Power absorption in electrically asymmetric dual frequency capacitiveradio frequency discharges

E. Schüngel,1,a� J. Schulze,1,2 Z. Donkó,2 and U. Czarnetzki11Institute for Plasma and Atomic Physics, Ruhr-University Bochum, 44780 Bochum, Germany2Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences,P.O. Box 49, 1525 Budapest, Hungary

�Received 13 October 2010; accepted 17 December 2010; published online 11 January 2011�

The symmetry of capacitive radio frequency discharges can be controlled via the electricalasymmetry effect by driving one electrode with a fundamental frequency and its second harmonic.In such electrically asymmetric discharges, the mean ion energies at both electrodes are controlledseparately from the ion flux by tuning the phase angle � between the harmonics at fixed voltageamplitudes. Here, the question why the ion flux is nearly independent of � is answered byinvestigating the power absorbed by the electrons Pe as a function of � and time experimentally, bya particle in cell simulation, and an analytical model. The dynamics of Pe is understood by themodel and is found to be strongly affected by the choice of �. However, on time average, Pe isnearly constant, independently of �. Thus, the ion flux remains approximately constant. In addition,it is shown that the absolute value of the individual voltages across the powered and groundedelectrode sheath vary linearly with the dc self-bias. However, their sum remains constant. Thisyields, in combination with the constancy of the ion flux, a constant power absorbed by the ions and,in conclusion, a total power absorption that is independent of �. © 2011 American Institute ofPhysics.�doi:10.1063/1.3535542�

I. INTRODUCTION

For many applications of capacitively coupled radio fre-quency �CCRF� discharges, separate control of ion energyand ion flux at a processing surface is of major importance.1

In “classical” dual frequency capacitive discharges operatedat two substantially different frequencies2–10 this separatecontrol is limited by the coupling of both frequencies2,3 andthe effect of secondary electrons.10,11

These limitations can be overcome by driving one elec-trode with a voltage waveform ���t ,��, which is the sum ofa fundamental frequency f and its second harmonic withfixed, but adjustable phase shift � between the driving har-monics:

���t,�� = 12�0�cos��rft + �� + cos�2�rft�� . �1�

Here �rf =2�f and �0 is the sum of the identical amplitudesof both harmonics. If such a driving voltage waveform isused, the electrical asymmetry effect �EAE�12–19 will allow tocontrol the mean ion energies at the electrodes separatelyfrom the ion fluxes in an almost ideal way by tuning � suchas demonstrated experimentally and by a particle in cell�PIC� simulation.13,15

In this work, we investigate and answer the questionwhy the ion flux at each electrode is nearly constant at fixed�0 regardless of � in such electrically asymmetric dis-charges. The article is structured in the following way: InSec. II, we introduce the methods used to investigate thepower absorption, i.e., experiment, PIC simulation, and

model. In Sec. III, the results are presented. This section isdivided into four parts. First, the energy balances for elec-trons and ions in electrically asymmetric dual frequency dis-charges are discussed. Second, the power absorption by theelectrons is analyzed based on the electron energy balance.Third, the power absorption by the ions is investigated basedon the ion energy balance and the results for the electronpower absorption. Fourth, the total absorbed power is dis-cussed based on the results for the electron and ion powerabsorption. Finally, conclusions are drawn in Sec. IV.

II. METHODS

We investigate an electrically asymmetric CCRF dis-charge such as shown in Fig. 1. The discharge is geometri-cally symmetric, i.e., the surface areas of the powered andgrounded electrode, Ap and Ag, are equal. The electrode gapis d. One electrode is driven by the voltage waveform���t ,�� with f =13.56 MHz to generate an argon plasma. Inthe experiment, the discharge is operated in a modifiedgaseous electronics conference �GEC� reference cell.13 Theplasma is confined between the electrodes by a glass cylin-der. The rf current is measured time resolved by a self-excited electron resonance spectroscopy �SEERS� sensorimplemented into the grounded electrode. The square of thismeasured current is proportional to the time resolved powerabsorbed by the plasma electrons.

Generally, it was found that in classical dual frequencydischarges, the matching quality can be affected by modula-tions of the discharge conditions.20 In our experimental studyof an electrically asymmetric dual frequency discharge, careis taken to always reach very good matching conditions.a�Electronic mail: edmund.schuengel@ep5.rub.de.

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Only a small change of the voltage amplitudes as a functionof � is observed and corrected. Therefore, it can be reason-ably assumed that the idealized voltage waveform given inEq. �1� is applied to the powered electrode. Similarly, match-ing effects are neglected in both the simulation and themodel.21

Such a discharge is also investigated by a one-dimensional �1d3v� PIC simulation complemented with aMonte Carlo treatment of collisions using the cross sectionsgiven in Refs. 22 and 23. At the infinite, planar, and parallelelectrodes, secondary electrons are emitted with a probabilityof �=0.1. Details of the simulation can be foundelsewhere.15 The power absorption of electrons and ions insuch electrically asymmetric discharges is analyzed theoreti-cally based on the analytical model of the EAE.14 The mainpart of this work is related to the analytical description of thepower absorption, which is supplemented and verified byexperimental and simulation results and finally yields theanswer why the ion flux in electrically asymmetric CCRFdischarges is constant as a function of �.

III. RESULTS

In this section, the power absorption of electrons andions in electrically asymmetric dual frequency CCRF dis-charges is investigated. First, we discuss the energy balancesfor electrons and ions in such a discharge. Based on theelectron energy balance, we investigate the time averagedelectron power absorption as a function of � and its effect onthe ion flux at the electrodes at different discharge conditionsin Sec. III B. In Sec. III C, we analyze the ion power absorp-tion based on the ion energy balance and the results for theelectron power absorption. Finally, we discuss the total ab-sorbed power in Sec. III D.

A. Energy balances for electrons and ions

Generally, the time averaged fluxes of electrons ���e��and ions ���i�� have to balance each other at the electrodes ofan electropositive CCRF discharge.24,25 The ion fluxes at thepowered and grounded electrode, �p

�i� and �g�i�, can be as-

sumed to be temporally constant. Thus,

��p�e�� = �p

�i�, �2�

��g�e�� = �g

�i�. �3�

Here, � . . . � indicates the temporal average over one period ofthe fundamental frequency.

The ions gain energy dominantly due to the accelerationtoward the electrodes by the voltage drop across thesheaths.1,26 There are different electron heating mechanismsin a CCRF discharge,1,5,7,9,16,19,27–36 e.g., Ohmic and stochas-tic heating. A detailed discussion of the different electronheating mechanisms is beyond the scope of this work.

The fluxes toward the electrodes are determined by thetime averaged power absorbed by the electrons, �Pe�, and bythe ions, �Pi�, via the energy balances1,24,25,29,30,32 for bothspecies, i.e., the energy gain mentioned above has to be bal-anced by the losses in the steady state:

�Pe� = e�p�i�Ap�E�c� + Ep

�e�� + e�g�i�Ag�E�c� + Eg

�e�� , �4�

�Pi� = e�p�i�ApEp

�i� + e�g�i�AgEg

�i�. �5�

Here, e is the elementary charge, E�c� is the collisional energyloss per electron-ion pair created, and Ep

�e� and Eg�e� are the

energy loss per electron lost from the discharge at the pow-ered and grounded electrode, respectively. Ep

�i� and Eg�i� is the

energy loss per ion lost from the discharge at the respectiveelectrode. �p,g

�i� is the ion flux at the respective electrode. Inthe following, we assume �p

�i�=�g�i�=��i�, Eg

�e�=Ep�e�=E�e�,

Ep�i�= ��sp�t��, and Eg

�i�= ��sg�t��, where �sp�t� ,�sg�t� is thevoltage drop across the sheath adjacent to the powered andgrounded electrode. The approximations for Ep

�i� and Eg�i� will

only be valid if ionization inside the sheaths can be ne-glected, which is the case under all conditions investigatedhere. Note that Ep

�i� can be significantly different from Eg�i� in

electrically asymmetric discharges. Under these assumptions,the electron and ion energy balances in a geometrically sym-metric discharge �Ap=Ag=A� are:

�Pe� = 2eA��i��E�c� + E�e�� , �6�

�Pi� = eA��i����sp�t�� + ��sg�t��� . �7�

Since E�c�+E�e� is constant independently of �, the ion flux isdirectly proportional to �Pe�. In the following, we demon-strate that �Pe�, and thus ��i�, is constant independently of �in electrically asymmetric dual frequency discharges.

B. Electron power absorption

We use the analytical model of the EAE �Ref. 14� toinvestigate the power absorption by the electrons in electri-cally asymmetric discharges theoretically. This model isbased on the following voltage balance of a CCRF discharge:

���t,�� + � = �sp + �sg + �b. �8�

Here, � is the dc self-bias and �b is the bulk voltage. Usingthe quadratic charge voltage relation for the sheaths in CCRFdischarges37 and neglecting �b �Ref. 15� yields

�̄��t,�� + �̄ = − q2�t� + �qt − q�t��2, �9�

with q�t�=Q�t� /Q0, where Q�t� is the space charge in thesheath at the powered electrode as a function of time and

Filter(lf block)

hf Matchhf

Generator

Filter(hf block)

lf Matchlf

Generator

High voltage probe

Poweredelectrode

Groundedelectrode

Bulkplasma

Sync.+

variablephase

Glasscylinder

SEERSsensor

FIG. 1. Experimental setup of the electrically asymmetric dual frequencydischarge used in this work.

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Q0Ap�2e0n̄sp�0 is a normalization constant. 0 is the di-

electric constant and n̄sp is the spatially averaged ion densityin the sheath at the powered electrode.14 qt is the total un-compensated charge in the discharge normalized by Q0; it isassumed to be temporally constant.18 �̄��t ,�� is the appliedvoltage normalized by its amplitude �0 and �̄ is the dc self-bias normalized by �0. The symmetry parameter is definedas the ratio of the maximum sheath voltages that drop acrossthe sheath at the grounded electrode, �̂sg, and powered elec-trode, �̂sp, �Ref. 14�:

= � �̂sg

�̂sp

� Ap

Ag�2 n̄sp

n̄sg

. �10�

In a geometrically symmetric discharge, depends only onthe ratio of the mean ion densities n̄sp and n̄sg in both sheaths.

If the low floating potentials during the sheath collapseat both electrodes are neglected, the normalized dc self-bias�̄ and the total charge qt are found to be14

�̄ = −�̃m1��� + �̃m2���

�0�1 + �, �11�

qt =��̃m1��� − �̃m2����0�1 + �

. �12�

Here, �̃m1��� and �̃m2��� are the maximum and minimum ofthe applied voltage waveform, which are functions of �.They can be approximated in the range 0° �90° as

�̃m1��� �0�1 + cos����

2, �13�

�̃m2��� −�0�1 + sin����

2. �14�

Based on Eq. �11�, a dc self-bias is generated via the EAEas an almost linear function of � for 0° �90°,13–17

which allows controlling the mean ion energies at theelectrodes.13–15

The balance �9� can be solved for q�t�:

q�t� =− qt + �qt

2 − �1 − ���̄ + �̄��t��1 −

. �15�

Based on Eq. �15�, the electron conduction current densityje�t�=−�Q0 /Ap���q /�t� and, finally, the power absorbed bythe electrons Pe�t ,� ,�= �je

2�t� /��Apl can be calculated.Here, l is the bulk length and �� n̄b is the bulkconductivity1,32 for a given pressure, where n̄b is an effectiveplasma density in the bulk. Using Eqs. �11� and �12�, thefollowing result for Pe�t ,� ,� is obtained:

Pe�t,�,�

=Pe,0

4

�1 + ���̇��t,���2

��̃m1��� − 2�̃m2��� − �1 − 2����t,����0

.

�16�

The dot represents the differentiation with respect to =�rft. Pe,0=2�0�RF

2 0en̄spApl /�� n̄sp / n̄b is approximatelyconstant at high pressures and changes as a function of � atlow pressures �see Fig. 2�, mainly due to the variation ofn̄sp���. Based on Eq. �16�, the time averaged value of Pe�t�can now be calculated using Pe,0 and as input parametersfrom the PIC simulation:

�Pe� = Pe,0�q̇2� . �17�

At high pressures in geometrically symmetric discharges, thesymmetry parameter is close to unity for all � �Ref. 14� andPe,0 is almost constant �see Fig. 2�. In this case, the timeaveraged power absorbed by the plasma electrons is deter-mined from Eq. �16� using ��̇��t ,�� /�0�=5 /8:

�Pe���� = Pe,0p0��� with

�18�

p0��� =5�0

16��̃m1��� − �̃m2����.

At low pressures, Pe,0 and are no longer constants, butchange as functions of �. Nevertheless, an approximate valuefor the total power absorbed by the electrons can be calcu-lated analytically by a linear expansion of Eq. �16� around=1:

�Pe���� = Pe,0�p0��� + � − 1�p1���� with

�19�

p1��� =5�0��̃m1��� + 3�̃m2���� − 3�0

2 cos�2��

32��̃m1��� − �̃m2����2.

Equation �19� will be a good approximation if 0.51.5.As shown in Fig. 2, typically remains within these limits ingeometrically symmetric electrically asymmetric dual fre-quency discharges. Although p0��� and p1��� are nominallyfunctions of �, they are effectively constants �p00.2 and

FIG. 2. �Color online� Symmetry parameter and P̄e,0, i.e., Pe,0 normalizedby its phase angle averaged value, as a function of � at high pressure�100 Pa, �0=240 V, d=2 cm� and at low pressure �2.66 Pa, �0=630 V,d=6.7 cm� resulting from PIC simulations.

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p1−0.1 as shown in Fig. 3�. Generally, it can be shown bythe analytical model that �Pe���� is symmetric around�=45°, i.e., �Pe����= �Pe����=90°−�� with 0�45°.

Figure 4 shows Pe�t ,�� resulting from the experiment,the PIC simulation, and the analytical model �Eq. �16�� athigh pressure �=1∀��. According to the four extrema of�̇��t ,��, each plot shows four maxima. While the heights ofthe first and third maximum remain almost constant for all �,the second and fourth maxima vary oppositely in the range0° �90°.

As a result of this compensation, �Pe� remains almostconstant for all � �see Fig. 5�a��. The slight decrease at�45° in comparison to �Pe� at �=0° and �=90° is causedby a small variation of the total charge qt��� due to a varia-tion of ��̃m1− �̃m2� as a function of � �Eq. �12��. Also at lowpressures, �Pe� is approximately constant �Fig. 5�b��. Basedon Eq. �6�, these results demonstrate why the ion flux at bothelectrodes is constant as a function of � using fixed appliedvoltage amplitudes in electrically asymmetric dual frequencyCCRF discharges.

C. Ion power absorption

Based on Eq. �7� and ��i����=const. the power absorbedby the ions can now be investigated by analyzing the abso-lute values of the time averaged sheath voltages ��sp�t��and ��sg�t���0. For symmetry reasons, they can be ex-pressed as functions of the dc self-bias � in the followingway:

��̄sp�t�� = f��̄�, ��̄sg�t�� = f�− �̄� . �20�

Without loss of generality, f��̄� can be expressed by a powerexpansion:

f��̄� = �n=0

�

an�̄n. �21�

Here, all voltages are normalized by the constant appliedvoltage amplitude �0, i.e., �̄sp�t�=�sp�t� /�0 and �̄sg�t�=�sg�t� /�0. The dc self-bias corresponds to the sum of thetime averaged sheath voltages under the assumption of a

negligible voltage drop across the plasma bulk, as can beseen by taking the time average of Eq. �8�. Therefore, thefollowing relation between the normalized self-bias �̄ andthe function f��� is valid:

FIG. 3. �Color online� �̃m1��� /�0 �Eq. �13��, �̃m2��� /�0 �Eq. �14��, p0 �Eq.�18��, and p1 �Eq. �19�� as a function of � calculated by the model.

FIG. 4. �Color online� Power absorbed by the electrons as a function of �and time within one low frequency period. The color scale shows the rela-tive power in percent of the maximum value. �a� Experiment, 100 Pa,d=1 cm, �0=100 V; �b� PIC simulation, 100 Pa, d=2 cm, �0=240 V;and �c� analytical model, =1.

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�̄ = − ��̄sp�t�� + ��̄sg�t�� , �22�

=− �n=0

�

an�̄n + �n=0

�

an�− �̄�n, �23�

=− 2�n=0

�

a2n+1�̄2n+1. �24�

Here, only odd powers contribute. Equation �24� is only ful-filled if a1=−1 /2 and a2n+1=0∀n�1.

With the symmetry considerations made above, we canassume a symmetric discharge at =1 and �̄=0. This means,

that the absolute value of the mean sheath voltages in frontof the powered and grounded electrode are the same:��̄sp�t��= ��̄sg�t�� due to Eq. �22�. Therefore, we can findf��̄=0�=a0 analytically by solving Eq. �9� in the case ofsymmetry

a0 = ��̄sp�t�� = ��̄sg�t�� �25�

=�q2� =qt

2���4

+1

16qt2���

. �26�

Here, ��̄�2 �t ,���=1 /4∀� is used. As mentioned above, the

total charge qt��� depends weakly on � �see Eq. �12��.In case of =1, the dc self-bias is expected to vanish at

�=45° due to Eq. �11�. With the approximations of the maxi-mum �̃m1 /�0 and minimum �̃m2 /�0 of �̄� shown in Fig. 3,we find qt

2��=45°�= �1+�2� / �2�2�. This yields a00.29.Therefore, the normalized sheath voltages are analyticallyfound to be:

��̄sp� − 0.29 + 0.5�̄ , �27�

��̄sg� 0.29 + 0.5�̄ . �28�

The linear functions of Eqs. �27� and �28� are compared toresults of the PIC simulation in Fig. 6. The slope is very wellreproduced. The model predicts a0=0.29, which is slightlysmaller than the value a00.33 found in the simulations.This might be caused by small temporal variations of qt,which are neglected in the model.18

In conclusion, ��sp�+ ��sg� is a constant. Thus, accord-ing to Eq. �7�, the power absorbed by the ions is, via the ionflux, simply proportional to �Pe� and also constant as a func-tion of � in electrically asymmetric dual frequency dis-charges such as observed by PIC simulations before.15

FIG. 5. �Color online� Time averaged power absorbed by the electrons as afunction of � resulting from experiment, simulation, and model under sev-eral conditions. For all �, the period averaged absorbed power is normalizedby the mean value. �a� High pressure and �b� low pressure �Pe,0��� and ���from the PIC simulation are used as input parameters in the model�.

FIG. 6. �Color online� Absolute value of the normalized mean sheath volt-age in front of the powered and grounded electrode as a function �̄�PIC simulation, 100 Pa, d=2 cm, �0=240 V�. The symbols are simulationresults and the lines correspond to model fits using ��̄sp��−0.5�̄ , ��̄sg��0.5�̄ , ��̄sp�+ ��̄sg�=const.

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D. Total power absorption

The time averaged total power absorbed in the discharge�Pabs� is the sum of the powers absorbed by electrons andions

�Pabs� = �Pe� + �Pi� . �29�

Based on Eq. �29� and the results of Secs. III B and III C forthe power absorbed by electrons and ions, respectively, thetotal absorbed power is also approximately constant as afunction of �. This might be important for practical purposes,since usually the power supplied by the generator rather thanthe voltage applied to the electrode is monitored.

IV. CONCLUSIONS

We investigated the power absorption by electrons andions in electrically asymmetric dual frequency capacitivelycoupled radio frequency discharges experimentally, by a PICsimulation, and by an analytical model. Based on the analyti-cal model of the EAE, we have shown that the power ab-sorbed by the electrons is nearly constant as a function of thephase shift � between the driving harmonics at high and lowpressures. These theoretical results have been verified experi-mentally and by PIC simulations. Based on the energy bal-ance for the electrons, we have demonstrated that a constantelectron power absorption causes the ion flux at the elec-trodes to remain constant as a function of �. In this way, thequestions why the ion flux is approximately constant as afunction of � and why the ion energy can be controlled sepa-rately from the ion flux in an almost ideal way in electricallyasymmetric dual frequency discharges have been answered.Based on the ion energy balance and an analysis of the timeaveraged sheath voltages, we have found that a constant elec-tron power absorption, i.e., a constant ion flux, causes thepower absorption of the ions to remain constant, too. Thus,the total power absorbed in the discharge is also nearly con-stant as a function of �.

ACKNOWLEDGMENTS

The support by the Hungarian Fund for ScientificResearch �Grants OTKA K77653 and OTKA-IN-85261�, theAlexander von Humboldt Foundation, and the Ruhr-University Research Department Plasma are gratefullyacknowledged.

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