Nonlinear electron resonance heating in capacitive radio frequency dischargesThomas Mussenbrock and Ralf Peter Brinkmann Citation: Applied Physics Letters 88, 151503 (2006); doi: 10.1063/1.2194824 View online: http://dx.doi.org/10.1063/1.2194824 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/88/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Measurements of time average series resonance effect in capacitively coupled radio frequency discharge plasma Phys. Plasmas 18, 103509 (2011); 10.1063/1.3646317 Self-excitation of the plasma series resonance in radio-frequency discharges: An analytical description Phys. Plasmas 13, 123503 (2006); 10.1063/1.2397043 Driving frequency effect on the electron energy distribution function in capacitive discharge under constantdischarge power condition Appl. Phys. Lett. 89, 161506 (2006); 10.1063/1.2363945 A nonlinear global model of a dual frequency capacitive discharge Phys. Plasmas 13, 083501 (2006); 10.1063/1.2244525 Stochastic heating in single and dual frequency capacitive discharges Phys. Plasmas 13, 053506 (2006); 10.1063/1.2203949

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Nonlinear electron resonance heating in capacitive radiofrequency discharges

Thomas Mussenbrocka� and Ralf Peter BrinkmannRuhr-Universität Bochum, Lehrstuhl für Theoretische Elektrotechnik, Universitätsstraße 150,D-44801 Bochum, Germany

�Received 6 January 2006; accepted 28 February 2006; published online 12 April 2006�

Technological processing plasmas are frequently operated at relatively low gas pressure��10 Pa�. A characteristic feature of this regime is that collisions of the electrons with the atoms ormolecules of the neutral background are relatively rare, and the so-called collisional or Ohmicheating ceases to be an effective mechanism of energy deposition into the plasma. Experimentsindicate that at low pressure an alternative mechanism of electron heating exists which can sustainthe plasma. Despite 30 years of intense research, the exact nature of this “anomalous” heatingmechanism is still under discussion. The two standard models are known as “stochastic heating” and“pressure heating,” respectively. This work proposes a third explanation of anomalous electronheating and suggests that, in the last analysis, all three mechanisms may contribute to the observedeffect. © 2006 American Institute of Physics. �DOI: 10.1063/1.2194824�

In order to achieve better control over the ion energy,radio frequency plasmas for material processing are fre-quently operated at relatively low pressure ��10 Pa�.1 In thisregime, the mean free path for ion-neutral collisions is muchlarger than the Debye length and the plasma boundarysheaths are practically collisionless. The accelerated ionsreach the substrate therefore with a narrow energy distribu-tion and practically no angular spread �beamlike distribu-tion�. Such a characteristic has several technologicaladvantages.2

The disadvantage of the low pressure regime is that alsothe collisions of the electrons with the atoms or molecules ofthe neutral background are rare, and the standard collisional�Ohmic� heating ceases to be an effective mechanism of en-ergy deposition into the plasma. Experimental evidence,however, exists that at low pressure there is another mecha-nism of electron heating which can sustain a rf discharge.Early work was conducted by Popov and Godyak3 on a sym-metric capacitively coupled discharge in mercury vapordriven at frequencies of 40.8–110 MHz. They showed that atlow pressure the effective power dissipation substantially ex-ceeds the value expected from mere Ohmic heating. Specifi-cally, at a gas pressure of p=0.19 mTorr=0.025 Pa it was 30times larger than expected.

Despite a 30 year research effort, the exact mechanismbehind this phenomenon is still under discussion. Two stan-dard explanations are known as Fermi acceleration and pres-sure heating, respectively. The concept of Fermi accelerationwas first applied to capacitive rf discharges by Godyak4 inthe late 1970s. �See also the review by Lieberman andGodyak.5� The basic idea is quite simple; it rests on the in-sight that the interaction of an electron with a modulatedboundary sheath can be seen as a reflection at a moving wall,v→−v+2u. Averaging over all phases, this should lead to anet increase of the average particle energy. Unfortunately, theconsistent evaluation of the concept is very complicated. Animportant contribution was made by Kaganovich et al.6 whoshowed that Fermi heating can be kinetically understood in

terms of an effective diffusion along the total energy axis.The term “pressure heating” was coined by Turner.7 He

showed the existence of a powerful heating mechanism as-sociated with pressure effects that arise during the expansionand contraction phases of the boundary sheath. In Turner’sopinion, the non-Ohmic heating in capacitively coupled dis-charges is not intrinsically kinetic and can be understood alsoin a fluid dynamic representation. His conclusions were sup-ported by Gozadinos et al.8

A recent attempt to reanalyze the situation was made byBrinkmann and Hamme.9 They concentrated on the so-calledtransition regime �1 Pa� p�10 Pa� which bridges the gapbetween the previously treated cases of the local regime�p�10 Pa�, where only Ohmic heating is present, and thenonlocal regime �p�10 Pa�, which exhibits only non-Ohmicheating. They claimed the existence of two boundary relatedelectron heating mechanisms and tentatively proposed thecoexistence of Fermi heating and pressure heating. Objec-tions to this interpretation were raised by Lieberman.10

This article will discuss the possibility of yet anotheranomalous heating mechanism of capacitively driven plas-mas at low pressure. Our scenario is independent of the othermechanism discussed above, and may very well be presentsimultaneously. We propose the enhanced Ohmic dissipationof a capacitively coupled plasma via the selfexcitiation of theplasma series resonance by the nonlinearities of the bound-ary sheath. The scenario will be termed “nonlinear electronresonance heating” �NERH�. �It is not to be confused withthe direct excitation of the discharge on the plasma seriesresonance as considered by Qui et al.11�

Consider a capacitive discharge driven by a strong sinu-soidal rf voltage, and do not assume from the outset—incontrast to most models—that the rf current is sinusoidal aswell. Instead, assume that the nonlinear behavior of theboundary sheath will make the current strongly nonhar-monic, with Fourier components present up to considerableorder. Some of the harmonics will be in resonance with theplasma series resonance. With falling pressure their excita-tion will be less and less damped, and ultimately an oscilla-tory rf current will develop with a rms value considerablya�Electronic mail: mussenbrock@g.mail.com

APPLIED PHYSICS LETTERS 88, 151503 �2006�

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above the one expected from the sinusoidal approximation.Of course, this will also lead to a drastically enhanced Ohmicdissipation. Let us see if this idea survives a closer scrutiny.

We assume that the Debye length is small compared withthe reactor dimension and describe the bulk and the sheathby separate models. In the bulk, quasineutrality holds and thecurrent is carried by electron conduction. To model the rfbehavior we neglect the ion current contribution, as well asthe effects of finite electron temperature and plasma chemis-try �ionization or attachment�. This leads to the cold plasmaapproximation, i.e., to a generalized Ohm’s law which takesinto account electron acceleration by the field and momen-tum loss due to collisions with neutrals, and an equation ofcontinuity,

�j

�t= �0�pe

2 E − �cej, � · j = 0. �1�

For a homogeneous plasma bulk which has the shape of aregular cylinder of length lB and an electrode area A, we findan equation which relates the rf current I= I�t� to the totalvoltage drop VB=VB�t�,

dI

dt=

A

lB�0�pe

2 VB − �ceI . �2�

In the boundary sheath electrons are absent and the rf currentis carried by displacement. The sheath can thus be modeledas a capacitor, with a nonlinear voltage-charge characteristicVS=VS�Q�t��. We assume that VS can be adequately de-scribed by a quadratic nonlinearity, even when the sheathmodulation is not small,

VS =lS

�0A�Q +

Q2

Q0� . �3�

Here, lS denotes the sheath width and Q0 is a form parameter.The connection between sheath and bulk is twofold. First,the current from the bulk charges the sheath, I=dQ /dt, and

second, Kichhoff’s law holds VB+VS= V̂ cos �rft. We employnormalized quantities t�=�rft, Q�=Q /Q0, I�= I / ��rfQ0�, and

V̂�= V̂ / ��0A�pe2 /Q0lB�rf

2 �, and introduce the dimensionlessparameters �= �lB / lS�1/2�pe /�rf and �=�ce /�rf. Dropping theprime, we obtain the following nonlinear system of differen-tial equations:

dQ

dt= I,

dI

dt= − �I − �2Q − �2Q2 + V̂ cos t . �4�

A more compact form can be found in abstract notation. Wedefine a state vector �z�= �Q , I�T built of the sheath chargeand the bulk current. The dynamical equations can then berepresented by a matrix equation, i.e.,

d�z�dt

= TC�z� + TD�z� + TNL�z��z� + V̂ cos t�e� . �5�

The matrices in this dynamic equation are the conservativepart TC of the dynamics, the dissipative part TD, and thenonlinear part TNL. The external excitation is given by

V̂ cos t�e�. In order to find the analytic solution to the dy-namic equation �5� we need the eigenvector relation of thematrix TC+TD. As it is neither Hermitian nor anti-Hermitian,we distinguish between left eigenvectors Ll� and right eigen-vectors �Rl� to the eigenvalues pl. �The eigenvectors are cho-

sen orthogonal.� The dynamic equation �5� can now besolved with a triple series ansatz which combines a power

expansion in the amplitude of the applied voltage V̂, a Fou-rier expansion in the periodic time t, and a finite expansion inthe right eigenvectors �Rl� of TC+TD:

�z� = =1

k=−

l=1

2

zk,l��V̂eikt�Rl� . �6�

The unknown coefficients zk,l�� are found by the following

recursion formulas:

zk,l�1� =

��k�,1

2

Ll�e�ik − pl

for = 1 and l = 1,2, �7�

�8�

The solution of the nonlinear system of equations shows thepostulated form of the rf current �Fig. 1�. Due to the nonlin-ear interaction of bulk and sheath, the plasma series reso-nance is visibly excited. Its amplitude is controlled by thevalue of the electron-neutral collision frequency �ce. Theresonance effect increases therefore with decreasing pres-sure, and so does the rms value of the calculated rf current.

It is important to note that there exists robust experimen-tal evidence for the self-excitation of the plasma series reso-nance. Klick, for example, reported the observation of higherharmonics using a rf current sensor which was integratedinto the grounded wall of an asymmetric argon rfdischarge.12

The results show that the increasing rms value of the rfcurrent for small collision frequency �pressure� leads also todrastically enhanced dissipation. The normalized dissipationvalue per area for the NERH model is depicted as a functionof the normalized collision frequency in Fig. 2 �solid line�.For comparison, the corresponding values under the sinu-soidal assumption �dotted line� and for the traditional “sto-chastic heating” �dashed line�—based on the model by Lie-berman and Godyak5—are also given. Above �=5�p�10 Pa�, all three curves scale virtually linearly with thecollision frequency, and the difference decreases. The curves,in fact, converge for very large values of �. In the low pres-sure regime ���1�, however, only the sinusoidal approxima-tion scales linearly with � and vanishes for �→0. The NERHvalue, in contrast, remains finite, and corresponds thus to theexperimental observations of Popov and Godyak.3 Note thatalso the stochastic heating model shows similar behavior, butits deviation from the linear model is less pronounced. Forthe particular simulation parameters given in Fig. 1 theNERH dissipation even shows a slight increase for verysmall collision frequencies �very low pressures�. Experimen-tal results to that effect were recently reported byFranz and Klick.13 This remarkable behavior needs furtherinvestigation.

In summary, we have presented an alternative explana-tion for the observed anomalous electron heating in capaci-

151503-2 T. Mussenbrock and R. P. Brinkmann Appl. Phys. Lett. 88, 151503 �2006�

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tive rf discharges at low pressures. Of course, our proposedmechanism, NERH, is conceptually a form of Ohmic dissi-pation. The model differs from other calculations only by notassuming from the outset that the rf current is sinusoidal.Taking instead full account of the dynamic interaction of thenonlinear sheath behavior with the current transport in thebulk, we found a strong self-excitation of the series reso-

nance and consequently a strongly enhanced Ohmic dissipa-tion. Based on our model, we propose that the traditionalview of the dissipation at low pressure should be corrected.

In the final analysis, the observed enhanced heating maywell have three independent causes that act simultaneously:Fermi heating, pressure heating, and the described nonlinearelectron resonance heating. A self-consistent kinetic modelthat includes all three effects is clearly called for. �Thismodel should also cover the characteristics of the drivingsource. Quantitative modifications of the conclusions mayarise, for example, when the finite internal resistance of thepower supply is taken into account.�

The authors gratefully acknowledge the support pro-vided by Deutsche Forschungsgemeinschaft �DFG� via theSonderforschungsbereich SFB 591 “Universal behavior ofplasmas far from equilibrium.”

1M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Dischargesand Materials Processing �Wiley, New York, 1994�.

2M. Kratzer, R. P. Brinkmann, W. Sabisch, and H. Schmidt, J. Appl. Phys.90, 2169 �2001�.

3O. A. Popov and V. A. Godyak, J. Appl. Phys. 57, 53 �1985�.4V. A. Godyak, Sov. J. Plasma Phys. 2, 78 �1976�.5M. A. Lieberman and V. A. Godyak, IEEE Trans. Plasma Sci. 26, 955�1998�.

6I. D. Kaganovich, V. I. Kolobov, and L. D. Tsendin, Appl. Phys. Lett. 69,3818 �1996�.

7M. M. Turner, Phys. Rev. Lett. 75, 1312 �1995�.8G. Gozadinos, M. M. Turner, and D. Vender, Phys. Rev. Lett. 87, 135004�2001�.

9R. P. Brinkmann and F. Hamme �unpublished�.10M. A. Lieberman �private communication�.11W. D. Qui, K. J. Bowers, and C. K. Birdsall, Plasma Sources Sci. Technol.

12, 57 �2003�.12M. Klick, J. Appl. Phys. 79, 3445 �1996�.13G. Franz and M. Klick, J. Vac. Sci. Technol. A 23, 917 �2005�.

FIG. 1. Normalized rf currents �solid lines� in an Ar discharge based on theanalytical solution of the nonlinear matrix equation for different normalizedcollision frequencies �gas pressures�: �=9.46 �p=20 Pa�, �=2.36

�p=5 Pa�, and �=0.47 �p=1 Pa�. Simulation parameters are V̂=200 V atf rf=13.56 MHz, ne=109 cm−3, lB=5.7 cm, lS=1.0 cm, and A=160 cm2. Thedashed lines are rf currents from the corresponding sinusoidalapproximation.

FIG. 2. Normalized power deposition as a function of the normalizedcollision frequency � for the nonlinear model �solid line�, the traditional“stochastic” heating model �Ref. 5� �dashed line�, and the sinusoidalapproximation �dotted line�.

151503-3 T. Mussenbrock and R. P. Brinkmann Appl. Phys. Lett. 88, 151503 �2006�

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