Kinetic two-dimensional modeling of inductively coupled plasmas based on a hybrid kinetic approach

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 27, NO. 5, OCTOBER 1999 1297

Kinetic Two-Dimensional Modelingof Inductively Coupled Plasmas

Based on a Hybrid Kinetic ApproachUwe Kortshagen and Brian G. Heil

Abstract—In this paper, we present a two-dimensional (2-D)kinetic model for low-pressure inductively coupled discharges.The kinetic treatment of the plasma electrons is based on ahybrid kinetic scheme in which the range of electron energiesis divided into two subdomains. In the low energy range theelectron distribution function is determined from the traditionalnonlocal approximation. In the high energy part the completespatially dependent Boltzmann equation is solved. The schemeprovides computational efficiency and enables inclusion of elec-tron–electron collisions which are important in low-pressurehigh-density plasmas. The self-consistent scheme is complementedby a 2-D fluid model for the ions and the solution of the complexwave equation for the RF electric field. Results of this model arecompared to experimental results. Good agreement in terms ofplasma density and potential profiles is observed. In particular,the model is capable of reproducing the transition from on-axis tooff-axis peaked density profiles as observed in experiments whichunderlines the significant improvements compared to modelspurely based on the traditional nonlocal approximation.

Index Terms—Electron kinetics, hybrid plasma model, induc-tively coupled discharge.

I. INTRODUCTION

I N RECENT years the design cycle for semiconductor man-ufacturing equipment has shortened to about 18 months.

The ever shorter period of development and the need for cost-effective design of new plasma processing tools has created de-mand for efficient, engineering-type plasma modeling. Plasmamodels which can be used as predictive tools for dischargedesign have to be both comprehensive and computationallyefficient. Models have to provide an accurate description ofcharged and neutral species production and transport, RFand DC electric field profiles, as well as plasma-substrateinteraction. A model useful for engineering design has tobe capable of addressing two-dimensional (2-D) [1]–[10] oreven three-dimensional (3-D) [11] discharge geometries. Afurther requirement for engineering discharge models is thatthey should be executable on workstation type computers in“reasonable” CPU times of the order of some minutes to afew hours.

Manuscript received December 2, 1998; revised April 7, 1999. This workwas supported by the National Science Foundation under Grant ECS-9713137and supported in part by the University of Minnesota, SupercomputingInstitute for Digital Simulation and Advanced Computation.

The authors are with the Department of Mechanical Engineering,University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]).

Publisher Item Identifier S 0093-3813(99)08118-7.

One of the most computer-intensive problems in dischargemodeling is the description of the plasma electrons. On theone hand an accurate knowledge of the electron distribu-tion function (EDF) is required for a meaningful treatmentof the electron-induced plasma chemistry and the chargedparticle transport. On the other hand, the computation ofthe electron distribution function presents the problem withthe highest dimensionality in the plasma model. The factthat the EDF in typical low pressure processing plasmassignificantly deviates from a Maxwellian distribution and thatmany heating mechanisms related to anomalous skin effect[12], [13] and stochastic heating [7], [14]–[16] are kinetic innature requires a kinetic, i.e., energy-resolved treatment of theEDF. This adds another dimension, “the energy dimension,”to the spatial complexity of the problem. Currently, mostmodeling approaches try to circumvent this problem by usinga fluid approach to describe the plasma electrons [1]–[4], [6],[8]–[11]. However, in particular, in the case of low pressurehigh density discharges, which operate in a pressure rangeof only 1–2 Pa, the typical electron mean free path can beof the order of the discharge dimensions thus making a fluidapproach highly questionable.

Several years ago the “nonlocal approximation” (NLA)attracted considerable attention as an approximation whichenables computationally efficient kinetic treatment of the elec-tron. The NLA was originally developed by Bernstein andHolstein [17] and Tsendin [18]. Within the past few years ithas been successfully applied to the modeling of low pressureglow discharges [14], [19]–[35]. The basic idea of the NLA isto describe the EDF in terms of the total energy of electrons(kinetic plus potential energy in the ambipolar space chargefield). A consideration of the different time scales for spatialmotion of the electrons in the discharge and of their “motion”in total energy space leads to the conclusion that: a) the EDFcan be considered as a spatially uniform function of totalenergy and b) that this function can be computed from aspatially averaged form of the Boltzmann equation which is aone-dimensionalordinary differential equation. The spatiallyresolvedEDF of kinetic energyis then determined using a“generalized Boltzmann relation” for non-Maxwellian EDF’s.The applicability of the NLA is determined by the so calledenergy relaxation length of the electrons. If is large com-pared to typical discharge dimensions, the NLA provides anaccurate description of the electron kinetics. For more detailson the NLA the reader is referred to the reviews [5], [36].

0093–3813/99$10.00 1999 IEEE

1298 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 27, NO. 5, OCTOBER 1999

For modern plasma processing tools, however, large scaleplasma uniformity is of increasing importance. Future plasmatools will have to provide plasma conditions uniform to withina few percent over areas of 300 or 450 mm in diameter. Evenlarger uniform plasmas are needed for the processing of flatpanel displays. For such large scale discharges the traditionalNLA will be applicable only at extremely low pressure whichmay not be of practical interest in actual production processes.Moreover, the fact that the EDF within the applicable rangeof the traditional NLA is determined only by spatial averagesof the RF electric field is undesirable since it limits theeffectiveness of tailoring the RF field configuration in orderto obtain the desired plasma density profile. Hence, for anumber of reasons one can conclude that the modern dischargetools are explicitly designed to operate in a regime in whichthe traditional NLA is no longer strictly applicable. For thisregime, in which slight to modest deviations from the fullynonlocal behavior of the EDF persist, an extended approachhas been proposed by Kolobov and Hitchon [25]. For reasonswhich will become obvious in the following section we willrefer to this approach as the “hybrid approach.” We havedeveloped a fully self-consistent, 2-D, kinetic plasma modelbased on this “hybrid approach.” While still preserving most ofthe computational efficiency of the traditional NLA, our modelis physically more appropriate for the modeling of large-scalelow-pressure high-density discharges than models based on thetraditional NLA. The model is capable of describing featuresobserved in actual experiments which could not be predictedwithin the framework of the traditional NLA.

This paper is organized as follows. In the next sectionwe give a brief outline of the kinetic plasma model for thelow-pressure ICP based on the “hybrid kinetic model.” Sincewe will verify the capabilities of our model by comparisonsto an actual experiment, we will give a brief description ofthe experiment used in Section III. In Section IV we presentresults of our models and a critical comparison to experimentalresults. Conclusions are presented in the last section of thepaper.

II. THE HYBRID 2-D KINETIC MODEL OF THE ICP

The plasma model presented in this section mainly consistsof three modules which are coupled in an iterative numer-ical scheme to achieve a self-consistent plasma description.The model addresses an inductively coupled plasma (ICP) atlow pressure. We assume that the plasma can be consideredazimuthally symmetric so that the model can be formulatedin two spatial dimensions for the radius and the height

. The electron kinetics is treated in terms of the “hybridapproach.” For the ions a simple fluid model is used. Themodel is complemented by an electromagnetic module for thedetermination of the RF induced electric field.

A. The Hybrid Kinetic Model

The hybrid kinetic approach is based on the solution of theelectron Boltzmann equation. We make the standard assump-tions that the two-term expansion of the EDF into sphericalharmonics [37] and the effective field approximation [38], [39]

are applicable. In this case the main part of the distributionfunction will be isotropic in velocity space and stationary.Temporal fluctuations are limited to higher order terms in thespherical harmonics expansion. We chose a formulation in totalelectron energy in volts

(1)

where is the electron mass, the elementary charge,the electron velocity, and the stationary ambipolarpotential. In the total energy representation the kinetic equationfor the main part of the EDF reads

(2)

The details of the derivation of this equation have beenpresented in a previous review paper [5]. The electron velocityhas to be considered as a function of the total energy andposition: . Thesame is true for the momentum transfer collision frequency

. The symbolis the effective field strength for Joule electron heating by theRF field with the amplitude and the angular frequency

. We currently do not consider stochastic heating in thisstudy. is the collision operator. It involvesterms for inelastic (excitation) collisions, ionization, elasticcollisions, and Coulomb collisions. The inelastic collision termis given by

(3)

where the kinetic energy hasbeen used for clarity. The sum is extended over all inelasticexcitation processes. The are the respective collisionfrequencies and the respective threshold energies. Theionization term in the approximation that the excess energyof the primary electron is evenly distributed on the primaryand the released electron is [40]

(4)

The sum is extended over all ionization processes. Theare the respective ionization frequencies and the

respective ionization thresholds. For both ionization as well asexcitation, only processes involving gas atoms or moleculesin the ground state are considered here. The inclusion of

KORTSHAGEN AND HEIL: KINETIC 2-D MODELING OF INDUCTIVELY COUPLED PLASMAS 1299

Fig. 1. Scheme of the partition of the domain of integration used in thehybrid kinetic scheme.

collisions with excited atoms or molecules would require acollisional-radiative model to determine the concentration ofexcited species which is beyond the scope of the current study.

The elastic collision term reads

(5)

with , the atom mass, the gas tempera-ture, and the Boltzmann constant.

The collision term for electron–electron collisions is givenby a Fokker–Planck term [41]

(6)

with

(7)

(8)

(9)

is the dielectric constant and is the Coulomb loga-rithm [41].

In principle, the kinetic (2) can be solved numerically. In ourcase of a 2-D geometry in and , (2) is a 3-D partial differ-ential equation. However, there are two aspects which highlycomplicate the efficient numerical solution of this equation: a)the Fokker–Planck term (6) introduces a nonlinearity into theproblem and b) the domain of integration is irregular with acurved boundary at the space-charge potential whichconstitutes the minimal total energy at each position ,i.e., ; see Fig. 1.

It is worthwhile to consider some physical arguments onhow to solve (2) by approximation. As mentioned above, thesolution of (2) can be highly simplified if the NLA is applica-ble by solving a spatially averaged kinetic equation which isonly an ordinary differential equation. This approximation isjustified if the energy relaxation length is large comparedto the discharge dimensions over the entire energy range ofinterest for the EDF. If we move to larger discharges and/orhigher pressures this argument may not be fulfilled for theentire energy range but it may still hold at least for part of it.The energy relaxation length strongly depends on the type ofcollisions which can occur at a given energy

for elastic collisions

for inelastic collisions.

(10)

In the energy range in which only elastic collisions arepossible, exceeds the electron mean free pathby thesquare root of the mass ratio which is a number ofthe order of 100. The reason for this is that the energy lossof electrons in elastic collisions is very small. Supposedly,vibrational and rotational excitation collisions behave similarlyas elastic collisions due to the relatively small energy loss forthe electrons involved in these processes. However, this topichas not been studied in the literature up to now. In electronexcitation “inelastic” collisions, the energy loss is of the orderof the entire kinetic energy and the energy relaxation is muchfaster. The energy relaxation length in this range is thus muchshorter and it scales with the mean free path for inelasticcollisions . The idea of the “hybrid approach” is to make useof this physical division of the electron energy range into twoparts and to treat these two parts with different approximationsto solve the kinetic (2) (see Fig. 1).

1) The low energy “elastic range“ with less than thesmallest excitation threshold energy: in this rangeat pressures of a few Pa is of the order of several meters.The EDF can be considered fully nonlocal, meaning thatthe EDF is a spatially uniform function of total energy.It can be determined within the traditional NLA solvingonly a spatially averaged kinetic equation which is anordinary differential equation. In this case it is easy toinclude Coulomb collisions, as discussed in more detailbelow, which is mandatory for high-density plasmas.

2) The high energy “inelastic range“ withgreater than thesmallest excitation threshold energy: in this energyrange will be smaller than the typical dischargedimensions so that the full kinetic equation (2) has tobe solved. However, the domain of integration for thisenergy range is typically regular with straight bound-aries. Also, for not too high degrees of ionization it canbe justified to neglect Coulomb collisions as discussedbelow. This allows us to treat the full kinetic equation(2) as a linear equation which strongly simplifies itssolution.

The inclusion of Coulomb collisions requires some dis-cussion. The influence of Coulomb collisions on the EDF isdifferent for the “elastic” and the “inelastic” energy range of


the EDF for two reasons. The Coulomb cross sections scaleswith 1/ which makes it particularly effective at low kineticenergies. Also, in the “elastic” range Coulomb collisions haveto compete only with energy-conserving, elastic collisions.In the “inelastic” range the EDF formation is dominated byinelastic collisions which strongly change the energy of thecolliding electron. Numerical studies [42] of the effect ofCoulomb collisions showed that modifications of the lowenergy part of the EDF can be observed for degrees ofionization as low as 10 or even smaller. In the “inelastic”range Coulomb collisions start to affect the EDF formationat degrees of ionization larger than 10. For this reason,we choose to include Coulomb collisions in the treatmentof the “elastic” range using the traditional NLA, but weneglect Coulomb collisions in the “inelastic” range. Thisapproximation limits the applicability to degrees of ionizationnot much larger than 10 . While this condition is fulfilledin the comparison to our experiments presented below, werealize the need for a more consistent inclusion of Coulombcollisions in future work.

We also want to point out that the main reason for usingthe “hybrid approach” is to efficiently obtain more accurateionization profiles, which allow the study of plasma uniformitythan would be possible with the traditional nonlocal approach.The “hybrid approach” is much more efficient and simplerto implement than the numerical solution of the full kineticequation (2). It also seems physically more appropriate thansolving the complete kinetic equation (2) over the entire energyrange but neglecting Coulomb collisions. However, one hasto realize that some physical information is lost by usingthe traditional NLA for the low energy part of the EDF. Inparticular, information about the electron fluxes is not directlyaccessible. If this information is required one either has toresort to solving the full equation (2) or to calculating the firstorder correction of the traditional NLA which would yieldinformation about the electron flux [43]. However, this is notthe focus of the current paper.

Fig. 2 shows experimental results obtained in our exper-iment (for details see below) which strongly support theassumptions made above. The figure shows two EDF’s inargon measured at different positions in the discharge. It isclearly seen that both EDF’s plotted as functions of the totalenergy coincide very well at low energies proving the spatialuniformity of the EDF, i.e., the applicability of the NLA forthe low energy part of the EDF. At energies above the firstexcitation threshold of argon at 11.55 eV the EDF’s startto diverge and to exhibit a spatial nonuniformity. The EDFat the position of the highest plasma potential was measuredclose to the induction coil in a region of high RF field. It thusshows a higher population of the high energy part than thesecond EDF which was measured on the axis of the dischargewhere the RF field is close to zero. Exactly the same hybridbehavior has been observed in [44] in a capacitive discharge.

Our practical implementation of the hybrid model usesthe following scheme. For the low energy part the EDF isdetermined using the traditional NLA. The spatially averagedform of (2) is solved for the spatially uniform part of the EDF

Fig. 2. Measured electron distribution functions at 2.0 Pa. The solid line isthe EDF at the position of the maximum plasma potential (r = 9 cm, z = 4

cm), the dashed line is an EDF on axis (r = 0 cm, z = 1 cm).

. For our particular case, this equation is

(11)

where is the collision operator including theFokker–Planck operator.The overlined quantities representaverages performed over the part of the volume whichcan be accessed by electrons with a given total energy, i.e.,for which . This means the average of a quantity

is defined as

(12)

Here represents the total volume of the discharge. Theaccessible volume is defined by

or in (13)

The boundary of is thus given by (seeFig. 1). The concrete form of the averaged coefficients canbe found in [29]. The computation of the spatial averagesrequires the knowledge of the space charge potentialand the RF electric field . Initially, guessed profilesfor these quantities are used which are successively improvedin an iterative scheme. Equation (1) is solved for the entireenergy range considered, i.e., the “elastic” and the “inelastic”range. This is necessary since the evaluation of the Coulombcollisions operator also requires knowledge of the high en-ergy part of the EDF. The boundary conditions for (11) are

, and , where is the maximumenergy considered. We typically solve (11) on a grid with 1000node points and eV. After having foundover the entire energy range only the part with is usedfurther.

For the high energy part of the EDF ( ) the com-plete kinetic equation (2) is solved using simultaneous over-relaxation (SOR). The boundary conditions for this equation


have essentially been described by Busch and Kortshagen [45].Briefly:

1) boundary at : ;2) the three “wall boundaries” at , (plasma

height), (plasma radius): for energiesless thanthe potential energy at the wall

where is the direction normal to the wall. For energiesabove

(14)

with , and the solid angle of theloss cone in velocity space given by [45], [46]

(15)

with the potential at the sheath boundary. Condition(14) basically states that the fraction of the thermal fluxof electrons scattered into the loss cone, which leads toloss of electrons to the wall (left-hand side), has to bebalanced by spatial flux toward this position (right-handside). In our model we assume nonconducting walls. Thewall potential is found from the requirement of locallybalanced electron flux (left-hand side of (14) integratedover energy) with ion flux which is found from the ionfluid model discussed below [32]. Also, the potential atthe sheath boundary is found from this model. While(15) is certainly an approximation which relies on thesomewhat arbitrary definition of a sheath boundary, ithas been proven useful and accurate in comparison withMonte Carlo calculations of the EDF [46];

3) boundary : = from nonlocalsolution to ensure continuity of the electron flux inenergy space. (We have also experimented with thealternative boundary condition atwhich typically also yields a smooth transition of theEDF slope since deviations from the nonlocal behaviorof the EDF set in gradually for energies only slightlyabove .);

4) boundary at maximum : .

Since we expect only slight deviations from the fullynonlocal EDF, we use the nonlocal EDF as a startingpoint for the iterative solution. Equation (2) is then solved inthree dimensions—radial and axial direction as well as totalenergy on a mesh of 32 32 50 points, respectively.In particular, the discretization in energy witheV appears relatively coarse. However, we have tested moreaccurate discretizations with eV and 125 points inenergy directions without having found significant changes inthe results. In our practical implementation of the hybrid modelwe divide the energy range not exactly atbut at the slightlylower energy of 11.0 eV. As an approximation to simplifythe numerical treatment we currently consider the coefficients

and as slowly varying in coordinateand energy space, respectively, so that their derivatives can be

neglected compared to those of and . We willrelax this assumption in the future. The EDF determinedfrom this model is used to compute the spatial profile of theionization frequency.

Since the vast majority of the electrons at any time isfound in the elastic energy range which is described by thenonlocal EDF, we use the “generalized Boltzmann relation”between electron density and ambipolar potential known fromthe traditional NLA [5]

(16)

B. Ambipolar Potential and RF Field

The electron kinetics model is complemented by an ion fluidmodel and the wave equation for the RF electric field. Thespace charge potential in the plasma is found from a fluidmodel for the ions. For a high-density plasma the assumptionof a quasi-neutral plasma bulk is well fulfilled. If one limitsthe consideration to the description of the quasi-neutral plasmabulk up to the sheath boundary, the solution of the Poissonequation may be avoided and be replaced by using the quasi-neutrality conditions . For pressures of a fewPa and discharge dimensions of several tens of centimeters,the ion motion is usually collision-dominated. Thus, assumingmobility-limited ion motion, the ion momentum and continuityequation can be combined to

(17)

where denotes the field-dependent ion mobility (cf. [29])and denotes the local ionization frequency. Thisequation is solved subject to the boundary condition that theBohm criterion has to be fulfilled at the sheath boundary

. Here is the ion meanfree path, denotes the Boltzmann constant, and

is the so called screeningtemperature [47]. For simplicity we use the screening temper-ature found from the nonlocal EDF . The plasma density

for a given potential can be found from (16).Equation (17) is thus a 2-D nonlinear differential equation forthe ambipolar potential .

The RF electric field for an azimuthally symmetric ICPhas only an azimuthal component , which can bedetermined from the complex wave equation. Assuming aharmonic time-dependence of the electric field we get

(18)

Here is the vacuum speed of light, is the vacuumpermeability. The symbol is the coil current densityand is the kinetic conductivity of the plasma [29].The solution of this equation is strongly simplified by the


assumption of metallic walls at , , and, so that zero boundary conditions can be used at these

boundaries. On axis ( ) the azimuthal field is also zero.Both (17) and (18) are solved on a 6464 mesh in radial

and axial direction using a multigrid solver [48].The iterative numerical scheme used to find a self-consistent

solution of this set of equations has been described in greatdetail in [29]. We present only a brief sketch of this scheme.The computation starts with a rather unsophisticated guessfor the plasma density profile: a Bessel function variationin radial and a cosine profile in axial direction. We simplyuse a flat potential profile as an initial guessfor the potential. (We use this starting potential, since wedo not want to bias the algorithm to converge to a certainsolution. Even though we have not observed multiple solutionsalso using other starting potentials, the existence of othersolutions seems at least to be possible. Multiple solutionsmight manifest in actual experiments as mode transitionsor jumps between different plasma density profiles.) Usingthe initial plasma density profile we find the RF electricfield profile by solving (18) assuming a cold-plasma Lorentzconductivity. A self-consistent set of EDF, potential, and RFfield profile is found in an iterative scheme which uses threenested loops. In the innermost loop the EDF is determined bysolving the “hybrid” model for a given coil current. Here, theSOR scheme to solve (2) is iterated until the relative changein the ionization frequency between two successive iterationsat the middle of the – plane is less than 10. Then thecomplete ionization profile is calculated. The potential profileresulting from this ionization profile is found from (17). Thepurpose of the innermost loop is to adjust the coil currentsuch that the Bohm criterion is fulfilled at the sheath boundary.Once this is achieved, the resulting potential replaces the initialpotential profile in the next-outer loop. The loop of updatingthe potential is repeated until the sum over the squared changesbetween two successive potentials taken at all 4096 potentialnodes is less than 1.0. Finally, if a self-consistent potentialprofile has been found the RF field profile is iterated in theoutermost loop using the same convergence criterion as forthe potential profiles.

The use of the various approximations described aboveenables a very efficient modeling of the discharge. Typicalcomputation times for a given set of parameters are of theorder of a few minutes up to 20 min on a Pentium II classcomputer.

III. EXPERIMENTAL SETUP

Since the emphasis of this paper is on the plasma modelthe experiment is only briefly described. The measurementshave been performed in an inductively coupled plasma whichis sustained in a Pyrex chamber with 14 cm inner radius. Thetop wall of the chamber is a flat 2.2 cm thick Pyrex plate.The bottom plate is a grounded, movable sheet metal whichenables the use of various discharge heights. In this study theheight was set to 7 cm. A punched hole pattern in the metalplate allows pumping with a 1000 liter/s turbo pump as wellas inlet of argon gas. The plasma is produced using a flat one-turn induction coil with an inner radius of 11 cm and an outer

radius of 13 cm. The coil is Faraday shielded to eliminateelectrostatic coupling to the plasmas.

A Langmuir probe with 5 mm length and 0.125 mmdiameter is introduced into the discharge through a radial slit inthe bottom plate. A – probe manipulator enables computer-controlled 2-D positioning of the probe in the discharge. In thisstudy the probe was moved in an area of 0–12 cm (fromaxis) and 1–5 cm (from bottom plate) with a resolutionof 1 cm in each direction. At each position 10 000 probecharacteristics were sampled and averaged with a fast 16 bitA/D card. The EDF is obtained using the Druyvesteyn formula[49] and absolute units. Integration of the EDF over energyyields the electron density [50]. Results previously obtainedwith this experiment have been reported in [51].

IV. RESULTS

Before discussing results of our model we want to pointout that our measurements in Fig. 2 clearly demonstrate thenonMaxwellian character of the EDF and that they underlinethe importance of a kinetic treatment of the electrons.

In Fig. 3 we present a comparison of measured and calcu-lated electron density profiles at pressures of 0.67, 1.3, and2.6 Pa (5, 10, and 20 mtorr). At the lowest pressure themeasurements show an almost uniform density profile overthe range accessible to our Langmuir probe. While even atthis low pressure a slight off-axis maximum of the densityis observed, the overall variation of the density between theaxis and the position of the maximum is less than 15%. Withincreasing pressure the experimental profiles become morenonuniform and the position of the density maximum shiftsto slightly larger radii. At 1.3 Pa the variation between themaximum and the on-axis density is about 25%, at 2.6 Pait is already more than a factor of two. The results of ourmodel reflect the general trend reasonably well. The modelproduces slightly more uniform density profiles at 0.67 and1.3 Pa. While the calculated profile at 0.67 Pa peaks on axis,a slight shift of the maximum from the axis is observed at1.3 Pa. The overall variation of the density over the rangeaccessible to the Langmuir probe is about the same as observedin the measurements. The best agreement between model andmeasurements is found at the highest pressure of 2.6 Pa forwhich a purely nonlocal model would work the worst. Boththe position of the density maximum as well as the variation ofthe density between maximum and on-axis position is almostidentical.

The observation from Fig. 3 deserves some interpretation.At the lowest pressure of 0.67 Pa our model almost yieldsthe same results as a model based on the traditional NLA[29]. In particular, the classical NLA tends to always produceon-axis maximum density profiles. An estimate of the energyrelaxation length at 0.67 Pa shows that cm up toenergies of about 20 eV. This estimate shows that the resultsof our model which show nonlocal behavior are at least notunreasonable. However, one should keep in mind that theenergy relaxation length describes only deviations of the EDFfrom the nonlocal case due to the action of collisions but notdue to the electric field [5]. At 0.67 Pa our calculation yields


(a) (b)

(c)

Fig. 3. Electron density profiles at: (a) 0.67 Pa, (b) 1.3 Pa, and (c) 2.0 Pa. The upper graphs show experimental data, the lower graphs show results of themodel. The dashed lines in the lower graphs mark the region assessable to the Langmuir probe. The electron densities are given in cm�3.

electric field strengths of up to 0.7 V/cm in the plasma inwhich the electron mean free path is of the order of 10 cm.The electric field in principle could cause deviations from thenonlocal EDF as discussed in [5]. However, this effect shouldbe accounted for in the hybrid approach. On the other hand,

it is possible that the RF electric field causes a considerableanisotropy of the EDF so that the two-term approximationused in our calculations is no longer accurate. This effecthas to be studied in the future. Of course, it is also possiblethat these deviations between experiment and measurement are


(a) (b)

(c)

Fig. 4. Plasma potential profiles in volt at: (a) 0.67 Pa, (b) 1.3 Pa, and (c) 2.0 Pa. The upper graphs show experimental data, and the lower graphsshow results of the model.

simply caused by some inaccuracies resulting from the variousapproximations used in our model. In discussing these details,however, one should keep in mind that the agreement betweenour model and the experiment is rather reasonable. It shouldalso be pointed out that the hybrid model presents a significant

advance over models based on the traditional NLA since it iscapable of describing the shift of the density maximum toan off-axis position with increasing pressure. As mentionedabove, a model based on the traditional NLA would typicallyalways produce on-axis maxima of the density profile.


(a) (b)

(c)

Fig. 5. Profiles of the mean kinetic energy in eV at: (a) 0.67 Pa, (b) 1.3 Pa, and (c) 2.0 Pa. The upper graphs show experimental data, and thelower graphs show results of the model.

Fig. 4 shows a comparison of measured and calculatedplasma potential profiles. First, the very good agreementbetween the calculated and measured absolute values of theplasma potential should be pointed out. The maximum valuesof the potentials related to the wall potential are: 19.47 V

(exp.) and 17.4 V (model) at 0.67 Pa, 17.75 V (exp.) and 16.8V (model) at 1.3 Pa, and 16.86 V (exp.) and 16.85 V (model)at 2.6 Pa. Again, the best agreement between experiment andmodel is found at the highest pressure of 2.6 Pa. The profiles ofthe potential mainly reflect the electron density profiles since


(a) (b)

(c)

Fig. 6. Contour plots of EDF’s at a constant total energy of 17 eV at: (a) 0.67 Pa, (b) 1.3 Pa, and (c) 2.0 Pa. The upper graphs show experimental data,and the lower graphs show results of the model. The EDF is given in cm�3eV�3=2.

electron density and potential are related to each other by thegeneralized Boltzmann relation (16). Hence, the profile at 0.67Pa is also centered on-axis and the maximum shifts to anoff-axis position with increasing pressure.

The experimental and calculated profiles of the mean kineticenergy of the EDF’s are plotted in Fig. 5. Experiment and

model show the same trend of an increasing mean kineticenergy toward the region of high RF electric field. The generaltrend of a slight increase of the mean energy if moving awayfrom the position of the maximum plasma potential can beexplained with a “nonlocal” cutting of the low energy part ofthe EDF which has a slightly lower “temperature” than the part


(a) (b)

(c)

Fig. 7. Calculated profiles of the ionization frequency in s�1 at: (a) 0.67 Pa, (b) 1.3 Pa, and (c) 2.0 Pa.

of the EDF between 5–10 eV (compare Fig. 2). The biggestdiscrepancy between model and experiment is found at 0.67Pa for which the experimental mean energies are about 20%higher than the calculated values. The best agreement is againfound at 2.0 Pa. It has to be noted that we estimate the error inthe measured mean energies to be about 10% due to the limitednumber of current–voltage points recorded and the inaccuracyinvolved in the integration of the EDF.

Fig. 6 gives a comparison of contour plots of experimentaland calculated EDF’s at a giventotal energyof 17 eV. Theexperimental and theoretical results for the lowest pressureof 0.67 Pa show that the EDF is still basically nonlocal,i.e., spatially uniform in terms of total energy. However, thedeviations from nonlocality are obviously large enough tocause the off-axis density maximum at 0.67 Pa. The exactprofile, in particular, of the measured EDF should not byoverinterpreted, since the overall variation of the EDF overthe accessible range of the Langmuir probe is only about 10%at this pressure which is certainly within the experimentalerrors of the EDF measurement. At higher pressures of 1.3and 2.6 Pa the EDF’s demonstrate increasing deviations fromthe nonlocal EDF with an increase of the EDF toward theregion of the highest electric field. The overall variations ofthe EDF’s observed in the experiment and in the model arevery similar.

Fig. 7 shows the computational results for the ionizationfrequency profiles. At the lowest pressure the ionization fre-quency is rather uniform across the discharge cross sectionwhich is consistent with the mostly nonlocal behavior ofthe computed EDF. With increasing pressure the ionization

is more and more concentrated in regions of high plasmapotential close to the position of the coil. Most likely asmall positive feedback enhances the ionization at the plasmapotential maximum. The plasma potential maximum off-axisis a consequence of the ionization starting to become localizedin regions of high electric fields. However, once the plasmapotential maximum has shifted to an off-axis position itbecomes at least in part self-sustaining since the electronsgain kinetic energy when approaching the plasma potentialmaximum which enhances the ionization at this position.

V. CONCLUSIONS

In this paper, we have presented an efficient, 2-D kineticmodel based on a hybrid kinetic approach to the solutionof the Boltzmann equation. In this approach the domain ofintegration of the Boltzmann equation is divided into twosubdomains: the low energy “elastic” range is treated withinthe traditional nonlocal approximation, and the high energyinelastic range is treated by solving the complete kineticequation derived from the Boltzmann equation as a partialdifferential equation in two space dimensions and in totalenergy. Coulomb collisions are taken into account for the lowenergy, nonlocal part of the EDF in which these collisionsare effective since they compete only with “weak” elasticcollisions. In the high energy part Coulomb collisions areneglected since the Coulomb cross section is smaller and theCoulomb collisions compete with “strong” inelastic collisions[42]. This approach offers two main advantages: a) the highenergy domain, in which the complete kinetic equation isintegrated, typically has a regular shape which simplifies the


numerical formulation and b) the nonlinearity introduced byCoulomb collisions can be avoided in the treatment of the full,multidimensional kinetic equation. It is only included in themuch simpler problem of solving the nonlocal kinetic equationwhich is one-dimensional. The use of the hybrid model andof other physically motivated approximations enables greatcomputational efficiency of the model: complete kinetic, self-consistent 2-D simulations can be performed in computationtimes of less than 20 min on Pentium II class computers.Its efficiency makes this model an interesting candidate forengineering-type discharge calculations.

Comparisons of our model with an actual experimentshowed reasonable agreement. The best agreement wastypically found at the highest pressure considered, a pressureat which a model based on the traditional NLA for thegiven discharge dimensions would certainly fail. Our modelproved well capable of describing effects which are related toincreasing deviations from a fully nonlocal EDF such as theshift of the maxima of plasma potential and plasma density tooff-axis positions with increasing pressures. The model thus isdefinitely an advance compared to models based on the tradi-tional NLA which would not be able to describe these effects.

As we have pointed out the development of such a modelis overdue due to the fact that modern large-scale plasmaprocessing equipment is explicitly designed to operate in aregime with modest deviations from the nonlocality. Operationin this regime is mandatory in order to gain maximum controlover plasma uniformity in the discharge. In conclusion, thehybrid kinetic model presented seems to be a promisingapproach to achieve rapid, fully kinetic modeling of large-scale low-pressure discharges. Further efforts are needed toinclude Coulomb collisions more consistently and to study theeffect of collisionless heating.

REFERENCES

[1] J. D. Bukowski, D. B. Graves, and P. Vitello, “Two-dimensional fluidmodel of an inductively coupled plasma with comparison to experi-mental spatial profiles,”J. Appl. Phys., vol. 80, no. 5, pp. 2614–2623,1996.

[2] G. DiPeso, V. Vahedi, D. W. Hewett, and T. D. Rognlien, “Two-dimensional self-consistent fluid simulation of radio frequency inductivesources,”J. Vac. Sci. Technol.,vol. 12, no. 4, pp. 1387–1396, 1994.

[3] P. Vitello, J. N. Bardsley, G. DiPeso, and G. J. Parker, “Modelingan inductively coupled plasma reactor with chlorine chemistry,”IEEETrans. Plasma Sci.,vol. 24, pp. 123–124, Feb. 1996.

[4] D. J. Economou, T. J. Bartel, R. S. Wise, and D. P. Lymberlopoulos,“Two-dimensional direct simulation Monte Carlo (dsmc) of reactiveneutral and ion flow in a high density plasma reactor,”IEEE Trans.Plasma Sci.,vol. 23, pp. 581–590, Aug. 1995.

[5] U. Kortshagen, C. Busch, and L. D. Tsendin, “On simplifying ap-proaches to the solution of the Boltzmann equation in spatially inho-mogeneous plasmas,”Plasma Sources Sci. Technol., vol. 5, no. 1, pp.1–17, 1996.

[6] D. P. Lymberopoulos and D. J. Economou, “Two-dimensional simulationof polysilicon etching with chlorine in a high density plasma reactor,”IEEE Trans. Plasma Sci.,vol. 23, pp. 573–580, Aug. 1995.

[7] S. Rauf and M. J. Kushner, “Model for noncollisional heating ininductively coupled plasma processing sources,”J. Appl. Phys.,vol. 81,no. 9, pp. 5966–5974, 1997.

[8] R. A. Stewart, P. Vitello, and D. B. Graves, “Two-dimensional fluidmodel of high density inductively coupled plasma sources,”J. Vac. Sci.Technol.,vol. 12, no. 1, pp. 478–485, 1994.

[9] V. Vahedi, M. A. Lieberman, G. DiPeso, T. D. Rognlien, and D. Hewett,“Analytic model of power deposition in inductively coupled plasmasources,”J. Appl. Phys.,vol. 78, no. 3, pp. 1446–1458, 1995.

[10] R. S. Wise, D. P. Lymberopoulos, and D. J. Economou, “Rapid 2-Dself-consistent simulation of inductively coupled plasma and compar-ison with experimental data,”Appl. Phys. Lett.,vol. 68, no. 18, pp.2499–2501, 1996.

[11] M. J. Kushner, W. Z. Collison, M. J. Grapperhaus, J. P. Holland,and M. S. Barnes, “A 3-D model for inductively coupled plasmaetching reactors: Azimuthal symmetry, coil properties, and comparisonto experiments,”J. Appl. Phys.,vol. 80, no. 3, pp. 1337–1344, 1996.

[12] I. D. Kaganovich, I. V. Kolobov, and L. D. Tsendin, “Stochastic electronheating in bounded radio-frequency plasmas,”Appl. Phys. Lett.,vol. 69,no. 25, pp. 3818–3820, 1996.

[13] V. A. Godyak and V. I. Kolobov, “Negative power absorption ininductively coupled plasmas,”Phys. Rev. Lett.,vol. 79, no. 23, pp.4589–4592, 1997.

[14] I. D. Kaganovich and L. D. Tsendin, “Low–pressure rf discharge in thefree–flight regime,”IEEE Trans. Plasma Sci.,vol. 20, pp. 86–92, Apr.1992.

[15] V. A. Godyak and V. I. Kolobov, “Effect of collisionless heating onelectron energy distribution in an inductively coupled plasma,”Phys.Rev. Lett.,vol. 81, no. 2, pp. 369–372, 1998.

[16] V. A. Godyak, R. B. Piejak, B. M. Alexandrovich, and V. I. Kolobov,“Experimental evidence of collisionless power absorption in inductivelycoupled plasmas,”Phys. Rev. Lett.,vol. 80, no. 15, pp. 3264–3267,1998.

[17] I. B. Bernstein and T. Holstein, “Electron energy distributions instationary discharges,”Phys. Rev.,vol. 94, no. 6, pp. 1475–1482, 1954.

[18] L. D. Tsendin, “Energy distribution of electrons in a weakly ionizedcurrent-carrying plasma with a transverse inhomogeneity,”Sov. Phys.,vol. 39, no. 5, pp. 805–810, 1974.

[19] L. D. Tsendin and Yu. B. Golubovskii, “Positive column of a low-density, low-pressure discharge—I: Electron energy distribution,”Sov.Phys. Tech. Phys.,vol. 22, no. 9, pp. 1066–1073, 1977.

[20] Yu. Benke and Yu. B. Golubovskii, “Kinetic description of confinedelectrons in gas discharge plasmas,”Opt. Spectrosc.,vol. 73, no. 1, pp.37–46, 1992.

[21] J. Behnke, Yu. B. Golubovskij, S. U. Nisimov, and I. A. Porokhova,“Self-consistent kinetic description of the positive column of a dischargewith direct and step-wise ionization,”Sov. Phys. Tech. Phys.,vol. 39,no. 1, pp. 33–39, 1994.

[22] L. D. Tsendin, “Anode region of a glow discharge,”Sov. Phys. Tech.Phys.,vol. 31, no. 2, pp. 169–175, 1986.

[23] V. I. Kolobov and L. D. Tsendin, “Analytic model of the cathode regionof a short glow discharge in light gases,”Phys. Rev.,vol. 46, no. 12,pp. 7837–7852, 1992.

[24] V. I. Kolobov, D. F. Beale, L. J. Mahoney, and A. E. Wendt, “Nonlocalelectron kinetics in a low-pressure inductively coupled radio-frequencydischarge,”Appl. Phys. Lett., vol. 65, no. 5, pp. 537–539, 1994.

[25] V. Kolobov and W. N. G. Hitchon, “Electron distribution function ina low–pressure inductively coupled plasma,”Phys. Rev.,vol. 52, no. 1,pp. 972–980, 1995.

[26] V. Kolobov, G. J. Parker, and W. N. G. Hitchon, “Modeling of nonlocalelectron kinetics in low pressure inductively coupled plasma,”Phys.Rev.,vol. 53, no. 1, pp. 1110–1124, 1996.

[27] U. Kortshagen, I. Pukropski, and M. Zethoff, “On the spatial variationof the electron distribution function in a rf inductively coupled plasma:Experimental and theoretical study,”J. Appl. Phys.,vol. 76, no. 4, pp.2048–2058, 1994.

[28] U. Kortshagen and L. D. Tsendin, “Fast 2-D self-consistent kineticmodel of an inductively coupled rf plasma,”Appl. Phys. Lett.,vol. 65,no. 11, pp. 1355–1357, 1994.

[29] U. Kortshagen, I. Pukropski, and L. D. Tsendin, “Experimental in-vestigation and fast 2-D self-consistent modeling of a low pressureinductively coupled rf discharge,”Phys. Rev.,vol. 51, no. 6, pp.6063–6078, 1995.

[30] U. Kortshagen and M. Zethoff, “Ion energy distribution functions in aplanar inductively coupled rf discharge,”Plasma Sources Sci. Technol.,vol. 4, no. 4, pp. 541–550, 1995.

[31] G. Muumken and U. Kortshagen, “On the radial distribution andnonambipolarity of charged particle fluxes in a nonmagnetized planarinductively coupled plasma,”J. Appl. Phys.,vol. 80, no. 12, pp.6639–6645, 1996.

[32] U. Kortshagen, “Kinetic modeling of the charging of nonconductingwalls in a low pressure inductively coupled plasma,”J. Vac. Sci.Technol.,vol. 16, no. 1, pp. 300–305, 1998.

[33] S. V. Berezhnoi, I. D. Kaganovich, L. D. Tsendin, and V. A. Schweigert,“Fast modeling of the low-pressure capacitively coupled radio-frequencydischarge based on the nonlocal approach,”Appl. Phys. Lett.,vol. 69,no. 16, pp. 2341–2343, 1996.


[34] S. V. Berezhnoi, I. D. Kaganovich, and L. D. Tsendin, “Fast modelingof the low-pressure radio-frequency collisional capacitively coupleddischarge and investigation of the formation of a non-Maxwellianelectron distribution function,”Plasma Sources Sci. Technol.,vol. 7,no. 3, pp. 268–281, 1998.

[35] I. D. Kaganovich and L. D. Tsendin, “The space-time-averaging pro-cedure and modeling of the rf discharge—Part II: Model of collisionallow-pressure rf discharge,”IEEE Trans. Plasma Sci.,vol. 20, pp. 66–75,Apr. 1992.

[36] V. I. Kolobov and V. A. Godyak, “Non–local electron kinetics incollisional gas discharge plasmas,”IEEE Trans. Plasma Sci.,vol. 23,pp. 503–531, Aug. 1995.

[37] I. P. Shkarofsky, T. W. Johnston, and M. P. Bachynski,The ParticleKinetics of Plasmas. Reading, MA: Addison-Wesley, 1966.

[38] L. D. Tsendin, “Electron energy distribution in rf electric field,”Sov.Phys. Tech. Phys.,vol. 22, no. 8, pp. 925–928, 1977.

[39] R. Winkler, H. Deutsch, J. Wilhelm, and Ch. Wilke, “Electron kinetics ofweakly ionized hf plasmas—I: Direct treatment and Fourier expansion,”Beitr. Plasmaphys.,vol. 24, no. 3, pp. 285–302, 1984.

[40] S. Yoshida, A. V. Phelps, and L. C. Pitchford, “Effect of electronsproduced by ionization on calculated electron-energy distributions,”Phys. Rev.,vol. 27, no. 6, pp. 2858–2867, 1983.

[41] H. Dreicer, “Electron velocity distributions in a partially ionized gas,”Phys. Rev.,vol. 117, no. 2, pp. 343–354, 1960.

[42] U. Kortshagen and H. Schluter, “On the influence of Coulomb collisionson the electron energy distribution function in surface wave producedplasmas argon plasmas,”J. Phys. D, Appl. Phys.,vol. 25, no. 4, pp.644–651, 1992.

[43] G. J. Parker and D. Uhrlandt, “Investigation of nonlocal approach inpositive column discharges,”Bull. Amer. Phys. Soc.,vol. 43, no. 5, p.1479, 1998.

[44] V. A. Godyak and R. B. Piejak, “Paradoxical spatial distribution of theelectron temperature in a low pressure rf discharge,”Appl. Phys. Lett.,vol. 63, no. 23, pp. 3137–3139, 1993.

[45] C. Busch and U. Kortshagen, “Numerical solution of the spatiallyinhomogeneous Boltzmann equation and verification of the nonlocalapproach,”Phys. Rev.,vol. 51, no. 1, pp. 280–288, 1995.

[46] U. Kortshagen, G. J. Parker, and J. E. Lawler, “Comparison of nonlocalcalculations and Monte Carlo simulations of the electron distributionfunction in a positive column,”Phys. Rev.,vol. 54, no. 6, pp. 6746–6761,1996.

[47] K.-U. Riemann, “The Bohm criterion and sheath formation,”J. Phys.D, Appl. Phys.,vol. 24, no. 4, pp. 493–518, 1991.

[48] W. Hackbusch,Multi-Grid Methods and Applications. Berlin, Ger-many: Springer, 1985.

[49] M. J. Druyvesteyn, “Der Niedervoltbogen,”Z. Phys., vol. 64, pp.781–798, 1930.

[50] V. A. Godyak, R. B. Piejak, and B. M. Alexandrovich, “Measurementsof electron energy distributions in low-pressure rf discharges,”PlasmaSources Sci. Technol.,vol. 1, no. 1, pp. 36–58, 1992.

[51] B. Heil and U. Kortshagen, “Two-dimensional mapping of electrondistribution functions in low pressure ICP,”IEEE Trans. Plasma Sci.,vol. 27, pp. 56–57, Feb. 1998.

Uwe Kortshagen received the Diploma degree inphysics in 1988 and the Ph.D. degree in physics in1991 from the University of Bochum, Germany.

He continued to work at the University ofBochum as a Research Associate. He receivedthe venia legendi (Habilitation) for experimentalphysics in 1995. From 1995–1996 he worked as aVisiting Scholar in the Department of Physics at theUniversity of Wisconsin-Madison with a fellowshipof the Alexander-von-Humboldt Foundation. Since1996, he has been an Assistant Professor at the

University of Minnesota, Minneapolis. His research interests are in diagnosticsand modeling of low-pressure plasmas as well as in nanoparticle formation inprocessing plasmas. He recently received the NSF Career Award to performresearch on the photo-detachment from nanometer-sized particles.

Brian G. Heil received the Bachelor of Physicsdegree in 1993 from the University of Minnesota,Minneapolis, where he is currently working towardthe M.S. degree in mechanical engineering.

After graduating in 1993, he was an ApplicationsEngineer for the America X-ray Instrument Corpo-ration in Santa Clara, CA. From 1996 to 1998, asa student, he performed research at the HoneywellTechnology Center for several cold cathode andMEMS programs. His work at the university is inthe area of diagnostics and modeling of low pressure

inductively coupled plasmas.

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Kinetic two-dimensional modeling of inductively coupled plasmas based on a hybrid kinetic approach