma so that any effects of finite size of the plasma could not be predicted. FIG. 1. Sketch of an inductive discharge with a coaxial solenoi- In the near-collisionless operating regime, electron ‘‘col- dal coil. A time-varying magnetic field Bz induces a solenoidal lisions’’ with plasma boundaries ͑potential barriers͒ are more electric field E␪ . An electrostatic potential ␾(r) is generated by the frequent than collisions with gas species. Under these condi- excess positive space charge. The depth of the potential well ␾w tions the finite dimensions of the plasma become an impor- ensures the absence of a net charge flow to the dielectric wall of the tant consideration. The momentum gained by electrons from chamber. the electromagnetic forces in one place can be transferred by thermal motion to another place where the momentum may ics. An analysis of the anomalous skin effect in ICPs is be- lead or lag the phase of the electric field, depending on the yond the scope of this paper. It is known that penetration of field frequency and transit time of electrons. Phase correla- electromagnetic fields into a bounded discharge plasma may tions and transit time resonances have a marked influence on be accompanied by nonmonotonic field profiles, resonance the penetration of electromagnetic fields into a bounded phemonena, etc. ͓14,17͔. A review of classical and recent plasma ͓12͔. The influence of finite-size effects on electron works on the anomalous skin effect can be found in Ref. heating was analyzed in ͓13͔ for a planar rf discharge. It was ͓18͔. found that the mechanism of heating depends on the velocity The structure of the paper is as follows. Section II pre- component affected by the electron interaction with electro- sents an analysis of collisionless particle dynamics. The axial magnetic forces. The rf magnetic field in the ICP may have a symmetry of the problem enables one to rely on the strict major impact on this interaction. constancy of the canonical angular momentum to separate Most papers on ICPs have neglected the influence of the the radial and azimuthal motions. In Sec. III we develop a oscillatory magnetic field B on electron motion. The possible nonlocal approach to the solution of the Boltzmann equation influence of the B field on the anomalous skin effect in in- for electrons in a near-collisionless ICP. Section IV describes ductive discharges was mentioned by Demirkhanov et al. the technique of dynamic Monte Carlo ͑DMC͒ simulation. ͓14͔ more than 30 years ago. Recently, Cohen and Rognlien The results of the DMC simulations in argon are presented in ͓15͔ have analyzed the influence of the B field on electron Sec. V for a wide range of discharge conditions. Section VI heating in ICPs. They pointed out that the Lorentz force due contains a discussion of the results. to the B field may exceed the electric force in the ICP. The Lorentz force changes the direction of the velocity kick an II. COLLISIONLESS DYNAMICS electron acquires in the skin layer. As a result, electron heat- OF CHARGED PARTICLES ing in the near-collisionless regime may be substantially different if the oscillatory magnetic field is accounted for in Consider an inductively coupled plasma that is produced theoretical analysis or numerial modeling. Gibbons and in a dielectric tube of radius R inserted into a long solenoidal Hewett ͓16͔ have observed in a PIC simulation that both coil ͑Fig. 1͒. The electric and magnetic fields induced by the electron velocity components in the plane orthogonal to B rf current in the coil generally have both axial and circum- are affected by collisionless heating. Only the azimuthal ferential components ͓19͔. The axial component of the elec- component v␪ would have changed if the B field were ne- tric field Ez , which is due to the rf potential across the coil glected. terminals, provides capacitive coupling. It can be eliminated The purpose of the present paper is to analyze electron or substantially reduced either by a metal screen or by an dynamics and the mechanism of electron heating in weakly electrolyte enveroping the discharge vessel ͓20͔. A time- collisional ICPs accounting for the oscillatory magnetic field varying magnetic field Bz gives rise to a solenoidal electric and the finite dimensions of the plasma. The problem is field E␪ according to Faraday’s law. The E␪ field imparts treated analytically and numerically by using ͑a͒ integration kinetic energy on electrons and an inductive discharge can be of equations of single-particle motion, ͑b͒ approximate solu- sustained. Such electrodeless discharges ͑sometimes called tion to the Boltzmann equation, and ͑c͒ dynamic Monte ‘‘ring discharges’’͒ have a long and interesting history ͓2͔. Carlo simulations. We have chosen a simple cylindrical sys- However, the low-pressure operating regime has not been tem with prescribed profiles of the fields rather than solving thoroughly studied yet. the Maxwell equations self-consistently with electron kinet- We are interested in the case when the mean free path of 3410 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55 plasma species exceeds the tube radius R, so that the particle motion is almost collisionless. Charged particles are accelerated in the azimuthal direction by the inductive electric field E␪ ͑Fig. 1͒. The Bz field produces a Lorentz force acting in the radial direction. When averaged over the rf period, this force attracts particles towards the axis regardless of the particle’s charge. This ‘‘radiation pressure’’ effect is well known in high-temperature plasmas ͓21͔. In addition to the rf fields, a substantial static space-charge electric field Er is built up in the discharge. This field accelerates positive ions towards the wall and confines the majority of electrons in the plasma providing equality of charge flow to the dielectric wall of the chamber. The ion motion is a free fall. Electrons can escape the plasma after acquiring enough kinetic energy to overcome the potential barrier near the wall. At steady state, each electron during its lifetime must produce on average an electron-ion pair in ionization events.

FIG. 2. Schematic of the effective potential ⌿ (r) ͓Eq. ͑6͔͒ for A. Equations of motion 0 p␪Ͼ0 and for p␪ϭ0. rmin and rmax indicate coordinates of turning It is convenient to represent the space-charge field Er by points for an electron with energy ␧Ќ . its scalar potential ␾ and to use the vector potential A for 2 2 describing the alternating fields generated by the coil current. p␪ q͗A␪͘ ⌿ ͑p ,r͒ϭ ϩ ϩ␾͑r͒ ͑6͒ By virtue of azimuthal symmetry of the problem, the only 0 ␪ 2mqr2 2m nonvanishing component of A is the azimuthal component A␪(r,t). Magnetic and electric rf fields are recovered from and an alternating part tϭE␪ . Theseץ/ A␪ץrϭrBz and Ϫץ/(rA␪)ץ the relations fields do not affect the particle motion along the z axis. Col- p␪A␪ q lisionless motion in the plane normal to B is governed by ⌿ ϭϪ Ϫ ͑A2Ϫ͗A2͒͘. ͑7͒ z 1 mr 2m ␪ ␪ the Hamiltonian ͓21͔ 2 2 Here A2 denotes the time-averaged value of A2(r,t) ͑the pr 1 p␪ϪqrA␪ ͗ ␪͘ ␪ Hϭ ϩ ϩq␾͑r͒, ͑1͒ 2m 2m ͩ r ͪ time-average of A␪ is zero͒. The first term on the right-hand side of Eq. ͑6͒ is due to the centrifugal force, i.e., an effective force in the r direction resulting from particle motion in where m and qϭϮe are the mass and charge of a particle, the ␪ direction. The second term on the right-hand side of r and ␪ are cylindrical coordinates, and p and p are ca- r ␪ Eq. ͑6͒ is the pondermotive or Miller force that describes nonical momenta. Since the Hamiltonian ͑1͒ is independent ‘‘radiation pressure’’ ͓21͔. Radiation pressure forces both of ␪, the canonical angular momentum p is an invariant of ␪ electrons and ions from regions of high rf field into regions the motion: of weaker field, i.e., towards the tube axis. Since the Miller force is inversely proportional to particle mass, its influence 2˙ p␪ϭmr ␪ϩqrA␪ϭconst. ͑2͒ on ions is typically negligible compared to the electrostatic force, which accelerates ions towards the wall. Strict constancy of p allows separating the radial and azi- ␪ For electrons, the alternating part of the potential ⌿1 may muthal motions. The angle ␪(t) can be found from Eq. ͑2͒ if be considered as a small perturbation at relatively high fre- r(t) and A␪ are known. On the other hand, the radial motion quencies. In the high-frequency limit, the perpendicular en- is independent of ␪: ergy of electrons

mr˙ϭp , ͑3͒ ˙2 r ␧Ќϭmr /2ϩe⌿0͑p␪ ,r͒ ͑8͒

.⌿ is a constant of the motion and there is no electron heatingץ p˙ ϭϪq , ͑4͒ r The potential ⌿0 conﬁnes the majority of electrons in theץ r plasma. Setting vrϭ0 determines, for each set of ␧Ќ and where p␪ , the coordinates of two turning points rmin and rmax ͑Fig. 2͒. The period of bounce oscillations 2 2 1 p␪ϪqrA␪ mv␪ ⌿͑p ,r,t͒ϭ ϩ␾͑r͒ϭ ϩ␾͑r͒ rmax dr ␪ 2mq ͩ r ͪ 2q T͑␧Ќ ,p␪͒ϭ2 ͵ ͑9͒ ͑5͒ rmin ͱ2͓␧ЌϪe⌿0͑p␪ ,r͔͒/m is an effective potential. It is useful to separate ⌿ into a and the bounce frequency ⍀(␧Ќ ,p␪)ϭ2␲/T are functions of time-independent part electron energy ␧Ќ and angular momentum p␪ . 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3411

B. Low-density plasmas Electromagnetic fields in ICPs are spatially inhomogeneous even in the absence of a skin effect. To distinquish finite-size effects and the field shielding by the plasma, let us consider the collisionless skin depth ␦ϭc/␻ p , where ␻ p is the electron plasma frequency and c is the speed of light. For 10 Ϫ3 neϭ10 cm , ␦ϭ5 cm, which is equal to the radius of the tube R considered in this paper. At low plasma density, when ␦ϾR, the skin effect is negligible. ͑Simple estimates indicate that the ambipolar diffusion regime is established long before the field shielding begins.͒ In any case, however, E␪ must vanish on the axis due to azimuthal symmetry of the problem. In the absence of skin effect, the rf current in the coil with angular frequency ␻ produces electromagnetic fields with the vector potential

A␪ϭA0͑r͒sin␻tϭ͑B0r/2͒sin␻t, ͑10͒ FIG. 3. Radial electron position as a function of time for reso- which corresponds to a spatially uniform magnetic field nance conditions Bϭ1G,␻ϭ7ϫ106 s Ϫ1 ͑1.1 MHz͒, and ␻ /2␻ϭ1.27, p ϭ0.1m␻R2, Rϭ5 cm. A rectangular potential B ϭB sin␻t ͑11͒ L ␪ z 0 well was assumed. The time modulation of the amplitude of bounce oscillations corresponds to phase oscillations of the electron energy and linearly varying electric field with respect to the electric field.

E␪ϭϪ͑E0r/R͒cos␻t ͑12͒ This resonance may have a major impact on gas breakdown in inductive discharges. of amplitude E0ϭB0␻R/2. The ratio of the electric force The resonance between bounce oscillations and rf fields eE␪ to the magnetic ͑Lorentz͒ force evB is ␻r/2v. This ratio can rapidly accelerate electrons. The amplitude of electron decreases with a decrease of ␻ and can be rather small at low oscillations increases with time ͑Fig. 3͒. Under breakdown frequencies even at rϭR. The magnetic field can therefore conditions (␾ϭ0), electrons would escape to the wall when have considerable impact on electron dynamics. the amplitude of their bounce oscillations equals the tube Consider collisionless electron motion under the influence radius. In a discharge, the potential ␾(r) is nonparabolic in of the rf fields ͑11͒ and ͑12͒ and an electrostatic field the vicinity of the wall due to the presence of a space-charge ErϭϪd␾/dr. A similar problem was treated by Weibel sheath. Equation ͑14͒ is no longer valid at that location. Due ͓22͔, who analyzed stable orbits of charged particles in elec- to electron reflection from the sheath, the bounce frequency tromagnetic fields of a circular waveguide ͓when A0(r)isa ⍀ becomes a function of electron energy. That detunes the Bessel function͔ for ␾ϭ0. In our case, the first term of ⌿1 frequency, generally leading to phase oscillations with re- gives no contribution to the force and the equations of mo- spect to the field ͑see Fig. 3͒. The variations of the bounce tion ͑3͒–͑5͒ are reduced to amplitudes shown in Fig. 3 correspond to variations of electron energy with time. Under the conditions of Fig. 3, the 2 2 p␪ ␻L eEr limited energy excursions correspond to regular electron mo- r¨ϭ Ϫ rsin2␻tϪ , ͑13͒ m2r3 ͩ 2 ͪ m tion. Under certain conditions, due to the nonlinearity of oscillations ͓the dependence of ⍀ on ␧Ќ in Eq.͑9͔͒, the motion where ␻LϭeB0 /m is the Larmor frequency. For the particu- of electrons may become chaotic. That corresponds to the larly simple case Er(r)ϭE0r/R, Eq. ͑13͒ is equivalent to a onset of collisionless heating. We shall illustrate the origin of set of two completely decoupled equations for xϭrcos␪ and the chaos and the appearance of collisionless heating for the yϭrsin␪, both of the Mathieu type more practical case of a thin skin layer, which is analyzed in the next subsection. 2 ␻L ͕x¨ ,y¨ ͖ϭϪ⍀2 1Ϫ cos2␻t ͕x,y͖. ͑14͒ ͩ 8⍀2 ͪ C. High-density plasmas For high-density plasmas with ␦ӶR, the rf fields are lo- Equation ͑14͒ describes a linear oscillator under the influ- calized within a thin skin layer near the wall. Electromag- ence of a harmonic external force of frequency 2␻. The netic forces act impulsively rather than continuously on 2 1/2 frequency of natural oscillations ⍀ϭ(␻L/8ϩeE0 /mR) is bouncing electrons. Since ␦ӶR, one can neglect the time independent of energy. A parametric resonance is known to ␶ϭ2␦/vr electrons spend in the skin layer compared to the occur when the frequency of the force (2␻) is close to bounce time tnϩ1Ϫtn between two sequential interactions. In k⍀, where k is an integer. For ⍀ϭ␻L /ͱ8 ͑no electrostatic the presence of an electrostatic potential in the plasma, only field͒, the basic resonance (kϭ1) takes place in the interval sufficiently energetic electrons are capable of overcoming ͓22͔ the potential barrier and reach the skin layer. For these electrons one can neglect the influence of the electrostatic poten- 1.15Ͻ͑␻L/2␻͒Ͻ1.89. ͑15͒ tial for calculation of the bounce period from Eq. ͑9͒: 3412 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55

2 where ⌬␧ is the magnitude of the energy kick. The phase of R p␪ Tϭ 2m␧ Ϫ . ͑16͒ the rf field appears to be random for electrons if small ␧ ͱ Ќ R Ќ ͩ ͪ changes of electron energy in the skin layer result in consid- Due to the two-dimensional nature of the problem, the period erable changes in phase. In the vicinity of the maximum of T(␧), where T is weakly energy dependent, small changes in T is a function of two variables p␪ and ␧Ќ . The period T vanishes for particles with p ϭ0 moving along a circle of T(␧) also result in a small change in the phase over the r period of bounce oscillations. The heating becomes second radius rϷR. Another limiting case, namely, p ϭ0, re- ␪ order with respect to ⌬␧. On the contrary, for T 0, where sembles a planar geometry. For p Þ0, the period T has a ␪ the derivative dT/d␧ tends to infinity, even small→ changes in maximum with respect to ␧ . Due to strict constancy of the Ќ energy may result in considerable changes in phase; the de- canonical momentum p , a change in ␧ in the skin layer ␪ Ќ nominator of Eq. ͑21͒ becomes zero. Far from these pecu- corresponds to a change in radial electron velocity. Thus, in liarities, which are specific to cylindrical geometry, inequal- the limiting case ␦ӶR, electron interactions with the rf fields ity ͑21͒ requires that ␻T⌬␧/␧Ͼ1. This condition, well affect only the radial part of electron energy. known for a planar case ͓24͔, states that for small-amplitude Let us construct a mapping relating the values of variables kicks ⌬␧, the driving frequency ␻ must be sufficiently large between sequential interactions. Let ␰ designate the phase n so that the field could change many times during the bounce of the field (␰ ϭ␻t ) at the moment of the nth electron n n period. For small p ͑as in the planar case͒, the bounce pe- reflection from the plasma boundary at rϭR. The phases of ␪ riod T decreases with an increase in energy and the motion the field between two sequential interactions are given by of energetic electrons is more regular. Equation ͑21͒ imposes ␰ ϭ␰ ϩ␻T͑␧ ͒. ͑17͒ restrictions on the maximum energy electrons can achieve by nϩ1 n nϩ1 the collisionless heating mechanism. Integrating the equations of motion in the skin layer yields To proceed further, we have to calculate the energy kick per single pass through the skin layer. For the sake of sim- plicity, let us assume, as in ͓11,15͔, an exponential decay of ␧nϩ1ϭ␧nϩQ͑␰n͒, ͑18͒ the fields within the skin layer: A␪ϭA0exp͓Ϫ(RϪr)/␦͔. For ˙ where ␧nϩ1ϭ␧(tnϩ1Ϫ␶) and ␧nϭ␧(tnϪ␶) are radical elec- QӶ␧, integrating along unperturbed trajectories rϭϮvrt, tron energies before the nϩ1 and n kicks, respectively. The we obtain ͓11,15͔ energy change in the skin layer Q is proportional to the r and electron velocity ␻␶ץ/ ⌿1ץpower ͑the product of force e r˙) evaluated along a trajectory ͓23͔ Q͑p ,p ,␰͒ϭ␻ ␦sin␰͑2p /Rϩm␻ ␦cos␰͒ , ␪ r L ␪ L 1ϩ͑␻␶͒2 22 t ϩ␶ ͑ ͒ …⌿1„r͑t͒,tץ n Qϭe ͵ r˙ dt. ͑19͒ rץ tnϪ␶ where ␻LϭeB0 /m is the Larmor frequency in the magnetic field B ϭA /␦. Two points are worth mentioning regarding The difference equations ͑17͒ and ͑18͒ are equivalent to the 0 0 Eq. ͑22͒. ͑a͒ Evidently, the maximal energy can be acquired equations of motions ͑3͒ and ͑4͒. The value of the derivative if the electric field remains in the same direction during the d␰ /d␰ defines the degree of phase correlations. The in- nϩ1 n entire electron transit through the skin layer. However, since equality the amplitude of the induced electric field (E␪ϰ␻) decreases with ␻, the optimal conditions for the energy gain take place d␰nϩ1 Ϫ1 у1 ͑20͒ at ␻␶ϭ1 when Q reaches a maximum with respect of ␻␶. ͯ d␰ ͯ n ͑b͒ The term of Q linear with respect to ␻L␦ vanishes for p␪ϭ0 due to symmetry of the problem. serves as a rough criterion of local phase instability ͓23͔.If The important difference of our analysis from ͓11͔, where Eq. ͑20͒ holds true, electron trajectories become chaotic in the rf magnetic field was neglected, is in the direction of the spite of the fact that the equations of motion contain no velocity kick. Accounting for the magnetic field results in a random forces. In some sense, the chaotic component of the velocity kick normal to the plasma boundary, similar to that motion remains weak compared to the regular component: it taking place in a capacitively coupled rf discharge ͓15͔. The manifests itself on larger spatial and time scales in compari- direction of the velocity kick greatly influences the dynamics son to the regular bounce motion. Chaotic dynamics corre- of bouncing electrons; for instance, in the planar case, kicks sponds to ‘‘slow’’ electron diffusion along the ‘‘energy parallel to the boundary ͑as in ͓11͔͒ imply the absence of axis’’ from regions in phase space where particles are abun- collisionless heating ͓13͔. Contrary to the capacitively dant to regions where particle density is low. Since the elec- coupled plasma, electrons entering the skin layer with zero tron distribution function ͑EDF͒ usually decreases with en- azimuthal velocity experience no energy kick in the linear ergy, such a diffusion corresponds to collisionless electron approximation: the first term in Eq. ͑22͒ vanishes for heating. p␪ϭ0. In the linear approximation, the mapping ͑17͒ and Using Eqs. ͑20͒ and ͑16͒ and taking into account only the ͑18͒ can be written in the form linear term of ⌿1 in Eq. ͑7͒, one obtains from Eq. ͑20͒

2 ␻T⌬␧ ͉͑p␪ /R͒ Ϫm␧͉ 4␻L ͱ n у1, ͑21͒ ϭ ϩ Psin␰ E , ͑23͒ ␧ 2m␧Ϫ͑p /R͒2 Enϩ1 En ␻ n ϩ1 ␪ En 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3413

FIG. 5. Characteristic frequencies for an argon discharge for FIG. 4. Regions of regular above and chaotic below solid ͑ ͒ ͑ electrons with an energy of 5 eV, a plasma density of 1011 cm Ϫ3, lines͒ electron dynamics for the case of a thin skin layer. The solid and the tube radius Rϭ5 cm: angular frequency ␻, bounce fre- lines indicate the maximal values of the radial electron energy quency ⍀, transport collision frequency ␯, and the frequency of that can be reached in collisionless heating for a particular value ofE Coulomb interaction among electrons ␯ . the angular momentum P. ee into two parts can be performed for the free-flight regime as 2R ͱ nϩ1 2␻L nϩ1Ϫ1 well ͓26,27͔. In rf discharges, one can separate rapidly and ␰ ϭ␰ ϩ E ϩ Pcos␰ E , nϩ1 n 2 n 2 ␦ nϩ1ϩP ␻ ϩ1 slowly varying parts of the EDF. For trapped electrons, the E ͑ nϩ1 ͒ ͱ nϩ1 E E ͑24͒ slowly varying part turns out to be almost isotropic. Further- more, by performing an appropriate spatial averaging, the 2 2 where Pϭp␪ /m␻␦R and ϭ2␧/m(␻␦) ϪP are dimen- kinetic equation for the isotropic part is reduced to an ordi- sionless momentum and energy,E respectively. Since the nary differential equation. Thus, for the majority of elec- transformation from n to nϩ1 is generated by Hamilton’s trons, the principal part of the EDF turns out to be a function equations, the mapping ͑23͒ and ͑24͒ must be area preserv- of a sole variable, the total electron energy. ing. The last term in Eq. ͑24͒ accounts for the first-order change in phase to ensure area preservation ͓24͔. In variables A. Nonlocal approach and P, the boundary of chaos defined by Eq. ͑21͒ becomes E It is convenient to use invariants of the particle motion as 2 independent variables in the Boltzmann equation. Taking ad- 4␻LR Pͱ ͉P Ϫ ͉ E E Ͼ1. ͑25͒ vantage of the azimuthal symmetry of our problem and using ␻␦ ͑ ϩ1͒͑ ϩP2͒2 E E the canonical momentum p␪ as an independent variable, the Boltzmann equation for electrons can be written in the form This boundary is shown in Fig. 4 for typical discharge con- 25 7 Ϫ1 7 Ϫ1 ͓ ͔ ditions, ␻ϭ8.5ϫ10 s ͑13.56 MHz͒, ␻Lϭ1.7ϫ10 s f ץ f ץ f ץ -B ϭ1G͒,␦ϭ1 cm, and Rϭ5 cm. The dashed line corre) 0 ˙ sponds to the maximum of T( ), where heating becomes ϩvr ϩpr ϭS, ͑26͒ prץ rץ tץ second order with respect to ⌬␧E. Since collisionless heating affects only the radial part of electron energy, electrons can where S is a collision operator and p˙ is given by Eq. 4 . be heated up to radial energies m(␻␦)2/2ϳ2 eV for the r ͑ ͒ The electron distribution function f in Eq. ͑26͒ is a function conditions of Fig. 4. An increase of ␻L or a decrease of ␻ leads to an increase of these energies. of pr ,p␪ ,pz ,r, and t. Since there is no variation with z and ␪, these coordinates have been omitted. Also, the term p vanishes because p˙ ϭ0 ͓see Eq. ͑2͔͒. Accordingץ/ f ץ ˙p III. BOLTZMANN EQUATION FOR ELECTRONS ␪ ␪ ␪ to Eq. ͑26͒, electron motion along z is coupled to the motion The electron distribution function in a rf discharge is a in the (x,y) plane solely due to collisions. function of time, three electron velocities, and ͑at least one͒ The ICP is a weakly ionized plasma. For the majority of spatial coordinate. In general, the solution of the Boltzmann electrons, the frequency of electron collisions with neutral equation for the EDF is a formidable numerical task. For that atoms exceeds the frequency of Coulomb interactions among reason, the development of reliable approximations is valu- electrons ͑see Fig. 5͒. In the pressure range of interest, the able. The traditional two-term Legendre expansion of the bounce frequency ⍀ exceeds the momentum transfer colli- EDF is valid only for the collisional regime ␭ӶR ͓25͔. How- sion frequency ␯. In argon ͑as in many other gases͒ the total ever, due to the large difference between momentum and frequency of inelastic collisions ␯* ͑which includes excita- energy relaxation rates of electrons, a separation of the EDF tion and ionization of atoms͒ is small compared to ␯ in the 3414 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55

energy range of interest. The difference between ␯ and ␯* where the angular brackets designate time averaging over a ͑i.e., the difference between momentum and energy relax- rf period ation rates͒ can be used to simplify the Boltzmann equation. ␻ 2␲/␻ Let us rewrite Eq. ͑26͒ using the transverse energy ␧Ќ ͗ f ͘ϭ f 0ϭ fdt. ͑32͒ ͓Eq. ͑8͔͒ as an independent variable 2␲ ͵0 f ץ ⌿ץ f ץ f ץ 1 For the majority of electrons, the difference, f 0ϪF0,isex- ϩvr Ϫe vr ϭϪ␯͑fϪF0͒ϩS*. ␧Ќ pected to be rather small, so that the energy relaxation termץ rץ ͪ rץ ͩ tץ ␧ Ќ S* in Eq. 31 may exceed ( f ϪF ). ͑27͒ ͑ ͒ ␯ 0 0 The first two terms of Eq. ͑30͒ are recognized as the total time derivative in configuration space. Thus the solution of Now, the EDF is a function of ␧Ќ ,p␪ ,pz ,r, and t, while the Eq. ͑30͒ can be found by integration along electron trajecto- radial electron velocity vrϭͱ2(␧ЌϪe⌿0)/m is a function of independent variables. The first term on the right-hand ries ͓12͔ side of Eq. ͑27͒ describes momentum transfer in collisions ϱ f0ץ ⌿1ץ with gas species. The scattering is assumed to be isotropic f ϭe ds eϪ␯sv ͑tϪs͒ ͓r͑tϪs͒,tϪs͒] . 1 ͵ r r ␧Ќץ ץ and F0 designates the isotropic part of the EDF 0 ͑33͒ 1 F ͑␧͒ϭ fd⍀ , ͑28͒ Having expressed the alternating part f in terms of the main 0 4␲ ͵ v 1 part f 0, we may substitute Eq. ͑33͒ into Eq. ͑31͒ to obtain an which is assumed time independent for reasons discussed equation for f 0 below. Integration in Eq. ͑28͒ is performed over angles in ץ f ץ velocity space and ϭ ϩp2/2m denotes the total energy. 0 ␧ ␧Ќ z vr Ϫvr J␧͑␧Ќ ,r,p␪͒ϭϪ␯͑f0ϪF0͒ϩS*, ␧Ќץ ͪ rץ ͩ The term S* on the right-hand side of Eq. ͑27͒ describes ␧Ќ inelastic processes such as excitation and ionization of at- ͑34͒ oms, excitation of molecular vibrations and rotations, energy loss in elastic collisions with atoms, and Coulomb interac- where tions among electrons. These processes can be divided into fץ ⌿ץ ⌿ץ ϱ quasielastic and substantially inelastic. Quasielastic pro- 2 Ϫ␯s 1 1 0 J␧ϭe ds e vr͑tϪs͒ ͓r͑tϪs͒,tϪs͔ ␧Ќץ ʹ rץ rץ cesses, such as excitation of molecular vibrations and rota- ͵0 ͳ tions, energy loss in elastic collisions with atoms, and Cou- ͑35͒ lomb interactions among electrons, can be written in a Fokker-Planck form ͓25͔. Substantially inelastic processes is the electron flux along the ␧Ќ axis caused by electron correspond to removing high-energy electrons and replacing heating. them by low-energy ones. For instance, excitation of a single Consider trapped electrons with ␧ЌϽe␾w . The bounce atomic level with energy ␧* is described by motion of these electrons ͓described by the first term of Eq. ͑34͔͒ occurs much more rapidly than their displacement ␧Ј along the energy axis; the first term in Eq. ͑34͒ dominates. S*ϭϪ␯*͑␧͒fϩ ␯*͑␧Ј͒f͑␧Ј͒. ͑29͒ This term vanishes if f does not depend explicitly on r, i.e., 1 1 ͱ␧ 1 0 if f 0ϭF(␧Ќ ,p␪ ,pz). An equation for F can be obtained by In this process a high-energy electron loses an energy quan- dividing Eq. ͑34͒ by vr and integrating over the discharge tum ␧* and reappears at energy ␧Јϭ␧Ϫ␧*. Let ␯* desig- cross section accessible to electrons with energy ␧Ќ . This nate the frequency of all substantially inelastic processes in- procedure of spatial averaging results in ͓26͔ cluding ionization. For most gases ␯ӷ␯* in the considered ¯ Fץ ץ energy range. Also, for plasma densities typical to ICPs, D͑␧Ќ ,p␪͒ ϭ¯␯͑FϪF0͒ϪS*, ͑36͒ ␧Ќץ ␧Ќץ ␯ӷ␯ee for the majority of electrons ͑see Fig. 5͒. Thus the inelastic collision term S* in Eq. ͑27͒ is usually small com- where pared to ␯ f .

The EDF may be expressed as the sum of a rapidly vary- rmaxdr ϱ 2 Ϫ␯s ing part f 1 and a slowly varying part f 0. The latter is almost Dϭe ⍀ ͵ ͵ ds e rmin vr 0 time independent for ␻ӷ␯* ͓25͔. The equation for f 1 is ⌿ץ ⌿ץ f f f 1 1 ϫ vr͑tϪs͒ ͓r͑tϪs͒,tϪs͔ ͑37͒ 0 ץ ⌿1ץ 1 ץ 1 ץ ʹ rץ rץ ϩvr Ϫe vr ϭϪ␯f1 . ͑30͒ ͳ ␧Ќץ rץ ͪ rץ ͩ tץ ␧Ќ is the energy diffusion coefﬁcient and ⍀ is the bounce fre- The equation for f 0(␧Ќ ,p␪ ,pz ,r)is quency for an electron with energy ␧Ќ and canonical momentum p␪ . The overbar denotes spatial averaging: f1ץ ⌿1ץ f0ץ v Ϫ e v ϭϪ␯͑f ϪF ͒ϩS*, r r 0 0 rmax g ʹ␧Ќץ rץ rͪ ͳץͩ ␧Ќ ¯gϭ⍀ dr. ͑38͒ ͵ v ͑31͒ rmin r 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3415

Equation ͑36͒ describes a complex process of energy re- distribution between three degrees of freedom. Heating produces an electron ﬂux along the ␧Ќ axis towards high energies and thus causes a departure of the EDF from an isotropic one. Elastic collisions tend to restore the isotropy of the EDF. They act on a time scale that is long compared to the bounce time, but short compared to the electron lifetime, resulting in an almost isotropic distribution of trapped electrons. When ␯ӷ␯* holds true, Eq. ͑36͒ can be simpliﬁed even further. The EDF can be written as the sum of an isotropic part F0 and a small anisotropic addition F1ӶF0: F(␧Ќ ,p␪ ,pz)ϭF0(␧)ϩF1(␧Ќ ,p␪ ,pz). The equation for the isotropic part has the form

d dF0 D ͑␧͒ ϭϪ¯S*, ͑39͒ d␧ ␧ d␧ where D␧(␧) is the energy diffusion coefficient averaged over p␪ and pz : FIG. 6. Phase shift ␣ between oscillations of the EDF f 1 and oscillations of the accelerating force ͓Eq. ͑43͔͒. 1 D ͑␧͒ϭ D͑␧ ,p ͒dp dp . ͑40͒ ␧ 2m␧ ͵͵2 2 Ќ ␪ ␪ z B. Energy diffusion coefficient for a thin skin layer p␪ϩpzϽ2m␧ The induced electric field in a cylindrical ICP is spatially The energy diffusion coefficient D␧ contains all information inhomogeneous even in the absence of the skin effect ͑see about electron heating and determines microscopic character- Fig. 1͒. The magnetic field becomes spatially inhomoge- istics of the electron ensemble such as the principal part of neous only due to the field shielding by the plasma. For the EDF, F0, and macroscopic quantities such as the rate of ␦ϽR, we consider only the linear term of ⌿1 in Eq. ͑7͒, power deposition into the plasma. which gives the principal contribution to the electromagnetic Thus, similar to the collisional case ␭Ͻ␦ ͓3,4͔, electrons force for electrons with nonzero p␪ . In the limiting case of a are separated into two groups with respect to total energy: thin skin layer ␦ӶR, retaining only the linear term of Q in trapped electrons with ␧Ͻe␾w and ‘‘free’’ electrons with Eq. ͑22͒, integration of Eq. ͑33͒ can be carried out to give the ␧Ͼe␾w . For trapped electrons, the kinetic equation is re- oscillating part of the EDF in the form duced to an ordinary differential equation, Eq. ͑39͒. For free ϱ f0ץ electrons, the more general equation, Eq. ͑34͒, should be Ϫ␯tn f 1͑␧Ќ ,p␪ ,t͒ϭ⌬␧ ͚ e sin␻͑tϪtn͒. ͑42͒ ␧Ќnϭ0ץ used. These electrons are capable of escaping the discharge and their EDF depends explicitly on radial position. More- over, at ␭ϾR, the EDF of free electrons becomes notably Here ⌬␧ is the energy kick in one pass through the skin layer anisotropic. While in the collisional case ␭ϽR considerable and tn designates the time of the nth electron interaction with anisotropy of the EDF appears only at a distance approxi- the skin layer. Consider the hybrid regime when chaotization mately equal to ␭ near the wall ͑due to the absence of elec- of electron motion is due to collisions ͓13͔. For this regime, tron flux from the wall͒, in the free-flight case the EDF an- using the approximation tnϷ2␲n/⍀,wefind isotropy manifests itself everywhere. Since only those fץ ␧⌬ electrons for which vr(␧Ќ ,p␪ ,R)Ͼ0 can escape to the wall, 0 f1ϭ ͕⌽1͑x,y͒sin␻tϪ⌽2͑x,y͒cos␻t͖, ͑43͒ ␧Ќץ the velocity of an escaping electron must satisfy the condi- 2 tion where xϭ2␲␻/⍀, yϭ2␲␯/⍀, and functions ⌽1 and ⌽2 p␪ are defined as ␧ Ϫ Ͼe␾ . ͑41͒ Ќ mR2 w expyϪcosx ⌽1ϭ , ͑44͒ Thus electrons with p␪Ϸ0 that move away from the axis can coshyϪcosx escape if ␧ЌϾe␾w . Since these electrons are present only in the vicinity of the axis, the EDF at rϷ0 decreases rapidly sinx ⌽2ϭ . ͑45͒ with energy starting at ␧ЌϷe␾w . For larger r, such a de- coshyϪcosx crease of the EDF begins at higher ␧Ќ . Consequently, the EDF at ␧ЌϾe␺w becomes a complicated function of radial The function f 1 oscillates with frequency ␻ and with a phase position and angles. While free electrons are primarily re- shift ␣ϭarctan(⌽2 /⌽1) with respect to the oscillations of the rϰsin␻t. The phase shift ␣ is small at highץ/ ⌿1ץ sponsible for the dc component of the electron current and force give considerable contribution to the energy balance at low pressures, when ␯ӷ⍀. At low pressures, in the near- pressures, their contribution to average quantities such as collisionless regime (␯Ӷ⍀) the phase shift ␣ is a periodic electron density, temperature, and power deposition is, in function of x with a period 2␲ ͑see Fig. 6͒. Under resonance general, negligible. conditions, the amplitude of f 1 may be quite large due to 3416 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55

In the case of frequent collisions ␯ӷ⍀, the energy diffusion coefﬁcient ͑46͒ is particularly simple

Dϭ͑⌬␧͒2⍀. ͑49͒

It is interesting to note that the collision frequency ␯ does not appear explicitly in Eq. ͑49͒ even though collisions are responsible for phase randomization. For collisionless heating, the energy diffusion coefficient can be obtained using a random-phase approximation ͓13͔. The resulting expression for D coincides with Eq. ͑49͒. Al- though the mechanism of phase randomization is different in the collisionless and hybrid regimes, the heating rate described by Eq. ͑49͒ turns out to be the same. This is due to the fact that, for the considered hybrid regime, only electrons at a distance ␭ from the skin layer participate in the heating process. The energy diffusion coefficients given by Eqs. ͑46͒–͑49͒ are valid for both planar and cylindrical coordinates. To calculate the EDF in the cylindrical case, we have to average D over angles according to Eq. ͑40͒. Introducing cylindrical coordinates P and ␽ in the (p␪ ,pz) plane, we obtain, for the FIG. 7. Function ⌽1 ͓Eq. ͑43͔͒ that describes phase correlations for hybrid heating regime. Strong resonances between bounce os- particularly simple case ͑49͒, cillations and rf fields are observed in the near-collisonless regime 2 2 2 ͑␻L␦͒ m͑␻␦͒ m͑␻␦͒ (␯Ӷ⍀)at␻ϭk⍀, where kϭ0,1,2,.... D ϭ ͱ2m␧Y , ͑50͒ ␧ R 2 ͫ 2␧ ͬ strong correlations of the field phases and large energy ex- cursion compared to energy kick ⌬␧ per single pass through where Y(x) is defined as the skin layer. 1 2␲ 1 P3cos2␽͑1ϪP2sin2␽͒ͱ1ϪP2 The second term of f , shifted in phase by ␲/2 with re- 1 Y͑x͒ϭ d␽ dP 2 2 spect to the force, produces no work when averaged over one 2 ͵0 ͵0 ͑1ϪP ϩx͒ period. Only the first term contributes to the power absorp- 2 tion in the plasma. From Eqs. ͑37͒ and ͑43͒, the energy dif- 1 Ϫ23ϩ15x ͑Ϫ3Ϫ6xϩ5x ͒arctan1/ͱx ϭ Ϫ . fusion coefficient for the hybrid heating regime is 16 ͩ 3 ͱx ͪ 1 ͑51͒ 2 Dϭ ͑⌬␧͒ ⍀⌽1͑x,y͒. ͑46͒ 2 Expression ͑50͒ differs considerably from the energy diffusion coefficient In limiting cases, this energy diffusion coefficient was obtained in Ref. ͓13͔ for a planar geometry. The energy diffu- ͑eE␭͒2␯3 D ϭ , ͑52͒ sion coefficient is a product of a single energy kick in the ␧ 6͑␯2ϩ␻2͒ skin layer, the bounce frequency ⍀, and the function ⌽1, which describes the phase correlations between successive which describes Joule heating ͓3͔. The principal difference is kicks. At yӶ1, function ⌽1(x,y) exhibits resonances at that Joule heating is local ͓the energy diffusion coefficient xϭ2␲k, where k is an integer ͑see Fig. 7͒. The energy dif- ͑52͒ is a function of the local value of the electric field͔, fusion coefficient while non-Joule heating is nonlocal. The energy diffusion coefficient ͑50͒ is determined by the entire profile of the rf 2 ͑⌬␧͒ 1 ␯⍀ fields in the skin layer. Also, contrary to Eq. ͑52͒, D in Eq. Dϭ ⍀ 1ϩ ␧ 2 ͫ ␲ ␯2ϩ͑2⍀/␲͒2͓1Ϫcos͑2␲␻/⍀͔͒ͬ ͑50͒ does not depend explicitly on gas pressure: the collision ͑47͒ frequency does not appear in Eq. ͑50͒. The energy diffusion coefficient ͑50͒ exhibits a maximum as a function of the is anomalously large for resonant particles with ␻Ϸk⍀. For product ␻␦ and increases monotonically as a function of ␧ ␯ 0, D tends to a ␦ function. Far from resonances, the ͑see Fig. 8͒. → second term in the set of large square brackets of Eq. ͑47͒ is The EDF calculated from Eq. ͑39͒ with D given by Eq. 2 ␧ small. At low excitation frequencies ␻ Ӷ2␯⍀, the energy ͑50͒ is shown in Fig. 9. The EDF is found with neglect of the diffusion coefficient ͑46͒ also reaches a maximum due to electrostatic potential in the plasma ͑rectangular potential phase correlations of the rf field with respect to electron mo- well͒ and no Coulomb interactions among electrons. In a tion semilogarithmic plot versus energy, the EDF is concave at low energies and convex at high energies that can be char- 2 2 ͑⌬␧͒ ␯⍀ acterized by three temperatures. The concave shape of the Dϭ . ͑48͒ ␻2ϩ␯2 EDF at low energies is due to a decrease of the energy dif- 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3417

Alternating electric and magnetic fields were taken in the form ͑11͒ and ͑12͒. The profile of the electrostatic potential 4 was taken as ␾(r)ϭ␾0(r/R) . An infinitely thin sheath with a potential drop ⌬␾ϭ␾wϪ␾0 was assumed to exist near the wall. The wall potential ␾w with respect to the plasma potential at the axis ͑equal to zero͒ was found self-consistently to equalize the number of ionizations and the number of electrons lost to the wall. The electron distribution function and the heating rate in a steady state were computed for different gas pressures and driving frequencies. The collision processes are described using the DMC technique ͓31,32͔. The DMC method has some advantages compared to the null-collision Monte Carlo method ͓33,34͔, which is widely used to calculate electron velocity distribution functions ͑EVDFs͒. The null-collision method requires the electron free-flight distribution as an input to the simulation. In contrast, the DMC method does not require knowl- edge of the free-flight distribution. In fact, the free-flight dis-

FIG. 8. Energy diffusion coefﬁcient D␧ as a function of the total tribution is an output of the DMC simulation. In addition, the electron energy for a thin skin layer ͓Eq. ͑50͔͒. The discharge con- DMC method requires fewer random numbers per time step 7 Ϫ1 ditions are ␦ϭ1 cm, Rϭ5 cm, B0ϭ2G,␻ϭ8.5ϫ10 s , and a to describe the electron collision processes, as compared to rectangular potential well for ␾(r). the null-collision method. The DMC method is a transparent solution to the Boltzmann equation. The essence of the fusion coefﬁcient with electron energy. The convex shape in method consists in using the set of probability functions the inelastic energy range ␧Ͼ␧* is due to energy loss in ␲ inelastic collisions. Such three-temperature EDFs have re- C ij͑v0͒ϭ⌬tnjv02␲ ␴ij͑v0,␹Ј͒sin␹Јd␹Ј, ͑53͒ cently been reported for ICPs in argon ͓28͔. P ͵0

v͑v͉v ͒ϭ␦ uϪg͑v ,␧ ,␹͒ , ͑54͒ IV. PARTICLE-IN-CELL–DYNAMIC MONTE CARLO Pij 0 „ 0 ij … SIMULATION ␲ ␹ The particle-in-cell–dynamic Monte Carlo simulation was ij͑␹͉v0͒ϭ␴ij͑v0,␹͒sin␹ ␴ij͑v0,␹Ј͒sin␹Јd␹Ј, P Ͳ͵0 employed to assess the electron-heating efficiency and to cal- ͑55͒ culate the electron distribution function in weakly collisional regimes. The well-known particle-in-cell method ͓29,30͔ C v ␹ where ij , ij , and ij correspond to the electron collision was used to ascribe particle attributes onto a grid. The simu- probability,P P the conditionalP probability for an electron to lations were performed for prescribed profiles of the fields. have speed v after collision, and the probability for an electron to be scattered into angle ␹, respectively. The index i represents the collision process ͑i.e., excitation, ionization, etc.͒ that involves an electron and a particle j that has a collision cross section ␴ij(v0,␹), a function of electron speed before the collision v0, and the scattering angle ␹. The heavy species are assumed motionless. The ␦ function in Eq. ͑54͒ ensures that momentum and energy are both conserved during the collision. These conservation laws require the postcollision electron speed to be

1/2 2m 2␧ij g͑v ,␧ ,␹͒ϭ v2 1Ϫ ͑1Ϫcos␹͒ Ϫ , 0 ij ͫ 0ͩ m ͪ m ͬ j ͑56͒

where m is the mass of the electron, m j is the mass of particle j, and ␧ij is the energy lost by the electron in the collision of type i with particle j. The free motion of the electrons ͑i.e., the ﬂight of an electron between collisions͒ is described by explicitly integrating Newton’s equations of motion

FIG. 9. EDF in argon calculated from Eq. ͑39͒ with the energy 1 t0ϩ⌬t diffusion coefﬁcient ͑50͒. The discharge conditions are the same as v͑t ϩ⌬t͒ϭv͑t ͒ϩ F͑t͒dt, ͑57͒ 0 0 m ͵ in Fig. 8. t0 3418 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55

TABLE I. Energy lost in inelastic collisions energy carried to the wall ͑per electron in a unit time͒, and Ein Ew the ratio of electron escapes to the number of inelastic collisions ␬ for different pressures and driving frequencies at Bϭ1G.

t0ϩ⌬t dependent electron velocity distribution function f (v,r,␰)is r͑t0ϩ⌬t͒ϭr͑t0͒ϩ u͑t͒dt, ͑58͒ computed on discrete volume elements ⌬v ⌬v ⌬v ⌬r⌬␰ ͵t r ␪ z 0 located around the velocity v, the radial position r, and the where v and r are the electron velocity and position vector, phase angle ␰. As the simulation advances in time, the ap- respectively, t is the current time, ⌬t is a small time incre- propriate (vr ,v␪ ,vz ,r,␰) bins of the EVDF are updated. ment, and F is the force acting on the electrons. The particles were initially given positions and velocities An algorithm can now be devised based on Eqs. ͑53͒– chosen randomly. At the beginning of the simulation the wall ͑55͒ to describe the collisional motion of an electron in a potential was set to a relatively high value ͑exceeding the plasma. The probability that the electron will not suffer a ionization threshold͒ to ensure that the electron ensemble collision is given by will not decrease with time. Thereafter, the wall potential was adjusted periodically ͑every 10–80 rf cycles depending on conditions͒ to maintain a constant, within limits, number NC C ϭ1Ϫ ij , ͑59͒ of electrons. The procedure was as follows. After an electron P ͚i, j P move, the new location of the electron was tested to deter- mine whether the electron had reached the plasma boundary. whereas the probability for a collision of type i ͑ionization, 2 excitation, or elastic͒ with a particle j is given by Eq. ͑53͒.A Then, if the radial energy of the electron mvr /2 was greater random number uniformly distributed in the interval ͓0,1͔ than the potential drop in the sheath e⌬␾, the electron was dictates whetherY the electron suffers no collision lost, otherwise it was specularly reflected. In between wall- potential adjustements statistics were collected with regard to р NC ͑60͒ the energy of the electrons striking the plasma boundary. The Y P resulting vector reflected the electron energy distribution or suffers a collision of type i with particle j, function ͑EEDF͒ of the electrons penetrating the sheath. The vector containing the electron energies was then sorted in kϪ1 k ascending order. If the number of ionizations was not equal NC C NC C ϩ l р р ϩ l , ͑61͒ to the number of electrons escapes ͑highly likely͒ the wall P l͚ϭ1 P Y P l͚ϭ1 P potential was adjusted based on the net electron number change. If there was an electron number deficit the wall po- where each value of l corresponds to a unique pair (i, j). tential was set to a higher value to confine electrons. If Ϫ is Once the collision type has been determined, the energy of N the electron is revised according to the collision characteris- the electron-number deficit, the new potential was determined by simply moving Ϫ notches up ͑with respect to tics ͑e.g., elastic and inelastic͒, by using Eq. ͑54͒. The veloc- N ity of the electron is then updated based on the scattering and energy͒ the EEDF sorted vector from the notch closest to the azimuthal angles, with probability distributions given by Eq. current wall potential. If there was an electron number sur- ͑55͒. For the simulation reported in this paper, only electron plus, the wall potential was set to a lower value to allow more electrons to escape. If ϩ is the electron-number sur- collisions with neutral atoms were considered and the scat- N tering was assumed to be isotropic. plus, the new wall potential was determined by simply moving ϩ notches down ͑with respect to energy͒ from the The chamber geometry studied here is axisymmetric. Ad- N vantage is taken of this symmetry, so that all quantities are notch closest to the current wall potential. Naturally, this functions of only one spatial coordinate ͑the radial position results in a wall potential that fluctuates with time. If the of a particle͒. However, the logic associated with the three- statistics are adequate, the wall potential fluctuations are kept dimensional particle motion is exact ͓35͔. Cylindrical coor- to a minimum. After the establishment of a dynamic steady dinates are used to describe the electron motion in velocity state, the characteristics of the electron ensemble were re- space. During the PIC-DMC simulation all three electron corded to calculate the EDF and the ionization, loss, and heating rates. velocity components ͑i.e., vr , v␪ , and vz) and two spatial coordinates (r and ) are recorded. As electrons are moved ␪ V. RESULTS OF THE DMC SIMULATIONS forward in time, their velocity components are calculated at specific times of the rf cycle ͓phase angle ␰ϭ␻t(mod2␲)͔ Argon-gas pressures in the range 0.1–10 mtorr, excitation and statistics are accumulated. Numerically, the time- frequencies in the range 2–40 MHz, and magnetic induction 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3419

FIG. 11. Radial profiles of electron density and ‘‘temperature’’ for the conditions of Fig. 10. The radial increase of temperature FIG. 10. EEDF from DMC simulations as a function of total corresponds to the concave EEDF in Fig. 10. energy ␧ for different radial positions, with an argon pressure of 1 7 Ϫ3 mTorr, B0ϭ1G͑no skin effect͒, and ␻ϭ8.5ϫ10 cm . The elec- rent density vs phase ␰ at different r ͓Fig. 12͑b͔͒. The current 4 trostatic potential is ␾(r)ϭ␾0(r/R) , where ␾0ϭ5V. density is shifted with respect to the E␪ field by ϳ␲/2 as is expected for ␯Ӷ␻. The phase shift does not depend signifi- fields of 1 or 2 G were examined for a chamber radius cantly on the radial position. The radial distribution of the rf Rϭ5 cm. The results of simulations are summarized in Table current does not show significant anomalies, which are typi- I for the case of a low-density plasma with uniform inductive cal of the anomalous skin effect ͓10,18͔. Under these condi- field Bϭ1G͑no skin effect͒. One observes the following tions, the distance a thermal electron travels during the field trends. For 13.56 and 40.7 MHz, the energy lost in inelastic period, lϳv/␻Ϸ1.4 cm, is small compared to the character- collisions in and the energy carried to the wall w both istic scale of the E␪ field inhomogeneity, approximately increase withE increasing pressure. For 2 MHz thereE is a equal to R. Significant current diffusion off the skin layer maximum of in and w with respect to pressure. For 0.3 and and formation of multiple current layers with a phase shift of 1 mtorr, andE decreaseE with an increase of ␻, while at the current density and the field are expected to occur at Ein Ew 10 mtorr in and w increase with ␻. The ratio of electron lϾ␦ ͓18͔. escapes toE the numberE of inelastic collisions ␬ decreases with Figure 13 illustrates the anisotropy degree of the EDF. increasing pressure, except for the 2-MHz case. The higher The dashed curve shows the distribution f z(␧z ,r) found by 2 the heating rate, the longer the EDF ‘‘tail,’’ and a larger part sampling electrons with a given energy ␧zϭmvz /m, parallel of energy is lost in collisions compared to energy carried to to the magnetic field regardless of vr and v␪ . The solid the wall. At higher pressures more energy is lost for excita- curve shows the distribution fЌ(␧Ќ ,r) perpendicular to the tion than for ionization. field, which was found by collecting electrons with given Figure 10 shows the calculated EEDF as a function of 2 2 vr ϩv␪ irrespective of the value of vz . Both distributions total electron energy for different radial positions. The am- ␧ f z and fЌ are at rϭ0.1R. It is seen that f z and fЌ are close to plitude of time modulation of the EEDF in DMC simulations each other for trapped electrons, with ␧Ͻe␾w , but differ was found to be negligible, as expected for ӷ *. Namely, ␻ ␯ considerably for free electrons, with ␧Ͼe␾w . One may con- the EEDFs at different phases of the field ␰ coincide with clude that the EDF of trapped electrons is almost isotropic, each other within the accuracy of the numerical simulation. while the EDF of free electrons is notably anisotropic. The The EEDFs shown in Fig. 10 are normalized according to tail of fЌ is strongly depleted due to the escape of electrons with ␧ЌϾe␾w to the wall as discussed in Sec. III. ϱ Figure 14 shows the distribution f (␧ ,r) at different f ͱ␧Ϫe␾ r͒d␧ϭn r͒/n , ͑62͒ Ќ Ќ ͵ ͑ e͑ 0 radial positions. It is seen that the tail of the distribution e␾͑r͒ decays with energy more rapidly near the axis, in accord with the discussion in Sec. III. Overall, the DMC simulation where ne(r) is the electron density at position r and n0 is the electron density on the axis. results corroborate the theory presented in earlier sections. Figure 11 shows the radial profiles of electron density and temperature for the conditions of Fig. 10. The temperature VI. DISCUSSION increases towards the wall since the EEDF is concave in the elastic energy range, which contains most of the electrons Electron heating in a gas discharge is a two-step process ͓36͔. that includes ͑a͒ electron interactions with electromagnetic Figure 12 shows the radial distribution of the rf current fields and ͑b͒ transfer of the directed energy gained from the density for different field phases ␰ ͓Fig. 12͑a͔͒ and the cur- field into the energy of thermal motion. An electron can be 3420 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55

FIG. 13. Electron distributions fЌ(␧Ќ ,r) ͑solid line͒ and f z(␧z ,r) ͑dashed line͒ at rϭ0.1R. The discharge conditions are the same as in Fig. 10.

heating rate is determined by the local value of the electric field. The energy diffusion coefficient given by Eq. ͑52͒ is 2 the product of a single kick in energy ⌬␧ϭ(eEeff␭) and the frequency of collisions ␯. When the electron mean free path exceeds the thickness of the rf sheath or the skin layer, electron heating becomes nonlocal. The processes of electron interaction with rf fields and ‘‘randomizing collisions’’ are now separated in space. When the electron mean free path exceeds the discharge dimensions ‘‘collisions’’ with plasma boundaries ͑potential barriers͒ occur more frequently than collisions with gas species. The finite dimensions of the plasma become an important consideration under these conditions since phase correlations of the rf fields could result in FIG. 12. Azimuthal rf current density versus ͑a͒ the radial posi- large energy excursions compared to the energy change in tion r for different phases of the rf field ͑labels near the curves͒ and the single interaction with the rf sheath or skin layer. On the ͑b͒ the field phases for different r. The discharge conditions are the same as in Fig. 10. accelerated or decelerated by the electric field depending on whether it moves along or against the direction of the force. The result of successive interactions depends upon phase correlations. Collisions play a twofold role. They change the direction of the electron motion and thus ͑i͒ transfer directed kinetic energy acquired from the fields into kinetic energy of random motion and ͑ii͒ randomize the field phase between succesive interactions. The average energy gain per collision is the small net difference between large actual gains and losses. We have demonstrated that phase randomization can also occur without collisions so that collisionless heating ex- ists in the ICP. The common feature of different heating regimes is the statistical nature of the heating. In all cases, heating represents a random walk of an electron along the energy axis, which is described in terms of diffusion ͑energy diffusion͒. Joule heating dominates at high gas pressures when the fields do not change appreciably over the mean free path of electrons. The Joule heating is therefore local; processes ͑a͒ FIG. 14. Electron distribution fЌ(␧Ќ ,r) for different radial po- and ͑b͒ referred to above take place at the same point and the sitions. The discharge conditions are the same as in Fig. 10. 55 ELECTRON KINETICS AND NON-JOULE HEATING IN . . . 3421 other hand, under certain conditions, electron dynamics becomes chaotic even without collisions with particles. The entire discharge volume participates in the heating process under these nonlocal conditions even though the heating fields are clearly localized. The heating regime where electron interactions with the rf field and collisions are spatially separated is referred to as non-Joule heating. We have further distinguished collisionless and hybrid heating regimes as belonging to non-Joule heating. The specific feature of collisionless heating is that it can impart energy into one direction. However, if the heating process is slow compared to collisions ͑as it typically occurs in gas discharges͒, the EDF should be almost isotropic. An analysis of electron heating and formation of the EDF requires simulations of electron kinetics on a long time scale compared to the period of the rf field, the bounce time, and the intercollision time 1/␯. The electron energy spectrum is established on a time scale that is of the order of the energy relaxation time ϳ1/␯*. During its lifetime, an average electron undergoes many elastic collisions. Moreover, during its lifetime, an average electron must generate one electron-ion FIG. 15. Electron distribution fЌ(␧Ќ ,r) for a thin skin layer: 7 Ϫ1 pair to maintain a steady state. The fields in a discharge are ␦ϭR/5, B0ϭ2 G, and ␻ϭ8.5ϫ10 s . established in such a way that during their lifetime slow electrons are heated up to energy ϳe␾w . The shape of the EDF is determined by a balance of electron fluxes along the ͓37,27,38͔ and has been recently reported for ICPs at low- energy axis. Heating produces a duffusive flux of electrons power input ͑hence relatively low plasma density where from the low-energy region where particles are abundant to Coulomb interactions are not too strong͓͒28͔. Manifestation the high-energy region where the particle density is low. In- of the nonlocal electron kinetics is therefore rather common elastic collisions produce a sink of electrons at high energies for all low-pressure plasmas regardless of the particular and a source of electrons at low energies. In the absence of mechanism of electron heating and discharge maintenance. Coulomb interactions, the EDF may substantially differ from a Maxwellian. VII. CONCLUSION The oscillatory magnetic field has a large impact on col- We have studied electron kinetics in a nearly collisionless lisionless electron heating. The Lorentz force changes the cylindrical ICP taking into account the influence of the os- direction of electron diffusion in velocity space. In the hy- cillatory magnetic field and the finite dimensions of the brid regime, however, the influence of the B field is not so plasma. An analysis of single-particle dynamics revealed that critical. In fact, DMC simulations of low-density ICP ͑no electron motion may become chaotic even without collisions skin effect͒ with and without the B field reveal a surprisingly with gas particles. We have distinguished collisionless heat- small difference in the heating rate at a frequency 13.56 ing from hybrid heating. In the hybrid heating regime, colli- MHz and argon pressure 1 mtorr. With a decrease of ␻, the sions with particles are important for randomization of elec- influence of the B field may become more important. If we tron motion. We have developed a nonlocal approach to the assume that the same electric field E␪ is required to sustain a solution of the electron Boltzmann equation in a free-flight discharge for different ␻, then a larger rf current and B field regime when the traditional two-term Legendre expansion is would be necessary at lower ␻ since E␪ϰ␻B. The higher not valid. We have calculated the energy diffusion coeffi- B field produces higher ‘‘radiation pressure,’’ preventing cient for hybrid heating regimes and identified resonance electrons from entering the skin layer of high E␪ . Thus, at phenomena caused by the finite dimensions of the plasma. lower ␻ the heating rate with account of the B field should We have used the Dynamic Monte Carlo simulations to cal- be lower than that without the B field. This was in fact ob- culate the EDF in a wide range of discharge conditions. The served in our simulations. results of the DMC simulations have been compared to theo- The heating rate and the electron energy spectrum depend retical analysis. Our studies indicate that the EDF of trapped on the entire profiles of electromagnetic and electrostatic electrons with total energy below the wall potential is almost fields in the discharge. A pronounced skin effect, together isotropic and is a function solely of total energy ␧, while the with weak Coulomb interactions among electrons, results in EDF of free electrons with ␧Ͼe␾w is notably anisotropic the following phenomenon. Due to the presence of the elec- and depends on the radial position. trostatic potential in the plasma, slow electrons are confined in the vicinity of the potential maximum near the discharge ACKNOWLEDGMENTS center. These electrons cannot reach the skin layer near the wall and be heated. In the absence of Coulomb interactions, We are grateful to V.A. Godyak, A.J. Lichtenberg, and the mean energy of these electrons can be very low and a L.D. Tsendin for reading and commenting on the manuscript. sharp peak of the EDF can be formed at low energies ͑see This work was supported in part by NSF Grant No. CTS- Fig. 15͒. This phemonenon is well known for CCPs 9216023. 3422 V. I. KOLOBOV, D. P. LYMBEROPOULOS, AND D. J. ECONOMOU 55

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