harmonic content of electron-impact source functions in

JOURNAL OF APPLIED PHYSICS VOLUME 92, NUMBER 2 15 JULY 2002

Harmonic content of electron-impact source functions in inductivelycoupled plasmas using an ‘‘on-the-fly’’ Monte Carlo technique

Arvind Sankarana) and Mark J. Kushnerb)

University of Illinois, 1406 W. Green St., Urbana, Illinois 61801

~Received 4 February 2002; accepted for publication 24 April 2002!

Electron temperatures in low-pressure~,10s mTorr! inductively coupled plasma~ICP! reactorsoperating at 10s MHz do not significantly vary during the radio frequency~rf! cycle. There can be,however, considerable modulation of electron-impact source functions having high-thresholdenergies due to modulation of the tail of the electron energy distributions~EEDs!. In manyinstances, it is convenient to use cycle-averaged values for these quantities in models due to thecomputational burden of computing and storing spatial and time-dependent EEDs. In this paper an‘‘on-the-fly’’ ~OTF! Monte Carlo technique is described to address these time-dependent plasmaparameters. The OTF method directly computes moments of the EEDs during advancement of thetrajectories of the pseudoparticles, thereby reducing computational complexities. The method canalso be used to directly calculate the harmonic components of excitation, which can subsequently beused to reconstruct the time-dependent source functions. The OTF technique was incorporated intoa two-dimensional plasma equipment model to investigate the time dependence of electron-impactsource functions in low-pressure ICP systems. We found that even harmonics dominated the sourcefunctions for high-threshold processes, and that the harmonic content decreased with increasingfrequency and increased with increasing pressure. We also observed axial pulses of excitation andincreasing harmonic content at low pressures which are attributed to nonlinear Lorentz forceacceleration and nonlocal transport. ©2002 American Institute of Physics.@DOI: 10.1063/1.1487455#

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I. INTRODUCTION

The trend in plasma processing as used for microetronics fabrication is towards systems operating at lowpressures~,10s mTorr!.1 Inductively coupled plasma~ICP!reactors operating at these pressures are typically thougas providing a continuous wave~cw! source of excitation.2–6

This is a good approximation for properties which largedepend on the average electron energy or temperature wvalue is typically considered constant during the radio fquency ~rf! cycle.7 As a consequence, multidimensionmodels for ICP reactors have traditionally used temperatuor electron energy distributions~EEDs! averaged over the rcycles to determine rate coefficients and source functionselectron-impact processes.2,4,5 Although the electron temperatures in low-pressure ICPs, a quantity dependent onbulk of the EED, may not significantly vary during thecycle, the tail of the EED may, in fact, have phase-dependproperties.8–11As a result, rate coefficients and source funtions for electron-impact reactions having high-thresholdergies may have a phase dependence.8,10

The phase dependence of electron-impact processesexcited systems has been extensively previously investigin the context of capacitively coupled discharg~CCPs!.12–16These systems, in addition to having excitatiwhich is dominated by sheath heating, typically operate

a!Department of Chemical Engineering; electronic mail: asankara@uiucb!Department of Electrical and Computer Engineering; electronic m

[email protected]

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higher pressures~100s mTorr!. As the rate of equilibration ofEEDs with the excitation source scales with the collisifrequency, source functions in CCPs operating at higpressures would be expected to exhibit more harmonichavior than ICPs operating at lower pressure. For examGraves developed a hybrid model for rf excited CCPswhich solutions were expanded in a Fourier series in tiand linear finite element basis functions in space.12 He foundsignificant modulation in space and time in rates of ioniztion and excitation. Sommereret al. investigated a similarCCP system having an interelectrode spacing of 4 cm oating in 100 mTorr helium at 13.56 MHz.13 The ionizationrates were found to peak near the sheath-bulk boundarying the cathodic phase while the expanding sheath acceates the electrons that had migrated toward the electrodeing the anodic cycle. The difference in collisionality betwethe bulk and tail electrons explained phase differencesexcitation rates produced by low- and high-energy electroMeyyappan and Govindan also investigated time-dependexcitation in one-dimensional CCP reactors.8,14 They found,for example, that in argon at 100s mTorr, ionization rates hmaxima twice a cycle with a location which propagated inthe plasma with speeds of'53107 cm/s. Surendraet al.referred to this phenomenon as a moving ionizat‘‘pulse.’’ 15 Petrovicet al.16 measured optical emission fromCCPs sustained in SF6 /N2 at frequencies of 20 kHz to 20MHz at pressures of 0.05–1 Torr. They observed similar stially dependent harmonic emission with propagation speof 33106 cm/s.

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© 2002 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

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737J. Appl. Phys., Vol. 92, No. 2, 15 July 2002 A. Sankaran and M. J. Kushner

The harmonic content discussed in these previous intigations results from, to a large degree, linear procesThese processes are periodic accelerations of electronelectrostatic fields by either oscillating sheaths resultingstochastic~collisionless! heating or oscillating electric fieldin the bulk plasma producing collisional heating. In lowpressure, electromagnetically driven devices, such as ICthe opportunity presents itself for both these linear mecnisms and for nonlinear processes, in particularvW 3BW Lor-entz forces (FL). These accelerations result from electrohaving an azimuthal component of velocity produced byazimuthal inductively coupled electric field~force FE! andthe dominantly radial rf magnetic field combining to produan axial acceleration having a second-harmocomponent.11,17,18~These effects have also been investigain the context of pondermotive forces.19,20! For example,Godyak has found thatFL /FE can exceed unity at frequencies of a few megahertz~where the rf magnetic field islarger! and pressures lower than 10 mTorr~where collisionalmixing of velocity components is less important!. The strongaxial force may, in fact, be sufficiently large to deplete loenergy electrons in the skin layer and so produce an elecstatic potential.11 These effects can be measured as a secharmonic component in the plasma potential.18 The second-harmonic component of the nonlinear Lorentz force~NLF!scales as 1/(vnm), wherenm is the total electron momentumtransfer collision frequency~including elastic and inelasticprocesses! andv is the fundamental driving frequency of thelectromagnetic field.21

Recent investigations of phase resolved emission frICPs sustained on Ar and O2 at pressures of 15–300 mTohave also shown significant harmonic content. Tadoket al.8 made tomographic measurements of emission frAr(3p5) ~radiative lifetime 90 ns! and O(3p3P) ~radiativelifetime 34 ns! at 13.56 MHz in a cylindrically symmetricICP. The scaling of the harmonic content for the Ar plasm~decreasing with increasing pressure! was different from thatfor the oxygen discharges~increasing with increasing pressure!. Tadokoroet al. suggest thatEW 3BW drift, produced bythe quasi-dc radial electrostatic field and the axial rf mnetic field, produced periodic accelerations in the azimutdirection.

The purpose of the work reported here was to investigthe time dependence of excitation rates in low pressure,systems. Our specific interest was the time-harmonic conof these rates and their two-dimensional spatial dependeThis latter behavior should provide insight to noncollisionelectron transport and NLF. In order to capture the dynamof the EEDs, a kinetic Monte Carlo~MC! simulation wasused as the electron transport module of a plasma equipmmodel. In doing so, a MC technique called ‘‘on-the-fly~OTF! was developed wherein statistics are collected forharmonic content of moments of the EED as opposedcollecting statistics on the time dependence of the EEDself. The model and the OTF technique are described inII. The model was used to investigate ICPs sustainedAr/N2 gas mixtures while varying excitation frequency, presure, and mole fraction of N2 and those results are discussin Sec. III. We found that similar to CCP systems, the ra


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of high-threshold processes such as ionization had signifiharmonic content, whereas low-threshold processes, sucvibrational excitation, were well represented by time avaged values.

II. DESCRIPTION OF MODEL

The OTF method, developed to investigate harmonicsexcitation in ICPs, was implemented into a two-dimensiosimulator, the Hybrid Plasma Equipment Model~HPEM!.The HPEM will first be briefly described followed by a description of the OTF method. The HPEM is a modular dsigned to numerically investigate low-pressure and lotemperature plasma processing reactors in two dimensiand is described in detail in Refs. 22, 23, and referenlisted therein. The main modules are the electromagnemodule~EMM!, electron energy transport module~EETM!,and the fluid kinetics module~FKM!. The HPEM iterates onthese coupled modules to generate a cycle-to-cycle qusteady state. Inductively coupled electromagnetic fieldscomputed in the EMM. These fields are used in the EETMgenerate electron-transport coefficients and electron-imsource functions. These values are produced as a functioposition by using either the electron Monte Carlo simulati~EMCS! or by solving the electron energy equation couplwith a solution of Boltzmann’s equation obtained usingtwo-term spherical harmonic expansion.~In this paper, theEMCS is employed.! The transport coefficients and sourcfunctions are transferred to the FKM which solves separfluid continuity, momentum, and energy equations forneutral and ion species, while coupling a semi-implicit sotion of Poisson’s equation for the electric potential withdrift-diffusion formulation for the electron density. The desities, conductivity, and the time-dependent electrostfields obtained from the FKM are then transferred to tEMM and EETM. The modules are iterated until a coverged solution is obtained.

The specifics of the algorithms employed in the EMChave recently been described in detail in Ref. 23 and sobe only briefly summarized here. Electron pseudoparticare initially distributed in the reactor with a spatial distribtion provided by the electron density computed in the FKThe trajectories of the pseudoparticles are advanced uthe electromagnetic fields from the EMM and the electstatic fields from the FKM.

drW

dt5vW,

dvW

dt5

qe

me~EW1vW3BW !, ~1!

where vW, EW , and BW are the electron velocity, local electrifield, and magnetic field, respectively. EW contains contribu-tions from the inductively coupled electric field from thEMM and the electrostatic field from the FKM. In the absence of an externally applied magnetostatic field, BW containscontributions from only the electromagnetic field. Time steare chosen to be less than 0.01 of the rf period, 0.01 oflocal electron cyclotron frequency, the time to cross halflocal computational cell or the time to the next collisiowhichever is smaller. The integration scheme is a secoorder predictor-corrector method. Several hundred to a


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738 J. Appl. Phys., Vol. 92, No. 2, 15 July 2002 A. Sankaran and M. J. Kushner

thousand particles are integrated in time for 10–100 scycles each iteration through the HPEM. Statistics onlocation and energy of each particle are recorded with evupdate of a pseudoparticle’s trajectory using finite-sized pticle ~FSP! techniques.24 The statistics are weighted by thtime step used for the integrating the trajectory.

Collisions are included with anisotropic scattering, icorporating null collision cross sections to account for eergy and spatial dependencies in the collision frequencFor purposes of collecting statistics, the electron enerange is divided into discrete energy bins with widths whare generally smaller at lower energies to resolve more cplex collision cross sections, and larger at higher energwhere cross sections tend to be smoother. The total collifrequencyn i within energy bini is computed by summing alpossible collisions within the energy range

n i5S 2« i

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where « i is the average energy within the bin,s i jk is thecross section at energyi, for speciesj and collision processk,andNj is the number density of speciesj. The time betweenthe collisions is randomly determined using the maximcollision frequency for the energy range of interest,Dt521/n ln(r), r 5(0,1). At the time of a collision, the reaction that occurs is randomly chosen from all the possiprocesses for that energy bin. The velocity of the electronadjusted based on the type of collision it undergoes. Ifcollision is null then the electron’s trajectory is left unaltere

The statistics for computing spatially dependent, timaveraged EEDs are updated with each move of a pseudoticle. The update of the raw statisticsFil for energy bini andmesh pointl is

Fil →Fil 1(j

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where the summation is over particlesj, wj is the weightingof the particle,« j is its energy,D« i is the width of the energybin, rW j is the location of the particle, andDrW i is the width ofthe mesh cell. The weightingwj is a product of at least twofactors; the relative number of electrons each pseudoparrepresents and the time step used to advance the trajecak is spatial weighting for the particle’s and neighborincells to distribute the particle’s contribution according to Fprinciples. The EEDsf i l are obtained from the raw statisticFil by requiring the normalization constantAl at each spatialocation to yield

Al(i

Fil 5(i

f i l « i1/2D« i51. ~4!

Given thef i l at the end of the EMCS, the electron tempeture, collision frequency and electron-impact rate coefficieare computed as a function of position.~The electron tem-perature is defined by convention to beTe52/3 ^«&.! The


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electron impact rate coefficient (kml) for electron-impactprocessm and locationl is then obtained from

kml5E0

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or equivalently,

kml5( iFil v ismi

( iFil, ~6!

wherev i is the electron speed.This method of producing rate coefficients or sour

functions produces spatially dependent but time-averavalues. To obtain time-dependent values,Fil would need tobe additionally binned by phase in the rf cycle. In doing sthere are implications with respect to memory requiremeand computational burden to obtain acceptable statistics,ticularly for three-dimensional applications. To address thissues, a method of sampling particles was developedwhich one needs to calculate only moments of the distrition function and this is done ‘‘on-the-fly’’~OTF!. As a re-sult, in contrast to the conventional Monte Carlo method,time-dependent distribution functions are not calculatedplicitly and thus are not stored either. The basic algorithjust described for the conventional EMCS are retained. Othe method of collecting statistics is different.

In the OTF method the collection of raw statisticsFil isreplaced by a continuing summation whose asymptotic vain time produces thekml described by Eq.~6!. That is, at timesteps, we compute

Dkmls 5(

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1

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~7a!

Dwls5(

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1

2DrW l 1kD2rW j G , ~7b!

whereDkmls andDwl

s are the incremental updates to the racoefficient and total weighting. The rate coefficient for prcessm at times is then obtained from

kmls 5

kmls21wml

s211Dkmls

wls where wl

s5wls211Dwl

s . ~8!

At any given instant during the EMCS, values for the racoefficients are immediately available based on the statistaken to date. Although continuously updating the rate coficients in this manner is not absolutely critical to obtainithe time-dependent values discussed later, it is conveniethat the rate coefficients are available at any time for use,example, by another processor during a computationallyallel implementation of the model. As time progresses amore statistics are gathered,kml simply becomes more accurate. The OTF method has potential advantages over theventional EMCS by being more accurate than calculatand storing intermediate values of the EED followed by cculating the rate coefficients. Although the OTF bins p


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ticles in space, it avoids discretization or the binning of telectrons in energy and time, and so can utilize the full ctinuum nature of the EED. The method is advantageous fa computational standpoint as well. Since moments ofdistribution function are less sensitive to statistical noise tthe EED itself, fewer particles are required in the simulati

One of the features of the OTF technique is the abilityadditionally calculate the Fourier components of telectron-transport coefficients. From these components,can then reconstruct the time dependence of electron-imreactions. To calculate these Fourier components the cosponding harmonic terms are incorporated into Eq.~7!. ~Thefundamental frequencyv is taken to be the driving frequencof the antenna.! For example the Fourier component for thn-th harmonic at frequencynv is obtained by replacing Eq~7a! with

Dknmls 5(

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The normalization factor is obtained in the same fashionin Eq. ~7b!. This procedure produces a complex rate coecient. The amplitudeSnml

I and phasefnmlI of the source func-

tion for then-th harmonic for theI -th iteration through theEMCS is then obtained from

SnmlI 5@e# I 21uknmluNI 21,l , fnml

I 5tan21Im~knml!

Re~knml!, ~10!

where@e# I 21,l andNI 21,l are the electron density and densof the gas collision partner at the end of the previous itetion. The time-dependent electron impact source functiused in the continuity equations for plasma species incurrent iteration are then

SmlI ~ t !5maxF0,(

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wherenm is the number of harmonics computed. The soufunction for any given electron-impact event should alwabe positive. The maximum function in Eq.~11! is used toaccount for noise in the EMCS which might result in the suof the phase weighted amplitudes being negative.

The classic definition of skin depth is for a plasmawhich there are no sources of ionization. Although thoconditions do not strictly apply here, we nevertheless useterminology skin depth~or skin layer! to denote the distancin the amplitude of the electric field decays by 1/e.25

For validation purposes, results from the HPEM usithe conventional Monte Carlo method and the OTF tenique were compared for an ICP reactor using a 5 mTorrAr/N2590/10 gas mixture. The reactor has a four-turn flantenna and a 6-cm substrate to window gap, as showFig. 1. Gas is injected through a shower head nozzlepumped annularly from around the substrate. The final retor averaged gas temperature was 415 K for a power desition of 650 W at 13.56 MHz.~These are the ‘‘base case


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operating conditions.! The resulting time-averaged electrodensities and time-averaged source functions for electimpact ionization of argon are shown in Fig. 1.~The sourcefunction for ionization is@e#kI@Ar# wherekI is the rate co-efficient for direct electron-impact ionization of the grounstate.! The spatial distributions and magnitudes of the eltron densities are essentially the same for the conventioMC and OTF methods, with the OTF providing a few pecent higher density, resulting from a few percent highsource function. The spatial distribution of the source funtion has small differences at small values, largely a resulthis statistical noise. Due to the more collisional nature ofAr/N2 gas mixture compared to only argon, the electrodensity peaks in the vicinity of the maximum in the sourfunction, as opposed to the pure Ar case which, for otherwsimilar operating conditions, would have an on-axis peak.Te

is also more peaked in the region of largest power deposi~under the coils! compared to the pure Ar case where telectron temperature has less severe gradients. The presof N2 reduces both the energy relaxation distance~due toexcitation of the low-lying vibrational and rotational state!and reduces the electron thermal conductivity, therebystricting the peak in electron temperature.

III. HARMONIC CONTENT OF SOURCE FUNCTIONS

In this section, the harmonic content of electron-impasource functions produced in ICPs sustained in Ar/N2 gasmixtures will be discussed. The rational for this choicegases is that vibrational excitation of N2 has low-thresholdenergies~,1 eV! and is produced in large part by the bulkthe EED. Electron-impact processes with Ar, particulaionization, are high-threshold reactions and hence are setive to the tail of the EED. We can expect two classesharmonic behavior, both of which are likely to have an evharmonic component. The first dominantly results from clisional heating by the azimuthal electric field,Eu . The spa-tial location of this harmonic excitation should be where tEu is largest, in the skin layer. Since acceleration is amuthal~in this case, parallel to the top dielectric!, even non-collisional heating may not stray far from the skin layer. Tmodulation of the EEDs in this manner, is dependent onrf frequency and the energy-dependent collision frequenand generally scales asnm /v. The tail of the EED, usuallybeing more collisional than the bulk~in this case, by a factorof two or more!, is expected to be more modulated than tbulk distribution in the limit that heating is collisional, anso high-threshold processes should have more modulaThe second source of modulation results from NLF. Its snature is that NLF acceleration will be dominantly in thaxial direction with a second-harmonic component.

As diagnostics for the differences between bulk andbehavior, we will examine electron source functions forbrational excitation of N2(v51) with threshold energyD«50.29 eV, and ionization of Ar with thresholdD«516 eV.The approximate trends for expected behavior may bedicted by the scalings shown in Fig. 2. Here, the tomomentum-transfer collision frequencies~elastic and inelas-tic! for the bulk and tail of the EED are plotted as a functi


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FIG. 1. ~Color! Time-averaged electron density in an ICP reactor obtained using~a! conventional MCS method and~b! OTF method; and time-averagesource functions for electron-impact ionization of ground-state argon using~c! conventional MCS method and~d! OTF method. Operating conditions arAr/N2590/10, 5 mTorr, 13.56 MHz, and power deposition of 650 W.

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of average energy for Ar/N2590/10. These results were obtained by solving the zero-dimensional, time-independBoltzmann’s equation using a two-term spherical harmoexpansion while varying the electric field/gas number den

FIG. 2. Collision frequency and specific power loss rate as a functionaverage electron energy for an Ar/N2580/20 mixture at 5 mTorr. The tail ofthe EED is typically more collisional than the bulk for both momentutransfer and power loss.


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(E/N). The tail of the EED was arbitrarily defined as cotaining those electrons having energies greater than twiceaverage energy. Over the range of expected average ene~4–6 eV!, the collision frequency of electrons in the tail othe EED is significantly larger than that in the bulk. Thhigher-collision frequency also translates into a shorter mfree path. For rf frequencies of a few megahertz both theand the bulk are sufficiently collisional that both segmentsthe EED should have modulation. For rf frequencies of 1MHz, it is likely that only the tail will be modulated. Thesecond parameter in Fig. 2 isn«5(1/«)(d«/dt), the charac-teristic frequency at which electrons of a given energy lothat energy. At low-average energies, the division betwthe bulk and the tail of the EED sweeps across the vibtional excitation cross sections of N2 , resulting in nonmono-tonic behavior forn« . At higher-average energies, it is agathe tail electrons which lose their energy at a higher rate,so would be expected to be more modulated during thecycle.

As a point of departure, the time dependence~as de-picted by sequential phasesf during the rf cycle! of thesource functions for ionization of Ar when including fou

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harmonics for the base case conditions are shown in Fig.obtained with the OTF method. Note that only the first hof the cycle is shown as, discussed below, the even harmics dominate.~For this figure and the results which follow,phase of zero corresponds to when the electric field adjato the dielectric and the antenna current cross zero.! The totalelectron-impact source functions shown here are the sumthe time-averaged value~the zeroth harmonic! and the har-monic amplitudes multiplied by their corresponding phas

FIG. 3. ~Color! Source functions for electron-impact ionization of argon fthe base case conditions~Ar/N2590/10, 5 mTorr, 650 W, 13.56 MHz! fordifferent times during the rf cycle, as indicated by the phase notation in efigure. As even harmonics dominate, results are shown only for the firstof the cycle. Ionization rates are significantly modulated with nonlocalhavior demonstrated by pulses which propagate across the reactor.


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The ratios of the magnitude of the spatially averaged hmonic amplitudes compared to the time-averaged valuSn /S0 are shown in Table I. The source function for Ar1 isnear its maximum value, located under the coils, betweenphasesf50 andp/10. The source function in this volume iheavily modulated, reaching its minimum value appromately p/2 later in phase. The volume averagedSn /S0 forthe second and fourth harmonics are 0.22 and 0.05, restively, compared to 0.02 for the first and third harmonicThe location and periodicity of this excitation would impthat the harmonic heating has a large collisional compondue to the twice a period peak in the magnitude ofEu . Thereis substantial evidence for noncollisional or long-mean-frpath transport. For example, note the local extrema inionization source function which propagates axially acrothe reactor, resulting from electrons having been accelerfrom a volume in the skin depth. This propagating excitatilikely results from electrons accelerated by NLF and retaing their energy through noncollisional transport acrossreactor.

The amplitudes of the Fourier components as a functof position for the Ar1 source function are shown in Fig. 4The electromagnetic skin depth is approximately 2 cmthese conditions. The amplitude of the zeroth componenthe time-averaged source function shown in Fig. 1~d!. Itsmaximum is located under the coils where the inductivcoupled electric field is largest and decreases at larger rdue to a falloff in electron density, the transport of higenergy electrons from the smaller volume of the electromnetic skin depth to the larger volume of the periphery of treactor and to the depletion of NLF accelerated electroThe source function is most highly modulated near its mamum and in the path of the axial excitation pulse as it progates across the reactor. In these regionsS2 /S0'0.5. Thefirst and third harmonics should, in principle, be negligibgiven the 2v content of the acceleration by the inductivecoupled electric field and NLF. We do see, however, spatiadiverse, though somewhat numerically noisy, amplitudesthe first and third components, with maxima directly undthe coils. These amplitudes, as well as much of the fouharmonic, result from the nonlinear dependence of excitarates on excursions of the tail of the EED which in tugenerate Fourier components, as discussed below. Forreason alone, the amplitude of the odd components shscale withnm /v. The peak amplitude of the second hamonic, being more than half that of the time-averaged vasignifies that the tail of the EED is highly modulated in thelectromagnetic skin depth, a consequence of both csional processes, and noncollisional transport and NLF h

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TABLE I. Ratio of amplitude of harmonicsSn to time averaged amplitudeS0 .

Sn/S0

Harmonic Ar ionization N2 (v51) excitation

1 0.019 0.0072 0.222 0.0413 0.017 0.0054 0.049 0.008


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FIG. 4. ~Color! Amplitudes of the spatially dependent Fourier coefficients,Sn for harmonic n, for ionization of argon for the base case conditio~Ar/N2590/10, 5 mTorr, 650 W, 13.56 MHz!. The even harmonics dominate, having amplitudes approximately 20% that of the time-averaged valueS0 .

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ing which accelerate~or ‘‘launch’’ ! electrons out of the skindepth. The peak amplitude of the fourth component, isproximately 0.15 that of the zeroth, and occurs deeper inskin depth where the electric field is largest. Another posscontribution to the fourth harmonic is backscattered electrwhich, having been initially accelerated forward by the avancing electric field, are back scattered and acceleratsecond time during the same half cycle.

Significant harmonic content in ionization rates of Awas also observed by Oh and Makabe in their particle-in-simulations of a 5 cmdiameter, cylindrically symmetric, solenoidally driven ICP.10 At a pressure of 300 mTorr and frequency of 6.78 MHz, the ionization rate was 100% modlated in time at the second harmonic. At 27.12 MHz, tmodulation was still significant whereas at 100 MHz, littmodulation was observed.

The source functions for vibrational excitation of N2(v51) are shown in Fig. 5 as a function of position for diffeent phases during the rf cycle. There is significantly lemodulation in both space and time for this low-threshoprocess compared to the higher-threshold ionization procThe volume averaged fractional harmonic content,Sn /S0 are0.041 and 0.008 for the more dominant second and foharmonics, and less than 0.007 for the first and third harmics. This lower degree of modulation reflects the fact t


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vibrational excitation, with its 0.29 eV threshold, largely rsponds to the modulation in the bulk portion of the EEDue to the lower degree of collisionality of these electrocompared to the tail, they retain their energy for larger frations of the rf cycle after acceleration, and so producelarger dc component of the excitation rates.

To further illustrate the noncollisional nature of the NLheating which launches energetic electrons to locatideeper than the electromagnetic skin depth, simulations wperformed for a plasma reactor with a taller plasma regwhile keeping other conditions the same as the base cThe time dependence of the ionization source functionsthis reactor are shown in Fig. 6. The amplitude ofEu isshown in Fig. 7. The field is absorbed and decays to negible values~, 10 mV/cm! 3–4 cm above the substrate. Thionization source function in the skin depth builds througout the positive half of the cycle and reaches its peakproximately at the zero crossing which implies some sstantial contribution from noncollisional heating. The pulof excitation which propagates across the reactor appeaoriginate close to the peak in the rf cycle. The ionizatipulse propagates at approximately 3.63108 cm/s, corre-sponding to an electron energy of 35–40 eV. The pupropagates across the reactor until these energetic elec


ct

ionit-g.Th

-w

ula-and

the rfent

zi-ld,

n ifi

gy

orn ofreby


impact onto the substrate, crossing regions where the elefield is essentially zero~see Fig. 7!.

The temporal dynamics of the ionization source functare significantly different when the rf magnetic field is omted from the electron acceleration terms, as shown in Fifor what would otherwise be the base case conditions.ionization source functions with the rf magnetic field~Fig. 3!and without the rf magnetic field~Fig. 8! both have substantial ionization beyond the electromagnetic skin depth. Ho

FIG. 5. ~Color! Source functions for vibrational excitation of N2(v51) offor the base case conditions~Ar/N2590/10, 5 mTorr, 650 W, 13.56 MHz!for different times during the rf cycle, as indicated by the phase notatioeach figure. As even harmonics dominate, results are shown only for thehalf of the cycle. There is little modulation of this low threshold enerprocess.


ric

8e

-

ever the case without the rf magnetic field has less modtion both in the electromagnetic skin depth and beyond,lacks the distinct ‘‘ballistic’’ ionization component whichpropagates across the reactor as seen for the case withmagnetic field. We therefore attribute the ballistic componof the ionization source to thevW 3BW acceleration resultingfrom the rf magnetic field.

In the plasma volume directly under the coil, the amuthal component of the inductively coupled electric fieEu , is roughly parallel to the coils~see Fig. 7!. Although the

nrst

FIG. 6. ~Color! Source functions for electron-impact ionization of argon fthe conditions of Fig. 3 in a reactor with an extended height. Propagatiothe excitation pulse occurs beyond the electromagnetic skin depth, thedemonstrating noncollisional transport.


dc

ur

e

Athithngage

o

po

stedilerere

o-lethn

ow

-n-

ta-

ld-

po-t

asi

or

ess


rf magnetic field has both radial and axial components,rectly under the coils within a few centimeters of the dieletric the rf magnetic field,Br5(1/v)(]Eu /]z), consistsdominantly of the radial component which also has contoroughly parallel to the coils. If we approximate]Eu /]z'Eu /d, where d5@me /(e2m0ne)#1/2 is the conventionalelectromagnetic skin depth, then the axial acceleration duthe vW 3BW force is approximatelyaz'qEunu /vd, wherevu

is the net azimuthal component of the electron velocity.sufficiently low pressure, there is the expectation thatelectron energy distribution will have some anisotropy wnet velocity in the azimuthal direction. For example, duriexecution of the electron MCS we calculated the avervalue ofvu /v for in the first 1–2 cm of the skin depth for thbase case conditions and foundvu /v'0.25. This value pro-duces a ratio of the axial to azimuthal accelerationaz /au5FL /FE'vu /vd'5ne

1/2«1/2/ f for antenna frequencyf where the electron density in the electromagnetic skin dehas units cm23 and the average electron energy has unitseV. For base case conditions (ne51011 cm23,«53 eV,f513.56 MHz), FL /Fu'0.2– 0.25, which qualitativelyagrees with the measurements by Godyak.17

The collisional component of heating will be largewhen the EED is equilibrated with the inductively couplelectric field. There is the expectation that the rate of equbration with the electric field will scale with the ratio of thelectron energy relaxation collision frequency to the rf fquency. For collision frequencies which are large compato the rf frequencies~high pressure, smallv!, electrons maycome into equilibrium with the rf electric field, and so prduce high-harmonic content. For large rf frequencies, etrons average over rf periods, thereby responding to onlytime-averaged field, and so we should expect low-harmocontent. On the other hand, NLF maximizes at both lfrequency~largeBrf! and low pressure~small nm! producinga largevu .

FIG. 7. Amplitude of the inductively coupled electric field for the base cconditions in the extended reactor. The electric field is only 10s mV/cmthe vicinity of the substrate.


i--

s

to

te

e

f

thf

i-

-d

c-e

ic

Even for conditions where nonlocal transport is not important and heating is largely collisional, the harmonic cotent may not simply scale withnm /v, particularly at largevalues ofnm /v due to the nonlinear dependence of excition rates onE/N. Consider the situation wherenm /v@1where there is the expectation that the local-fieapproximation~LFA! would be valid. Electron-impact ratecoefficients for high-threshold events typically depend exnentially onE/N. Therefore, a transition to LFA behavior a

en

FIG. 8. ~Color! Source functions for electron-impact ionization of argon fthe base case conditions~Ar/N2590/10, 5 mTorr, 650 W, 13.56 MHz! at

different times during a rf cycle when excluding thenW 3BW forces resultingfrom the rf magnetic field. The propagation of the ionization pulse is lpronounced.


utru

en

o-r arfor

n

do-

prects

ly

r

eil

icshon

fi-re

rouli

cisf

ng

alHz,so-ase30

de-

sses

one is

re

lue,Nro-

e.re,


largenm /v will retain a large second harmonic content balso necessitates growth in higher harmonics to reconsthis exponential dependence.

To demonstrate this sensitivity of the harmonic contof high-threshold events to increasingnm /v compared toTe , the following procedure was followed. The zerdimensional electron energy equation was integrated foAr/N2580/20 mixture over a sufficiently large number ofcycles to achieve a quasisteady state. A Fourier transfwas performed of the time dependence ofTe and the ratecoefficient for ionization of argon,kion , over the last fewcycles. The ratioSn /S0 for harmonics<10 were then com-pared. The energy equation we solved was

dS 3

2nekbTeDdt

5q2ne@E sin~vt !#2

mevm~Te!

23

2nekb(

i

2me

MiNikm~Te2Ti !

2ne(i , j

Niki j « i j , ~12!

wherekb is Boltzmann’s constant,km is the rate coefficientfor momentum transfer for speciesi having densityNi , andmassMi , andki j is the rate coefficient for inelastic collisioj for speciesi having energy loss« i j . The electric field wasoscillated atv510 MHz having anE/N amplitude of 500Td ~1 Td510217 V2cm2!. Rate coefficients were obtaineby solving Boltzmann’s equation for the EED using a twterm spherical harmonic expansion over a large rangeE/N. A lookup table for the rate coefficients usingTe as thefree parameter was then created from the results. Thiscedure and the immediately following discussion is intendto demonstrate scaling laws and not intended to map direto the ICPs of interest. The choice of 500 Td corresponda few V/cm at 5–20 mTorr, a typicalE/N in the skin depth.

Values ofSn /S0 for Te and kion for pressures of 5 and100 mTorr obtained from analysis of Eq.~12! are shown inFig. 9. At 5 mTorr, there is significant harmonic content onfor the second harmonic reflecting the low value ofnm /v~about 0.3 for the bulk and 3 for the tail of the EED! whichresults in the EED averaging the electric field over thecycle.26 In general, the harmonic content of bothTe andkion

track each other. At 100 mTorr, bothTe and,kion have sig-nificantly more harmonic content, dominantly at the evharmonics~nm /v is about 6 for the bulk and 60 for the taof the EED!. In all even harmonics,Sn /S0 for kion is larger~by more than an order of magnitude at higher harmon!that for Te . Sn /S0 for kion retain these large values througthe 20th harmonic. Askion has an exponential dependencethe electron temperature, small harmonic excursions ofTe

are amplified in a highly nonlinear fashion, requiring signicant Fourier coefficients extending to high harmonics toconstruct its time dependence.

Reactor averaged harmonic content of the Ar1 sourcefunction obtained with the OTF method as a function offrequency is shown in Fig. 10 where we have retained fharmonics in the analysis. The individual harmonic amp


tct

t

n

m

of

o-dlyto

f

n

-

fr

-

tudes are shown in Fig. 10~a! and the sum of the harmoniamplitudes, ignoring the contributions of phase factors,shown in Fig. 10~b!. The operating conditions are 5 mTorr oAr/N2590/10 and a power deposition of 650 W. Increasifrequency decreases bothnm /v and 1/(nmv), and so oneshould expect lower harmonic content for both collisionand NLF processes. For example, at a frequency of 10 Mnm /v'0.15, which is sufficient for significant amplitudefor the second and fourth harmonics from collisional prcesses. The individual amplitudes of the harmonics decreto less than 3% of the time-averaged value at aroundMHz. In this regime,nm /v'0.05 which is sufficiently smallthat electrons respond to the time-averaged field. NLFcreases with increasingv due to the reduction inBrf and solarge harmonic content is not expected from these proceat high frequency. The increase in harmonic content at lowvlikely contains contributions from both collisional~linear!and NLF processes.

The harmonic content of the ionization source functias obtained with the OTF method as a function of pressurshown in Fig. 11 for 13.56 MHz, Ar/N2590/10, and a powerdeposition of 650 W. The individual harmonic amplitudes ashown in Fig. 11~a! and their sum is shown in Fig. 11~b!.

FIG. 9. The ratio of the harmonic amplitudes to the time-averaged vaSn /S0 , for electron temperature and ionization of argon in an Ar/2

590/10 gas mixture at 10 MHz obtained by integrating the zedimensional electron energy equation.~a! 5 mTorr and~b! 100 mTorr. Theharmonic content of bothTe and kion increase with increasing pressurAlthough Te andkion have similar harmonic content at the lower pressuthe harmonic content forkion greatly exceedsTe at the higher pressure.


edo

inrellitrsLo

othn

n

s

h2

t oifie

olu

lury lue,

ry-

lue,


~The sum of the harmonics may be greater than one duthe neglect of phase factors.! For these results, we computeFourier coefficients through the eight harmonic. Basedonly collisional ~linear! arguments which usenm /v as ascaling parameter~which scales with pressure!, one shouldexpect a monotonic increase in harmonic content whencreasing pressure. This trend is seen at pressures greate15 mTorr. At lower pressures, there is also a net increasharmonic content, which is counter intuitive based on cosional arguments. However, at lower pressures, the elecmean free path increases and collision frequency decreathereby increasing the magnitude of second-harmonic Nacceleration. We can attribute the increase in harmonic ctent at low pressures to this effect.

The effect of gas composition on harmonic contentthe source functions was also investigated. To this end,HPEM-OTF was used to perform parametrizations as a fution of frequency were performed for an Ar/N2560/40 gasmixture and the results are shown in Fig. 12. The harmocontent varied with frequency similarly to the Ar/N2

590/10 gas mixture, decreasing with decreasing valuevm /v. As the collision frequency for Ar/N2560/40 gas mix-tures is about 50% larger than the leaner gas mixture, higharmonics have significant amplitude to approximatelyMHz, compared to about 18 MHz for the leaner mixture.

The contributions to the second-harmonic componenexcitation between linear processes and NLF was quantin the following manner. Parametrizations of the HPEMOTF were performed over a pressure range of 1–10 mTand frequency range of 2–10 MHz. Reactor averaged va

FIG. 10. The ratio of the harmonic amplitudes to the time-averaged vaS2 /S0 , for ionization of argon for the base case conditions except for vaing frequency.~a! Individual harmonics and~b! sum of the harmonic ampli-tudes~ignoring phase factors!.


to

n

-thatin-ones,Fn-

fe

c-

ic

of

er5

fd

-rres

e,-FIG. 11. The ratio of the harmonic amplitudes to the time-averaged vaS2 /S0 , for ionization of argon for the base case conditions except for vaing pressure.~a! Individual harmonics and~b! sum of the harmonic ampli-tudes~ignoring phase factors!.

FIG. 12. The ratio of the harmonic amplitudes to the time-averaged vaS2 /S0 , for ionization of argon as a function of frequency for Ar/N2

560/40 at 5 mTorr and 650 W.~a! Individual harmonics and~b! sum of theharmonic amplitudes~ignoring phase factors!.


es

no

re

-

her

forhar-to

ithis

ndnte

wo-od,m-ar-

ion-ingksld

s ini-

an-chttrib-

r-ell

ofntc

enon-on-

ichib-

un-r.

E.Sci.

. A

-


of S2 /S0 for ionization of argon and the time averagednm

for an Ar/N2560/40 gas mixture were computed. Valuwere obtained when including and excluding thevW 3BW termsin the acceleration for electrons in the MCS.S2 /S0 was thenplotted as a function ofnm /v and 1/(nmv).

S2 /S0 as a function ofnm /v for arbitrary combinationsof pressure and frequency is shown in Fig. 13~a!. The expec-tation is thatS2 /S0 should scale withnm /v if collisionalprocesses dominate. In all cases,S2 /S0 is larger when in-cluding thevW 3BW terms than when excluding them. Wheincluding thevW 3BW terms, there appears to be no scalingS2 /S0 with nm /v. When excluding thevW 3BW terms,S2 /S0

appears to be well correlated withnm /v for large values ofnm /v. The dashed line in Fig. 13~a! is a logarithmic fit ofS2 /S0 when excluding thevW 3BW terms. Those points whichdepart from the fit tend to be at the extremes of either psure of frequency.

S2 /S0 as a function of 1/(nmv) for arbitrary combina-tions of pressure and frequency@same cases as for Fig. 13~a!#is shown in Fig. 13~b!. The expectation is thatS2 /S0 shouldscale with 1/(nmv) if NLF processes dominate. When in

FIG. 13. Reactor averagedS2 /S0 for ionization of argon in Ar/N2560/40gas mixtures for arbitrary combinations of pressure~1–10 mTorr! and rffrequency~2–10 MHz! as a function of~a! nm /v and~b! 1/(nmv). Results

are shown when including and excludingvW 3BW terms in the electron accel

eration. The dashed lines are logarithmic fits to the data withoutvW 3BW in ~a!

and withvW 3BW in ~b!. Harmonic content at high values of 1/(nmv) is attrib-utable to nonlinear Lorentz forces and at high values ofnm /v to collisionalprocesses.


f

s-

cluding the vW 3BW terms, S2 /S0 is well correlated with1/(nmv). The dashed line in Fig. 13~b! is a logarithmic fit ofS2 /S0 when including thevW 3BW terms. Those points whichdepart from the fit again tend to be at the extremes of eitpressure or frequency. When excluding thevW 3BW terms,S2 /S0 appears to have no correlation with 1/(nmv).

Based on the scalings from Fig. 13, it appears thatthe pressure and frequency ranges investigated, evenmonic content of excitation rates is largely attributableNLF, and an appropriate scaling factor is 1/(nmv). In theabsence of the NLF, even harmonic content scales wnm /v, though the magnitude of that harmonic contentsmaller.

IV. CONCLUDING REMARKS

To resolve the harmonic content of excitation rates asource functions in low-pressure plasma sources, a MoCarlo technique was developed and implemented into a tdimensional plasma equipment model. In the OTF methmoments of the EED and their harmonics are directly coputed during integration of the trajectories of the pseudopticles. Harmonic content of source functions for Ar/N2 gasmixtures at,10s mTorr powered at,10s MHz was inves-tigated. We found that high threshold processes such asization have significant even harmonic content indicatmodulation of the tail of the EED with the half cycle peaof the inductively coupled electric field. Lower-threshoprocesses, such as vibrational excitation of N2 , have signifi-cantly less harmonic content due, in part, to the electronthe bulk of the EED being less collisional. We found evdence for significant noncollisional heating and long mefree-path transport in the form of pulses of ionization whiaxially propagate across the reactor. These pulses are auted to nonlinear Lorentz forces producingvW 3BW accelera-tion resulting from the rf magnetic field. The scaling of hamonic content was, on a reactor averaged basis, wcorrelated with 1/(nmv) when includingvW 3BW acceleration,with more harmonic content occurring with larger values1/(nmv). When excluding the NLF terms, harmonic contescaled withnm /v. When varying only frequency, harmonicontent increased with decreasingv, an effect which couldbe attributed to both NLF and collisional processes. Whchanging only pressure, we found increases in harmonic ctent at both high and low pressures. The increase in harmics at high pressure is attributed to collisional heating whscales withnm /v. The increase at lower pressures is attruted to increasing NLF heating which scales as 1/(nmv).

ACKNOWLEDGMENTS

This work was supported by the National Science Fodation ~CTS99-74962!, Applied Materials, SemiconductoResearch Corporation, and Sandia National Laboratories

1J. T. C. Lee, N. Layadi, K. V. Guinn, H. L. Maynard, F. P. Klemens, D.Ibbotson, I. Tepermeister, P. O. Egan, and R. A. Richardson, J. Vac.Technol. A14, 2510~1996!.

2G. diPeso, V. Vahedi, D. Hewitt, and T. Rognlien, J. Vac. Sci. Technol12, 1387~1994!.


em

pl

. E

S

ys

a

a

-

ma


3C. Lee, D. B. Graves, M. A. Lieberman, and D. W. Hess, J. ElectrochSoc.141, 1546~1994!.

4J. K. Lee, L. Meng, Y. K. Shin, H. J. Lee, and T. H. Chung, Jpn. J. ApPhys.36, 5714~1997!.

5I. D. Kagnovich, B. N. Ramamruthi, and D. J. Economou, Phys. Rev64, 036 402~2001!.

6E. Meeks, P. Ho, A. Ting, and R. J. Buss, J. Vac. Sci. Technol. A16, 2227~1998!.

7J. T. Gudmundsson, T. Kimura, and M. A. Lieberman, Plasma SourcesTechnol.8, 22 ~1999!.

8M. Tadokoro, H. Hirata, N. Nakano, Z. L. Petrovic, and T. Makabe, PhRev. E57, 43 ~1998!.

9K. Hou, S. Nakagarni, and T. Makabe, Thin Solid Films386, 239 ~2001!.10J.-S. Oh and T. Makabe, Jpn. J. Appl. Phys.39, 1358~2000!.11V. A. Godyak, B. Alexandrovich, and V. I. Kolobov, Phys. Rev. E64,

026 406~2001!.12D. B. Graves, J. Appl. Phys.62, 88 ~1987!.13T. J. Sommerer, W. N. G. Hitchon, and J. E. Lawler, Phys. Rev. Lett.63,

2361 ~1989!.14M. Meyyappan and T. R. Govindan, IEEE Trans. Plasma Sci.PS-19, 122

~1991!.


.

.

ci.

.

15M. Surendra, D. B. Graves, and I. J. Morey, Appl. Phys. Lett.56, 1022~1990!.

16Z. L. Pertovic, F. Tochikubo, S. Kakuta, and T. Makabe, J. Appl. Phys.73,2163 ~1993!.

17V. A. Godyak, Bulg. J. Phys.27, 13 ~2000!.18V. A. Godyak, B. Alexandrovich, R. Piejak, and A. Smolyakov, Plasm

Sources Sci. Technol.9, 541 ~2000!.19R. H. Cohen and T. D. Rognlien, Plasma Sources Sci. Technol.5, 442

~1996!.20V. A. Godyak, R. Piejak, B. Alexandrovich, and A. Smolyakov, Plasm

Sources Sci. Technol.10, 459 ~2001!.21R. B. Piejak and V. A. Godyak, Appl. Phys. Lett.76, 2188~2000!.22R. Kinder and M. J. Kushner, J. Vac. Sci. Technol. A19, 76 ~2001!.23R. Kinder and M. J. Kushner, J. Appl. Phys.90, 3699~2001!.24C. K. Birdsall and A. B. Langdon,Plasma Physics by Computer Simula

tion ~McGraw-Hill, New York, 1985!, Sec. 8-5.25N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics

~San Francisco Press, San Francisco, 1986!, p. 152.26M. J. Pinheiro, G. Gousset, A. Granier, and C. M. Ferreira, Plas

Sources Sci. Technol.7, 524 ~1998!.


harmonic content of electron-impact source functions in

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