Two-dimensional simulation of inductive–capacitive transition instability in an electronegative

plasma

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Plasma Sources Sci. Technol. 21 (2012) 045014 (11pp) doi:10.1088/0963-0252/21/4/045014

Two-dimensional simulation ofinductive–capacitive transition instabilityin an electronegative plasmaE Kawamura1,3, M A Lieberman1, A J Lichtenberg1 and D B Graves2

1 Department of Electrical Engineering, University of California, Berkeley, CA 94720, USA2 Department of Chemical and Biomolecular Engineering, University of California, Berkeley,CA 94720, USA

E-mail: emi119@comcast.net

Received 18 April 2012, in final form 27 June 2012Published 31 July 2012Online at stacks.iop.org/PSST/21/045014

AbstractPlasma instabilities are observed in low-pressure inductive discharges in the transition betweenlow density capacitively driven and high density inductively driven discharges when attachinggases are used. A two-dimensional hybrid fluid-analytic simulation is used to determine thespace- and time-varying densities of electrons, positive and negative ions, and neutral species,and electron and neutral gas temperatures. The simulation includes both the capacitive andinductive coupling of the source coils to the plasma and the neutral gas dissociation andheating. The plasma is described using the time-dependent fluid equations, along with ananalytical sheath model. The simulation is applied to an experiment in Cl2, in which gaps inthe electron and positive ion densities versus power curves were observed, with our numericalresults indicating the existence of an inductive–capacitive transition instability, correspondingapproximately to the observed gaps. The fluid calculation captures various features that are notincluded in previous global instability models. A method is developed to match the numericalresults to the global model formalism, which predicts the existence of the unstable mode, asnumerically found. The time and space variations can be used to improve the global modelformalism.

(Some figures may appear in colour only in the online journal)

1. Introduction

Plasma instabilities have been observed in low-pressureinductive discharges, in the transition between low densitycapacitively driven and high density inductively drivendischarges when attaching gases such as SF6, Ar/SF6 mixtures,O2, CF4 and Cl2 are used [1–10]. Oscillations of chargedparticle densities, electron temperature, plasma potential andlight, with frequencies usually from sub-kilohertz to tens ofkilohertz, are seen for gas pressures between a few mTorrand some tens of mTorr, with discharge powers in therange 75–1200 W, within the inductive–capacitive transition.The region of instability generally increases as the plasmabecomes more electronegative and the frequency of plasma

3 Author to whom any correspondence should be addressed.

oscillations generally increases as the power and pressureincrease. A volume-averaged (global) model of the instabilitywas previously developed, for a discharge containing time-varying densities of electrons, positive ions and negative ions,and time-varying electron temperature, but with time-invariantneutral densities [2–4, 7]. The particle and energy balanceequations were integrated to produce the dynamical behavior.The model agreed qualitatively with experimental observation.However, the experimental configurations and the spatialvariations of plasma quantities can only be approximated witha global model. Some differences that generally were observedwere that the range of pressures over which the instabilityoccurred was larger in the experiments than the global model,and that the electron densities and consequently the frequencyof the instability were somewhat higher in the experiments. Inaddition, the condition of matching the experimental plasma

0963-0252/12/045014+11$33.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

impedance to the power source, by means of a match box, wascomplicated, and differences caused by the matching remainunresolved.

A fast two-dimensional (2D) hybrid fluid-analyticalinductive reactor model [11], developed using the finiteelements simulation tool COMSOL, has been previouslyused to closely approximate experiments under steady-stateconditions. Both inductive and capacitive coupling of thesource coils to the plasma are included in the model. A bulkfluid plasma model, which solves the time-dependent plasmafluid equations for the ion continuity and the electron energybalance, is coupled to an analytical sheath model. The fluidmodel was designed to predict the changing plasma parameterswith input parameters, and was used to model an experimentwith a Cl2 feedstock gas in which gaps in the electron andpositive ion densities versus power curves appeared at thetransitions between capacitive and inductive operation. Thesegaps are indicative of the presence of instabilities. In thispaper, we use a modified version of the simulation softwareto examine the time-varying instability.

The fluid codes can be thought of as an intermediate stepbetween the experimental observations of unstable behaviorand the global model that have been previously used to examinethem, as the fluid simulations more closely approximateexperiments than global models. Also, more informationis available from the fluid code than from experimentalmeasurements, such that the connection between the fluidresults and the global model can be made tighter. In particular,the complexities of matching networks can be eliminated usingcurrent sources to drive the coil set, thus making comparisonsmore transparent. In experiments, the global model predictionsimprove as the information connecting the experiments to themodel improves [5].

In section 2 we outline the fluid method and present theresults of the instability, connecting them to the gaps in theexperiment they approximate. In section 3, the global modelis summarized, the method of connecting the model to thefluid results is presented and global predictions are given. Insection 4 we discuss the neutral dynamics during the instability.Conclusions and further comments are given in section 5.

2. Fluid simulation

2.1. Model

The fast hybrid-analytical 2D transformer-coupled plasma(TCP) reactor model, developed using the finite elementssimulation tool COMSOL, consists of four basic parts: (1)an electromagnetic (EM) model which includes both theinductive and capacitive coupling of the external coils tothe plasma through a dielectric window; (2) a plasma fluidmodel described by the 2D time-dependent equations for ioncontinuity and electron energy balance; (3) an analytical sheathmodel which approximates a vacuum sheath of variable sheaththickness as a fixed-width sheath of varying dielectric constant;and (4) a gas flow model which determines the steady-statecomposition, pressure, temperature and velocity of the reactivegas. By solving for both the capacitive and inductive fields, the

center of

(κ = 1)

0.1

0.1

0.2

0.3

0.20

symmetry (r = 0)

(κ = κ )

(κ = 4)

quartz window

wafer chuck

coils

inlet1

z

r (m)

conducting walls(m)

quartz spacer

bulk plasma

sheath

air

0.4

outlet

2 3 4

p

(κ = κ )s

(κ = 4)

Figure 1. The model geometry of the 2D axisymmetric TCP reactor.

ratio of inductive to capacitive power is calculated as modelparameters are varied, which is a key quantity when exploringthe capacitive–inductive transition instability. A detaileddescription of the different parts of the simulation and themethod of their solution can be found in [11]. Modificationsrequired for the instability analysis are outlined in appendixA. A simulation time for a typical stable TCP reactor is about70 min for a chlorine discharge and about 30 min for an argondischarge, using a workstation with a 2.2 GHz CPU and 4 GBof memory. For instability studies, however, the time steps inthe simulation must be shortened to capture the fast changesin electron density and temperature. The total time followingthe instability must also be longer to determine the completeoscillatory dynamics. The instability simulations usually takeseveral days for a single case.

In figure 1, the TCP reactor, chosen to be similar to theexperiment [12], is shown. It has an axisymmetric cylindricalgeometry with center of symmetry at r = 0 (z-axis). The gasinlet is modeled by a 2.54 cm diameter hole at the radial centerwhile the gas outlet is modeled by a 2.54 cm thick annularregion near the radial edge at the bottom of the reactor. Theouter walls are perfect conductors. The 10 cm radius wafer

2

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

1e+14 1e+15 1e+16 1e+17Center n (m )

0

1

2

3

4

5

6

R (

ohm

s)

e -3

capacitive

e

inductive

I =7.5A

Gap

I =8Arf rf

Figure 2. Discharge resistance Re due to electron power absorbed,versus electron density ne at 15 mTorr Cl2. Input current Irf wasvaried from 4 to 17.5 A, and the gap occurred between 7.5 and 8.0 A.

electrode is insulated from the outer walls by a 1.9 cm widequartz dielectric spacer. A 4 turn stove-top coil set with coilslabeled 1, 2, 3 and 4 is placed 0.32 cm above a 1.9 cm thickquartz dielectric window. The coils have a 1 cm by 1 cm crosssection and are centered at r = 4, 6, 8 and 10 cm. One endof the innermost coil (coil 1) is connected to a radio frequency(rf) current source while one end of the outermost coil (coil 4)is connected to ground. The air chamber above the dielectricwindow has radius of 18.5 cm and height of 19 cm while theplasma chamber below the dielectric window has the sameradius and a height of 20 cm. All dimensions are chosen tocorrespond to the experiment [12].

Although the experiment includes a matching network, theexact details of this network are not given in [12] to which weare comparing our results. We rather use a simpler excitationof an input current Irf , to which we parametrize our results.Matching networks can be included in the simulations andalso in the global models [3–5], but increase the parameterspace. Global models are simpler and also better understoodwith the simpler current input [4]. In the experiment, agap in the density versus power results was found between2 and 20 mTorr, with the most pronounced effects at 5 and10 mTorr. In the simulations, we mainly examine results at10 and 15 mTorr, which is the region of greatest numericalinstability. Other pressures are also examined to delineate theunstable pressure range.

2.2. Results

Using the common driving frequency of 13.56 MHz at ourbase case of 15 mTorr and 100 sccm Cl2, we vary the inputcurrent Irf from 4 to 17.5 A, obtaining, in figure 2, values of thedischarge resistance Re = 2Pe/I

2rf versus the electron density

ne, needed for the global model theory. Here Pe is the time-averaged electron power absorbed by the discharge and Re isthe corresponding discharge resistance. There is a gap whichoccurs between input currents Irf = 7.5 A and Irf = 8.0 A,corresponding to a region where stable equilibria could notbe found, i.e. regions of oscillatory behavior. The gap in the

0 0.0005 0.001 0.0015 0.0026e+16

8e+16

1e+17

1.2e+17

n (

m )

0 0.0005 0.001 0.0015 0.0020

5e+15

1e+16

1.5e+16

n (

m )

0 0.0005 0.001 0.0015 0.002t (s)

22.12.22.32.42.5

T (

V)

(a)

(b)

(c)

-e

e-3

-3

tt

tt

1

2

34

t t

t

t1

2

3

4

t

t t t

1

23

4

Figure 3. Oscillations of the discharge center (a) Cl− densityn− (m−3), (b) electron density ne (m−3) and (c) electron temperatureTe(V ) at Irf = 7.75 A for 15 mTorr, 100 sccm Cl2. The labels t1 to t4mark four different times during an oscillation period.

electron density ne is from ne ≈ 1.4 × 109 to 7.5 × 109 cm−3,while the ne gap observed for the experiment at 10 mTorr isfrom ne ≈ 1.0 × 109 to 7 × 109 cm−3 [12, figure 4]. Thecorresponding jump in power absorbed by the plasma Pabs inthe 15 mTorr simulation is 59 to 153 W. If we assume (as we didin [11]) that Pabs/Prf = 0.75, where Prf is the power deliveredby the rf source, then the Pabs gap in the 10 mTorr experimentis about 45 to 113 W [12, figure 4]. Thus, the simulationconducted at 15 mTorr shows similar gaps in ne and Pabs asthe experiment conducted at 10 mTorr. Mostly, because of thedifference in pressure and also due to differences in reactor coilconfiguration, we did not expect exact agreement for the Pabs

and ne gap between the 15 mTorr simulation and the 10 mTorrexperiment. (Since we did not know the exact coil circuit ofthe experiment, we used a standard coil configuration.)

Figure 3 shows the oscillations of the discharge center(a) Cl− density n−, (b) electron density ne and (c) electrontemperature Te at Irf = 7.75 A for 15 mTorr, 100 sccm Cl2.The ion temperatures are fixed at 0.052 V in these simulations.After an initial transient period, the oscillations become regularin time and amplitude. The physics can be qualitativelyunderstood as follows: when ne is high, n− builds up dueto attachment. When ne collapses, due to the instability,n− decays, due to an imbalance between the generation andloss, but on a slower time scale. The electron temperature Te

spikes to allow ne to recover from its depressed state. Thesemechanisms will become clearer in the following section wherethe results are compared with the global model.

In figure 3(c), we see small ∼0.02 V blips in Te(t) (e.g.between t2 and t3), which are much smaller than the large∼0.3 V spikes seen at the beginning of each oscillation cycle.These small blips are only seen in the 2D simulation and not inany global model, so at present, there is no theory that explainsthese blips. We only note that the small Te blips appear in anoscillation cycle when ne(t) has its sharpest positive gradientwhile the large Te spikes appear when ne(t) has its sharpestnegative gradient.

3

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

Figure 4. Relative electron density ne/ne max profiles at four times t1to t4 during the instability oscillation at Irf = 7.75 A for 15 mTorr,100 sccm Cl2.

The 2D hybrid fluid-analytical code allows the study of theaxisymmetric 2D profiles of the temperatures and densities. Infigure 3, the labels t1 to t4 mark four different times during anoscillation period: t1 = 5.87 × 10−4 s, t2 = 8.16 × 10−4 s,t3 = 8.89 × 10−4 s, and t4 = 1.00 × 10−3 s. Figures 4, 5 and6 give the 2D axisymmetric profiles at these four times for therelative electron density ne/ne max, the relative Cl− ion densityn−/n− max and the electron temperature Te(V ). The gray-scales go from maxima (black) to minima (white), indicatinghigher densities and Te near the coil, and also the significantspatial variations during the instability. The electronegativityα = n−/ne profiles for the four times are given in figure 7, Notefor all of these 2D axisymmetric figures that the left-hand axisis the axis of symmetry.

These figures are well worth considering in some detail asbearing both on the experimental results, where the quantitiesare not easy to measure, and on the global models, where all ofthe spatial variations are suppressed. The most salient featureof the charged particle densities and the electron temperatureTe is that the major portion of the particles and the highestTe is close to the driving coils. The implication is thatthe losses are higher than those calculated from the globalmodel assumptions, which, in turn affects the instability asdescribed theoretically [4, equation (50)]. There is sometime and space modification of the concentration of densitiesnear the coil, most notable for the relative electron densityne/ne max, particularly in figure 4(a), which occurs when thedischarge center ne(t) is at a minimum, as seen in figure 3(b).The generally higher charged particle densities near the coilsare physically understood from the higher coil driving field

Figure 5. Relative Cl− density n−/n− max profiles at four times t1 tot4 during the instability oscillation at Irf = 7.75 A for 15 mTorr,100 sccm Cl2.

Figure 6. Electron temperature Te (V ) profiles at four times t1 to t4during the instability oscillation at Irf = 7.75 A for 15 mTorr,100 sccm Cl2.

there, and the losses are higher due to the proximity of thedenser region to the upper wall. The electrons spread morefrom the instability, itself, which ejects most of the electronscreated during the inductive part of the cycle, and therefore

4

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

Figure 7. Electronegativity α = n−/ne profiles at four times t1 to t4during the instability oscillation at Irf = 7.75 A for 15 mTorr,100 sccm Cl2.

is most prominent at the end of the ejection period at timet = t1. Another effect worth noting is that the heating ismost pronounced away from the axis, due to the axi-symmetry,which is most easily observed in the ring-shaped Te profile, andalso somewhat in the n−/n− max profile. It is not noticeable inne/ne max because of the instability.

The very high αmax ≈ 160 observed in figure 7(a) occursafter the electrons have been both expelled and dispersed, whilethe negative ions have been only moderately affected in spaceand time. The more complex structures seen in the α profile(figures 7(b) and (c)) occur as the electron density recovers.Both ne and n− decrease when approaching the bottom right-hand corner, which is farthest away from the source coils. Therelative n− gradient from the source coils to the lower right-hand corner does not differ significantly between figures 5(a)–(d). However, the relative ne gradient from the source coils tothe lower right-hand corner is much sharper in figures 4(b) and(c) compared with figures 4(a) and (d). Thus, at t = t2 andt = t3, ne is falling more rapidly than n− when approachingthe bottom right-hand corner, accounting for the peaks in α

seen in figures 7(b) and (c).The above effects are physically interesting in their own

right. They may also be relevant to the use of the discharge inapplications. Finally, the results can be used in refining globalmodels, which we discuss in section 3.

The instability is quite robust. Increasing the relativedielectric constant of the window from 4 to 9 gives similarinstability results at 15 mTorr at the slightly lower current of7 A. Results at a lower pressure of 10 mTorr are similar tothose of the 15 mTorr base case, but the size of the oscillations

has been reduced. This indicates a shift in the most unstablepressure from that observed experimentally, which has amaximum gap region at 10 mTorr, but we do not consider this tobe significant. More significant is the range of pressures wheregaps in the equilibrium values of ne and Pabs were observedexperimentally, compared with the reduced pressure rangewhere an instability was found in the simulation. Althoughthe gap in ne and Pabs space is large, it corresponds to asmall change in the driving current Irf , which is explained insection 3 from the global model. In the experiment [12], a gapin the ne versus Prf curve was seen at 2, 5, 10 and 20 mTorr,becoming smaller at the low and high values. In our simulation,in addition to the instabilities observed at 10 and 15 mTorr,borders of instability were observed at 7.5 and 20 mTorr, but thesimulations at 5 mTorr were stable. These results are consistentwith previous work in which experiments were found to beunstable over a larger pressure range than the correspondingglobal models in SF6/Ar mixtures [3–5]. As mentionedpreviously, a current source is used in the simulation, so theeffect of matching networks is not investigated.

3. Global model and comparison with simulations

3.1. Basic model of the dynamics

We have previously developed a global model for the instability[2–4], and we summarize the main points in the following.We assume a weakly dissociated discharge with time-varyingquantities of electrons, positive ions, negative ions and electrontemperature Te, with the positive ion densities related to thedensities of negative ions and electrons by quasi-neutrality.The negative ion fraction α = n−/ne is sufficiently large(α � 5) that the electron density is essentially uniform withinthe bulk plasma, with the negative ions approximated by aparabolic profile. Volume averages of all densities are used,with the sheaths, which exclude negative ions, assumed smallcompared with the plasma dimensions. The model is based onthree first order differential equations for particle and powerbalance. The two particle balance equations are

dne

dt= Kizneng − Kattneng − �eA/V (1)

dn−dt

= Kattneng − Krecn2− (2)

where V = πR2l and A = 2πR(R + l) are the plasma volumeand the surface loss area, respectively, for a cylindrical plasmaof radius R and length l, with sheaths neglected. Electronsare created by ionization and lost by attachment and flux tothe walls. Negative ions are created by attachment on groundstate neutrals (density ng) and are destroyed in the plasmavolume by recombination with positive ions, with n+ ≈ n−for the usual highly electronegative case. The ionization andattachment rate constants Kiz and Katt have been calculatedas a function of Te for assumed Maxwellian electrons, andthe positive and negative ion recombination rate constant Krec

is taken to be independent of Te. The electron flux �e to thewall is a function of the plasma potential � (with respect to thewalls) through the Boltzmann relation, which is determined by

5

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

Irf

Figure 8. TCP circuit model used for global model.

the approximate equality of electron and ion fluxes, �e ≈ �+.The positive ion flux �+, derived from a one-dimensional low-pressure diffusion model [13, chapter 10], is given by

�+ = hln+uBi (3)

where, for high electronegativity, uBi = (eT−/M+)1/2 is the

ion Bohm velocity, and the edge-to-center positive ion densityratio hl can be approximated as

hl = 3

2

(1 +

l

(2π)1/2λi

)−1

, (4)

with λi the ion–neutral mean-free path.The third first order differential equation for electron

energy balance is

d

dt

(3

2eneTe

)V = Pe − Ploss (5)

where

Ploss =nenge(KizEiz + KattEatt + KexcEexc)V + �ee(� + 2Te)A

(6)

is the electron power loss. The terms proportional to neng givethe power losses for electron–neutral collisions, where Eiz, Eatt

and Eexc are the threshold energies. The term proportional to �

gives the power losses for positive ions that accelerate across asheath potential resulting from electron trapping, and the termproportional to 2Te gives the kinetic energy carried to the wallsby electrons. Pe is the electron power absorbed, consisting ofboth inductive and capacitive parts. To describe the inductivedischarge with capacitive coupling, we use a simplified lumpedelement circuit driven by an rf current source Irf , as shown infigure 8. The absorbed electron power can be approximated asPe = 1

2I 2rfRe, with

Re(ne) = RL

nen1/20

n3/2e + n

3/20

+ (ω2LC)2RC

nc

ne(7)

where RL, RC , n0, L, C, and nc are constants (although RL

and RC are functions of pressure) chosen from experimental

parameters (see [3]), or from the simulation (see appendix B).From (7) we see that Pe is a function of ne, the capacitive part isinversely proportional to ne in simplified heating models [13,chapter 11], and the inductive part has a maximum at ne =n0 [13, chapter 12]. We have modified the form used previously[3, 4] to make Re ∝ n

−1/2e at high electron densities, which is

the correct variation resulting from the skin effect, as discussedin [10].

Differential equations (1), (2) and (5) can be integratednumerically, together with the subsidiary conditions, toproduce the dynamical behavior. However, to understandthe physics more easily, it is useful to obtain a reduced setof equations by noting that typically there are three timescales. The fastest time scale is for changes in electron energy(equation (5)), the next fastest is for changes in electron density(equation (1)), and the slowest is for changes in negativeion density (equation (2)). Using this ordering, we suppressthe fastest process by considering that the electron energy isinstantaneously equilibrated in (5)

d

dt

(3

2eneTe

)= 0, (8)

i.e. Pe = Ploss, which is solved for the highly electron-temperature-sensitive term Kiz. When substituted in (1), thisyields

dne/dt = (Pe − Kene − K−n−)/(eEizV ) (9)

and repeating (2)

dn−/dt = Kattneng − Krecn2−. (10)

In (9) the coefficients are

Ke = Kattnge(Eiz + Eatt)V + KexcngeEexcV

+ hluBie(Eiz + � + 2Te)A (11)

and

K− = hluBie(Eiz + � + 2Te)A. (12)

By setting the left-hand side (lhs) of (9) and (10) to zero,

Pe − Kene − K−n− = 0 (13)

Kattneng − Krecn2− = 0 (14)

we obtain the equilibrium from the intersection in the n−–ne

phase plane of (13) and (14) The existence of an instabilitycan then be deduced from the approximate motion of thetrajectory about the equilibrium, using the separation of timescales discussed earlier, or more exactly by integrating (9)and (10) from some initial condition in the neighborhood ofthe equilibrium. The physics and resulting motion will beexplained more completely when comparing the results withan unstable numerical case. A complete exposition of thedynamics is given in [4].

6

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

Figure 9. n− versus ne at Irf = 7.75 A for 15 mTorr Cl2.

3.2. Results and comparison with simulation

Because of the detailed diagnostics available from the numericsin section 2, all of the constants can be obtained that arerequired to solve the equations of motion, as described above.Particular care must be taken to properly match the electronresistance Re to the simulation results, as this turns out to be asensitive parameter. In particular, since the shape of Re(ne)

in (say) figure 2 depends sensitively on the minimum andmaximum of Re, these values are important in constructingRe. We present the procedure for determining these quantitiesin appendix B, show how well it approximates the simulationresult, and discuss the consequences of a mismatch. Herewe take the analytic form as matched to figure 2 to comparethe resulting predicted instability with the instability observedfrom the simulation. After matching Pe(Re) from figure 2 toobtain the constants in (7), we plot n− versus ne on log–logscales in figure 9, for (13) and (14) as dark lines. Theintersection is the equilibrium point. Theory [3, 4] tells usthat this equilibrium is unstable. Superposed on this plot, witha lighter line is the n− versus ne phase space trajectory as foundfrom the simulation results in figures 5 and 4. The trajectorymoves between the capacitive and inductive negative slopesleaving the dne/dt = 0 curve approximately at the minimumand maximum, as predicted theoretically. The result is quiteclose, despite the fact that the trajectory is from the simulation,rather than from the global model, which would be obtained bynumerically integrating (9) and (10). There is a slight shift inequilibrium densities, which is not significant. Note that (14)gives a simple n− ∝ n

1/2e scaling, and in order to have a single

unstable root, there must be a single positive slope crossingof dne/dt = 0, implying that the crossing occurs where, from(13), n− is less than proportional to n

1/2e .

Further insight can be obtained from a theoreticalformulation developed in [4]. In figure 10 we plot thenormalized electron density X = ne/n0, with n0 the maximumdensity of the inductive part of Re as given by (7), versus anormalized current parameter given by

b = Keno

0.5RLI 2rf

. (15)

Between the dark lines is the region where equilibria can beunstable, for various values of a normalized electronegativity

Figure 10. X = ne/n0 versus b for Irf = 6–10 A for 15 mTorr Cl2.

parameter α0 = n−0/n0, given for the simulated case infigure 9. The values of ne versus Irf are found followinga dashed curve or curves, which correspond to the differentvalues of α0 shown in the figure. The actual simulation resultsare given by the squares, which proceed from small Irf (largeb) in the capacitive region, through the unstable region intothe inductive region. The circle is the equilibrium point infigure 9. The point labeled ‘6’ is the sixth simulation point inthe Re sequence, which is used in the matching as describedin appendix B. The reason for the narrow range of unstableIrf , with ne varying over a broad range, can be understood interms of the shape of the constant α0 curves. From figure 9, wesee that the region of instability lies between the minimum andthe maximum of the dne/dt = 0 curve, corresponding to thelower and upper boundaries (dark lines) of the unstable regionin figure 10. As we see in figure 9, this part of the curve israther flat, and therefore the rather large percentage increasein ne across the region corresponds to only a small percentagechange in n−. The consequence is that a small increase in Irf

causes the α0 dashed curves to traverse the unstable region. Amore complete account with the mathematical development isgiven in [4].

4. Neutral dynamics

Understanding the neutral dynamics is important fordiagnostics, and may be relevant for applications even thoughthey do not affect the instability. The global model, presentedin section 3, does not include neutral variations. Also, wefound that the oscillations in plasma parameters (as observedin section 2) occur even when the neutral parameters are heldconstant during the entire fluid simulation.

The hybrid fluid-analytical model was originallydeveloped to study steady-state discharges so the steady-stateneutral equations and the time-dependent plasma equationsare solved in a sequential manner [11]. That is, foreach simulation cycle, the steady-state equations for neutralpressure, velocity, temperature and species concentrations aresolved first, and the resulting neutral parameters are heldconstant while the time-dependent plasma fluid equations aresolved. Then, the steady-state equations for the neutralsare solved again with the updated plasma parameters heldconstant, and the cycle repeats. This sequential method gives

7

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

us the correct neutral parameters for steady-state dischargesand is much faster and simpler than solving the time-dependentneutral equations simultaneously with the time-dependentplasma equations. However, this sequential method does notaccurately follow the neutral dynamics during an instabilityand instead overestimates the neutral oscillations.

One way to observe this is to study the Cl number densitynCl. In a global model nCl is approximately given by

dnCl

dt= KdissnenCl2 − 1

4γrecvClnCl

A

V, (16)

whereγrec = γCl

1 − γCl/2(17)

is the effective surface recombination coefficient proposed byChantry [15], and γCl is the wall recombination probability forCl. Kdiss is the reaction rate coefficient for Cl2 dissociation,and vCl = (8kBTg/(πMCl))

1/2 is the thermal Cl velocity.Corr et al [16] found the best agreement between model andexperiment was obtained for γCl = 0.02. A more accuratemethod requiring no adjustable parameters is to use a γCl

which is a function of the Cl to Cl2 ratio, as was performed byThorsteinsson and Gudmundsson [17]. Since our simulationswere mostly run in a regime where γCl is a slowly varyingfunction of the Cl to Cl2 ratio with a value between 0.01 to0.05, we decided to just use a constant value of 0.02.

For the steady-state solution used in our sequentialmethod, the lhs of (16) is set to zero (i.e. dnCl/dt = 0)so that nCl becomes directly proportional to ne. For asteady-state discharge this poses no problem, and we get thecorrect solution. However, for an unstable discharge with anoscillating ne, the sequential approach forces nCl to followthe ne oscillations, thus overestimating the oscillations in nCl.Similar global analyses show that other neutral parameterswill also have exaggerated oscillations under the sequentialmethod.

We can still obtain the correct 2D axisymmetric time-averaged neutral parameter profiles by taking a time averageof the neutral parameter profiles obtained from the sequentialapproach over an instability period. This can be seen by takingthe time average of both sides of (16) over an instability period.The lhs goes to zero for an oscillatory function, so we getthe same time-averaged equation as that for the steady-stategas solution used in the sequential method. In figure 11,we show the 2D axisymmetric plots of time-averaged (a) gastemperature Tg (K), (b) molar fraction of Cl, (c) Cl2 densitynCl2 (m−3), and (d) Cl density nCl (m−3). These plots wereobtained by averaging over 14 equally spaced points withinan instability cycle. Since the gas is heated by the plasmaand cools at the walls, the gas temperature Tg is greatest nearthe plasma chamber center and lowest near the walls. The Cldensity is high, as expected, where the electron density is high(see figure 4), through the dissociation process. It is interestingto note in figure 11(d), the effect of the flow of Cl2 gas at theinlet in the upper left-hand corner of the plasma chamber. Theturbulent appearance near the inlet is a local numerical artifactcaused by lack of mesh refinement. However, it does notaffect the overall simulation, and increasing mesh refinement

Figure 11. Time-averaged (a) gas temperature Tg (K), (b) molarfraction of Cl, (c) Cl2 density nCl2 (m−3), and (d) Cl density nCl

(m−3) over an instability cycle at Irf = 7.75 A for 15 mTorr,100 sccm Cl2.

would have led to much longer running times. We also notefrom figure 11(b) that the Cl2 dissociation is weak since themaximum molar fraction of Cl is only about 17%. Despiau-Pujo and Chabert [10] also saw a similar low dissociation ofabout 16% to 17% in their global model of instabilities in lowpressure (3 to 7 mTorr) inductive chlorine discharges.

The sequential method is not only useful in obtainingthe time-averaged axisymmetric 2D profiles of the neutralparameters, but also in obtaining the volume-averaged timeoscillations of the neutral parameters. For each neutralparameter X, we can define a corresponding volume-integratedquantity Y ≡ ∫

VX dV , where V is the discharge volume.

Then, for each Y , we can write an ordinary differential equation(ODE) of the form

dY

dt= RY (t) − KY (t)Y (t), (18)

where KY (t)Y (t) and RY (t) are source or sink terms for Y

that do or do not depend explicitly on Y . For example, ifX is equal to the Cl number density nCl, then Y is equalto the total Cl number NCl in the discharge volume, RY (t)

includes rates of volume reactions between Cl2, electrons andions that affect Cl number, and KY (t)Y (t) include volume andsurface reactions with Cl that affect Cl number. In our example,RY (t) is dominated by the Cl2 volume dissociation rate, andKY (t)Y (t) is dominated by the Cl surface recombination rate.During the 2D fluid simulations, we collect the data forthe volume-integrated quantities Y (t), RY (t) and KY (t)Y (t).The coefficient KY (t) is obtained by dividing the data forKY (t)Y (t) with the data for Y (t). Once these data arecollected, we solve the ODE in (18) using the time-averaged

8

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

data shown in figure 11 for initial conditions. We solve forX(t) equal to (i) the Cl density nCl, (ii) the Cl2 density nCl2 ,and (iii) the mass-weighted gas temperature ρTg, where ρ ≡MClnCl +MCl2nCl2 is the mass density of the gas. The solutionsshow that over an instability cycle, the neutral parametersoscillate about their time-averaged values with tiny oscillationamplitudes. In all three cases, X/X � 1%, where X isthe peak-to-peak oscillation amplitude of X and X is the time-averaged value of X.

In order to test that the instabilities are independent ofany neutral oscillations, we conducted 2D fluid simulationsin which the neutral parameters were held fixed to the time-averaged quantities described above and shown in figure 11.In this case, we still observed the oscillations in the plasmaparameters as shown in figure 3 with the same instability periodof about 0.5 ms. As a further check, we again collected datato solve the ODE in (18) for X(t) equal to (i) nCl, (ii) nCl2

and (iii) ρTg. We found that the neutral parameters oscillatedabout their time-averaged values with oscillation amplitudesthat were larger than in the sequential method, but still verysmall with X/X � 1%. Hence, the neutral parameters areessentially constant and show little variation over an instabilitycycle, showing that the chemistry time scale is much longerthan the instability period. The same conclusion was reachedby Despiau-Pujo and Chabert [10] whose global model showedlittle variation (�3%) in nCl and nCl2 during an instability at5 mTorr; (Tg was assumed fixed in their model). The neutraloscillations shown by Despiau-Pujo and Chabert at 5 mTorrare larger than the ones we observed at 15 mTorr, mostlybecause their instability period of ≈1.2 ms was larger than ourinstability period of ≈0.5 ms; i.e. their instability period wascloser to the chemistry time scale, resulting in larger neutraloscillations.

5. Conclusions and further discussion

We have used a 2D hybrid fluid-analytical modeling technique,employing the finite elements tool COMSOL, to approximatea TCP reactor experiment in chlorine [12]. In previous work[11], the simulation was compared with the experimentalequilibrium results, obtaining good agreement when theexperiment was stable. Here we have extended the simulation,increasing the ability to follow fast time dependences, inorder to examine experimental results which showed gaps indensity versus power. We found that the gaps corresponded tounstable behavior, previously observed in other experimentswith electronegative gases, and understood qualitatively froma global model [2–8, 10]. A global model can, however, onlyroughly approximate an experiment, so leaving considerablequantitative differences between the experiment and the globalmodel predictions. The hybrid fluid-analytical simulations canapproximate an experiment more closely and therefore producea better quantitative correspondence. Furthermore, theadditional diagnostic information obtained in the simulationcan be used to improve the global model. Thus the2D simulation serves as a useful intermediary between anexperiment and the explanatory theory.

In our simulation, considerable effort was made to havethe configuration approximate the experiment. The resultsshow quite good agreement between the size of the electrondensity gap found experimentally and from the simulation. Thepressure range of the gap is somewhat shifted toward higherpressures and is somewhat smaller in the simulation. Thesedifferences are also seen, more pronounced, in global models.We have not significantly modified the global model from itsbasic form to incorporate various effects in the simulation, butthe coefficients in the global model can be made to conformquite closely to the results of the simulation, which leads togood agreement as seen in figures 9 and 10. In the discussion,given below, we will point out a few areas in which the globalmodel can be improved, in response to the simulation results.

In exploring the matching of the global model to thesimulation (see appendix B for a more complete account),we found that some of the global model approximations canbe improved. In particular the approximation that Ploss islinear with respect to both ne and n− was found to deviatesignificantly at low ne. This can be understood theoreticallyfrom the use of approximations such as (3) and (4), rather thanmore exact relations [13, 14]. However, such modificationsare at the cost of increased model complexity, so would beemployed only if significant disparities exist between theresults of simulations and a global model.

An improved approximation for the inductive power thathas been incorporated into the present global model is Re ∝n

−1/2e for ne > n0, which corresponds more closely to the

skin effect than our previous approximation. The form ∝ n−1e

used for the capacitive power could also be improved, but thishas not been explored. It was shown in [10], examining Cl2

feedstock gas, as in this paper, that the chemistry of chlorinecan be quite important in determining the unstable region. Thisis automatically included in our simulation. The chemistrycan be incorporated into the global model, as in [10], if thedifference between the simulation and global model is large.This does not appear to be the case for the comparisons givenhere, probably because the procedure described in appendix Bincorporates some of the effects of the additional chemistry.

In section 2, we presented 2D plots of the charged particledensities at different times during the oscillation, and brieflydiscussed the implications for global models. Of particularimportance was the concentration of the charged particledensities near the source coils. This tends to increase walllosses, which has a significant influence on the instability. Wealready have some solutions for wall losses that incorporatesteeper gradients in the density [13, 14]. Guided by thesimulations, these effects can be incorporated in the globalmodels by increasing the hl factor given in (4). Accordingto [4], this will lead to increasingly unstable behavior.

Our approach of solving the neutral steady-state equationsand then the time-dependent plasma equations in a sequentialmanner exaggerates the neutral oscillations. However, wefound that we can still obtain the correct time-averaged 2Daxisymmetric neutral profiles as well as the correct volume-averaged neutral oscillations by post-processing the sequentialsimulation data. We found that the neutral oscillations were�1% about the average values. Thus, indicating that the

9

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

chemistry time scale is much longer than the instability period.We also found that the neutral dynamics do not significantlyaffect the instability since the plasma parameter oscillationsobserved in the 2D fluid simulations are very similar whetherthe neutral parameters are allowed to vary or are held constantat their time-averaged values.

A topic not investigated in any detail is the effect of amatching network. Such a network affects the instability,usually increasing the stability, but in some cases can leadto additional unstable regions [3]. A general discussion of thetheory, including a matching network, within a global model,was given in [3], and the mathematical formalism presentedin [4]. In [5], we attempted to improve the comparison ofthe global model with the experiment by including matchingeffects. Significant improvements were found, but these weremostly due to a better characterization of the discharge. Someanomalies were also observed in the theoretical results, notseen experimentally. For our comparisons in this paper wechose not to include a matching network, both because thematching network was not characterized for a comparison, andbecause it theoretically complicates the comparisons and theirphysical interpretations. We leave this for future work.

Acknowledgments

This work was partially supported by the Department of EnergyOffice of Fusion Energy Science Contract DE-SC000193 andby California industries and University of California DiscoveryGrant ele07-10283 under the IMPACT program. Discussionsof this work with P Chabert are gratefully acknowledged.

Appendix A. 2D hybrid fluid-analytical simulationcode details

The 2D hybrid fluid-analytical TCP model used in thesimulations is described in detail in [11]. The model solves forboth the inductive transverse electric (TE) fields and capacitivetransverse magnetic (TM) fields generated by the source coils,as well as their couplings to the plasma. This allowed us toinvestigate the E to H transition and any associated instabilities.A quasi-neutral bulk plasma fluid model, which solves theion continuity and electron temperature balance equations,is coupled with an analytical sheath model which solves forsheath parameters such as sheath heating (of both ions andelectrons), as well as the sheath thickness and sheath voltage.A gas flow model solves for neutral species concentrations, gaspressure, temperature and velocity. The code uses Matlab andCOMSOL, a commercial finite elements partial differentialequation (PDE) solver.

We found that when simulating higher pressure andmore highly electronegative discharges, where there can besteep density gradients near the plasma–sheath boundary, ouroriginal code would encounter numerical instabilities. Theproblem was that nothing prevented the COMSOL PDE solverfrom setting the dependent variables to negative numbers whenattempting to satisfy the boundary conditions. Since wewere using the electron temperature Te and the ion densitiesas our dependent variables, we would encounter numerical

i i

I I I

v v

VV

1

21

1

1

2

2

2

Generator

v v

i i

I I

V V43

3 4

3 4

3 4

Coil set 1, 2, 3 and 4

rf

Figure 12. TCP coil circuit driven by a specified current Irf from therf generator.

instabilities when the PDE solver non-physically set thesequantities to negative numbers.

The solution was to keep the same PDE equations, butto change the dependent variables to logarithms of Te andthe ion densities. This way, even if the PDE solver varieddependent variables such as log Te and log n− from −infinity toinfinity in an attempt to match boundary conditions, the actualphysical quantities Te and n− would never go negative. Wealso changed our original boundary condition that the negativeion density n− = 0 at the plasma–sheath boundary to themore physically accurate condition that the negative ion flux�− = 0 at the plasma–sheath boundary. These changes madethe code more robust and allowed us to simulate higher pressureand more highly electronegative discharges, as well as followinstabilities.

Another change from the previous set of simulations in[11] is that, in this study, we use all four coils of the stove-topcoil set instead of only the first two inner coils. The coil circuitis shown in figure 12 for a specified input current Irf from the rfgenerator. The inductive loop voltages V l = (V1, V2, V3, V4)

and inductive loop currents Il = (I1, I2, I3, I4) are coupledwith the capacitive coil voltages vc = (v1, v2, v3, v4) andcapacitive coil currents ic = (i1, i2, i3, i4).

From Kirchoff’s voltage law applied to the circuit infigure 12, we obtain the relation

Vl = Mvc (A.1)

where

M =

1 −1 0 00 1 −1 00 0 1 −10 0 0 1

. (A.2)

From Kirchoff’s current law, we find similarly

ic = −MTIl + Iin (A.3)

where Iin ≡ (Irf , 0, 0, 0).However, when doing the simulations in [11], we used a

different M matrix where the second row was (0 1 0 0) insteadof (0 1 −1 0). Physically, this meant that in the previous study,we were only using the two innermost coils of the four-coilcircuit. Thus, the input current Irf required to generate aspecific Pabs is much higher in the previous study than inthe current one. Also, the capacitive coupling for the twoinnermost coils is much lower than that for all four coils,explaining why no E to H instabilities were observed in theprevious study.

10

Plasma Sources Sci. Technol. 21 (2012) 045014 E Kawamura et al

Table 1. Simulation data for 15 mTorr, 100 sccm Cl2 (base case).

N Irf (A) ne (m−3) Pe cap (W) Pi cap (W) Pe ind (W) n− (m−3)

1 4.00 4.81 × 1014 12.19 5 2.04 4.03 × 1016

2 5.00 6.21 × 1014 13.4 7.92 4.16 4.26 × 1016

3 6.00 8.07 × 1014 14.02 11.88 7.93 4.72 × 1016

4 7.00 1.14 × 1015 13.33 17.39 15.61 5.33 × 1016

5 7.50 1.56 × 1015 11.5 21.39 25.73 5.82 × 1016

6a 7.75 4.58 × 1015 7.5 24.64 73.87 8.01 × 1016

7 8.00 7.6 × 1015 3.5 27.88 122 1.02 × 1017

8 8.25 1.0 × 1016 2.94 29.28 151.78 1.12 × 1017

9 8.50 1.2 × 1016 2.65 30.84 175.9 1.2 × 1017

10 9.00 1.58 × 1016 2.35 33.89 217.28 1.31 × 1017

11 10.00 2.37 × 1016 2.1 40.43 289.47 1.45 × 1017

12 12.50 4.28 × 1016 2.08 60.83 455.33 1.71 × 1017

13 15.00 6.48 × 1016 2.23 84.96 617.32 1.87 × 1017

14 17.50 8.98 × 1016 2.43 111.85 781.62 1.97 × 1017

a Linearly interpolated from the two neighboring rows.

Appendix B. Global model parameters

Table 1 gives the 15 mTorr, 100 sccm Cl2 base case equilibriumdata from the 2D hybrid fluid-analytical simulations. Thedensities ne and n− are those at the center of the discharge.The values of all quantities in row 6, at the unstable 7.75 Acurrent, are interpolated from the two neighboring rows. Theparameters of the global model are extracted from these data asfollows. The electron inductive and capacitive resistances aredetermined from the data in the table using Re ind = 2Pe ind/I

2rf

and Re cap = 2Pe cap/I2rf . (The sum Re = Re ind+Re cap is plotted

versus ne in figure 2.) From (7), the inductive resistance isassumed to have the form Re ind = RLnen

1/20 /(n

3/2e + n

3/20 ),

and the two parameters RL and n0 are determined using thedata in two rows. Also from (7), the capacitive resistance isassumed to have the form Re cap = Ke cap/ne, and the singleparameter Ke cap is determined using the data in a single row.The choice of rows 5 and 11 for the inductive resistance androw 4 for the capacitive resistance gives good fits to the dataover the region of interest for the instability near the inductive–capacitive transition (row 6).

The ratio of Katt/Krec, which appears in the global modelequation (14), is determined from the values of n− and ne atthe unstable equilibrium point (row 6). The two parametersKe and K−, which appear in the global model equation (13),Pe = Kene + K−n−, are found using an average value K− ofK− from the data in the table, expressing

K−(Ke) =N∑

j=1

Pej − Kenej

n−j

(B.1)

where Ke is taken to be constant and N is the number of rows.Inserting this into (13), we obtain

Ke = Pe − n−∑N

j=1 (Pej /n−j )

ne − n−∑N

j=1 nej /n−j

(B.2)

which we evaluate at the unstable equilibrium (row 6). ThenK− at the unstable equilibrium is given by K− = (Pe −Kene)/n−. The results for our discharge with R = 0.185 m

and L = 0.2 m are RL = 8.78 , n0 = 2.92 × 1016 m−3,Ke cap = 1.14 × 1015 m−3, Katt/Krec = 3.0 × 10−3, Ke =1.13 × 10−14 W m3, and K− = 3.71 × 10−16 W m3. Thesevalues are used in the global model to construct the solutionsgiven in figures 9 and 10.

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