Thin Solid Films 518 (2010) 6694–6699

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A simple analysis on the abnormal behavior of the argon metastable density in aninductively coupled Ar plasma

Min Park a, Hong-Young Chang a, Shin-Jae You b,⁎, Jung-Hyung Kim b, Dae-Jin Seong b, Yong-Hyeon Shin b

a Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Republic of Koreab Center for Vacuum Technology, Korea Research Institute of Standards and Science, Daejeon, 305-306, Republic of Korea

⁎ Corresponding author.E-mail address: sjyou@kriss.re.kr (S.-J. You).

0040-6090/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.tsf.2010.03.135

a b s t r a c t

a r t i c l e i n f o

Available online 2 April 2010

Keywords:ICPPlasmaEEDFMetastableModeHysteresisGlobal modelMultistep ionization

The abnormal behavior of the argon metastable density during the E–H mode transition in argon ICP dischargewas investigated. Lots of investigations including global models expected that during and after the modetransition of ICP discharge the argon metastable density increases with applied rf power (i.e. electron density).However, recentmeasurement ofmetastable density [A.M. Daltrini, S.A. Moshkalev, T.J. Morgan, R.B. Piejak,W.G.Graham, Appl. Phys. Lett. 92 (2008) 061504] revealed that the argon metastable density decreases with theapplied power during and after the mode transition. This result may not be explained by the previous globalmodel which is based on the assumption of theMaxwellian electron energy distribution function (EEDF). In thispaper, to explain this abnormal behaviorwith simplemanners, a simple globalmodel taking account of the effectof the non-Maxwellian EEDFs is proposed. The calculated result from the proposed globalmodel showed that theargon metastable density can exhibit an abnormal behavior with electron density which is in good agreementwith the measurement results, indicating the close coupling of electron kinetics and the behavior of metastabledensity. The proposed simple analysis is expected to provide qualitative kinetic insight to understand thebehavior of the metastable density in various plasma discharges which typically exhibit non-Maxwelliandistribution.

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1. Introduction

Inductively coupled discharges have emerged as promising high-density plasma sources for materials processing applications due totheir excellent characteristics in semiconductor manufacturing: it canproducehigh-density plasma in lowpressure and control the ion energyand flux independently [3]. In spite of their advantages in the process,there are problems in application of ICP discharge to new opened nano-scale plasmaprocess. Themain problemof ICP for newplasmaprocess isthe non-linear discharge operating characteristics against the appliedpower which is originated from the electron density jump andhysteresis, i.e. E–H mode transition and hysteresis. Therefore, to makea secure ICP operation with processing linearity for the successfulfrontier plasma processes, it is necessary to understand the physics andfind the key player for the mode transition and hysteresis.

A number of studies investigated different mechanisms for themode transition and hysteresis [4–10], and in particular, since Lee etal. [10] showed that the multistep ionization via the metastable statesis an important process for the mode transition and hysteresis, themetastable atom density and its evolution with external parameters(especially power) have been an interesting issue in research on E–Hmode transition and hysteresis of ICP. Recently, Daltrini et al. [1,9] and

Graham et al. [2] conducted the direct measurement of the argonmetastable density during the mode transition of ICP by laser inducedfluorescence technique and revealed that there was an abnormalbehavior of the argon metastable density with the power (electrondensity) during (and after) the E–H mode transition. The argonmetastable density increased with applied power(electron density) inthe E-mode, however, during or after the mode transition the argonmetastable density decreased. In order to explain this abnormalbehavior of the metastable density with the electron density, themodel developed by Lee et al. [11,12] was used ,which is the self-consistent global model including the multi-step ionization in theargon ICP, and the relation between the metastable density and theelectron density was calculated. However, unlike the experimentalmeasurements of Daltrini et al. and Graham et al., the calculated resultshowed that the argonmetastable density increasedmonotonically asthe electron density (applied power) increased. This indicates thatbecause the Lee et al. model is based on the assumption of theMaxwellian electron energy distribution function (EEDF), whenEEDFs are non-Maxwellian the model may not be applicable.

The measured abnormal behavior of the metastable densityagainst the plasma power (electron density) during (and after) themode transition has not been understood yet clearly. In this study,to explain this abnormal behavior of the metastable density duringthe mode transition, a simple global model taking account of theeffect of the non-Maxwellian EEDFs is proposed. Since the creation

Table 1Reactions and corresponding rate constants used in calculations [11]. Unit of K arem3 s−1

and A are s−1.

Reaction Rate constant

Ar+e elastic collision Kel=2.3×10−14Te1.61 exp(0.06(lnTe)2−0.12(lnTe)3)

Ar+e→Ar++2e Kgi=2.3×10−14Te0.68 exp(−15.76/Te)

Arm+e→Ar++2e Kmi=6.8×10−15Te0.67 exp(−4.2/Te)

Arr+e→Ar++2e Kri=6.8×10−15Te0.67 exp(−4.2/Te)

Arp+e→Ar++2e Kpi=1.8×10−13Te0.61 exp(−2.61/Te)

Ar+e→Arm+e Kgm=2.5×10−15Te0.74 exp(−11.56/Te)

Ar+e→Arr+e Kgr=2.5×10−15Te0.74 exp(−11.56/Te)

Ar+e→Arp+e Kgp=1.4×10−14Te0.71 exp(−13.2/Te)

Arm+e→Ar+e Kmg=4.3×10−16Te0.74

Arr+e→Ar+e Krg=4.3×10−16Te0.74

Arp+e→Ar+e Kpg=3.9×10−16Te0.71

Arm+e→Arr+e Kmr=2.0×10−13

Arm+e→Arp+e Kmp=8.9×10−13Te0.51 exp(−1.59/Te)

Arr+e→Arp+e Krp=8.9×10−13Te0.51 exp(−1.59/Te)

Arr+e→Arm+e Krm=3.0×10−13

Arp+e→Arm+e Kpm=1.5×10−13Te0.51

Arp+e→Arr+e Kpr=1.5×10−13Te0.51Arr→Ar+hν Ar,eff=8×106

Arp→Arm+hν Apm,eff=3×107

Arp→Arr+hν Apr,eff=3×107

Deffng 1.0×1020(m−1s−1)

6695M. Park et al. / Thin Solid Films 518 (2010) 6694–6699

and annihilation of metastable atoms is closely related to electronkinetics in the high-density ICP discharge, the evolution of the EEDFduring and after the mode transition was intensively considered.The calculated result from the proposed global model showed thatthe argon metastable density can exhibit an abnormal behavior withelectron density which is in good agreement with the previousmeasurement results [1,2], indicating the close coupling of electronkinetics and the behavior of metastable density. The proposedsimple analysis is expected to provide qualitative kinetic insight tounderstand the behavior of the metastable density in variousplasma discharges which typically exhibit non-Maxwelliandistribution.

2. Model

In this paper, the analysis is based on the global model establishedby Lee et al. However, since the Lee et al. model was assumed as aMaxwellian EEDF, in order to incorporate the effect of non-Maxwellian EEDF and its evolution during the mode transition intothe model, we make use of the experimental data measured by Seo etal. as input parameters and the argon metastable density isdetermined theoretically.

2.1. Global model

Lee et al. formulated the balance equation considering metastablestates(4sm), resonant states(4sr) and higher excited states(4p) andthe reactions concerned with them on the assumption of theMaxwellian EEDF. According to the study of Lee et al. [11,12], particlebalance equations of each excited state are expressed as follows.

dnm

dt= Kgmngne + Krmnrne + Kpmne + Apm;eff

� �np

− Kmr + Kmp + Kmg + Kmi

� �ne +

Deff

Λ2

� �nm = 0

ð1Þ

dnr

dt= Kgrngne + Kmrnmne + Kprne + Apr;eff

� �np

− Krp + Krm + Krg + Kri

� �ne + Ar;eff +

Deff

Λ2

� �nr = 0

ð2Þ

dnp

dt= Kgpngne + Kmpnmne + Krpnrne

− Kpm + Kpr + Kpg + Kpi

� �ne + Apm;eff + Apr;eff +

Deff

Λ2

� �np = 0

ð3Þ

where nm, nr and np are atom densities of 4s metastable states,resonance state and 4p state, respectively. All rate coefficients andeffective diffusion loss rate(Deff) are listed in the Table 1 and alsofollowed from [12]. Effective diffusion length(Λ) is given by,

1Λ2 =

πL

� �2+

χ01

R

� �2 ð4Þ

where χ01≈2.405 is the first zero of the zero-order Bessel function,L(=15 cm) is chamber length and R(=15 cm) is chamber radius.Chamber length and Chamber radius are corresponding to theexperimental set-up of Seo et al. [13].

Each excited state density can be obtained by solving aboveEqs. (1)–(3) and be expressed as a function of electron density (ne)

and electron temperature (Te). The solved metastable state density asa function of electron density (ne) and electron temperature (Te) is asfollows.

nm = ½ bcα Kgmne +1α

bKgpKpm + cKgrKrm

� �n2e

+1α

KgpKprKrm + KgrKrpKpm−KgmKprKrp

� �n3e�ng

ð5Þ

where

a = Kmr + Kmp + Kmg + Kmi

� �ne +

Deff

λ3 ;

b = Krp + Krm + Krg + Kri

� �ne + Arg;eff +

Deff

λ3 ;

c = Kmr + Kmp + Kmg + Kmi

� �ne + Apm;eff + Apr;eff +

Deff

λ3 ;

α = abc− aKprKrp + bKmpKpm + cKmrKrm

� �n2e :

− KmpKprKrm + KmrKrpKpm

� �n3e

Once ne and Te are known, the relation between metastable statedensity and electron density can be calculated from Eq. (5).

2.2. Electron energy distribution function

In order to incorporate the effect of the non-Maxwellian EEDFinto the model, the effective rate constants for any processes, suchas ionization or excitation embedded in the Eq. (5) are taken theform as

Keff = αK Tecð Þ + 1−αð ÞK Tehð Þ ð6Þ

in the case of bi-Maxwellian distribution [14]. Experimental inputparameters Tec and Teh are electron temperatures of cold and hotelectron groups and α is the ratio of density of cold electron group tothe total electron density(ne). These parameters are obtained from

Fig. 1. (a) The dependance of the relative metastable density, the ion density, andplasma emission intensity at 750.4nm on plasma power at 120mTorr (see Ref. [1]). (b)The calculated metastable density dependance on the electron density at 50mTorr.

Fig. 2. (a) The evolution of the EEPF against rf power during the mode transition at5mTorr (see Ref. [13]) and (b) the electron density (ne) and electron temperatures oflow energy electron group (Tec) and high energy electron group (Teh) obtained by thenumerical fitting of EEPFs from (a).

Fig. 3. The dependancy of the metastable density on the electron density calculatedfrom the modified model taking account of the effect of non-Maxwellian EEDF.

6696 M. Park et al. / Thin Solid Films 518 (2010) 6694–6699

the measured EEDF by the non-linear least square fitting based onEq. (7).

f εð Þ = 2ffiffiffiπ

p ne αT−3=2ec exp −ε= Tecð Þ + 1−αð ÞT−3=2

eh exp −ε= Tehð Þ� �

ð7Þ

The bi-Maxwellian distribution expressed as Eq. (7) [15] is thecharacteristics of the non-local electron kinetics in the low pressureregime where the energy relaxation length, λ , is longer than thedischarge scale length L. Godyak et al. have experimentally shown in thelowpressureCCPdischarge that twoMaxwellian distributions appear toco-exist with considerably different temperatures and densities [16].Likewise, in the case of E-mode in the low pressure ICP discharges, bi-Maxwellian distribution is typical due to the dominant capacitive powercoupling [13,17,18]. As the applied power increases, the power couplingchanges from capacitive to inductive and the EEDF evolves from a bi-Maxwellian to aMaxwellian, resulting in the corresponding evolution ofexperimental input parameters ne, Tec, Teh and α.

Fig. 4. The evolution of the EEDFs and the corresponding metastable density evolution simuMaxwellian EEDF (c) the evolution from bi-Maxwellian to Maxwellian EEDF, and (d) the ev

3. Result and discussion

Fig. 1(a) shows the abnormal behavior of the argon metastabledensity during the mode transition [1]. The metastable density

lated on assumptions of (a) the evolution of bi-Maxwellian EEDF, (b) the evolution ofolution of EEDF as (a), (c) and (b) in sequence.

6697M. Park et al. / Thin Solid Films 518 (2010) 6694–6699

6698 M. Park et al. / Thin Solid Films 518 (2010) 6694–6699

increases with the power (electron density) in the E-mode and take amaximum value around the E–H mode transition, then decreases inthe H-mode. Fig. 1(b) shows the calculation result of the self-consistent global model [11,12]. The metastable density monotoni-cally increases with the electron density (plasma power) contrary tothe measurement result. As mentioned in the introduction, thisindicates that the model on the assumption of the Maxwellian EEDFmay not describe the behavior of the argon metastable densitycorrectly.

The evolution of the EEDF during the mode transition of ICPdischarge has been measured by several groups [13,17,18] in the lowpressure regime, where the energy relaxation length, λ , is longer thanthe discharge scale length L such that characterized by the non-localelectron kinetics. Fig. 2(a) shows the evolution of the EEDF during themode transition of argon ICP at 5 mTorr [13]. As the applied powerincreases the EEDF evolves from a bi-Maxwellian distribution to aMaxwellian distribution. Fig. 2(b) shows the electron density(ne) andthe electron temperatures of low and high energy electron groups, Tecand Teh as a function of the plasma power. Tec and Teh were obtainedby the numerical fitting of the Fig. 2(a) as described in previoussection. By incorporating ne, Tec, Teh and α into the model, themetastable density with the electron density were calculated andshown in Fig. 3. The pressure condition of Daltrini et al. was 120mTorrwith the discharge length of 4cm. This is quite higher pressure thanthat of Seo et al. However, since there was no electrostatic shield used,more strong capacitive power coupling can exist and thus in the E-mode the discharge would have the similar characteristics of the lowpressure CCP's, i.e. bi-Maxwellian distribution. We consider that theEEDF evolution from a bi-Maxwellian to a Maxwellian would exist inDaltrini et al. experiment and the EEDF evolution is closely related tothe behavior of the metastable density.

Fig. 3 shows the calculated argon metastable density. Thecalculation was done with the experimental input parameters in theFig. 2(b). The calculated argonmetastable density by taking account ofthe non-Maxwellian EEDFs and its evolution shows the abnormalbehavior with the electron density, in good agreement with theprevious measured result in Fig. 1(a). The increasing metastabledensity is seen in the E-mode and then metastable density starts todecrease with the increase of the power (electron density) in thesame manner as the result of experiment. In particular, the maximumvalue of the metastable density were calculated around the E–Hmodetransition, quite analogous to the experimental result.

Fig. 4 shows the several evolutions of the EEDF simulated bychanging the parameters and the followed behaviors of argonmetastable densities. The evolution of the EEDF during the modetransition can be regarded as the evolution of two electron groupstoward each other. The temperature of low energy electron groupincreases, whereas that of high energy electron group decreases,and the proportion of both groups also changes and finally asingle Maxwellian distribution is established. Fig. 4(a) is the caseof a bi-Maxwellian distribution when the total electron density(ne) increases, whereas low energy electron group temperature(Tec), high energy electron group temperature (Teh) and theproportion of low energy electron density to the total electrondensity (α) assumed constants, resulting in the increasingmetastable density. This case corresponds to the evolution ofEEDF in the early E-mode. As shown in Fig. 2(a), the shape ofEEDFs in the early E-mode does not change distinguishably, onlythe electron density increases with the power. Therefore, themetastable density increase in the E-mode, shown in the Fig. 1(a),can be explained by the increase of the electron density withoutsevere change of the shape of EEDF.

Fig. 4(b) is the case of Maxwellian distribution when the electrondensity(ne) increases with a constant electron temperature(Te),showing the metastable density increase with the power. This casecorresponds to the evolution of EEDF in the H-mode at high density.

When the discharge reaches the high density with sufficiently highpower after the mode transition, a single Maxwellian would beestablished. The electron density increases with a little change ofelectron temperature with the applied power. The increase of themetastable density in the H-mode was not observed in theexperiment shown in Fig. 1(a), however, expected to be measured ifthe more power applied to the discharge, and this is the casecorresponding to the prediction of the global model of Lee et al. basedon the assumption of Maxwellian EEDF.

Fig. 4(c) is the case when EEDF evolves from a bi-Maxwellian toMaxwellian distribution. The electron density(ne) and low energyelectron group temperature(Tec) increase, whereas the proportion oflow energy electron density(α) and high energy electron grouptemperature (Teh) decrease, resulting in the metastable densitydecrease. The decrease of the metastable density can be seen by thechange in the shape of the EEDF and this corresponds to during oraround the E–H mode transition. Although the electron densityincreases, due to the reduction of the population of high energyelectrons which can generate the metastables and the increase of thepopulation of middle energy electrons which can annihilate themetastables, the metastable density can be decreased.

Fig. 4(d) is the case of every parameters, Tec, Teh, α and ne werechanged so as to generate the evolution of EEDF similar to theexperimental measurement during the mode transition. The behaviorof the calculated metastable density is quite similar to the experiment(Fig. 1(a)).

As summarizing all of the simulated results shown in Fig. 4, it issuggest that the evolution of the EEDF as the consequence of thechange of power coupling from capacitive to inductive can generatethe abnormal behavior of the metastable density.

4. Conclusion

The abnormal behavior of the argon metastable density during(and after) the E–H mode transition was investigated. Themetastable density was predicted by the model taking into accountof non-Maxwellian EEDF and its evolution during the transition.The calculated metastable density shows an abnormal behaviorwith electron density and is in good agreement with the previousmeasurement results. It is suggested that the electron kinetics isclosely coupled to the behavior of metastable density. Thesimulated EEDF evolution provides simple and comprehensiveexplanation for our suggestion. Further experimental and theoret-ical study in various discharge condition is needed to confirm oursuggestion.

Acknowledgement

This work is financially supported by the Ministry of KnowledgeEconomy [10031812-2008-11] and supported by KRISS. One of theauthors would like to thank PLASMART.INC for its technical andfinancial support for this work.

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